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diff --git a/Build/source/texk/web2c/mplibdir/mpmath.w b/Build/source/texk/web2c/mplibdir/mpmath.w new file mode 100644 index 00000000000..6c0ee6d00a6 --- /dev/null +++ b/Build/source/texk/web2c/mplibdir/mpmath.w @@ -0,0 +1,1925 @@ +% $Id: mpmath.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 32-bit integer math} +\pdfoutput=1 + +@ Introduction. + +@c +#include <w2c/config.h> +#include <stdio.h> +#include <stdlib.h> +#include <string.h> +#include <math.h> +#include "mpmath.h" /* internal header */ +@h + +@ @c +@<Declarations@>; + +@ @(mpmath.h@>= +#ifndef MPMATH_H +#define MPMATH_H 1 +#include "mplib.h" +#include "mpmp.h" /* internal header */ +@<Internal library declarations@>; +#endif + +@* Math initialization. + +@ Here are the functions that are static as they are not used elsewhere + +@<Declarations@>= +static void mp_scan_fractional_token (MP mp, int n); +static void mp_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_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_print_number (MP mp, mp_number n); +static char * mp_number_tostring (MP mp, mp_number n); +static void mp_slow_add (MP mp, mp_number *ret, mp_number x_orig, mp_number y_orig); +static void mp_square_rt (MP mp, mp_number *ret, mp_number x_orig); +static void mp_n_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_m_unif_rand (MP mp, mp_number *ret, mp_number x_orig); +static void mp_m_norm_rand (MP mp, mp_number *ret); +static void mp_m_exp (MP mp, mp_number *ret, mp_number x_orig); +static void mp_m_log (MP mp, mp_number *ret, mp_number x_orig); +static void mp_pyth_sub (MP mp, mp_number *r, mp_number a, mp_number b); +static void mp_n_arg (MP mp, mp_number *ret, mp_number x, mp_number y); +static void mp_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_number_from_int(mp_number *A, int B); +static void mp_set_number_from_boolean(mp_number *A, int B); +static void mp_set_number_from_scaled(mp_number *A, int B); +static void mp_set_number_from_boolean(mp_number *A, int B); +static void mp_set_number_from_addition(mp_number *A, mp_number B, mp_number C); +static void mp_set_number_from_substraction (mp_number *A, mp_number B, mp_number C); +static void mp_set_number_from_div(mp_number *A, mp_number B, mp_number C); +static void mp_set_number_from_mul(mp_number *A, mp_number B, mp_number C); +static void mp_set_number_from_int_div(mp_number *A, mp_number B, int C); +static void mp_set_number_from_int_mul(mp_number *A, mp_number B, int C); +static void mp_set_number_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_number_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_scaled(mp_number A); +static int mp_number_to_boolean(mp_number A); +static int mp_number_to_int(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_fraction_to_round_scaled (mp_number *x); +static void mp_number_make_scaled (MP mp, mp_number *r, mp_number p, mp_number q); +static void mp_number_make_fraction (MP mp, mp_number *r, mp_number p, mp_number q); +static void mp_number_take_fraction (MP mp, mp_number *r, mp_number p, mp_number q); +static void mp_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_free_scaled_math (MP mp); +static void mp_scaled_set_precision (MP mp); + +@ And these are the ones that {\it are} used elsewhere + +@<Internal library declarations@>= +void * mp_initialize_scaled_math (MP mp); +void mp_set_number_from_double(mp_number *A, double B); +void mp_pyth_add (MP mp, mp_number *r, mp_number a, mp_number b); +double mp_number_to_double(mp_number A); + +@ + +@d coef_bound 04525252525 /* |fraction| approximation to 7/3 */ +@d fraction_threshold 2685 /* a |fraction| coefficient