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+% $Id$
+%
+% This file is part of MetaPost;
+% the MetaPost program is in the public domain.
+% See the <Show version...> code in mpost.w for more info.
+
+% Here is TeX material that gets inserted after \input webmac
+
+\font\tenlogo=logo10 % font used for the METAFONT logo
+\font\logos=logosl10
+\def\MF{{\tenlogo META}\-{\tenlogo FONT}}
+\def\MP{{\tenlogo META}\-{\tenlogo POST}}
+\def\pct!{{\char`\%}} % percent sign in ordinary text
+\def\psqrt#1{\sqrt{\mathstrut#1}}
+
+
+\def\title{Math support functions for decNumber based math}
+\pdfoutput=1
+
+@ Introduction.
+
+@c
+#include <w2c/config.h>
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+#include <math.h>
+#include "mpmathdecimal.h" /* internal header */
+#define ROUND(a) floor((a)+0.5)
+@h
+
+@ @c
+@<Declarations@>;
+
+@ @(mpmathdecimal.h@>=
+#ifndef MPMATHDECIMAL_H
+#define MPMATHDECIMAL_H 1
+#include "mplib.h"
+#include "mpmp.h" /* internal header */
+#define DECNUMDIGITS 1000
+#include "decNumber.h"
+@<Internal library declarations@>;
+#endif
+
+@* Math initialization.
+
+First, here are some very important constants.
+
+@d E_STRING "2.7182818284590452353602874713526624977572470936999595749669676277240766303535"
+@d PI_STRING "3.1415926535897932384626433832795028841971693993751058209749445923078164062862"
+@d fraction_multiplier 4096
+@d angle_multiplier 16
+
+@ Here are the functions that are static as they are not used elsewhere
+
+@<Declarations@>=
+#define DEBUG 0
+static void mp_decimal_scan_fractional_token (MP mp, int n);
+static void mp_decimal_scan_numeric_token (MP mp, int n);
+static void mp_ab_vs_cd (MP mp, mp_number *ret, mp_number a, mp_number b, mp_number c, mp_number d);
+/*|static void mp_decimal_ab_vs_cd (MP mp, mp_number *ret, mp_number a, mp_number b, mp_number c, mp_number d);|*/
+static void mp_decimal_crossing_point (MP mp, mp_number *ret, mp_number a, mp_number b, mp_number c);
+static void mp_decimal_number_modulo (mp_number *a, mp_number b);
+static void mp_decimal_print_number (MP mp, mp_number n);
+static char * mp_decimal_number_tostring (MP mp, mp_number n);
+static void mp_decimal_slow_add (MP mp, mp_number *ret, mp_number x_orig, mp_number y_orig);
+static void mp_decimal_square_rt (MP mp, mp_number *ret, mp_number x_orig);
+static void mp_decimal_sin_cos (MP mp, mp_number z_orig, mp_number *n_cos, mp_number *n_sin);
+static void mp_init_randoms (MP mp, int seed);
+static void mp_number_angle_to_scaled (mp_number *A);
+static void mp_number_fraction_to_scaled (mp_number *A);
+static void mp_number_scaled_to_fraction (mp_number *A);
+static void mp_number_scaled_to_angle (mp_number *A);
+static void mp_decimal_m_unif_rand (MP mp, mp_number *ret, mp_number x_orig);
+static void mp_decimal_m_norm_rand (MP mp, mp_number *ret);
+static void mp_decimal_m_exp (MP mp, mp_number *ret, mp_number x_orig);
+static void mp_decimal_m_log (MP mp, mp_number *ret, mp_number x_orig);
+static void mp_decimal_pyth_sub (MP mp, mp_number *r, mp_number a, mp_number b);
+static void mp_decimal_pyth_add (MP mp, mp_number *r, mp_number a, mp_number b);
+static void mp_decimal_n_arg (MP mp, mp_number *ret, mp_number x, mp_number y);
+static void mp_decimal_velocity (MP mp, mp_number *ret, mp_number st, mp_number ct, mp_number sf, mp_number cf, mp_number t);
+static void mp_set_decimal_from_int(mp_number *A, int B);
+static void mp_set_decimal_from_boolean(mp_number *A, int B);
+static void mp_set_decimal_from_scaled(mp_number *A, int B);
+static void mp_set_decimal_from_addition(mp_number *A, mp_number B, mp_number C);
+static void mp_set_decimal_from_substraction (mp_number *A, mp_number B, mp_number C);
+static void mp_set_decimal_from_div(mp_number *A, mp_number B, mp_number C);
+static void mp_set_decimal_from_mul(mp_number *A, mp_number B, mp_number C);
+static void mp_set_decimal_from_int_div(mp_number *A, mp_number B, int C);
+static void mp_set_decimal_from_int_mul(mp_number *A, mp_number B, int C);
+static void mp_set_decimal_from_of_the_way(MP mp, mp_number *A, mp_number t, mp_number B, mp_number C);
+static void mp_number_negate(mp_number *A);
+static void mp_number_add(mp_number *A, mp_number B);
+static void mp_number_substract(mp_number *A, mp_number B);
+static void mp_number_half(mp_number *A);
+static void mp_number_halfp(mp_number *A);
+static void mp_number_double(mp_number *A);
+static void mp_number_add_scaled(mp_number *A, int B); /* also for negative B */
+static void mp_number_multiply_int(mp_number *A, int B);
+static void mp_number_divide_int(mp_number *A, int B);
+static void mp_decimal_abs(mp_number *A);
+static void mp_number_clone(mp_number *A, mp_number B);
+static void mp_number_swap(mp_number *A, mp_number *B);
+static int mp_round_unscaled(mp_number x_orig);
+static int mp_number_to_int(mp_number A);
+static int mp_number_to_scaled(mp_number A);
+static int mp_number_to_boolean(mp_number A);
+static double mp_number_to_double(mp_number A);
+static int mp_number_odd(mp_number A);
+static int mp_number_equal(mp_number A, mp_number B);
+static int mp_number_greater(mp_number A, mp_number B);
+static int mp_number_less(mp_number A, mp_number B);
+static int mp_number_nonequalabs(mp_number A, mp_number B);
+static void mp_number_floor (mp_number *i);
+static void mp_decimal_fraction_to_round_scaled (mp_number *x);
+static void mp_decimal_number_make_scaled (MP mp, mp_number *r, mp_number p, mp_number q);
+static void mp_decimal_number_make_fraction (MP mp, mp_number *r, mp_number p, mp_number q);
+static void mp_decimal_number_take_fraction (MP mp, mp_number *r, mp_number p, mp_number q);
+static void mp_decimal_number_take_scaled (MP mp, mp_number *r, mp_number p, mp_number q);
+static void mp_new_number (MP mp, mp_number *n, mp_number_type t) ;
+static void mp_free_number (MP mp, mp_number *n) ;
+static void mp_set_decimal_from_double(mp_number *A, double B);
+static void mp_free_decimal_math (MP mp);
+static void mp_decimal_set_precision (MP mp);
+static void mp_check_decNumber (MP mp, decNumber *dec, decContext *context);
+static int decNumber_check (decNumber *dec, decContext *context);
+static char * mp_decnumber_tostring (decNumber *n);
+
+@ We do not want special numbers as return values for functions, so:
+
+
+@c
+int decNumber_check (decNumber *dec, decContext *context)
+{
+ int test = false;
+ if (context->status & DEC_Overflow) {
+ test = true;
+ context->status &= ~DEC_Overflow;
+ }
+ if (context->status & DEC_Underflow) {
+ test = true;
+ context->status &= ~DEC_Underflow;
+ }
+ if (context->status & DEC_Errors) {
+/*|fprintf(stdout, "DEC_ERROR %x (%s)\n", context->status, decContextStatusToString(context));|*/
+ test = true;
+ decNumberZero(dec);
+ }
+ context->status = 0;
+ if (decNumberIsSpecial(dec)) {
+ test = true;
+ if (decNumberIsInfinite(dec)) {
+ if (decNumberIsNegative(dec)) {
+ decNumberCopyNegate(dec, &EL_GORDO_decNumber);
+ } else {
+ decNumberCopy(dec, &EL_GORDO_decNumber);
+ }
+ } else { /* Nan */
+ decNumberZero(dec);
+ }
+ }
+ if (decNumberIsZero(dec) && decNumberIsNegative(dec)) {
+ decNumberZero(dec);
+ }
+ return test;
+}
+void mp_check_decNumber (MP mp, decNumber *dec, decContext *context)
+{
+ mp->arith_error = decNumber_check (dec, context);
+}
+
+
+
+
+@ There are a few short decNumber functions that do not exist, but
+make life easier for us:
+
+@d decNumberIsPositive(A) !(decNumberIsZero(A) || decNumberIsNegative(A))
+
+@c
+static decContext set;
+static decContext limitedset;
+static void checkZero (decNumber *ret) {
+ if (decNumberIsZero(ret) && decNumberIsNegative(ret))
+ decNumberZero(ret);
+}
+static int decNumberLess(decNumber *a, decNumber *b) {
+ decNumber comp;
+ decNumberCompare(&comp, a, b, &set);
+ return decNumberIsNegative(&comp);
+}
+static int decNumberGreater(decNumber *a, decNumber *b) {
+ decNumber comp;
+ decNumberCompare(&comp, a, b, &set);
+ return decNumberIsPositive(&comp);
+}
+static void decNumberFromDouble(decNumber *A, double B) {
+ char buf[1000];
+ char *c;
+ snprintf(buf,1000,"%-650.325lf",B);
+ c = buf;
+ while (*c++) {
+ if (*c == ' ') {
+ *c = '\0';
+ break;
+ }
+ }
+ decNumberFromString(A, buf, &set);
+}
+static double decNumberToDouble(decNumber *A) {
+ char *buffer = malloc(A->digits + 14);
+ double res = 0.0;
+ assert (buffer);
+ decNumberToString(A, buffer);
+ if (sscanf(buffer, "%lf", &res)) {
+ free(buffer);
+ return res;
+ } else {
+ free(buffer);
+ /*|mp->arith_error = 1;|*/
+ return 0.0; /* whatever*/
+ }
+}
+@ Borrowed code from libdfp:
+
+% x^3 x^5 x^7
+%arctan(x) = x - --- + --- - --- + ...
+% 3 5 7
+$$ \arctan(x) = x - {x^3\over3} + {x^5\over5} - {x^7\over7} + \ldots$$
+
+
+This power series works well, if $x$ is close to zero ($|x|<0.5$).
+If x is larger, the series converges too slowly,
+so in order to get a smaller x, we apply the identity
+
+% sqrt(1+x^2) - 1
+%arctan(x) = 2*arctan ---------------
+% x
+$$ \arctan(x) = 2\,\arctan{{\sqrt{1+x^2}-1}\over x}$$
+
+twice. The first application gives us a new $x$ with $x < 1$.
+The second application gives us a new x with $x < 0.4142136$.
+For that $x$, we use the power series and multiply the result by four.
