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authorDenis Bitouzé <dbitouze@wanadoo.fr>2021-02-25 18:23:07 +0000
committerDenis Bitouzé <dbitouze@wanadoo.fr>2021-02-25 18:23:07 +0000
commitc6101f91d071883b48b1b4b51e5eba0f36d9a78d (patch)
tree1bf7f5a881d7a4f5c5bf59d0b2821943dd822372 /Build/source/texk/web2c/mplibdir/mpmathdouble.w
parent07ee7222e389b0777456b427a55c22d0e6ffd267 (diff)
French translation for tlmgr updated
git-svn-id: svn://tug.org/texlive/trunk@57912 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Build/source/texk/web2c/mplibdir/mpmathdouble.w')
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-% $Id: mpmathdouble.w 2118 2017-02-15 17:49:54Z luigi $
-%
-% 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 IEEE double based math}
-\pdfoutput=1
-
-@ Introduction.
-
-@c
-#include <w2c/config.h>
-#include <stdio.h>
-#include <stdlib.h>
-#include <string.h>
-#include <math.h>
-#include "mpmathdouble.h" /* internal header */
-#define ROUND(a) floor((a)+0.5)
-@h
-
-@ @c
-@<Declarations@>;
-
-@ @(mpmathdouble.h@>=
-#ifndef MPMATHDOUBLE_H
-#define MPMATHDOUBLE_H 1
-#include "mplib.h"
-#include "mpmp.h" /* internal header */
-@<Internal library declarations@>;
-#endif
-
-@* Math initialization.
-
-First, here are some very important constants.
-
-@d PI 3.1415926535897932384626433832795028841971
-@d fraction_multiplier 4096.0
-@d angle_multiplier 16.0
-
-@ Here are the functions that are static as they are not used elsewhere
-
-@<Declarations@>=
-static void mp_double_scan_fractional_token (MP mp, int n);
-static void mp_double_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_double_ab_vs_cd (MP mp, mp_number *ret, mp_number a, mp_number b, mp_number c, mp_number d);
-static void mp_double_crossing_point (MP mp, mp_number *ret, mp_number a, mp_number b, mp_number c);
-static void mp_number_modulo (mp_number *a, mp_number b);
-static void mp_double_print_number (MP mp, mp_number n);
-static char * mp_double_number_tostring (MP mp, mp_number n);
-static void mp_double_slow_add (MP mp, mp_number *ret, mp_number x_orig, mp_number y_orig);
-static void mp_double_square_rt (MP mp, mp_number *ret, mp_number x_orig);
-static void mp_double_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_double_m_unif_rand (MP mp, mp_number *ret, mp_number x_orig);
-static void mp_double_m_norm_rand (MP mp, mp_number *ret);
-static void mp_double_m_exp (MP mp, mp_number *ret, mp_number x_orig);
-static void mp_double_m_log (MP mp, mp_number *ret, mp_number x_orig);
-static void mp_double_pyth_sub (MP mp, mp_number *r, mp_number a, mp_number b);
-static void mp_double_pyth_add (MP mp, mp_number *r, mp_number a, mp_number b);
-static void mp_double_n_arg (MP mp, mp_number *ret, mp_number x, mp_number y);
-static void mp_double_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_double_from_int(mp_number *A, int B);
-static void mp_set_double_from_boolean(mp_number *A, int B);
-static void mp_set_double_from_scaled(mp_number *A, int B);
-static void mp_set_double_from_addition(mp_number *A, mp_number B, mp_number C);
-static void mp_set_double_from_substraction (mp_number *A, mp_number B, mp_number C);
-static void mp_set_double_from_div(mp_number *A, mp_number B, mp_number C);
-static void mp_set_double_from_mul(mp_number *A, mp_number B, mp_number C);
-static void mp_set_double_from_int_div(mp_number *A, mp_number B, int C);
-static void mp_set_double_from_int_mul(mp_number *A, mp_number B, int C);
-static void mp_set_double_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_double_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_double_fraction_to_round_scaled (mp_number *x);
-static void mp_double_number_make_scaled (MP mp, mp_number *r, mp_number p, mp_number q);
-static void mp_double_number_make_fraction (MP mp, mp_number *r, mp_number p, mp_number q);