less than this is zeroed */ +@d half_fraction_threshold 1342 /* half of |fraction_threshold| */ +@d scaled_threshold 8 /* a |scaled| coefficient less than this is zeroed */ +@d half_scaled_threshold 4 /* half of |scaled_threshold| */ +@d near_zero_angle 26844 +@d p_over_v_threshold 0x80000 +@d equation_threshold 64 +@d tfm_warn_threshold 4096 + + +@c +void * mp_initialize_scaled_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.val = unity * 10; + mp_new_number (mp, &math->precision_max, mp_scaled_type); + math->precision_max.data.val = unity * 10; + mp_new_number (mp, &math->precision_min, mp_scaled_type); + math->precision_min.data.val = unity * 10; + /* here are the constants for |scaled| objects */ + mp_new_number (mp, &math->epsilon_t, mp_scaled_type); + math->epsilon_t.data.val = 1; + mp_new_number (mp, &math->inf_t, mp_scaled_type); + math->inf_t.data.val = EL_GORDO; + mp_new_number (mp, &math->warning_limit_t, mp_scaled_type); + math->warning_limit_t.data.val = fraction_one; + mp_new_number (mp, &math->one_third_inf_t, mp_scaled_type); + math->one_third_inf_t.data.val = one_third_EL_GORDO; + mp_new_number (mp, &math->unity_t, mp_scaled_type); + math->unity_t.data.val = unity; + mp_new_number (mp, &math->two_t, mp_scaled_type); + math->two_t.data.val = two; + mp_new_number (mp, &math->three_t, mp_scaled_type); + math->three_t.data.val = three; + mp_new_number (mp, &math->half_unit_t, mp_scaled_type); + math->half_unit_t.data.val = half_unit; + mp_new_number (mp, &math->three_quarter_unit_t, mp_scaled_type); + math->three_quarter_unit_t.data.val = 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.val = (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.val = fraction_one; + mp_new_number (mp, &math->fraction_half_t, mp_fraction_type); + math->fraction_half_t.data.val = fraction_half; + mp_new_number (mp, &math->fraction_three_t, mp_fraction_type); + math->fraction_three_t.data.val = fraction_three; + mp_new_number (mp, &math->fraction_four_t, mp_fraction_type); + math->fraction_four_t.data.val = fraction_four; + /* |angles| */ + mp_new_number (mp, &math->three_sixty_deg_t, mp_angle_type); + math->three_sixty_deg_t.data.val = three_sixty_deg; + mp_new_number (mp, &math->one_eighty_deg_t, mp_angle_type); + math->one_eighty_deg_t.data.val = one_eighty_deg; + /* various approximations */ + mp_new_number (mp, &math->one_k, mp_scaled_type); + math->one_k.data.val = 1024; + mp_new_number (mp, &math->sqrt_8_e_k, mp_scaled_type); + math->sqrt_8_e_k.data.val = 112429; /* $2^{16}\sqrt{8/e}\approx 112428.82793$ */ + mp_new_number (mp, &math->twelve_ln_2_k, mp_fraction_type); + math->twelve_ln_2_k.data.val = 139548960; /* $2^{24}\cdot12\ln2\approx139548959.6165$ */ + mp_new_number (mp, &math->coef_bound_k, mp_fraction_type); + math->coef_bound_k.data.val = coef_bound; + mp_new_number (mp, &math->coef_bound_minus_1, mp_fraction_type); + math->coef_bound_minus_1.data.val = coef_bound - 1; + mp_new_number (mp, &math->twelvebits_3, mp_scaled_type); + math->twelvebits_3.data.val = 1365; /* $1365\approx 2^{12}/3$ */ + mp_new_number (mp, &math->twentysixbits_sqrt2_t, mp_fraction_type); + math->twentysixbits_sqrt2_t.data.val = 94906266; /* $2^{26}\sqrt2\approx94906265.62$ */ + mp_new_number (mp, &math->twentyeightbits_d_t, mp_fraction_type); + math->twentyeightbits_d_t.data.val = 35596755; /* $2^{28}d\approx35596754.69$ */ + mp_new_number (mp, &math->twentysevenbits_sqrt2_d_t, mp_fraction_type); + math->twentysevenbits_sqrt2_d_t.data.val = 25170707; /* $2^{27}\sqrt2\,d\approx25170706.63$ */ + /* thresholds */ + mp_new_number (mp, &math->fraction_threshold_t, mp_fraction_type); + math->fraction_threshold_t.data.val = fraction_threshold; + mp_new_number (mp, &math->half_fraction_threshold_t, mp_fraction_type); + math->half_fraction_threshold_t.data.val = half_fraction_threshold; + mp_new_number (mp, &math->scaled_threshold_t, mp_scaled_type); + math->scaled_threshold_t.data.val = scaled_threshold; + mp_new_number (mp, &math->half_scaled_threshold_t, mp_scaled_type); + math->half_scaled_threshold_t.data.val = half_scaled_threshold; + mp_new_number (mp, &math->near_zero_angle_t, mp_angle_type); + math->near_zero_angle_t.data.val = near_zero_angle; + mp_new_number (mp, &math->p_over_v_threshold_t, mp_fraction_type); + math->p_over_v_threshold_t.data.val = p_over_v_threshold; + mp_new_number (mp, &math->equation_threshold_t, mp_scaled_type); + math->equation_threshold_t.data.val = equation_threshold; + mp_new_number (mp, &math->tfm_warn_threshold_t, mp_scaled_type); + math->tfm_warn_threshold_t.data.val = tfm_warn_threshold; + /* functions */ + math->from_int = mp_set_number_from_int; + math->from_boolean = mp_set_number_from_boolean; + math->from_scaled = mp_set_number_from_scaled; + math->from_double = mp_set_number_from_double; + math->from_addition = mp_set_number_from_addition; + math->from_substraction = mp_set_number_from_substraction; + math->from_oftheway = mp_set_number_from_of_the_way; + math->from_div = mp_set_number_from_div; + math->from_mul = mp_set_number_from_mul; + math->from_int_div = mp_set_number_from_int_div; + math->from_int_mul = mp_set_number_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_number_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_int = mp_number_to_int; + math->to_boolean = mp_number_to_boolean; + math->to_scaled = mp_number_to_scaled; + math->to_double = mp_number_to_double; + 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_fraction_to_round_scaled; + math->make_scaled = mp_number_make_scaled; + math->make_fraction = mp_number_make_fraction; + math->take_fraction = mp_number_take_fraction; + math->take_scaled = mp_number_take_scaled; + math->velocity = mp_velocity; + math->n_arg = mp_n_arg; + math->m_log = mp_m_log; + math->m_exp = mp_m_exp; + math->m_unif_rand = mp_m_unif_rand; + math->m_norm_rand = mp_m_norm_rand; + math->pyth_add = mp_pyth_add; + math->pyth_sub = mp_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_n_sin_cos; + math->slow_add = mp_slow_add; + math->sqrt = mp_square_rt; + math->print = mp_print_number; + math->tostring = mp_number_tostring; + math->modulo = mp_number_modulo; + math->ab_vs_cd = mp_ab_vs_cd; + math->crossing_point = mp_crossing_point; + math->scan_numeric = mp_scan_numeric_token; + math->scan_fractional = mp_scan_fractional_token; + math->free_math = mp_free_scaled_math; + math->set_precision = mp_scaled_set_precision; + return (void *)math; +} + +void mp_scaled_set_precision (MP mp) { +} + +void mp_free_scaled_math (MP mp) { + free_number (((math_data *)mp->math)->epsilon_t); + free_number (((math_data *)mp->math)->inf_t); + free_number (((math_data *)mp->math)->arc_tol_k); + 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)->fraction_half_t); + free_number (((math_data *)mp->math)->fraction_three_t); + free_number (((math_data *)mp->math)->fraction_four_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)->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)->twelvebits_3); + free_number (((math_data *)mp->math)->twentysixbits_sqrt2_t); + free_number (((math_data *)mp->math)->twentyeightbits_d_t); + free_number (((math_data *)mp->math)->twentysevenbits_sqrt2_d_t); + 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.val = 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_number_from_int(mp_number *A, int B) { + A->data.val = B; +} +void mp_set_number_from_boolean(mp_number *A, int B) { + A->data.val = B; +} +void mp_set_number_from_scaled(mp_number *A, int B) { + A->data.val = B; +} +void mp_set_number_from_double(mp_number *A, double B) { + A->data.val = (int)(B*65536.0); +} +void mp_set_number_from_addition(mp_number *A, mp_number B, mp_number C) { + A->data.val = B.data.val+C.data.val; +} +void mp_set_number_from_substraction (mp_number *A, mp_number B, mp_number C) { + A->data.val = B.data.val-C.data.val; +} +void mp_set_number_from_div(mp_number *A, mp_number B, mp_number C) { + A->data.val = B.data.val / C.data.val; +} +void mp_set_number_from_mul(mp_number *A, mp_number B, mp_number C) { + A->data.val = B.data.val * C.data.val; +} +void mp_set_number_from_int_div(mp_number *A, mp_number B, int C) { + A->data.val = B.data.val / C; +} +void mp_set_number_from_int_mul(mp_number *A, mp_number B, int C) { + A->data.val = B.data.val * C; +} +void mp_set_number_from_of_the_way(MP mp, mp_number *A, mp_number t, mp_number B, mp_number C) { + A->data.val = B.data.val - mp_take_fraction(mp, (B.data.val - C.data.val), t.data.val); +} +void mp_number_negate(mp_number *A) { + A->data.val = -A->data.val; +} +void mp_number_add(mp_number *A, mp_number B) { + A->data.val = A->data.val + B.data.val; +} +void mp_number_substract(mp_number *A, mp_number B) { + A->data.val = A->data.val - B.data.val; +} +void mp_number_half(mp_number *A) { + A->data.val = A->data.val/2; +} +void mp_number_halfp(mp_number *A) { + A->data.val = (A->data.val>>1); +} +void mp_number_double(mp_number *A) { + A->data.val = A->data.val + A->data.val; +} +void mp_number_add_scaled(mp_number *A, int B) { /* also for negative B */ + A->data.val = A->data.val + B; +} +void mp_number_multiply_int(mp_number *A, int B) { + A->data.val = B * A->data.val; +} +void mp_number_divide_int(mp_number *A, int B) { + A->data.val = A->data.val / B; +} +void mp_number_abs(mp_number *A) { + A->data.val = abs(A->data.val); +} +void mp_number_clone(mp_number *A, mp_number B) { + A->data.val = B.data.val; +} +void mp_number_swap(mp_number *A, mp_number *B) { + int swap_tmp = A->data.val; + A->data.val = B->data.val; + B->data.val = swap_tmp; +} +void mp_number_fraction_to_scaled (mp_number *A) { + A->type = mp_scaled_type; + A->data.val = A->data.val / 4096; +} +void mp_number_angle_to_scaled (mp_number *A) { + A->type = mp_scaled_type; + if (A->data.val >= 0) { + A->data.val = (A->data.val + 8) / 16; + } else { + A->data.val = -((-A->data.val + 8) / 16); + } +} +void mp_number_scaled_to_fraction (mp_number *A) { + A->type = mp_fraction_type; + A->data.val = A->data.val * 4096; +} +void mp_number_scaled_to_angle (mp_number *A) { + A->type = mp_angle_type; + A->data.val = A->data.val * 16; +} + + +@ Query functions + +@c +int mp_number_to_int(mp_number A) { + return A.data.val; +} +int mp_number_to_scaled(mp_number A) { + return A.data.val; +} +int mp_number_to_boolean(mp_number A) { + return A.data.val; +} +double mp_number_to_double(mp_number A) { + return (A.data.val/65536.0); +} +int mp_number_odd(mp_number A) { + return odd(A.data.val); +} +int mp_number_equal(mp_number A, mp_number B) { + return (A.data.val==B.data.val); +} +int mp_number_greater(mp_number A, mp_number B) { + return (A.data.val>B.data.val); +} +int mp_number_less(mp_number A, mp_number B) { + return (A.data.val<B.data.val); +} +int mp_number_nonequalabs(mp_number A, mp_number B) { + return (!