+
+
+
+@c
+static void decNumberAtan (decNumber *result, decNumber *x_orig, decContext *set)
+{
+ decNumber x, f, g, mx2, term;
+ int i;
+ decNumberCopy(&x, x_orig);
+ if (decNumberIsZero (&x)) {
+ decNumberCopy (result, &x);
+ return;
+ }
+ for (i=0; i<2; i++) {
+ decNumber y;
+ decNumberMultiply (&y, &x, &x, set); /* $y = x^2$ */
+ decNumberAdd (&y, &y, &one, set); /* $y = y+1$*/
+ decNumberSquareRoot (&y, &y, set); /* $y = sqrt(y)$ */
+ decNumberSubtract (&y, &y, &one, set); /* $y = y-1$ */
+ decNumberDivide (&x, &y, &x, set); /* $x = y/x$ */
+ if (decNumberIsZero (&x)) {
+ decNumberCopy (result, &x);
+ return;
+ }
+ }
+ decNumberCopy (&f, &x); /* $f(0) = x$ */
+ decNumberCopy (&g, &one); /*$ g(0) = 1$*/
+ decNumberCopy (&term, &x); /*$ term = x$*/
+ decNumberCopy (result, &x); /*$ sum = x $*/
+ decNumberMultiply (&mx2, &x, &x, set); /*$ mx2 = x^2$*/
+ decNumberMinus (&mx2, &mx2, set); /*$ mx2 = -x^2 $*/
+ for (i=0; i<2*set->digits; i++) {
+ decNumberMultiply (&f, &f, &mx2, set);
+ decNumberAdd (&g, &g, &two_decNumber, set);
+ decNumberDivide (&term, &f, &g, set);
+ decNumberAdd (result, result, &term, set);
+ }
+ decNumberAdd (result, result, result, set);
+ decNumberAdd (result, result, result, set);
+ return;
+}
+static void decNumberAtan2 (decNumber *result, decNumber *y, decNumber *x, decContext *set)
+{
+ decNumber temp;
+ if (!decNumberIsInfinite (x) && !decNumberIsZero (y)
+ && !decNumberIsInfinite (y) && !decNumberIsZero (x)) {
+ decNumberDivide (&temp, y, x, set);
+ decNumberAtan (result, &temp, set);
+ /* decNumberAtan doesn't quite return the values in the ranges we
+ * want for x < 0. So we need to do some correction */
+ if (decNumberIsNegative (x)) {
+ if (decNumberIsNegative (y)) {
+ decNumberSubtract(result, result, &PI_decNumber, set);
+ } else {
+ decNumberAdd(result, result, &PI_decNumber, set);
+ }
+ }
+ return;
+ }
+ if (decNumberIsInfinite (y) && decNumberIsInfinite (x)) {
+ /* If x and y are both inf, the result depends on the sign of x */
+ decNumberDivide(result, &PI_decNumber, &four_decNumber, set);
+ if (decNumberIsNegative (x) ) {
+ decNumber a;
+ decNumberFromDouble(&a, 3.0);
+ decNumberMultiply(result, result, &a, set);
+ }
+ } else if (!decNumberIsZero (y) && !decNumberIsInfinite (x) ) {
+ /* If y is non-zero and x is non-inf, the result is +-pi/2 */
+ decNumberDivide(result, &PI_decNumber, &two_decNumber, set);
+ } else { /* Otherwise it is +0 if x is positive, +pi if x is neg */
+ if (decNumberIsNegative (x)) {
+ decNumberCopy(result, &PI_decNumber);
+ } else {
+ decNumberZero(result);
+ }
+ }
+ /* Atan2 will be negative if y<0 */
+ if (decNumberIsNegative (y)) {
+ decNumberMinus(result, result, set);
+ }
+}
+
+@ And these are the ones that {\it are} used elsewhere
+
+@<Internal library declarations@>=
+void * mp_initialize_decimal_math (MP mp);
+
+@
+
+@d unity 1
+@d two 2
+@d three 3
+@d four 4
+@d half_unit 0.5
+@d three_quarter_unit 0.75
+@d coef_bound ((7.0/3.0)*fraction_multiplier) /* |fraction| approximation to 7/3 */
+@d fraction_threshold 0.04096 /* a |fraction| coefficient less than this is zeroed */
+@d half_fraction_threshold (fraction_threshold/2) /* half of |fraction_threshold| */
+@d scaled_threshold 0.000122 /* a |scaled| coefficient less than this is zeroed */
+@d half_scaled_threshold (scaled_threshold/2) /* half of |scaled_threshold| */
+@d near_zero_angle (0.0256*angle_multiplier) /* an angle of about 0.0256 */
+@d p_over_v_threshold 0x80000 /* TODO */
+@d equation_threshold 0.001
+@d tfm_warn_threshold 0.0625
+@d epsilon pow(2.0,-173.0) /* almost "1E-52" */
+@d epsilonf pow(2.0,-52.0)
+@d EL_GORDO "1E1000000" /* the largest value that \MP\ likes. */
+@d warning_limit "1E1000000" /* this is a large value that can just be expressed without loss of precision */
+@d DECPRECISION_DEFAULT 34
+
+@<Declarations@>=
+static decNumber zero;
+static decNumber one;
+static decNumber minusone;
+static decNumber two_decNumber;
+static decNumber three_decNumber;
+static decNumber four_decNumber;
+static decNumber fraction_multiplier_decNumber;
+static decNumber angle_multiplier_decNumber;
+static decNumber fraction_one_decNumber;
+static decNumber fraction_one_plus_decNumber;
+static decNumber PI_decNumber;
+static decNumber epsilon_decNumber;
+static decNumber EL_GORDO_decNumber;
+static decNumber **factorials = NULL;
+static int last_cached_factorial = 0;
+static boolean initialized = false ;
+@ @c
+void * mp_initialize_decimal_math (MP mp) {
+ math_data *math = (math_data *)mp_xmalloc(mp,1,sizeof(math_data));
+ /* various decNumber initializations */
+ decContextDefault(&set, DEC_INIT_BASE); /* initialize */
+ set.traps=0; /* no traps, thank you */
+ decContextDefault(&limitedset, DEC_INIT_BASE); /* initialize */
+ limitedset.traps=0; /* no traps, thank you */
+ limitedset.emax = 999999;
+ limitedset.emin = -999999;
+ set.digits = DECPRECISION_DEFAULT;
+ limitedset.digits = DECPRECISION_DEFAULT;
+ if (!initialized) {
+ initialized = true ;
+ decNumberFromInt32(&one, 1);
+ decNumberFromInt32(&minusone, -1);
+ decNumberFromInt32(&zero, 0);
+ decNumberFromInt32(&two_decNumber, two);
+ decNumberFromInt32(&three_decNumber, three);
+ decNumberFromInt32(&four_decNumber, four);
+ decNumberFromInt32(&fraction_multiplier_decNumber, fraction_multiplier);
+ decNumberFromInt32(&fraction_one_decNumber, fraction_one);
+ decNumberFromInt32(&fraction_one_plus_decNumber, (fraction_one+1));
+ decNumberFromInt32(&angle_multiplier_decNumber, angle_multiplier);
+ decNumberFromString(&PI_decNumber, PI_STRING, &set);
+ decNumberFromDouble(&epsilon_decNumber, epsilon);
+ decNumberFromString(&EL_GORDO_decNumber, EL_GORDO, &set);
+ factorials = (decNumber **)mp_xmalloc(mp,PRECALC_FACTORIALS_CACHESIZE,sizeof(decNumber *));
+ factorials[0] = (decNumber *)mp_xmalloc(mp,1,sizeof(decNumber));
+ decNumberCopy(factorials[0], &one);
+ }
+
+ /* alloc */
+ math->allocate = mp_new_number;
+ math->free = mp_free_number;
+ mp_new_number (mp, &math->precision_default, mp_scaled_type);
+ decNumberFromInt32(math->precision_default.data.num, DECPRECISION_DEFAULT);
+ mp_new_number (mp, &math->precision_max, mp_scaled_type);
+ decNumberFromInt32(math->precision_max.data.num, DECNUMDIGITS);
+ mp_new_number (mp, &math->precision_min, mp_scaled_type);
+ decNumberFromInt32(math->precision_min.data.num, 2);
+ /* here are the constants for |scaled| objects */
+ mp_new_number (mp, &math->epsilon_t, mp_scaled_type);
+ decNumberCopy(math->epsilon_t.data.num, &epsilon_decNumber);
+ mp_new_number (mp, &math->inf_t, mp_scaled_type);
+ decNumberCopy(math->inf_t.data.num, &EL_GORDO_decNumber);
+ mp_new_number (mp, &math->warning_limit_t, mp_scaled_type);
+ decNumberFromString(math->warning_limit_t.data.num, warning_limit, &set);
+ mp_new_number (mp, &math->one_third_inf_t, mp_scaled_type);
+ decNumberDivide(math->one_third_inf_t.data.num, math->inf_t.data.num, &three_decNumber, &set);
+ mp_new_number (mp, &math->unity_t, mp_scaled_type);
+ decNumberCopy(math->unity_t.data.num, &one);
+ mp_new_number (mp, &math->two_t, mp_scaled_type);
+ decNumberFromInt32(math->two_t.data.num, two);
+ mp_new_number (mp, &math->three_t, mp_scaled_type);
+ decNumberFromInt32(math->three_t.data.num, three);
+ mp_new_number (mp, &math->half_unit_t, mp_scaled_type);
+ decNumberFromString(math->half_unit_t.data.num, "0.5", &set);
+ mp_new_number (mp, &math->three_quarter_unit_t, mp_scaled_type);
+ decNumberFromString(math->three_quarter_unit_t.data.num, "0.75", &set);
+ mp_new_number (mp, &math->zero_t, mp_scaled_type);
+ decNumberZero(math->zero_t.data.num);
+ /* |fractions| */
+ mp_new_number (mp, &math->arc_tol_k, mp_fraction_type);
+ {
+ decNumber fourzeroninesix;
+ decNumberFromInt32(&fourzeroninesix, 4096);
+ decNumberDivide(math->arc_tol_k.data.num, &one, &fourzeroninesix, &set);
+ /* quit when change in arc length estimate reaches this */
+ }
+ mp_new_number (mp, &math->fraction_one_t, mp_fraction_type);
+ decNumberFromInt32(math->fraction_one_t.data.num, fraction_one);
+ mp_new_number (mp, &math->fraction_half_t, mp_fraction_type);
+ decNumberFromInt32(math->fraction_half_t.data.num, fraction_half);
+ mp_new_number (mp, &math->fraction_three_t, mp_fraction_type);
+ decNumberFromInt32(math->fraction_three_t.data.num, fraction_three);
+ mp_new_number (mp, &math->fraction_four_t, mp_fraction_type);