-static void mp_double_number_take_fraction (MP mp, mp_number *r, mp_number p, mp_number q);
-static void mp_double_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_double_from_double(mp_number *A, double B);
-static void mp_free_double_math (MP mp);
-static void mp_double_set_precision (MP mp);
-
-@ And these are the ones that {\it are} used elsewhere
-
-@<Internal library declarations@>=
-void * mp_initialize_double_math (MP mp);
-
-@
-
-@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 warning_limit pow(2.0,52.0) /* this is a large value that can just be expressed without loss of precision */
-@d epsilon pow(2.0,-52.0)
-
-@c
-void * mp_initialize_double_math (MP mp) {
- math_data *math = (math_data *)mp_xmalloc(mp,1,sizeof(math_data));
- /* alloc */
- math->allocate = mp_new_number;
- math->free = mp_free_number;
- mp_new_number (mp, &math->precision_default, mp_scaled_type);
- math->precision_default.data.dval = 16 * unity;
- mp_new_number (mp, &math->precision_max, mp_scaled_type);
- math->precision_max.data.dval = 16 * unity;
- mp_new_number (mp, &math->precision_min, mp_scaled_type);
- math->precision_min.data.dval = 16 * unity;
- /* here are the constants for |scaled| objects */
- mp_new_number (mp, &math->epsilon_t, mp_scaled_type);
- math->epsilon_t.data.dval = epsilon;
- mp_new_number (mp, &math->inf_t, mp_scaled_type);
- math->inf_t.data.dval = EL_GORDO;
- mp_new_number (mp, &math->warning_limit_t, mp_scaled_type);
- math->warning_limit_t.data.dval = warning_limit;
- mp_new_number (mp, &math->one_third_inf_t, mp_scaled_type);
- math->one_third_inf_t.data.dval = one_third_EL_GORDO;
- mp_new_number (mp, &math->unity_t, mp_scaled_type);
- math->unity_t.data.dval = unity;
- mp_new_number (mp, &math->two_t, mp_scaled_type);
- math->two_t.data.dval = two;
- mp_new_number (mp, &math->three_t, mp_scaled_type);
- math->three_t.data.dval = three;
- mp_new_number (mp, &math->half_unit_t, mp_scaled_type);
- math->half_unit_t.data.dval = half_unit;
- mp_new_number (mp, &math->three_quarter_unit_t, mp_scaled_type);
- math->three_quarter_unit_t.data.dval = three_quarter_unit;
- mp_new_number (mp, &math->zero_t, mp_scaled_type);
- /* |fractions| */
- mp_new_number (mp, &math->arc_tol_k, mp_fraction_type);
- math->arc_tol_k.data.dval = (unity/4096); /* quit when change in arc length estimate reaches this */
- mp_new_number (mp, &math->fraction_one_t, mp_fraction_type);
- math->fraction_one_t.data.dval = fraction_one;
- mp_new_number (mp, &math->fraction_half_t, mp_fraction_type);
- math->fraction_half_t.data.dval = fraction_half;
- mp_new_number (mp, &math->fraction_three_t, mp_fraction_type);
- math->fraction_three_t.data.dval = fraction_three;
- mp_new_number (mp, &math->fraction_four_t, mp_fraction_type);
- math->fraction_four_t.data.dval = fraction_four;
- /* |angles| */
- mp_new_number (mp, &math->three_sixty_deg_t, mp_angle_type);
- math->three_sixty_deg_t.data.dval = three_sixty_deg;
- mp_new_number (mp, &math->one_eighty_deg_t, mp_angle_type);
- math->one_eighty_deg_t.data.dval = one_eighty_deg;
- /* various approximations */
- mp_new_number (mp, &math->one_k, mp_scaled_type);
- math->one_k.data.dval = 1.0/64 ;
- mp_new_number (mp, &math->sqrt_8_e_k, mp_scaled_type);
- math->sqrt_8_e_k.data.dval = 1.71552776992141359295 ; /* $2^{16}\sqrt{8/e}\approx 112428.82793$ */
- mp_new_number (mp, &math->twelve_ln_2_k, mp_fraction_type);
- math->twelve_ln_2_k.data.dval = 8.31776616671934371292 *256; /* $2^{24}\cdot12\ln2\approx139548959.6165$ */
- mp_new_number (mp, &math->coef_bound_k, mp_fraction_type);
- math->coef_bound_k.data.dval = coef_bound;
- mp_new_number (mp, &math->coef_bound_minus_1, mp_fraction_type);
- math->coef_bound_minus_1.data.dval = coef_bound - 1/65536.0;
- mp_new_number (mp, &math->twelvebits_3, mp_scaled_type);