(abs(A.data.val)==abs(B.data.val))); +} + +@ 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 0x10000 /* $2^{16}$, represents 1.00000 */ +@d two (2*unity) /* $2^{17}$, represents 2.00000 */ +@d three (3*unity) /* $2^{17}+2^{16}$, represents 3.00000 */ +@d half_unit (unity/2) /* $2^{15}$, represents 0.50000 */ +@d three_quarter_unit (3*(unity/4)) /* $3\cdot2^{14}$, represents 0.75000 */ + +@d EL_GORDO 0x7fffffff /* $2^{31}-1$, the largest value that \MP\ likes */ +@d one_third_EL_GORDO 05252525252 + +@ 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. + +@d halfp(A) (integer)((unsigned)(A) >> 1) + +@ Here is a procedure analogous to |print_int|. If the output +of this procedure is subsequently read by \MP\ and converted by the +|round_decimals| routine above, it turns out that the original value will +be reproduced exactly. A decimal point is printed only if the value is +not an integer. If there is more than one way to print the result with +the optimum number of digits following the decimal point, the closest +possible value is given. + +The invariant relation in the \&{repeat} loop is that a sequence of +decimal digits yet to be printed will yield the original number if and only if +they form a fraction~$f$ in the range $s-\delta\L10\cdot2^{16}f<s$. +We can stop if and only if $f=0$ satisfies this condition; the loop will +terminate before $s$ can possibly become zero. + +@<Declarations@>= +static void mp_print_scaled (MP mp, int s); /* scaled */ +static char *mp_string_scaled (MP mp, int s); + +@ @c +static void mp_print_scaled (MP mp, int s) { /* s=scaled prints scaled real, rounded to five digits */ + int delta; /* amount of allowable inaccuracy, scaled */ + if (s < 0) { + mp_print_char (mp, xord ('-')); + s = -s; /* print the sign, if negative */ + } + mp_print_int (mp, s / unity); /* print the integer part */ + s = 10 * (s % unity) + 5; + if (s != 5) { + delta = 10; + mp_print_char (mp, xord ('.')); + do { + if (delta > unity) + s = s + 0100000 - (delta / 2); /* round the final digit */ + mp_print_char (mp, xord ('0' + (s / unity))); + s = 10 * (s % unity); + delta = delta * 10; + } while (s > delta); + } +} + +static char *mp_string_scaled (MP mp, int s) { /* s=scaled prints scaled real, rounded to five digits */ + static char scaled_string[32]; + int delta; /* amount of allowable inaccuracy, scaled */ + int i = 0; + if (s < 0) { + scaled_string[i++] = xord ('-'); + s = -s; /* print the sign, if negative */ + } + /* print the integer part */ + mp_snprintf ((scaled_string+i), 12, "%d", (int) (s / unity)); + while (*(scaled_string+i)) i++; + + s = 10 * (s % unity) + 5; + if (s != 5) { + delta = 10; + scaled_string[i++] = xord ('.'); + do { + if (delta > unity) + s = s + 0100000 - (delta / 2); /* round the final digit */ + scaled_string[i++] = xord ('0' + (s / unity)); + s = 10 * (s % unity); + delta = delta * 10; + } while (s > delta); + } + scaled_string[i] = '\0'; + return scaled_string; +} + +@ 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_slow_add (MP mp, mp_number *ret, mp_number x_orig, mp_number y_orig) { + integer x, y; + x = x_orig.data.val; + y = y_orig.data.val; + if (x >= 0) { + if (y <= EL_GORDO - x) { + ret->data.val = x + y; + } else { + mp->arith_error = true; + ret->data.val = EL_GORDO; + } + } else if (-y <= EL_GORDO + x) { + ret->data.val = x + y; + } else { + mp->arith_error = true; + ret->data.val = -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@> + +@ We need these preprocessor values + +@d TWEXP31 2147483648.0 +@d TWEXP28 268435456.0 +@d TWEXP16 65536.0 +@d TWEXP_16 (1.0/65536.0) +@d TWEXP_28 (1.0/268435456.0) + + +@c +static integer mp_make_fraction (MP mp, integer p, integer q) { + integer i; + if (q == 0) + mp_confusion (mp, "/"); +@:this can't happen /}{\quad \./@> + { + register double d; + d = TWEXP28 * (double) p / (double) q; + if ((p ^ q) >= 0) { + d += 0.5; + if (d >= TWEXP31) { + mp->arith_error = true; + i = EL_GORDO; + goto RETURN; + } + i = (integer) d; + if (d == (double) i && (((q > 0 ? -q : q) & 077777) + * (((i & 037777) << 1) - 1) & 04000) != 0) + --i; + } else { + d -= 0.5; + if (d <= -TWEXP31) { + mp->arith_error = true; + i = -EL_GORDO; + goto RETURN; + } + i = (integer) d; + if (d == (double) i && (((q > 0 ? q : -q) & 077777) + * (((i & 037777) << 1) + 1) & 04000) != 0) + ++i; + } + } +RETURN: + return i; +} +void mp_number_make_fraction (MP mp, mp_number *ret, mp_number p, mp_number q) { + ret->data.val = mp_make_fraction (mp, p.data.val, q.data.val); +} + + +@ 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@> + +@<Internal library declarations@>= +/* still in use by tfmin.w */ +integer mp_take_fraction (MP mp, integer q, int f); + +@ @c +integer mp_take_fraction (MP mp, integer p, int q) { /* q = fraction */ + register double d; + register integer i; + d = (double) p *(double) q *TWEXP_28; + if ((p ^ q) >= 0) { + d += 0.5; + if (d >= TWEXP31) { + if (d != TWEXP31 || (((p & 077777) * (q & 077777)) & 040000) == 0) + mp->arith_error = true; + return EL_GORDO; + } + i = (integer) d; + if (d == (double) i && (((p & 077777) * (q & 077777)) & 040000) != 0) + --i; + } else { + d -= 0.5; + if (d <= -TWEXP31) { + if (d != -TWEXP31 || ((-(p & 077777) * (q & 077777)) & 040000) == 0) + mp->arith_error = true; + return -EL_GORDO; + } + i = (integer) d; + if (d == (double) i && ((-(p & 077777) * (q & 077777)) & 040000) != 0) + ++i; + } + return i; +} +void mp_number_take_fraction (MP mp, mp_number *ret, mp_number p_orig, mp_number q_orig) { + ret->data.val = mp_take_fraction (mp, p_orig.data.val, q_orig.data.val); +} + + +@ 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@> + +@<Declarations@>= +static integer mp_take_scaled (MP mp, integer q, int f); + +@ @c +static integer mp_take_scaled (MP mp, integer p, int q) { /* q = scaled */ + register double d; + register integer i; + d = (double) p *(double) q *TWEXP_16; + if ((p ^ q) >= 0) { + d += 0.5; + if (d >= TWEXP31) { + if (d != TWEXP31 || (((p & 077777) * (q & 077777)) & 040000) == 0) + mp->arith_error = true; + return EL_GORDO; + } + i = (integer) d; + if (d == (double) i && (((p & 077777) * (q & 077777)) & 040000) != 0) + --i; + } else { + d -= 0.5; + if (d <= -TWEXP31) { + if (d != -TWEXP31 || ((-(p & 077777) * (q & 077777)) & 040000) == 0) + mp->arith_error = true; + return -EL_GORDO; + } + i = (integer) d; + if (d == (double) i && ((-(p & 077777) * (q & 077777)) & 040000) != 0) + ++i; + } + return i; +} +void mp_number_take_scaled (MP mp, mp_number *ret, mp_number p_orig, mp_number q_orig) { + ret->data.val = mp_take_scaled (mp, p_orig.data.val, q_orig.data.val); +} + + +@ 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.) + +@<Internal library ...@>= +/* still in use by svgout.w */ +int mp_make_scaled (MP mp, integer p, integer q); + +@ @c +int mp_make_scaled (MP mp, integer p, integer q) { /* return scaled */ + register integer i; + if (q == 0) + mp_confusion (mp, "/"); +@:this can't happen /}{\quad \./@> { + register double d; + d = TWEXP16 * (double) p / (double) q; + if ((p ^ q) >= 0) { + d += 0.5; + if (d >= TWEXP31) { + mp->arith_error = true; + return EL_GORDO; + } + i = (integer) d; + if (d == (double) i && (((q > 0 ? -q : q) & 077777) + * (((i & 037777) << 1) - 1) & 04000) != 0) + --i; + } else { + d -= 0.5; + if (d <= -TWEXP31) { + mp->arith_error = true; + return -EL_GORDO; + } + i = (integer) d; + if (d == (double) i && (((q > 0 ? q : -q) & 077777) + * (((i & 037777) << 1) + 1) & 04000) != 0) + ++i; + } + } + return i; +} +void mp_number_make_scaled (MP mp, mp_number *ret, mp_number p_orig, mp_number