+ decNumberFromInt32(math->fraction_four_t.data.num, fraction_four);
+ /* |angles| */
+ mp_new_number (mp, &math->three_sixty_deg_t, mp_angle_type);
+ decNumberFromInt32(math->three_sixty_deg_t.data.num, 360 * angle_multiplier);
+ mp_new_number (mp, &math->one_eighty_deg_t, mp_angle_type);
+ decNumberFromInt32(math->one_eighty_deg_t.data.num, 180 * angle_multiplier);
+ /* various approximations */
+ mp_new_number (mp, &math->one_k, mp_scaled_type);
+ decNumberFromDouble(math->one_k.data.num, 1.0/64);
+ mp_new_number (mp, &math->sqrt_8_e_k, mp_scaled_type);
+ {
+ decNumberFromDouble(math->sqrt_8_e_k.data.num, 112428.82793 / 65536.0);
+ /* $2^{16}\sqrt{8/e}\approx 112428.82793$ */
+ }
+ mp_new_number (mp, &math->twelve_ln_2_k, mp_fraction_type);
+ {
+ decNumberFromDouble(math->twelve_ln_2_k.data.num, 139548959.6165 / 65536.0);
+ /* $2^{24}\cdot12\ln2\approx139548959.6165$ */
+ }
+ mp_new_number (mp, &math->coef_bound_k, mp_fraction_type);
+ decNumberFromDouble(math->coef_bound_k.data.num,coef_bound);
+ mp_new_number (mp, &math->coef_bound_minus_1, mp_fraction_type);
+ decNumberFromDouble(math->coef_bound_minus_1.data.num,coef_bound - 1 / 65536.0);
+ mp_new_number (mp, &math->twelvebits_3, mp_scaled_type);
+ {
+ decNumberFromDouble(math->twelvebits_3.data.num, 1365 / 65536.0);
+ /* $1365\approx 2^{12}/3$ */
+ }
+ mp_new_number (mp, &math->twentysixbits_sqrt2_t, mp_fraction_type);
+ {
+ decNumberFromDouble(math->twentysixbits_sqrt2_t.data.num, 94906265.62 / 65536.0);
+ /* $2^{26}\sqrt2\approx94906265.62$ */
+ }
+ mp_new_number (mp, &math->twentyeightbits_d_t, mp_fraction_type);
+ {
+ decNumberFromDouble(math->twentyeightbits_d_t.data.num, 35596754.69 / 65536.0);
+ /* $2^{28}d\approx35596754.69$ */
+ }
+ mp_new_number (mp, &math->twentysevenbits_sqrt2_d_t, mp_fraction_type);
+ {
+ decNumberFromDouble(math->twentysevenbits_sqrt2_d_t.data.num, 25170706.63 / 65536.0);
+ /* $2^{27}\sqrt2\,d\approx25170706.63$ */
+ }
+ /* thresholds */
+ mp_new_number (mp, &math->fraction_threshold_t, mp_fraction_type);
+ decNumberFromDouble(math->fraction_threshold_t.data.num, fraction_threshold);
+ mp_new_number (mp, &math->half_fraction_threshold_t, mp_fraction_type);
+ decNumberFromDouble(math->half_fraction_threshold_t.data.num, half_fraction_threshold);
+ mp_new_number (mp, &math->scaled_threshold_t, mp_scaled_type);
+ decNumberFromDouble(math->scaled_threshold_t.data.num, scaled_threshold);
+ mp_new_number (mp, &math->half_scaled_threshold_t, mp_scaled_type);
+ decNumberFromDouble(math->half_scaled_threshold_t.data.num, half_scaled_threshold);
+ mp_new_number (mp, &math->near_zero_angle_t, mp_angle_type);
+ decNumberFromDouble(math->near_zero_angle_t.data.num, near_zero_angle);
+ mp_new_number (mp, &math->p_over_v_threshold_t, mp_fraction_type);
+ decNumberFromDouble(math->p_over_v_threshold_t.data.num, p_over_v_threshold);
+ mp_new_number (mp, &math->equation_threshold_t, mp_scaled_type);
+ decNumberFromDouble(math->equation_threshold_t.data.num, equation_threshold);
+ mp_new_number (mp, &math->tfm_warn_threshold_t, mp_scaled_type);
+ decNumberFromDouble(math->tfm_warn_threshold_t.data.num, tfm_warn_threshold);
+ /* functions */
+ math->from_int = mp_set_decimal_from_int;
+ math->from_boolean = mp_set_decimal_from_boolean;
+ math->from_scaled = mp_set_decimal_from_scaled;
+ math->from_double = mp_set_decimal_from_double;
+ math->from_addition = mp_set_decimal_from_addition;
+ math->from_substraction = mp_set_decimal_from_substraction;
+ math->from_oftheway = mp_set_decimal_from_of_the_way;
+ math->from_div = mp_set_decimal_from_div;
+ math->from_mul = mp_set_decimal_from_mul;
+ math->from_int_div = mp_set_decimal_from_int_div;
+ math->from_int_mul = mp_set_decimal_from_int_mul;
+ math->negate = mp_number_negate;
+ math->add = mp_number_add;
+ math->substract = mp_number_substract;
+ math->half = mp_number_half;
+ math->halfp = mp_number_halfp;
+ math->do_double = mp_number_double;
+ math->abs = mp_decimal_abs;
+ math->clone = mp_number_clone;
+ math->swap = mp_number_swap;
+ math->add_scaled = mp_number_add_scaled;
+ math->multiply_int = mp_number_multiply_int;
+ math->divide_int = mp_number_divide_int;
+ math->to_boolean = mp_number_to_boolean;
+ math->to_scaled = mp_number_to_scaled;
+ math->to_double = mp_number_to_double;
+ math->to_int = mp_number_to_int;
+ math->odd = mp_number_odd;
+ math->equal = mp_number_equal;
+ math->less = mp_number_less;
+ math->greater = mp_number_greater;
+ math->nonequalabs = mp_number_nonequalabs;
+ math->round_unscaled = mp_round_unscaled;
+ math->floor_scaled = mp_number_floor;
+ math->fraction_to_round_scaled = mp_decimal_fraction_to_round_scaled;
+ math->make_scaled = mp_decimal_number_make_scaled;
+ math->make_fraction = mp_decimal_number_make_fraction;
+ math->take_fraction = mp_decimal_number_take_fraction;
+ math->take_scaled = mp_decimal_number_take_scaled;
+ math->velocity = mp_decimal_velocity;
+ math->n_arg = mp_decimal_n_arg;
+ math->m_log = mp_decimal_m_log;
+ math->m_exp = mp_decimal_m_exp;
+ math->m_unif_rand = mp_decimal_m_unif_rand;
+ math->m_norm_rand = mp_decimal_m_norm_rand;
+ math->pyth_add = mp_decimal_pyth_add;
+ math->pyth_sub = mp_decimal_pyth_sub;
+ math->fraction_to_scaled = mp_number_fraction_to_scaled;
+ math->scaled_to_fraction = mp_number_scaled_to_fraction;
+ math->scaled_to_angle = mp_number_scaled_to_angle;
+ math->angle_to_scaled = mp_number_angle_to_scaled;
+ math->init_randoms = mp_init_randoms;
+ math->sin_cos = mp_decimal_sin_cos;
+ math->slow_add = mp_decimal_slow_add;
+ math->sqrt = mp_decimal_square_rt;
+ math->print = mp_decimal_print_number;
+ math->tostring = mp_decimal_number_tostring;
+ math->modulo = mp_decimal_number_modulo;
+ math->ab_vs_cd = mp_ab_vs_cd;
+ math->crossing_point = mp_decimal_crossing_point;
+ math->scan_numeric = mp_decimal_scan_numeric_token;
+ math->scan_fractional = mp_decimal_scan_fractional_token;
+ math->free_math = mp_free_decimal_math;
+ math->set_precision = mp_decimal_set_precision;
+ return (void *)math;
+}
+
+void mp_decimal_set_precision (MP mp) {
+ int i;
+ i = decNumberToInt32((decNumber *)internal_value (mp_number_precision).data.num, &set);
+ set.digits = i;
+ limitedset.digits = i;
+}
+
+void mp_free_decimal_math (MP mp) {
+ free_number (((math_data *)mp->math)->three_sixty_deg_t);
+ free_number (((math_data *)mp->math)->one_eighty_deg_t);
+ free_number (((math_data *)mp->math)->fraction_one_t);
+ free_number (((math_data *)mp->math)->zero_t);
+ free_number (((math_data *)mp->math)->half_unit_t);
+ free_number (((math_data *)mp->math)->three_quarter_unit_t);
+ free_number (((math_data *)mp->math)->unity_t);
+ free_number (((math_data *)mp->math)->two_t);
+ free_number (((math_data *)mp->math)->three_t);
+ free_number (((math_data *)mp->math)->one_third_inf_t);
+ free_number (((math_data *)mp->math)->inf_t);
+ free_number (((math_data *)mp->math)->warning_limit_t);
+ free_number (((math_data *)mp->math)->one_k);
+ free_number (((math_data *)mp->math)->sqrt_8_e_k);
+ free_number (((math_data *)mp->math)->twelve_ln_2_k);
+ free_number (((math_data *)mp->math)->coef_bound_k);
+ free_number (((math_data *)mp->math)->coef_bound_minus_1);
+ free_number (((math_data *)mp->math)->fraction_threshold_t);
+ free_number (((math_data *)mp->math)->half_fraction_threshold_t);
+ free_number (((math_data *)mp->math)->scaled_threshold_t);
+ free_number (((math_data *)mp->math)->half_scaled_threshold_t);
+ free_number (((math_data *)mp->math)->near_zero_angle_t);
+ free_number (((math_data *)mp->math)->p_over_v_threshold_t);
+ free_number (((math_data *)mp->math)->equation_threshold_t);
+ free_number (((math_data *)mp->math)->tfm_warn_threshold_t);
+ /* For sake of speed, we accept this memory leak. */
+ /* for (i = 0; i <= last_cached_factorial; i++) {*/
+ /* free(factorials[i]);*/
+ /* }*/
+ /* free(factorials); */
+ free(mp->math);
+}
+
+@ Creating an destroying |mp_number| objects
+
+@ @c
+void mp_new_number (MP mp, mp_number *n, mp_number_type t) {
+ (void)mp;
+ n->data.num = mp_xmalloc(mp,1,sizeof(decNumber));
+ decNumberZero(n->data.num);
+ n->type = t;
+}
+
+@
+
+@c
+void mp_free_number (MP mp, mp_number *n) {
+ (void)mp;
+ free(n->data.num);
+ n->data.num = NULL;
+ n->type = mp_nan_type;
+}
+
+@ Here are the low-level functions on |mp_number| items, setters first.