- math->twelvebits_3.data.dval = 1365 / 65536.0; /* $1365\approx 2^{12}/3$ */
- mp_new_number (mp, &math->twentysixbits_sqrt2_t, mp_fraction_type);
- math->twentysixbits_sqrt2_t.data.dval = 94906266 / 65536.0; /* $2^{26}\sqrt2\approx94906265.62$ */
- mp_new_number (mp, &math->twentyeightbits_d_t, mp_fraction_type);
- math->twentyeightbits_d_t.data.dval = 35596755 / 65536.0; /* $2^{28}d\approx35596754.69$ */
- mp_new_number (mp, &math->twentysevenbits_sqrt2_d_t, mp_fraction_type);
- math->twentysevenbits_sqrt2_d_t.data.dval = 25170707 / 65536.0; /* $2^{27}\sqrt2\,d\approx25170706.63$ */
- /* thresholds */
- mp_new_number (mp, &math->fraction_threshold_t, mp_fraction_type);
- math->fraction_threshold_t.data.dval = fraction_threshold;
- mp_new_number (mp, &math->half_fraction_threshold_t, mp_fraction_type);
- math->half_fraction_threshold_t.data.dval = half_fraction_threshold;
- mp_new_number (mp, &math->scaled_threshold_t, mp_scaled_type);
- math->scaled_threshold_t.data.dval = scaled_threshold;
- mp_new_number (mp, &math->half_scaled_threshold_t, mp_scaled_type);
- math->half_scaled_threshold_t.data.dval = half_scaled_threshold;
- mp_new_number (mp, &math->near_zero_angle_t, mp_angle_type);
- math->near_zero_angle_t.data.dval = near_zero_angle;
- mp_new_number (mp, &math->p_over_v_threshold_t, mp_fraction_type);
- math->p_over_v_threshold_t.data.dval = p_over_v_threshold;
- mp_new_number (mp, &math->equation_threshold_t, mp_scaled_type);
- math->equation_threshold_t.data.dval = equation_threshold;
- mp_new_number (mp, &math->tfm_warn_threshold_t, mp_scaled_type);
- math->tfm_warn_threshold_t.data.dval = tfm_warn_threshold;
- /* functions */
- math->from_int = mp_set_double_from_int;
- math->from_boolean = mp_set_double_from_boolean;
- math->from_scaled = mp_set_double_from_scaled;
- math->from_double = mp_set_double_from_double;
- math->from_addition = mp_set_double_from_addition;
- math->from_substraction = mp_set_double_from_substraction;
- math->from_oftheway = mp_set_double_from_of_the_way;
- math->from_div = mp_set_double_from_div;
- math->from_mul = mp_set_double_from_mul;
- math->from_int_div = mp_set_double_from_int_div;
- math->from_int_mul = mp_set_double_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_double_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_double_fraction_to_round_scaled;
- math->make_scaled = mp_double_number_make_scaled;
- math->make_fraction = mp_double_number_make_fraction;
- math->take_fraction = mp_double_number_take_fraction;
- math->take_scaled = mp_double_number_take_scaled;
- math->velocity = mp_double_velocity;
- math->n_arg = mp_double_n_arg;
- math->m_log = mp_double_m_log;
- math->m_exp = mp_double_m_exp;
- math->m_unif_rand = mp_double_m_unif_rand;
- math->m_norm_rand = mp_double_m_norm_rand;
- math->pyth_add = mp_double_pyth_add;
- math->pyth_sub = mp_double_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_double_sin_cos;
- math->slow_add = mp_double_slow_add;
- math->sqrt = mp_double_square_rt;
- math->print = mp_double_print_number;
- math->tostring = mp_double_number_tostring;
- math->modulo = mp_number_modulo;
- math->ab_vs_cd = mp_ab_vs_cd;
- math->crossing_point = mp_double_crossing_point;
- math->scan_numeric = mp_double_scan_numeric_token;
- math->scan_fractional = mp_double_scan_fractional_token;
- math->free_math = mp_free_double_math;
- math->set_precision = mp_double_set_precision;
- return (void *)math;
-}
-
-void mp_double_set_precision (MP mp) {
-}
-
-void mp_free_double_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);
- 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.dval = 0.0;
- n->type = t;
-}
-
-@
-
-@c
-void mp_free_number (MP mp, mp_number *n) {
- (void)mp;
- n->type = mp_nan_type;
-}
-
-@ Here are the low-level functions on |mp_number| items, setters first.