+
+@c
+void mp_set_decimal_from_int(mp_number *A, int B) {
+ decNumberFromInt32(A->data.num,B);
+}
+void mp_set_decimal_from_boolean(mp_number *A, int B) {
+ decNumberFromInt32(A->data.num,B);
+}
+void mp_set_decimal_from_scaled(mp_number *A, int B) {
+ decNumber c;
+ decNumberFromInt32(&c, 65536);
+ decNumberFromInt32(A->data.num,B);
+ decNumberDivide(A->data.num,A->data.num,&c, &set);
+}
+void mp_set_decimal_from_double(mp_number *A, double B) {
+ decNumberFromDouble(A->data.num, B);
+}
+void mp_set_decimal_from_addition(mp_number *A, mp_number B, mp_number C) {
+ decNumberAdd(A->data.num,B.data.num,C.data.num, &set);
+}
+void mp_set_decimal_from_substraction (mp_number *A, mp_number B, mp_number C) {
+ decNumberSubtract(A->data.num,B.data.num,C.data.num, &set);
+}
+void mp_set_decimal_from_div(mp_number *A, mp_number B, mp_number C) {
+ decNumberDivide(A->data.num,B.data.num,C.data.num, &set);
+}
+void mp_set_decimal_from_mul(mp_number *A, mp_number B, mp_number C) {
+ decNumberMultiply(A->data.num,B.data.num,C.data.num, &set);
+}
+void mp_set_decimal_from_int_div(mp_number *A, mp_number B, int C) {
+ decNumber c;
+ decNumberFromInt32(&c, C);
+ decNumberDivide(A->data.num,B.data.num,&c, &set);
+}
+void mp_set_decimal_from_int_mul(mp_number *A, mp_number B, int C) {
+ decNumber c;
+ decNumberFromInt32(&c, C);
+ decNumberMultiply(A->data.num,B.data.num,&c, &set);
+}
+void mp_set_decimal_from_of_the_way(MP mp, mp_number *A, mp_number t, mp_number B, mp_number C) {
+ decNumber c;
+ decNumber r1;
+ decNumberSubtract(&c,B.data.num, C.data.num, &set);
+ mp_decimal_take_fraction(mp, &r1, &c, t.data.num);
+ decNumberSubtract(A->data.num, B.data.num, &r1, &set);
+ mp_check_decNumber(mp, A->data.num, &set);
+}
+void mp_number_negate(mp_number *A) {
+ decNumberCopyNegate(A->data.num, A->data.num);
+ checkZero(A->data.num);
+}
+void mp_number_add(mp_number *A, mp_number B) {
+ decNumberAdd(A->data.num,A->data.num,B.data.num, &set);
+}
+void mp_number_substract(mp_number *A, mp_number B) {
+ decNumberSubtract(A->data.num,A->data.num,B.data.num, &set);
+}
+void mp_number_half(mp_number *A) {
+ decNumber c;
+ decNumberFromInt32(&c, 2);
+ decNumberDivide(A->data.num,A->data.num, &c, &set);
+}
+void mp_number_halfp(mp_number *A) {
+ decNumber c;
+ decNumberFromInt32(&c, 2);
+ decNumberDivide(A->data.num,A->data.num, &c, &set);
+}
+void mp_number_double(mp_number *A) {
+ decNumber c;
+ decNumberFromInt32(&c, 2);
+ decNumberMultiply(A->data.num,A->data.num, &c, &set);
+}
+void mp_number_add_scaled(mp_number *A, int B) { /* also for negative B */
+ decNumber b,c;
+ decNumberFromInt32(&c, 65536);
+ decNumberFromInt32(&b, B);
+ decNumberDivide(&b,&b, &c, &set);
+ decNumberAdd(A->data.num,A->data.num, &b, &set);
+}
+void mp_number_multiply_int(mp_number *A, int B) {
+ decNumber b;
+ decNumberFromInt32(&b, B);
+ decNumberMultiply(A->data.num,A->data.num, &b, &set);
+}
+void mp_number_divide_int(mp_number *A, int B) {
+ decNumber b;
+ decNumberFromInt32(&b, B);
+ decNumberDivide(A->data.num,A->data.num,&b, &set);
+}
+void mp_decimal_abs(mp_number *A) {
+ decNumberAbs(A->data.num, A->data.num, &set);
+}
+void mp_number_clone(mp_number *A, mp_number B) {
+ decNumberCopy(A->data.num, B.data.num);
+}
+void mp_number_swap(mp_number *A, mp_number *B) {
+ decNumber swap_tmp;
+ decNumberCopy(&swap_tmp, A->data.num);
+ decNumberCopy(A->data.num, B->data.num);
+ decNumberCopy(B->data.num, &swap_tmp);
+}
+void mp_number_fraction_to_scaled (mp_number *A) {
+ A->type = mp_scaled_type;
+ decNumberDivide(A->data.num, A->data.num, &fraction_multiplier_decNumber, &set);
+}
+void mp_number_angle_to_scaled (mp_number *A) {
+ A->type = mp_scaled_type;
+ decNumberDivide(A->data.num, A->data.num, &angle_multiplier_decNumber, &set);
+}
+void mp_number_scaled_to_fraction (mp_number *A) {
+ A->type = mp_fraction_type;
+ decNumberMultiply(A->data.num, A->data.num, &fraction_multiplier_decNumber, &set);
+}
+void mp_number_scaled_to_angle (mp_number *A) {
+ A->type = mp_angle_type;
+ decNumberMultiply(A->data.num, A->data.num, &angle_multiplier_decNumber, &set);
+}
+
+
+@* Query functions.
+
+@ Convert a number to a scaled value. |decNumberToInt32| is not
+able to make this conversion properly, so instead we are using
+|decNumberToDouble| and a typecast. Bad!
+
+@c
+int mp_number_to_scaled(mp_number A) {
+ int32_t result;
+ decNumber corrected;
+ decNumberFromInt32(&corrected, 65536);
+ decNumberMultiply(&corrected,&corrected,A.data.num, &set);
+ decNumberReduce(&corrected, &corrected, &set);
+ result = (int)floor(decNumberToDouble(&corrected)+0.5);
+ return result;
+}
+
+@
+
+@d odd(A) (abs(A)%2==1)
+
+@c
+int mp_number_to_int(mp_number A) {
+ int32_t result;
+ set.status = 0;
+ result = decNumberToInt32(A.data.num, &set);
+ if (set.status == DEC_Invalid_operation) {
+ set.status = 0;
+ /* |mp->arith_error = 1;| */
+ return 0; /* whatever */
+ } else {
+ return result;
+ }
+}
+int mp_number_to_boolean(mp_number A) {
+ uint32_t result;
+ set.status = 0;
+ result = decNumberToUInt32(A.data.num, &set);
+ if (set.status == DEC_Invalid_operation) {
+ set.status = 0;
+ /* |mp->arith_error = 1;| */
+ return mp_false_code; /* whatever */
+ } else {
+ return result ;
+ }
+}
+double mp_number_to_double(mp_number A) {
+ char *buffer = malloc(((decNumber *)A.data.num)->digits + 14);
+ double res = 0.0;
+ assert (buffer);
+ decNumberToString(A.data.num, buffer);
+ if (sscanf(buffer, "%lf", &res)) {
+ free(buffer);
+ return res;
+ } else {
+ free(buffer);
+ /* |mp->arith_error = 1;| */
+ return 0.0; /* whatever */
+ }
+}
+int mp_number_odd(mp_number A) {
+ return odd(mp_number_to_int(A));
+}
+int mp_number_equal(mp_number A, mp_number B) {
+ decNumber res;
+ decNumberCompare(&res,A.data.num,B.data.num, &set);
+ return decNumberIsZero(&res);
+}
+int mp_number_greater(mp_number A, mp_number B) {
+ decNumber res;
+ decNumberCompare(&res,A.data.num,B.data.num, &set);
+ return decNumberIsPositive(&res);
+}
+int mp_number_less(mp_number A, mp_number B) {
+ decNumber res;
+ decNumberCompare(&res,A.data.num,B.data.num, &set);
+ return decNumberIsNegative(&res);
+}
+int mp_number_nonequalabs(mp_number A, mp_number B) {
+ decNumber res, a, b;
+ decNumberCopyAbs(&a, A.data.num);
+ decNumberCopyAbs(&b, B.data.num);
+ decNumberCompare(&res, &a, &b, &set);
+ return !decNumberIsZero(&res);
+}
+
+@ Fixed-point arithmetic is done on {\sl scaled integers\/} that are multiples
+of $2^{-16}$. In other words, a binary point is assumed to be sixteen bit
+positions from the right end of a binary computer word.
+
+@ One of \MP's most common operations is the calculation of
+$\lfloor{a+b\over2}\rfloor$,
+the midpoint of two given integers |a| and~|b|. The most decent way to do
+this is to write `|(a+b)/2|'; but on many machines it is more efficient
+to calculate `|(a+b)>>1|'.
+
+Therefore the midpoint operation will always be denoted by `|half(a+b)|'
+in this program. If \MP\ is being implemented with languages that permit
+binary shifting, the |half| macro should be changed to make this operation
+as efficient as possible. Since some systems have shift operators that can
+only be trusted to work on positive numbers, there is also a macro |halfp|
+that is used only when the quantity being halved is known to be positive
+or zero.
+
+@ Here is a procedure analogous to |print_int|. The current version
+is fairly stupid, and it is not round-trip safe, but this is good
+enough for a beta test.
+
+@c
+char * mp_decnumber_tostring (decNumber *n) {
+ decNumber corrected;
+ char *buffer = malloc(((decNumber *)n)->digits + 14);
+ assert (buffer);
+ decNumberCopy(&corrected,n);
+ decNumberTrim(&corrected);
+ decNumberToString(&corrected, buffer);
+ return buffer;
+}
+char * mp_decimal_number_tostring (MP mp, mp_number n) {
+ return mp_decnumber_tostring(n.data.num);
+}
+
+
+@ @c
+void mp_decimal_print_number (MP mp, mp_number n) {
+ char *str = mp_decimal_number_tostring(mp, n);
+ mp_print (mp, str);
+ free (str);
+}
+
+
+
+
+@ Addition is not always checked to make sure that it doesn't overflow,
+but in places where overflow isn't too unlikely the |slow_add| routine
+is used.
+
+@c
+void mp_decimal_slow_add (MP mp, mp_number *ret, mp_number A, mp_number B) {
+ decNumberAdd(ret->data.num,A.data.num,B.data.num, &set);
+}
+
+@ The |make_fraction| routine produces the |fraction| equivalent of
+|p/q|, given integers |p| and~|q|; it computes the integer
+$f=\lfloor2^{28}p/q+{1\over2}\rfloor$, when $p$ and $q$ are
+positive. If |p| and |q| are both of the same scaled type |t|,
+the ``type relation'' |make_fraction(t,t)=fraction| is valid;
+and it's also possible to use the subroutine ``backwards,'' using
+the relation |make_fraction(t,fraction)=t| between scaled types.
+
+If the result would have magnitude $2^{31}$ or more, |make_fraction|
+sets |arith_error:=true|. Most of \MP's internal computations have
+been designed to avoid this sort of error.
+
+If this subroutine were programmed in assembly language on a typical
+machine, we could simply compute |(@t$2^{28}$@>*p)div q|, since a
+double-precision product can often be input to a fixed-point division
+instruction. But when we are restricted to int-eger arithmetic it
+is necessary either to resort to multiple-precision maneuvering
+or to use a simple but slow iteration. The multiple-precision technique
+would be about three times faster than the code adopted here, but it
+would be comparatively long and tricky, involving about sixteen
+additional multiplications and divisions.
+
+This operation is part of \MP's ``inner loop''; indeed, it will
+consume nearly 10\pct! of the running time (exclusive of input and output)
+if the code below is left unchanged. A machine-dependent recoding
+will therefore make \MP\ run faster. The present implementation
+is highly portable, but slow; it avoids multiplication and division
+except in the initial stage. System wizards should be careful to
+replace it with a routine that is guaranteed to produce identical
+results in all cases.
+@^system dependencies@>
+
+As noted below, a few more routines should also be replaced by machine-dependent
+code, for efficiency. But when a procedure is not part of the ``inner loop,''
+such changes aren't advisable; simplicity and robustness are
+preferable to trickery, unless the cost is too high.
+@^inner loop@>
+
+@c
+void mp_decimal_make_fraction (MP mp, decNumber *ret, decNumber *p, decNumber *q) {
+ decNumberDivide(ret, p, q, &set);
+ mp_check_decNumber(mp, ret, &set);
+ decNumberMultiply(ret, ret, &fraction_multiplier_decNumber, &set);
+}
+void mp_decimal_number_make_fraction (MP mp, mp_number *ret, mp_number p, mp_number q) {
+ mp_decimal_make_fraction (mp, ret->data.num, p.data.num, q.data.num);
+}
+
+@ @<Declarations@>=
+void mp_decimal_make_fraction (MP mp, decNumber *ret, decNumber *p, decNumber *q);
+
+@ The dual of |make_fraction| is |take_fraction|, which multiplies a
+given integer~|q| by a fraction~|f|. When the operands are positive, it
+computes $p=\lfloor qf/2^{28}+{1\over2}\rfloor$, a symmetric function
+of |q| and~|f|.
+
+This routine is even more ``inner loopy'' than |make_fraction|;
+the present implementation consumes almost 20\pct! of \MP's computation
+time during typical jobs, so a machine-language substitute is advisable.
+@^inner loop@> @^system dependencies@>
+
+@c
+void mp_decimal_take_fraction (MP mp, decNumber *ret, decNumber *p, decNumber *q) {
+ decNumberMultiply(ret, p, q, &set);
+ decNumberDivide(ret, ret, &fraction_multiplier_decNumber, &set);
+}
+void mp_decimal_number_take_fraction (MP mp, mp_number *ret, mp_number p, mp_number q) {
+ mp_decimal_take_fraction (mp, ret->data.num, p.data.num, q.data.num);
+}
+
+@ @<Declarations@>=
+void mp_decimal_take_fraction (MP mp, decNumber *ret, decNumber *p, decNumber *q);
+
+@ When we want to multiply something by a |scaled| quantity, we use a scheme
+analogous to |take_fraction| but with a different scaling.
+Given positive operands, |take_scaled|
+computes the quantity $p=\lfloor qf/2^{16}+{1\over2}\rfloor$.
+
+Once again it is a good idea to use a machine-language replacement if
+possible; otherwise |take_scaled| will use more than 2\pct! of the running time
+when the Computer Modern fonts are being generated.
+@^inner loop@>
+
+@c
+void mp_decimal_number_take_scaled (MP mp, mp_number *ret, mp_number p_orig, mp_number q_orig) {
+ decNumberMultiply(ret->data.num, p_orig.data.num, q_orig.data.num, &set);
+}
+
+
+@ For completeness, there's also |make_scaled|, which computes a
+quotient as a |scaled| number instead of as a |fraction|.
+In other words, the result is $\lfloor2^{16}p/q+{1\over2}\rfloor$, if the
+operands are positive. \ (This procedure is not used especially often,
+so it is not part of \MP's inner loop.)
+
+@c
+void mp_decimal_number_make_scaled (MP mp, mp_number *ret, mp_number p_orig, mp_number q_orig) {
+ decNumberDivide(ret->data.num, p_orig.data.num, q_orig.data.num, &set);
+ mp_check_decNumber(mp, ret->data.num, &set);
+}
+
+@
+@d halfp(A) (integer)((unsigned)(A) >> 1)
+
+@* Scanning numbers in the input.
+
+The definitions below are temporarily here
+
+@d set_cur_cmd(A) mp->cur_mod_->type=(A)
+@d set_cur_mod(A) decNumberCopy((decNumber *)(mp->cur_mod_->data.n.data.num),&A)
+
+@<Declarations...@>=
+static void mp_wrapup_numeric_token(MP mp, unsigned char *start, unsigned char *stop);
+
+@
+@d too_precise(a) (a == (DEC_Inexact+DEC_Rounded))
+@d too_large(a) (a & DEC_Overflow)
+@c
+void mp_wrapup_numeric_token(MP mp, unsigned char *start, unsigned char *stop) {
+ decNumber result;
+ size_t l = stop-start+1;
+ char *buf = mp_xmalloc(mp, l+1, 1);
+ buf[l] = '\0';
+ (void)strncpy(buf,(const char *)start, l);
+ set.status = 0;
+ decNumberFromString(&result,buf, &set);
+ free(buf);
+ if (set.status == 0) {
+ set_cur_mod(result);
+ } else if (mp->scanner_status != tex_flushing) {
+ if (too_large(set.status)) {
+ const char *hlp[] = {"I could not handle this number specification",
+ "because it is out of range.",
+ NULL };
+ decNumber_check (&result, &set);
+ set_cur_mod(result);
+ mp_error (mp, "Enormous number has been reduced", hlp, false);
+ } else if (too_precise(set.status)) {
+ set_cur_mod(result);
+ if (decNumberIsPositive((decNumber *)internal_value (mp_warning_check).data.num) &&
+ (mp->scanner_status != tex_flushing)) {
+ char msg[256];
+ const char *hlp[] = {"Continue and I'll round the value until it fits the current numberprecision",
+ "(Set warningcheck:=0 to suppress this message.)",
+ NULL };
+ mp_snprintf (msg, 256, "Number is too precise (numberprecision = %d)", set.digits);
+ mp_error (mp, msg, hlp, true);
+ }
+ } else { /* this also captures underflow */
+ const char *hlp[] = {"I could not handle this number specification",
+ "Error:",
+ "",
+ NULL };
+ hlp[2] = decContextStatusToString(&set);
+ mp_error (mp, "Erroneous number specification changed to zero", hlp, false);
+ decNumberZero(&result);
+ set_cur_mod(result);
+ }
+ }
+ set_cur_cmd((mp_variable_type)mp_numeric_token);
+}
+
+@ @c
+static void find_exponent (MP mp) {
+ if (mp->buffer[mp->cur_input.loc_field] == 'e' ||
+ mp->buffer[mp->cur_input.loc_field] == 'E') {
+ mp->cur_input.loc_field++;
+ if (!(mp->buffer[mp->cur_input.loc_field] == '+' ||
+ mp->buffer[mp->cur_input.loc_field] == '-' ||
+ mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class)) {
+ mp->cur_input.loc_field--;
+ return;
+ }
+ if (mp->buffer[mp->cur_input.loc_field] == '+' ||
+ mp->buffer[mp->cur_input.loc_field] == '-') {
+ mp->cur_input.loc_field++;
+ }
+ while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class) {
+ mp->cur_input.loc_field++;
+ }
+ }
+}
+void mp_decimal_scan_fractional_token (MP mp, int n) { /* n: scaled */
+ unsigned char *start = &mp->buffer[mp->cur_input.loc_field -1];
+ unsigned char *stop;
+ while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class) {
+ mp->cur_input.loc_field++;
+ }
+ find_exponent(mp);
+ stop = &mp->buffer[mp->cur_input.loc_field-1];
+ mp_wrapup_numeric_token (mp, start, stop);
+}
+
+
+@ We just have to collect bytes.
+
+@c
+void mp_decimal_scan_numeric_token (MP mp, int n) { /* n: scaled */
+ unsigned char *start = &mp->buffer[mp->cur_input.loc_field -1];
+ unsigned char *stop;
+ while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class) {
+ mp->cur_input.loc_field++;
+ }
+ if (mp->buffer[mp->cur_input.loc_field] == '.' &&
+ mp->buffer[mp->cur_input.loc_field+1] != '.') {
+ mp->cur_input.loc_field++;
+ while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class) {
+ mp->cur_input.loc_field++;
+ }
+ }
+ find_exponent(mp);
+ stop = &mp->buffer[mp->cur_input.loc_field-1];
+ mp_wrapup_numeric_token (mp, start, stop);
+}
+
+@ The |scaled| quantities in \MP\ programs are generally supposed to be
+less than $2^{12}$ in absolute value, so \MP\ does much of its internal
+arithmetic with 28~significant bits of precision. A |fraction| denotes
+a scaled integer whose binary point is assumed to be 28 bit positions
+from the right.
+
+@d fraction_half (fraction_multiplier/2)
+@d fraction_one (1*fraction_multiplier)
+@d fraction_two (2*fraction_multiplier)
+@d fraction_three (3*fraction_multiplier)
+@d fraction_four (4*fraction_multiplier)
+
+@ Here is a typical example of how the routines above can be used.
+It computes the function
+$${1\over3\tau}f(\theta,\phi)=
+{\tau^{-1}\bigl(2+\sqrt2\,(\sin\theta-{1\over16}\sin\phi)
+ (\sin\phi-{1\over16}\sin\theta)(\cos\theta-\cos\phi)\bigr)\over
+3\,\bigl(1+{1\over2}(\sqrt5-1)\cos\theta+{1\over2}(3-\sqrt5\,)\cos\phi\bigr)},$$
+where $\tau$ is a |scaled| ``tension'' parameter. This is \MP's magic
+fudge factor for placing the first control point of a curve that starts
+at an angle $\theta$ and ends at an angle $\phi$ from the straight path.
+(Actually, if the stated quantity exceeds 4, \MP\ reduces it to~4.)
+
+The trigonometric quantity to be multiplied by $\sqrt2$ is less than $\sqrt2$.
+(It's a sum of eight terms whose absolute values can be bounded using
+relations such as $\sin\theta\cos\theta\L{1\over2}$.) Thus the numerator
+is positive; and since the tension $\tau$ is constrained to be at least
+$3\over4$, the numerator is less than $16\over3$. The denominator is
+nonnegative and at most~6.
+
+The angles $\theta$ and $\phi$ are given implicitly in terms of |fraction|
+arguments |st|, |ct|, |sf|, and |cf|, representing $\sin\theta$, $\cos\theta$,
+$\sin\phi$, and $\cos\phi$, respectively.
+
+@c
+void mp_decimal_velocity (MP mp, mp_number *ret, mp_number st, mp_number ct, mp_number sf,
+ mp_number cf, mp_number t) {
+ decNumber acc, num, denom; /* registers for intermediate calculations */
+ decNumber r1, r2;
+ decNumber arg1, arg2;
+ decNumber i16, fone, fhalf, ftwo, sqrtfive;
+ decNumberFromInt32(&i16, 16);
+ decNumberFromInt32(&fone, fraction_one);
+ decNumberFromInt32(&fhalf, fraction_half);
+ decNumberFromInt32(&ftwo, fraction_two);
+ decNumberFromInt32(&sqrtfive, 5); /*$\sqrt{5}$*/
+ decNumberSquareRoot(&sqrtfive, &sqrtfive, &set);
+
+
+ decNumberDivide(&arg1,sf.data.num, &i16, &set); /* arg1 = sf / 16*/
+ decNumberSubtract(&arg1,st.data.num,&arg1, &set); /* arg1 = st - arg1*/
+ decNumberDivide(&arg2,st.data.num, &i16, &set); /* arg2 = st / 16*/
+ decNumberSubtract(&arg2,sf.data.num,&arg2, &set); /* arg2 = sf - arg2*/
+ mp_decimal_take_fraction (mp, &acc, &arg1, &arg2); /* acc = (arg1 * arg2) / fmul*/
+
+ decNumberCopy(&arg1, &acc);
+ decNumberSubtract(&arg2, ct.data.num, cf.data.num, &set); /* arg2 = ct - cf*/
+ mp_decimal_take_fraction (mp, &acc, &arg1, &arg2); /* acc = (arg1 * arg2 ) / fmul*/
+
+ decNumberSquareRoot(&arg1, &two_decNumber, &set); /* arg1 = $\sqrt{2}$*/
+ decNumberMultiply(&arg1, &arg1, &fone, &set); /* arg1 = arg1 * fmul*/
+ mp_decimal_take_fraction (mp, &r1, &acc, &arg1); /* r1 = (acc * arg1) / fmul*/
+ decNumberAdd(&num, &ftwo, &r1, &set); /* num = ftwo + r1*/
+
+ decNumberSubtract(&arg1,&sqrtfive, &one, &set); /* arg1 = $\sqrt{5}$ - 1*/
+ decNumberMultiply(&arg1,&arg1,&fhalf, &set); /* arg1 = arg1 * fmul/2*/
+ decNumberMultiply(&arg1,&arg1,&three_decNumber, &set); /* arg1 = arg1 * 3*/
+
+ decNumberSubtract(&arg2,&three_decNumber, &sqrtfive, &set); /* arg2 = 3 - $\sqrt{5}$*/
+ decNumberMultiply(&arg2,&arg2,&fhalf, &set); /* arg2 = arg2 * fmul/2*/
+ decNumberMultiply(&arg2,&arg2,&three_decNumber, &set); /* arg2 = arg2 * 3*/
+ mp_decimal_take_fraction (mp, &r1, ct.data.num, &arg1) ; /* r1 = (ct * arg1) / fmul*/
+ mp_decimal_take_fraction (mp, &r2, cf.data.num, &arg2); /* r2 = (cf * arg2) / fmul*/
+
+ decNumberFromInt32(&denom, fraction_three); /* denom = 3fmul*/
+ decNumberAdd(&denom, &denom, &r1, &set); /* denom = denom + r1*/
+ decNumberAdd(&denom, &denom, &r2, &set); /* denom = denom + r1*/
+
+ decNumberCompare(&arg1, t.data.num, &one, &set);
+ if (!decNumberIsZero(&arg1)) { /* t != r1*/
+ decNumberDivide(&num, &num, t.data.num, &set); /* num = num / t*/
+ }
+ decNumberCopy(&r2, &num); /* r2 = num / 4*/
+ decNumberDivide(&r2, &r2, &four_decNumber, &set);
+ if (decNumberLess(&denom,&r2)) { /* num/4 >= denom => denom < num/4*/
+ decNumberFromInt32(ret->data.num,fraction_four);
+ } else {
+ mp_decimal_make_fraction (mp, ret->data.num, &num, &denom);
+ }
+#if DEBUG
+ fprintf(stdout, "\n%f = velocity(%f,%f,%f,%f,%f)", mp_number_to_double(*ret),
+mp_number_to_double(st),mp_number_to_double(ct),
+mp_number_to_double(sf),mp_number_to_double(cf),
+mp_number_to_double(t));
+#endif
+ mp_check_decNumber(mp, ret->data.num, &set);
+}
+
+
+@ The following somewhat different subroutine tests rigorously if $ab$ is
+greater than, equal to, or less than~$cd$,
+given integers $(a,b,c,d)$. In most cases a quick decision is reached.