-
-@c
-void mp_set_double_from_int(mp_number *A, int B) {
- A->data.dval = B;
-}
-void mp_set_double_from_boolean(mp_number *A, int B) {
- A->data.dval = B;
-}
-void mp_set_double_from_scaled(mp_number *A, int B) {
- A->data.dval = B / 65536.0;
-}
-void mp_set_double_from_double(mp_number *A, double B) {
- A->data.dval = B;
-}
-void mp_set_double_from_addition(mp_number *A, mp_number B, mp_number C) {
- A->data.dval = B.data.dval+C.data.dval;
-}
-void mp_set_double_from_substraction (mp_number *A, mp_number B, mp_number C) {
- A->data.dval = B.data.dval-C.data.dval;
-}
-void mp_set_double_from_div(mp_number *A, mp_number B, mp_number C) {
- A->data.dval = B.data.dval / C.data.dval;
-}
-void mp_set_double_from_mul(mp_number *A, mp_number B, mp_number C) {
- A->data.dval = B.data.dval * C.data.dval;
-}
-void mp_set_double_from_int_div(mp_number *A, mp_number B, int C) {
- A->data.dval = B.data.dval / C;
-}
-void mp_set_double_from_int_mul(mp_number *A, mp_number B, int C) {
- A->data.dval = B.data.dval * C;
-}
-void mp_set_double_from_of_the_way(MP mp, mp_number *A, mp_number t, mp_number B, mp_number C) {
- A->data.dval = B.data.dval - mp_double_take_fraction(mp, (B.data.dval - C.data.dval), t.data.dval);
-}
-void mp_number_negate(mp_number *A) {
- A->data.dval = -A->data.dval;
- if (A->data.dval == -0.0)
- A->data.dval = 0.0;
-}
-void mp_number_add(mp_number *A, mp_number B) {
- A->data.dval = A->data.dval + B.data.dval;
-}
-void mp_number_substract(mp_number *A, mp_number B) {
- A->data.dval = A->data.dval - B.data.dval;
-}
-void mp_number_half(mp_number *A) {
- A->data.dval = A->data.dval/2.0;
-}
-void mp_number_halfp(mp_number *A) {
- A->data.dval = (A->data.dval/2.0);
-}
-void mp_number_double(mp_number *A) {
- A->data.dval = A->data.dval * 2.0;
-}
-void mp_number_add_scaled(mp_number *A, int B) { /* also for negative B */
- A->data.dval = A->data.dval + (B/65536.0);
-}
-void mp_number_multiply_int(mp_number *A, int B) {
- A->data.dval = (double)(A->data.dval * B);
-}
-void mp_number_divide_int(mp_number *A, int B) {
- A->data.dval = A->data.dval / (double)B;
-}
-void mp_double_abs(mp_number *A) {
- A->data.dval = fabs(A->data.dval);
-}
-void mp_number_clone(mp_number *A, mp_number B) {
- A->data.dval = B.data.dval;
-}
-void mp_number_swap(mp_number *A, mp_number *B) {
- double swap_tmp = A->data.dval;
- A->data.dval = B->data.dval;
- B->data.dval = swap_tmp;
-}
-void mp_number_fraction_to_scaled (mp_number *A) {
- A->type = mp_scaled_type;
- A->data.dval = A->data.dval / fraction_multiplier;
-}
-void mp_number_angle_to_scaled (mp_number *A) {
- A->type = mp_scaled_type;
- A->data.dval = A->data.dval / angle_multiplier;
-}
-void mp_number_scaled_to_fraction (mp_number *A) {
- A->type = mp_fraction_type;
- A->data.dval = A->data.dval * fraction_multiplier;
-}
-void mp_number_scaled_to_angle (mp_number *A) {
- A->type = mp_angle_type;
- A->data.dval = A->data.dval * angle_multiplier;
-}
-
-
-@ Query functions
-
-@c
-int mp_number_to_scaled(mp_number A) {
- return (int)ROUND(A.data.dval * 65536.0);
-}
-int mp_number_to_int(mp_number A) {
- return (int)(A.data.dval);
-}
-int mp_number_to_boolean(mp_number A) {
- return (int)(A.data.dval);
-}
-double mp_number_to_double(mp_number A) {
- return A.data.dval;
-}
-int mp_number_odd(mp_number A) {
- return odd((int)ROUND(A.data.dval * 65536.0));
-}
-int mp_number_equal(mp_number A, mp_number B) {
- return (A.data.dval==B.data.dval);
-}
-int mp_number_greater(mp_number A, mp_number B) {
- return (A.data.dval>B.data.dval);
-}
-int mp_number_less(mp_number A, mp_number B) {
- return (A.data.dval<B.data.dval);
-}
-int mp_number_nonequalabs(mp_number A, mp_number B) {
- return (!(fabs(A.data.dval)==fabs(B.data.dval)));
-}
-
-@ 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.