+The result is $+1$, 0, or~$-1$ in the three respective cases.
+
+@c
+void mp_ab_vs_cd (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig, mp_number c_orig, mp_number d_orig) {
+ decNumber q, r, test; /* temporary registers */
+ decNumber a, b, c, d;
+ decNumber ab, cd;
+ (void)mp;
+ decNumberCopy(&a, (decNumber *)a_orig.data.num);
+ decNumberCopy(&b, (decNumber *)b_orig.data.num);
+ decNumberCopy(&c, (decNumber *)c_orig.data.num);
+ decNumberCopy(&d, (decNumber *)d_orig.data.num);
+
+ decNumberMultiply (&ab, (decNumber *)a_orig.data.num, (decNumber *)b_orig.data.num, &set);
+ decNumberMultiply (&cd, (decNumber *)c_orig.data.num, (decNumber *)d_orig.data.num, &set);
+ decNumberCompare(ret->data.num, &ab, &cd, &set);
+ mp_check_decNumber(mp, ret->data.num, &set);
+ if (1>0)
+ return;
+
+
+ @<Reduce to the case that |a,c>=0|, |b,d>0|@>;
+ while (1) {
+ decNumberDivide(&q,&a,&d, &set);
+ decNumberDivide(&r,&c,&b, &set);
+ decNumberCompare(&test,&q,&r, &set);
+ if (!decNumberIsZero(&test)) {
+ if (decNumberIsPositive(&test)) {
+ decNumberCopy(ret->data.num, &one);
+ } else {
+ decNumberCopy(ret->data.num, &minusone);
+ }
+ goto RETURN;
+ }
+ decNumberRemainder(&q,&a,&d, &set);
+ decNumberRemainder(&r,&c,&b, &set);
+ if (decNumberIsZero(&r)) {
+ if (decNumberIsZero(&q)) {
+ decNumberCopy(ret->data.num, &zero);
+ } else {
+ decNumberCopy(ret->data.num, &one);
+ }
+ goto RETURN;
+ }
+ if (decNumberIsZero(&q)) {
+ decNumberCopy(ret->data.num, &minusone);
+ goto RETURN;
+ }
+ decNumberCopy(&a,&b);
+ decNumberCopy(&b,&q);
+ decNumberCopy(&c,&d);
+ decNumberCopy(&d,&r);
+ } /* now |a>d>0| and |c>b>0| */
+RETURN:
+#if DEBUG
+ fprintf(stdout, "\n%f = ab_vs_cd(%f,%f,%f,%f)", mp_number_to_double(*ret),
+mp_number_to_double(a_orig),mp_number_to_double(b_orig),
+mp_number_to_double(c_orig),mp_number_to_double(d_orig));
+#endif
+ mp_check_decNumber(mp, ret->data.num, &set);
+ return;
+}
+
+
+@ @<Reduce to the case that |a...@>=
+if (decNumberIsNegative(&a)) {
+ decNumberCopyNegate(&a, &a);
+ decNumberCopyNegate(&b, &b);
+}
+if (decNumberIsNegative(&c)) {
+ decNumberCopyNegate(&c, &c);
+ decNumberCopyNegate(&d, &d);
+}
+if (!decNumberIsPositive(&d)) {
+ if (!decNumberIsNegative(&b)) {
+ if ((decNumberIsZero(&a) || decNumberIsZero(&b)) && (decNumberIsZero(&c) || decNumberIsZero(&d)))
+ decNumberCopy(ret->data.num, &zero);
+ else
+ decNumberCopy(ret->data.num, &one);
+ goto RETURN;
+ }
+ if (decNumberIsZero(&d)) {
+ if (decNumberIsZero(&a))
+ decNumberCopy(ret->data.num, &zero);
+ else
+ decNumberCopy(ret->data.num, &minusone);
+ goto RETURN;
+ }
+ decNumberCopy(&q, &a);
+ decNumberCopy(&a, &c);
+ decNumberCopy(&c, &q);
+ decNumberCopyNegate(&q, &b);
+ decNumberCopyNegate(&b, &d);
+ decNumberCopy(&d, &q);
+} else if (!decNumberIsPositive(&b)) {
+ if (decNumberIsNegative(&b) && decNumberIsPositive(&a)) {
+ decNumberCopy(ret->data.num, &minusone);
+ goto RETURN;
+ }
+ if (decNumberIsZero(&c))
+ decNumberCopy(ret->data.num, &zero);
+ else
+ decNumberCopy(ret->data.num, &minusone);
+ goto RETURN;
+}
+
+@ Now here's a subroutine that's handy for all sorts of path computations:
+Given a quadratic polynomial $B(a,b,c;t)$, the |crossing_point| function
+returns the unique |fraction| value |t| between 0 and~1 at which
+$B(a,b,c;t)$ changes from positive to negative, or returns
+|t=fraction_one+1| if no such value exists. If |a<0| (so that $B(a,b,c;t)$
+is already negative at |t=0|), |crossing_point| returns the value zero.
+
+The general bisection method is quite simple when $n=2$, hence
+|crossing_point| does not take much time. At each stage in the
+recursion we have a subinterval defined by |l| and~|j| such that
+$B(a,b,c;2^{-l}(j+t))=B(x_0,x_1,x_2;t)$, and we want to ``zero in'' on
+the subinterval where $x_0\G0$ and $\min(x_1,x_2)<0$.
+
+It is convenient for purposes of calculation to combine the values
+of |l| and~|j| in a single variable $d=2^l+j$, because the operation
+of bisection then corresponds simply to doubling $d$ and possibly
+adding~1. Furthermore it proves to be convenient to modify
+our previous conventions for bisection slightly, maintaining the
+variables $X_0=2^lx_0$, $X_1=2^l(x_0-x_1)$, and $X_2=2^l(x_1-x_2)$.
+With these variables the conditions $x_0\ge0$ and $\min(x_1,x_2)<0$ are
+equivalent to $\max(X_1,X_1+X_2)>X_0\ge0$.
+
+The following code maintains the invariant relations
+$0\L|x0|<\max(|x1|,|x1|+|x2|)$,
+$\vert|x1|\vert<2^{30}$, $\vert|x2|\vert<2^{30}$;
+it has been constructed in such a way that no arithmetic overflow
+will occur if the inputs satisfy
+$a<2^{30}$, $\vert a-b\vert<2^{30}$, and $\vert b-c\vert<2^{30}$.
+
+@d no_crossing { decNumberCopy(ret->data.num, &fraction_one_plus_decNumber); goto RETURN; }
+@d one_crossing { decNumberCopy(ret->data.num, &fraction_one_decNumber); goto RETURN; }
+@d zero_crossing { decNumberCopy(ret->data.num, &zero); goto RETURN; }
+
+@c
+static void mp_decimal_crossing_point (MP mp, mp_number *ret, mp_number aa, mp_number bb, mp_number cc) {
+ decNumber a,b,c;
+ double d; /* recursive counter */
+ decNumber x, xx, x0, x1, x2; /* temporary registers for bisection */
+ decNumber scratch, scratch2;
+ decNumberCopy(&a, (decNumber *)aa.data.num);
+ decNumberCopy(&b, (decNumber *)bb.data.num);
+ decNumberCopy(&c, (decNumber *)cc.data.num);
+ if (decNumberIsNegative(&a))
+ zero_crossing;
+ if (!decNumberIsNegative(&c)) {
+ if (!decNumberIsNegative(&b)) {
+ if (decNumberIsPositive(&c)) {
+ no_crossing;
+ } else if (decNumberIsZero(&a) && decNumberIsZero(&b)) {
+ no_crossing;
+ } else {
+ one_crossing;
+ }
+ }
+ if (decNumberIsZero(&a))
+ zero_crossing;
+ } else if (decNumberIsZero(&a)) {
+ if (!decNumberIsPositive(&b))
+ zero_crossing;
+ }
+
+ /* Use bisection to find the crossing point... */
+ d = epsilonf;
+ decNumberCopy(&x0, &a);
+ decNumberSubtract(&x1,&a, &b, &set);
+ decNumberSubtract(&x2,&b, &c, &set);
+ /* not sure why the error correction has to be >= 1E-12 */
+ decNumberFromDouble(&scratch2, 1E-12);
+ do {
+ decNumberAdd(&x, &x1, &x2, &set);
+ decNumberDivide(&x, &x, &two_decNumber, &set);
+ decNumberAdd(&x, &x, &scratch2, &set);
+ decNumberSubtract(&scratch, &x1, &x0, &set);
+ if (decNumberGreater(&scratch, &x0)) {
+ decNumberCopy(&x2, &x);
+ decNumberAdd(&x0, &x0, &x0, &set);
+ d += d;
+ } else {
+ decNumberAdd(&xx, &scratch, &x, &set);
+ if (decNumberGreater(&xx,&x0)) {
+ decNumberCopy(&x2,&x);
+ decNumberAdd(&x0, &x0, &x0, &set);
+ d += d;
+ } else {
+ decNumberSubtract(&x0, &x0, &xx, &set);
+ if (!decNumberGreater(&x,&x0)) {
+ decNumberAdd(&scratch, &x, &x2, &set);
+ if (!decNumberGreater(&scratch, &x0))
+ no_crossing;
+ }
+ decNumberCopy(&x1,&x);
+ d = d + d + epsilonf;
+ }
+ }
+ } while (d < fraction_one);
+ decNumberFromDouble(&scratch, d);
+ decNumberSubtract(ret->data.num,&scratch, &fraction_one_decNumber, &set);
+RETURN:
+#if DEBUG
+ fprintf(stdout, "\n%f = crossing_point(%f,%f,%f)", mp_number_to_double(*ret),
+mp_number_to_double(aa),mp_number_to_double(bb),mp_number_to_double(cc));
+#endif
+ mp_check_decNumber(mp, ret->data.num, &set);
+ return;
+}
+
+
+@ We conclude this set of elementary routines with some simple rounding
+and truncation operations.
+
+
+@ |round_unscaled| rounds a |scaled| and converts it to |int|
+@c
+int mp_round_unscaled(mp_number x_orig) {
+ double xx = mp_number_to_double(x_orig);
+ int x = (int)ROUND(xx);
+ return x;
+}
+
+@ |number_floor| floors a number
+
+@c
+void mp_number_floor (mp_number *i) {
+ int round = set.round;
+ set.round = DEC_ROUND_FLOOR;
+ decNumberToIntegralValue(i->data.num, i->data.num, &set);
+ set.round = round;
+}
+
+@ |fraction_to_scaled| rounds a |fraction| and converts it to |scaled|
+@c
+void mp_decimal_fraction_to_round_scaled (mp_number *x_orig) {
+ x_orig->type = mp_scaled_type;
+ decNumberDivide(x_orig->data.num, x_orig->data.num, &fraction_multiplier_decNumber, &set);
+}
+
+
+
+@* Algebraic and transcendental functions.