-
-@d unity 1.0
-@d two 2.0
-@d three 3.0
-@d half_unit 0.5
-@d three_quarter_unit 0.75
-
-@d EL_GORDO (DBL_MAX/2.0-1.0) /* the largest value that \MP\ likes. */
-@d one_third_EL_GORDO (EL_GORDO/3.0)
-
-@ 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_double_number_tostring (MP mp, mp_number n) {
- static char set[64];
- int l = 0;
- char *ret = mp_xmalloc(mp, 64, 1);
- snprintf(set, 64, "%.17g", n.data.dval);
- while (set[l] == ' ') l++;
- strcpy(ret, set+l);
- return ret;
-}
-
-
-@ @c
-void mp_double_print_number (MP mp, mp_number n) {
- char *str = mp_double_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_double_slow_add (MP mp, mp_number *ret, mp_number x_orig, mp_number y_orig) {
- double x, y;
- x = x_orig.data.dval;
- y = y_orig.data.dval;
- if (x >= 0) {
- if (y <= EL_GORDO - x) {
- ret->data.dval = x + y;
- } else {
- mp->arith_error = true;
- ret->data.dval = EL_GORDO;
- }
- } else if (-y <= EL_GORDO + x) {
- ret->data.dval = x + y;
- } else {
- mp->arith_error = true;
- ret->data.dval = -EL_GORDO;
- }
-}
-
-@ 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
-double mp_double_make_fraction (MP mp, double p, double q) {
- return ((p / q) * fraction_multiplier);
-}
-void mp_double_number_make_fraction (MP mp, mp_number *ret, mp_number p, mp_number q) {
- ret->data.dval = mp_double_make_fraction (mp, p.data.dval, q.data.dval);
-}
-
-@ @<Declarations@>=
-double mp_double_make_fraction (MP mp, double p, double 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
-double mp_double_take_fraction (MP mp, double p, double q) {
- return ((p * q) / fraction_multiplier);
-}
-void mp_double_number_take_fraction (MP mp, mp_number *ret, mp_number p, mp_number q) {
- ret->data.dval = mp_double_take_fraction (mp, p.data.dval, q.data.dval);
-}
-
-@ @<Declarations@>=
-double mp_double_take_fraction (MP mp, double p, double 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_double_number_take_scaled (MP mp, mp_number *ret, mp_number p_orig, mp_number q_orig) {
- ret->data.dval = p_orig.data.dval * q_orig.data.dval;
-}
-
-
-@ 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
-double mp_double_make_scaled (MP mp, double p, double q) {
- return p / q;
-}
-void mp_double_number_make_scaled (MP mp, mp_number *ret, mp_number p_orig, mp_number q_orig) {
- ret->data.dval = p_orig.data.dval / q_orig.data.dval;
-}
-
-@ @<Declarations@>=
-double mp_double_make_scaled (MP mp, double p, double q);
-
-
-@
-@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) mp->cur_mod_->data.n.data.dval=(A)
-
-@<Declarations...@>=
-static void mp_wrapup_numeric_token(MP mp, unsigned char *start, unsigned char *stop);
-
-@ @c
-void mp_wrapup_numeric_token(MP mp, unsigned char *start, unsigned char *stop) {
- double result;
- char *end = (char *)stop;
- errno = 0;
- result = strtod ((char *)start, &end);
- if (errno == 0) {
- set_cur_mod(result);
- if (result >= warning_limit) {
- if (internal_value (mp_warning_check).data.dval > 0 &&
- (mp->scanner_status != tex_flushing)) {
- char msg[256];
- const char *hlp[] = {"Continue and I'll try to cope",
- "with that big value; but it might be dangerous.",
- "(Set warningcheck:=0 to suppress this message.)",
- NULL };
- mp_snprintf (msg, 256, "Number is too large (%g)", result);
-@.Number is too large@>;
- mp_error (mp, msg, hlp, true);
- }
- }
- } else if (mp->scanner_status != tex_flushing) {
- const char *hlp[] = {"I could not handle this number specification",
- "probably because it is out of range. Error:",
- "",
- NULL };
- hlp[2] = strerror(errno);
- mp_error (mp, "Enormous number has been reduced.", hlp, false);
-@.Enormous number...@>;
- set_cur_mod(EL_GORDO);
- }
- 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_double_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);
-}
-
-
-@ Input format is the same as for the C language, so we just collect valid
-bytes in the buffer, then call |strtod()|
-
-@c
-void mp_double_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 (0.5*fraction_multiplier)
-@d fraction_one (1.0*fraction_multiplier)
-@d fraction_two (2.0*fraction_multiplier)
-@d fraction_three (3.0*fraction_multiplier)
-@d fraction_four (4.0*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_double_velocity (MP mp, mp_number *ret, mp_number st, mp_number ct, mp_number sf,
- mp_number cf, mp_number t) {
- double acc, num, denom; /* registers for intermediate calculations */
- acc = mp_double_take_fraction (mp, st.data.dval - (sf.data.dval / 16.0),
- sf.data.dval - (st.data.dval / 16.0));
- acc = mp_double_take_fraction (mp, acc, ct.data.dval - cf.data.dval);
- num = fraction_two + mp_double_take_fraction (mp, acc, sqrt(2)*fraction_one);
- denom =
- fraction_three + mp_double_take_fraction (mp, ct.data.dval, 3*fraction_half*(sqrt(5.0)-1.0))
- + mp_double_take_fraction (mp, cf.data.dval, 3*fraction_half*(3.0-sqrt(5.0)));
- if (t.data.dval != unity)
- num = mp_double_make_scaled (mp, num, t.data.dval);
- if (num / 4 >= denom) {
- ret->data.dval = fraction_four;
- } else {
- ret->data.dval = mp_double_make_fraction (mp, 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
-}
-
-
-@ 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) {
- integer q, r; /* temporary registers */
- integer a, b, c, d;
- (void)mp;
-
- mp_double_ab_vs_cd(mp,ret, a_orig, b_orig, c_orig, d_orig);
- if (1>0)
- return ;
- /* TODO: remove this code until the end */
- a = a_orig.data.dval;
- b = b_orig.data.dval;
- c = c_orig.data.dval;
- d = d_orig.data.dval;
- @<Reduce to the case that |a,c>=0|, |b,d>0|@>;
- while (1) {
- q = a / d;
- r = c / b;
- if (q != r) {
- ret->data.dval = (q > r ? 1 : -1);
- goto RETURN;
- }
- q = a % d;
- r = c % b;
- if (r == 0) {
- ret->data.dval = (q ? 1 : 0);
- goto RETURN;
- }
- if (q == 0) {
- ret->data.dval = -1;
- goto RETURN;
- }
- a = b;
- b = q;
- c = d;
- 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
- return;
-}
-
-
-@ @<Reduce to the case that |a...@>=
-if (a < 0) {
- a = -a;
- b = -b;
-}
-if (c < 0) {
- c = -c;
- d = -d;
-}
-if (d <= 0) {
- if (b >= 0) {
- if ((a == 0 || b == 0) && (c == 0 || d == 0))
- ret->data.dval = 0;
- else
- ret->data.dval = 1;
- goto RETURN;
- }
- if (d == 0) {
- ret->data.dval = (a == 0 ? 0 : -1);
- goto RETURN;
- }
- q = a;
- a = c;
- c = q;
- q = -b;
- b = -d;
- d = q;
-} else if (b <= 0) {
- if (b < 0 && a > 0) {
- ret->data.dval = -1;
- return;
- }
- ret->data.dval = (c == 0 ? 0 : -1);
- 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 { ret->data.dval = fraction_one + 1; goto RETURN; }
-@d one_crossing { ret->data.dval = fraction_one; goto RETURN; }
-@d zero_crossing { ret->data.dval = 0; goto RETURN; }
-
-@c
-static void mp_double_crossing_point (MP mp, mp_number *ret, mp_number aa, mp_number bb, mp_number cc) {
- double a,b,c;
- double d; /* recursive counter */
- double x, xx, x0, x1, x2; /* temporary registers for bisection */
- a = aa.data.dval;
- b = bb.data.dval;
- c = cc.data.dval;
- if (a < 0)
- zero_crossing;
- if (c >= 0) {
- if (b >= 0) {
- if (c > 0) {
- no_crossing;
- } else if ((a == 0) && (b == 0)) {
- no_crossing;
- } else {