+\MP\ computes all of the necessary special functions from scratch, without
+relying on |real| arithmetic or system subroutines for sines, cosines, etc.
+
+@
+
+@c
+void mp_decimal_square_rt (MP mp, mp_number *ret, mp_number x_orig) { /* return, x: scaled */
+ decNumber x;
+ decNumberCopy(&x, x_orig.data.num);
+ if (!decNumberIsPositive(&x)) {
+ @<Handle square root of zero or negative argument@>;
+ } else {
+ decNumberSquareRoot(ret->data.num, &x, &set);
+ }
+ mp_check_decNumber(mp, ret->data.num, &set);
+}
+
+
+@ @<Handle square root of zero...@>=
+{
+ if (decNumberIsNegative(&x)) {
+ char msg[256];
+ const char *hlp[] = {
+ "Since I don't take square roots of negative numbers,",
+ "I'm zeroing this one. Proceed, with fingers crossed.",
+ NULL };
+ char *xstr = mp_decimal_number_tostring (mp, x_orig);
+ mp_snprintf(msg, 256, "Square root of %s has been replaced by 0", xstr);
+ free(xstr);
+@.Square root...replaced by 0@>;
+ mp_error (mp, msg, hlp, true);
+ }
+ decNumberZero(ret->data.num);
+ return;
+}
+
+
+@ Pythagorean addition $\psqrt{a^2+b^2}$ is implemented by a quick hack
+
+@c
+void mp_decimal_pyth_add (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig) {
+ decNumber a, b;
+ decNumber asq, bsq;
+ decNumberCopyAbs(&a, a_orig.data.num);
+ decNumberCopyAbs(&b, b_orig.data.num);
+ decNumberMultiply(&asq, &a, &a, &set);
+ decNumberMultiply(&bsq, &b, &b, &set);
+ decNumberAdd(&a, &asq, &bsq, &set);
+ decNumberSquareRoot(ret->data.num, &a, &set);
+ /*|if (set.status != 0) {|*/
+ /*| mp->arith_error = true;|*/
+ /*| decNumberCopy(ret->data.num, &EL_GORDO_decNumber);|*/
+ /*|}|*/
+ mp_check_decNumber(mp, ret->data.num, &set);
+}
+
+@ Here is a similar algorithm for $\psqrt{a^2-b^2}$. Same quick hack, also.
+
+@c
+void mp_decimal_pyth_sub (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig) {
+ decNumber a, b;
+ decNumberCopyAbs(&a, a_orig.data.num);
+ decNumberCopyAbs(&b, b_orig.data.num);
+ if (!decNumberGreater(&a,&b)) {
+ @<Handle erroneous |pyth_sub| and set |a:=0|@>;
+ } else {
+ decNumber asq, bsq;
+ decNumberMultiply(&asq, &a, &a, &set);
+ decNumberMultiply(&bsq, &b, &b, &set);
+ decNumberSubtract(&a, &asq, &bsq, &set);
+ decNumberSquareRoot(&a, &a, &set);
+ }
+ decNumberCopy(ret->data.num, &a);
+ mp_check_decNumber(mp, ret->data.num, &set);
+}
+
+
+@ @<Handle erroneous |pyth_sub| and set |a:=0|@>=
+{
+ if (decNumberLess(&a, &b)) {
+ char msg[256];
+ const char *hlp[] = {
+ "Since I don't take square roots of negative numbers,",
+ "I'm zeroing this one. Proceed, with fingers crossed.",
+ NULL };
+ char *astr = mp_decimal_number_tostring (mp, a_orig);
+ char *bstr = mp_decimal_number_tostring (mp, b_orig);
+ mp_snprintf (msg, 256, "Pythagorean subtraction %s+-+%s has been replaced by 0", astr, bstr);
+ free(astr);
+ free(bstr);
+@.Pythagorean...@>;
+ mp_error (mp, msg, hlp, true);
+ }
+ decNumberZero(&a);
+}
+
+
+@ Here is the routine that calculates $2^8$ times the natural logarithm
+of a |scaled| quantity;
+
+@c
+void mp_decimal_m_log (MP mp, mp_number *ret, mp_number x_orig) {
+ if (!decNumberIsPositive((decNumber *)x_orig.data.num)) {
+ @<Handle non-positive logarithm@>;
+ } else {
+ decNumber twofivesix;
+ decNumberFromInt32(&twofivesix, 256);
+ decNumberLn(ret->data.num, x_orig.data.num, &limitedset);
+ mp_check_decNumber(mp, ret->data.num, &limitedset);
+ decNumberMultiply(ret->data.num, ret->data.num, &twofivesix, &set);
+ }
+ mp_check_decNumber(mp, ret->data.num, &set);
+}
+
+@ @<Handle non-positive logarithm@>=
+{
+ char msg[256];
+ const char *hlp[] = {
+ "Since I don't take logs of non-positive numbers,",
+ "I'm zeroing this one. Proceed, with fingers crossed.",
+ NULL };
+ char *xstr = mp_decimal_number_tostring (mp, x_orig);
+ mp_snprintf (msg, 256, "Logarithm of %s has been replaced by 0", xstr);
+ free (xstr);
+@.Logarithm...replaced by 0@>;
+ mp_error (mp, msg, hlp, true);
+ decNumberZero(ret->data.num);
+}
+
+
+@ Conversely, the exponential routine calculates $\exp(x/2^8)$,
+when |x| is |scaled|.
+
+@c
+void mp_decimal_m_exp (MP mp, mp_number *ret, mp_number x_orig) {
+ decNumber temp, twofivesix;
+ decNumberFromInt32(&twofivesix, 256);
+ decNumberDivide(&temp, x_orig.data.num, &twofivesix, &set);
+ limitedset.status = 0;
+ decNumberExp(ret->data.num, &temp, &limitedset);
+ if (limitedset.status & DEC_Clamped) {
+ if (decNumberIsPositive((decNumber *)x_orig.data.num)) {
+ mp->arith_error = true;
+ decNumberCopy(ret->data.num, &EL_GORDO_decNumber);
+ } else {
+ decNumberZero(ret->data.num);
+ }
+ }
+ mp_check_decNumber(mp, ret->data.num, &limitedset);
+ limitedset.status = 0;
+}
+
+
+@ Given integers |x| and |y|, not both zero, the |n_arg| function
+returns the |angle| whose tangent points in the direction $(x,y)$.
+
+@c
+void mp_decimal_n_arg (MP mp, mp_number *ret, mp_number x_orig, mp_number y_orig) {
+ if (decNumberIsZero((decNumber *)x_orig.data.num) && decNumberIsZero((decNumber *)y_orig.data.num)) {
+ @<Handle undefined arg@>;
+ } else {
+ decNumber atan2val, oneeighty_angle;
+ ret->type = mp_angle_type;
+ decNumberFromInt32(&oneeighty_angle, 180 * angle_multiplier);
+ decNumberDivide(&oneeighty_angle, &oneeighty_angle, &PI_decNumber, &set);
+ checkZero(y_orig.data.num);
+ checkZero(x_orig.data.num);
+ decNumberAtan2(&atan2val, y_orig.data.num, x_orig.data.num, &set);
+#if DEBUG
+ fprintf(stdout, "\n%g = atan2(%g,%g)", decNumberToDouble(&atan2val),mp_number_to_double(x_orig),mp_number_to_double(y_orig));
+#endif
+ decNumberMultiply(ret->data.num,&atan2val, &oneeighty_angle, &set);
+ checkZero(ret->data.num);
+#if DEBUG
+ fprintf(stdout, "\nn_arg(%g,%g,%g)", mp_number_to_double(*ret),
+ mp_number_to_double(x_orig),mp_number_to_double(y_orig));
+#endif
+ }
+ mp_check_decNumber(mp, ret->data.num, &set);
+}
+
+
+@ @<Handle undefined arg@>=
+{
+ const char *hlp[] = {
+ "The `angle' between two identical points is undefined.",
+ "I'm zeroing this one. Proceed, with fingers crossed.",
+ NULL };
+ mp_error (mp, "angle(0,0) is taken as zero", hlp, true);
+@.angle(0,0)...zero@>;
+ decNumberZero(ret->data.num);
+}
+
+
+@ Conversely, the |n_sin_cos| routine takes an |angle| and produces the sine
+and cosine of that angle. The results of this routine are
+stored in global integer variables |n_sin| and |n_cos|.
+
+First, we need a decNumber function that calculates sines and cosines
+using the Taylor series. This function is fairly optimized.
+
+@d PRECALC_FACTORIALS_CACHESIZE 50
+
+@c
+static void sinecosine(decNumber *theangle, decNumber *c, decNumber *s)
+{
+ int n, i, prec;
+ decNumber p, pxa, fac, cc;
+ decNumber n1, n2, p1;
+ decNumberZero(c);
+ decNumberZero(s);
+ prec = (set.digits/2);
+ if (prec < DECPRECISION_DEFAULT) prec = DECPRECISION_DEFAULT;
+ for (n=0;n<prec;n++)
+ {
+ decNumberFromInt32(&p1, n);
+ decNumberFromInt32(&n1, 2*n);
+ decNumberPower(&p, &minusone, &p1, &limitedset);
+ if (n==0) {
+ decNumberCopy(&pxa, &one);
+ } else {
+ decNumberPower(&pxa, theangle, &n1, &limitedset);
+ }
+
+ if (2*n<last_cached_factorial) {
+ decNumberCopy(&fac,factorials[2*n]);
+ } else {
+ decNumberCopy(&fac,factorials[last_cached_factorial]);
+ for (i = last_cached_factorial+1; i <= 2*n; i++) {
+ decNumberFromInt32(&cc, i);
+ decNumberMultiply (&fac, &fac, &cc, &set);
+ if (i<PRECALC_FACTORIALS_CACHESIZE) {
+ factorials[i] = malloc(sizeof(decNumber));
+ decNumberCopy(factorials[i],&fac);
+ last_cached_factorial = i;
+ }
+ }
+ }
+
+ decNumberDivide (&pxa, &pxa, &fac, &set);
+ decNumberMultiply (&pxa, &pxa, &p, &set);
+ decNumberAdd (s, s, &pxa, &set);
+
+ decNumberFromInt32(&n2, 2*n+1);
+ decNumberMultiply (&fac, &fac, &n2, &set); /* fac = fac * (2*n+1)*/
+ decNumberPower(&pxa, theangle, &n2, &limitedset);
+ decNumberDivide (&pxa, &pxa, &fac, &set);
+ decNumberMultiply (&pxa, &pxa, &p, &set);
+ decNumberAdd (c, c, &pxa, &set);
+ /* |printf("\niteration %2d: %-42s %-42s",n,tostring(c), tostring(s));|*/
+ }
+}
+
+@ Calculate sines and cosines.