- one_crossing;
- }
- }
- if (a == 0)
- zero_crossing;
- } else if (a == 0) {
- if (b <= 0)
- zero_crossing;
- }
-
- /* Use bisection to find the crossing point... */
- d = epsilon;
- x0 = a;
- x1 = a - b;
- x2 = b - c;
- do {
- /* not sure why the error correction has to be >= 1E-12 */
- x = (x1 + x2) / 2 + 1E-12;
- if (x1 - x0 > x0) {
- x2 = x;
- x0 += x0;
- d += d;
- } else {
- xx = x1 + x - x0;
- if (xx > x0) {
- x2 = x;
- x0 += x0;
- d += d;
- } else {
- x0 = x0 - xx;
- if (x <= x0) {
- if (x + x2 <= x0)
- no_crossing;
- }
- x1 = x;
- d = d + d + epsilon;
- }
- }
- } while (d < fraction_one);
- ret->data.dval = (d - fraction_one);
-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
- 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) {
- int x = (int)ROUND(x_orig.data.dval);
- return x;
-}
-
-@ |number_floor| floors a number
-
-@c
-void mp_number_floor (mp_number *i) {
- i->data.dval = floor(i->data.dval);
-}
-
-@ |fraction_to_scaled| rounds a |fraction| and converts it to |scaled|
-@c
-void mp_double_fraction_to_round_scaled (mp_number *x_orig) {
- double x = x_orig->data.dval;
- x_orig->type = mp_scaled_type;
- x_orig->data.dval = x/fraction_multiplier;
-}
-
-
-
-@* 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_double_square_rt (MP mp, mp_number *ret, mp_number x_orig) { /* return, x: scaled */
- double x;
- x = x_orig.data.dval;
- if (x <= 0) {
- @<Handle square root of zero or negative argument@>;
- } else {
- ret->data.dval = sqrt(x);
- }
-}
-
-
-@ @<Handle square root of zero...@>=
-{
- if (x < 0) {
- 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_double_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);
- }
- ret->data.dval = 0;
- return;
-}
-
-
-@ Pythagorean addition $\psqrt{a^2+b^2}$ is implemented by a quick hack
-
-@c
-void mp_double_pyth_add (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig) {
- double a, b; /* a,b : scaled */
- a = fabs (a_orig.data.dval);
- b = fabs (b_orig.data.dval);
- errno = 0;
- ret->data.dval = sqrt(a*a + b*b);
- if (errno) {
- mp->arith_error = true;
- ret->data.dval = EL_GORDO;
- }
-}
-
-
-@ Here is a similar algorithm for $\psqrt{a^2-b^2}$. Same quick hack, also.
-
-@c
-void mp_double_pyth_sub (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig) {
- double a, b;
- a = fabs (a_orig.data.dval);
- b = fabs (b_orig.data.dval);
- if (a <= b) {
- @<Handle erroneous |pyth_sub| and set |a:=0|@>;
- } else {
- a = sqrt(a*a - b*b);
- }
- ret->data.dval = a;
-}
-
-
-@ @<Handle erroneous |pyth_sub| and set |a:=0|@>=
-{
- if (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_double_number_tostring (mp, a_orig);
- char *bstr = mp_double_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);
- }
- a = 0;
-}
-
-
-@ The subroutines for logarithm and exponential involve two tables.
-The first is simple: |two_to_the[k]| equals $2^k$.
-
-@d two_to_the(A) (1<<(unsigned)(A))
-
-@ Here is the routine that calculates $2^8$ times the natural logarithm
-of a |scaled| quantity; it is an integer approximation to $2^{24}\ln(x/2^{16})$,
-when |x| is a given positive integer.
-
-@c
-void mp_double_m_log (MP mp, mp_number *ret, mp_number x_orig) {
- if (x_orig.data.dval <= 0) {
- @<Handle non-positive logarithm@>;
- } else {
- ret->data.dval = log (x_orig.data.dval)*256.0;
- }
-}
-
-@ @<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_double_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);
- ret->data.dval = 0;
-}
-
-
-@ Conversely, the exponential routine calculates $\exp(x/2^8)$,
-when |x| is |scaled|.
-
-@c
-void mp_double_m_exp (MP mp, mp_number *ret, mp_number x_orig) {
- errno = 0;
- ret->data.dval = exp(x_orig.data.dval/256.0);
- if (errno) {
- if (x_orig.data.dval > 0) {
- mp->arith_error = true;
- ret->data.dval = EL_GORDO;
- } else {
- ret->data.dval = 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_double_n_arg (MP mp, mp_number *ret, mp_number x_orig, mp_number y_orig) {
- if (x_orig.data.dval == 0.0 && y_orig.data.dval == 0.0) {
- @<Handle undefined arg@>;
- } else {
- ret->type = mp_angle_type;
- ret->data.dval = atan2 (y_orig.data.dval, x_orig.data.dval) * (180.0 / PI) * angle_multiplier;
- if (ret->data.dval == -0.0)
- ret->data.dval = 0.0;
-#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
- }
-}
-
-
-@ @<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@>;
- ret->data.dval = 0;
-}
-
-
-@ 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|.
-
-@ Given an integer |z| that is $2^{20}$ times an angle $\theta$ in degrees,
-the purpose of |n_sin_cos(z)| is to set
-|x=@t$r\cos\theta$@>| and |y=@t$r\sin\theta$@>| (approximately),
-for some rather large number~|r|. The maximum of |x| and |y|
-will be between $2^{28}$ and $2^{30}$, so that there will be hardly
-any loss of accuracy. Then |x| and~|y| are divided by~|r|.
-
-@d one_eighty_deg (180.0*angle_multiplier)
-@d three_sixty_deg (360.0*angle_multiplier)
-
-@d odd(A) (abs(A)%2==1)
-
-@ Compute a multiple of the sine and cosine
-
-@c
-void mp_double_sin_cos (MP mp, mp_number z_orig, mp_number *n_cos, mp_number *n_sin) {
- double rad;
- rad = (z_orig.data.dval / angle_multiplier); /* still degrees */
- if ((rad == 90.0)||(rad == -270)){
- n_cos->data.dval = 0.0;
- n_sin->data.dval = fraction_multiplier;
- } else if ((rad == -90.0)||(rad == 270.0)) {
- n_cos->data.dval = 0.0;
- n_sin->data.dval = -fraction_multiplier;
- } else if ((rad == 180.0) || (rad == -180.0)) {
- n_cos->data.dval = -fraction_multiplier;
- n_sin->data.dval = 0.0;
- } else {
- rad = rad * PI/180.0;
- n_cos->data.dval = cos(rad) * fraction_multiplier;
- n_sin->data.dval = sin(rad) * fraction_multiplier;
- }
-#if DEBUG
- fprintf(stdout, "\nsin_cos(%f,%f,%f)", mp_number_to_double(z_orig),
-mp_number_to_double(*n_cos), mp_number_to_double(*n_sin));
-#endif
-}
-
-@ This is the 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;
- mp->randoms[(i * 21) % 55].data.dval = j;
- }
- mp_new_randoms (mp);
- mp_new_randoms (mp);
- mp_new_randoms (mp); /* ``warm up'' the array */
-
- ran_start((unsigned long) seed);
-
-
-}
-
-@ @c
-static double modulus(double left, double right);
-double modulus(double left, double right) {
- double quota = left / right;
- double frac,tmp;
- frac = modf(quota,&tmp);
- /* frac contains what's beyond the '.' */
- frac *= right;
- return frac;
-}
-void mp_number_modulo (mp_number *a, mp_number b) {
- a->data.dval = modulus (a->data.dval, b.data.dval);
-}
-
-
-
-@ 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) {
- double a;
- unsigned long int op;
- (void)mp;
- op = (unsigned)ran_arr_next();
- a = op/(MM*1.0);
- ret->data.dval = a;
-}
-
-
-
-@ 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_double_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_double_abs (&abs_x);
- mp_next_unif_random(mp, &u);
- y.data.dval = abs_x.data.dval * u.data.dval;
- 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_double_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_double_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_double_abs (&abs_x);
- } while (!mp_number_less(abs_x, u));
- mp_double_number_make_fraction (mp, &r, xa, u);
- mp_number_clone (&xa, r);
- mp_double_m_log (mp,&la, u);
- mp_set_double_from_substraction(&la, ((math_data *)mp->math)->twelve_ln_2_k, la);
- mp_double_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 is used only 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.
-
-@c
-void mp_double_ab_vs_cd (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig, mp_number c_orig, mp_number d_orig) {
- double ab, cd;
- (void)mp;
- ret->data.dval = 0 ;
- ab = a_orig.data.dval*b_orig.data.dval;
- cd = c_orig.data.dval*d_orig.data.dval;
- if (ab > cd )
- ret->data.dval = 1 ;
- else if (ab < cd )
- ret->data.dval = -1 ;
- return ;
-}
-