+@c
+void mp_decimal_sin_cos (MP mp, mp_number z_orig, mp_number *n_cos, mp_number *n_sin) {
+ decNumber rad;
+ double tmp;
+ decNumber one_eighty;
+ tmp = mp_number_to_double(z_orig)/16.0;
+
+#if DEBUG
+ fprintf(stdout, "\nsin_cos(%f)", mp_number_to_double(z_orig));
+#endif
+#if 0
+ if (decNumberIsNegative(&rad)) {
+ while (decNumberLess(&rad,&PI_decNumber))
+ decNumberAdd(&rad, &rad, &PI_decNumber, &set);
+ } else {
+ while (decNumberGreater(&rad,&PI_decNumber))
+ decNumberSubtract(&rad, &rad, &PI_decNumber, &set);
+ }
+#endif
+ if ((tmp == 90.0)||(tmp == -270)){
+ decNumberZero(n_cos->data.num);
+ decNumberCopy(n_sin->data.num,&fraction_multiplier_decNumber);
+ } else if ((tmp == -90.0)||(tmp == 270.0)) {
+ decNumberZero(n_cos->data.num);
+ decNumberCopyNegate(n_sin->data.num,&fraction_multiplier_decNumber);
+ } else if ((tmp == 180.0) || (tmp == -180.0)) {
+ decNumberCopyNegate(n_cos->data.num,&fraction_multiplier_decNumber);
+ decNumberZero(n_sin->data.num);
+ } else {
+ decNumberFromInt32(&one_eighty, 180 * 16);
+ decNumberMultiply(&rad, z_orig.data.num, &PI_decNumber, &set);
+ decNumberDivide(&rad, &rad, &one_eighty, &set);
+ sinecosine(&rad, n_sin->data.num, n_cos->data.num);
+ decNumberMultiply(n_cos->data.num,n_cos->data.num,&fraction_multiplier_decNumber, &set);
+ decNumberMultiply(n_sin->data.num,n_sin->data.num,&fraction_multiplier_decNumber, &set);
+ }
+#if DEBUG
+ fprintf(stdout, "\nsin_cos(%f,%f,%f)", decNumberToDouble(&rad),
+mp_number_to_double(*n_cos), mp_number_to_double(*n_sin));
+#endif
+ mp_check_decNumber(mp, n_cos->data.num, &set);
+ mp_check_decNumber(mp, n_sin->data.num, &set);
+}
+
+@ This is the {\tt http://www-cs-faculty.stanford.edu/~uno/programs/rng.c}
+with small cosmetic modifications.
+
+@c
+#define KK 100 /* the long lag */
+#define LL 37 /* the short lag */
+#define MM (1L<<30) /* the modulus */
+#define mod_diff(x,y) (((x)-(y))&(MM-1)) /* subtraction mod MM */
+/* */
+static long ran_x[KK]; /* the generator state */
+/* */
+static void ran_array(long aa[],int n) /* put n new random numbers in aa */
+ /* long aa[] destination */
+ /* int n array length (must be at least KK) */
+{
+ register int i,j;
+ for (j=0;j<KK;j++) aa[j]=ran_x[j];
+ for (;j<n;j++) aa[j]=mod_diff(aa[j-KK],aa[j-LL]);
+ for (i=0;i<LL;i++,j++) ran_x[i]=mod_diff(aa[j-KK],aa[j-LL]);
+ for (;i<KK;i++,j++) ran_x[i]=mod_diff(aa[j-KK],ran_x[i-LL]);
+}
+/* */
+/* the following routines are from exercise 3.6--15 */
+/* after calling |ran_start|, get new randoms by, e.g., "|x=ran_arr_next()|" */
+/* */
+#define QUALITY 1009 /* recommended quality level for high-res use */
+static long ran_arr_buf[QUALITY];
+static long ran_arr_dummy=-1, ran_arr_started=-1;
+static long *ran_arr_ptr=&ran_arr_dummy; /* the next random number, or -1 */
+/* */
+#define TT 70 /* guaranteed separation between streams */
+#define is_odd(x) ((x)&1) /* units bit of x */
+/* */
+static void ran_start(long seed) /* do this before using |ran_array| */
+ /* |long seed| selector for different streams */
+{
+ register int t,j;
+ long x[KK+KK-1]; /* the preparation buffer */
+ register long ss=(seed+2)&(MM-2);
+ for (j=0;j<KK;j++) {
+ x[j]=ss; /* bootstrap the buffer */
+ ss<<=1; if (ss>=MM) ss-=MM-2; /* cyclic shift 29 bits */
+ }
+ x[1]++; /* make x[1] (and only x[1]) odd */
+ for (ss=seed&(MM-1),t=TT-1; t; ) {
+ for (j=KK-1;j>0;j--) x[j+j]=x[j], x[j+j-1]=0; /* "square" */
+ for (j=KK+KK-2;j>=KK;j--)
+ x[j-(KK-LL)]=mod_diff(x[j-(KK-LL)],x[j]),
+ x[j-KK]=mod_diff(x[j-KK],x[j]);
+ if (is_odd(ss)) { /* "multiply by z" */
+ for (j=KK;j>0;j--) x[j]=x[j-1];
+ x[0]=x[KK]; /* shift the buffer cyclically */
+ x[LL]=mod_diff(x[LL],x[KK]);
+ }
+ if (ss) ss>>=1; else t--;
+ }
+ for (j=0;j<LL;j++) ran_x[j+KK-LL]=x[j];
+ for (;j<KK;j++) ran_x[j-LL]=x[j];
+ for (j=0;j<10;j++) ran_array(x,KK+KK-1); /* warm things up */
+ ran_arr_ptr=&ran_arr_started;
+}
+/* */
+#define ran_arr_next() (*ran_arr_ptr>=0? *ran_arr_ptr++: ran_arr_cycle())
+static long ran_arr_cycle(void)
+{
+ if (ran_arr_ptr==&ran_arr_dummy)
+ ran_start(314159L); /* the user forgot to initialize */
+ ran_array(ran_arr_buf,QUALITY);
+ ran_arr_buf[KK]=-1;
+ ran_arr_ptr=ran_arr_buf+1;
+ return ran_arr_buf[0];
+}
+
+
+
+@ To initialize the |randoms| table, we call the following routine.
+
+@c
+void mp_init_randoms (MP mp, int seed) {
+ int j, jj, k; /* more or less random integers */
+ int i; /* index into |randoms| */
+ j = abs (seed);
+ while (j >= fraction_one) {
+ j = j/2;
+ }
+ k = 1;
+ for (i = 0; i <= 54; i++) {
+ jj = k;
+ k = j - k;
+ j = jj;
+ if (k<0)
+ k += fraction_one;
+ decNumberFromInt32(mp->randoms[(i * 21) % 55].data.num, j);
+ }
+ mp_new_randoms (mp);
+ mp_new_randoms (mp);
+ mp_new_randoms (mp); /* ``warm up'' the array */
+
+ ran_start((unsigned long) seed);
+
+}
+
+@ @c
+void mp_decimal_number_modulo (mp_number *a, mp_number b) {
+ decNumberRemainder(a->data.num, a->data.num, b.data.num, &set);
+}
+
+
+@ To consume a random integer for the uniform generator, the program below will say `|next_unif_random|'.
+
+@c
+static void mp_next_unif_random (MP mp, mp_number *ret) {
+ decNumber a;
+ decNumber b;
+ unsigned long int op;
+ (void)mp;
+ op = (unsigned)ran_arr_next();
+ decNumberFromInt32(&a, op);
+ decNumberFromInt32(&b, MM);
+ decNumberDivide (&a, &a, &b, &set); /* a = a/b */
+ decNumberCopy(ret->data.num, &a);
+ mp_check_decNumber(mp, ret->data.num, &set);
+}
+
+
+@ To consume a random fraction, the program below will say `|next_random|'.
+
+@c
+static void mp_next_random (MP mp, mp_number *ret) {
+ if ( mp->j_random==0 )
+ mp_new_randoms(mp);
+ else
+ mp->j_random = mp->j_random-1;
+ mp_number_clone (ret, mp->randoms[mp->j_random]);
+}
+
+
+@ To produce a uniform random number in the range |0<=u<x| or |0>=u>x|
+or |0=u=x|, given a |scaled| value~|x|, we proceed as shown here.
+
+Note that the call of |take_fraction| will produce the values 0 and~|x|
+with about half the probability that it will produce any other particular
+values between 0 and~|x|, because it rounds its answers.
+
+@c
+static void mp_decimal_m_unif_rand (MP mp, mp_number *ret, mp_number x_orig) {
+ mp_number y; /* trial value */
+ mp_number x, abs_x;
+ mp_number u;
+ new_fraction (y);
+ new_number (x);
+ new_number (abs_x);
+ new_number (u);
+ mp_number_clone (&x, x_orig);
+ mp_number_clone (&abs_x, x);
+ mp_decimal_abs (&abs_x);
+ mp_next_unif_random(mp, &u);
+ decNumberMultiply (y.data.num, abs_x.data.num, u.data.num, &set);
+ free_number (u);
+ if (mp_number_equal(y, abs_x)) {
+ mp_number_clone (ret, ((math_data *)mp->math)->zero_t);
+ } else if (mp_number_greater(x, ((math_data *)mp->math)->zero_t)) {
+ mp_number_clone (ret, y);
+ } else {
+ mp_number_clone (ret, y);
+ mp_number_negate (ret);
+ }
+ free_number (abs_x);
+ free_number (x);
+ free_number (y);
+}
+
+
+
+@ Finally, a normal deviate with mean zero and unit standard deviation
+can readily be obtained with the ratio method (Algorithm 3.4.1R in
+{\sl The Art of Computer Programming\/}).
+
+@c
+static void mp_decimal_m_norm_rand (MP mp, mp_number *ret) {
+ mp_number ab_vs_cd;
+ mp_number abs_x;
+ mp_number u;
+ mp_number r;
+ mp_number la, xa;
+ new_number (ab_vs_cd);
+ new_number (la);
+ new_number (xa);
+ new_number (abs_x);
+ new_number (u);
+ new_number (r);
+
+ do {
+ do {
+ mp_number v;
+ new_number (v);
+ mp_next_random(mp, &v);
+ mp_number_substract (&v, ((math_data *)mp->math)->fraction_half_t);
+ mp_decimal_number_take_fraction (mp,&xa, ((math_data *)mp->math)->sqrt_8_e_k, v);
+ free_number (v);
+ mp_next_random(mp, &u);
+ mp_number_clone (&abs_x, xa);
+ mp_decimal_abs (&abs_x);
+ } while (!mp_number_less(abs_x, u));
+ mp_decimal_number_make_fraction (mp, &r, xa, u);
+ mp_number_clone (&xa, r);
+ mp_decimal_m_log (mp,&la, u);
+ mp_set_decimal_from_substraction(&la, ((math_data *)mp->math)->twelve_ln_2_k, la);
+ mp_ab_vs_cd (mp,&ab_vs_cd, ((math_data *)mp->math)->one_k, la, xa, xa);
+ } while (mp_number_less(ab_vs_cd,((math_data *)mp->math)->zero_t));
+ mp_number_clone (ret, xa);
+ free_number (ab_vs_cd);
+ free_number (r);
+ free_number (abs_x);
+ free_number (la);
+ free_number (xa);
+ free_number (u);
+}
+
+
+
+
+@ The following subroutine could be used in |norm_rand| and tests if $ab$ is
+greater than, equal to, or less than~$cd$.
+The result is $+1$, 0, or~$-1$ in the three respective cases.
+This is not necessary, even if it's shorter than the current |ab_vs_cd|
+and looks as a native implementation.
+
+@c
+/*
+|void mp_decimal_ab_vs_cd (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig, mp_number c_orig, mp_number d_orig) {|
+| decNumber a, b, c, d;|
+| decNumber ab, cd;|
+| (void)mp;|
+||
+| decNumberCopy(&a, (decNumber *)a_orig.data.num);|
+| decNumberCopy(&b, (decNumber *)b_orig.data.num);|
+| decNumberCopy(&c, (decNumber *)c_orig.data.num);|
+| decNumberCopy(&d, (decNumber *)d_orig.data.num);|
+||
+||
+| decNumberMultiply (&ab, (decNumber *)a_orig.data.num, (decNumber *)b_orig.data.num, &set);|
+| decNumberMultiply (&cd, (decNumber *)c_orig.data.num, (decNumber *)d_orig.data.num, &set);|
+| decNumberCompare(ret->data.num, &ab, &cd, &set);|
+| mp_check_decNumber(mp, ret->data.num, &set);|
+| return;|
+||
+|}|
+*/
+
+
+