q_orig) { + ret->data.val = mp_make_scaled (mp, p_orig.data.val, q_orig.data.val); +} + +@ The following function is used to create a scaled integer from a given decimal +fraction $(.d_0d_1\ldots d_{k-1})$, where |0<=k<=17|. + +@<Declarations@>= +static int mp_round_decimals (MP mp, unsigned char *b, quarterword k); + +@ @c +static int mp_round_decimals (MP mp, unsigned char *b, quarterword k) { /* return: scaled */ + /* converts a decimal fraction */ + unsigned a = 0; /* the accumulator */ + int l = 0; + (void)mp; /* Will be needed later */ + for ( l = k-1; l >= 0; l-- ) { + if (l<16) /* digits for |k>=17| cannot affect the result */ + a = (a + (unsigned) (*(b+l) - '0') * two) / 10; + } + return (int) halfp (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.val=(A) + +@<Declarations...@>= +static void mp_wrapup_numeric_token(MP mp, int n, int f); + +@ @c +static void mp_wrapup_numeric_token(MP mp, int n, int f) { /* n,f: scaled */ + int mod ; /* scaled */ + if (n < 32768) { + mod = (n * unity + f); + set_cur_mod(mod); + if (mod >= fraction_one) { + if (internal_value (mp_warning_check).data.val > 0 && + (mp->scanner_status != tex_flushing)) { + char msg[256]; + const char *hlp[] = {"It is at least 4096. 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 (%s)", mp_string_scaled(mp,mod)); +@.Number is too large@>; + mp_error (mp, msg, hlp, true); + } + } + } else if (mp->scanner_status != tex_flushing) { + const char *hlp[] = {"I can\'t handle numbers bigger than 32767.99998;", + "so I've changed your constant to that maximum amount.", + NULL }; + 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 +void mp_scan_fractional_token (MP mp, int n) { /* n: scaled */ + int f; /* scaled */ + int k = 0; + do { + k++; + mp->cur_input.loc_field++; + } while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class); + f = mp_round_decimals (mp, (unsigned char *)(mp->buffer+mp->cur_input.loc_field-k), (quarterword) k); + if (f == unity) { + n++; + f = 0; + } + mp_wrapup_numeric_token(mp, n, f); +} + + +@ @c +void mp_scan_numeric_token (MP mp, int n) { /* n: scaled */ + while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class) { + if (n < 32768) + n = 10 * n + mp->buffer[mp->cur_input.loc_field] - '0'; + mp->cur_input.loc_field++; + } + if (!(mp->buffer[mp->cur_input.loc_field] == '.' && + mp->char_class[mp->buffer[mp->cur_input.loc_field + 1]] == digit_class)) { + mp_wrapup_numeric_token(mp, n, 0); + } else { + mp->cur_input.loc_field++; + mp_scan_fractional_token(mp, n); + } +} + +@ 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 01000000000 /* $2^{27}$, represents 0.50000000 */ +@d fraction_one 02000000000 /* $2^{28}$, represents 1.00000000 */ +@d fraction_two 04000000000 /* $2^{29}$, represents 2.00000000 */ +@d fraction_three 06000000000 /* $3\cdot2^{28}$, represents 3.00000000 */ +@d fraction_four 010000000000 /* $2^{30}$, represents 4.00000000 */ + +@ 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. Hence the fixed-point calculations below +are guaranteed to stay within the bounds of a 32-bit computer word. + +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_velocity (MP mp, mp_number *ret, mp_number st, mp_number ct, mp_number sf, + mp_number cf, mp_number t) { + integer acc, num, denom; /* registers for intermediate calculations */ + acc = mp_take_fraction (mp, st.data.val - (sf.data.val / 16), sf.data.val - (st.data.val / 16)); + acc = mp_take_fraction (mp, acc, ct.data.val - cf.data.val); + num = fraction_two + mp_take_fraction (mp, acc, 379625062); + /* $2^{28}\sqrt2\approx379625062.497$ */ + denom = + fraction_three + mp_take_fraction (mp, ct.data.val, + 497706707) + mp_take_fraction (mp, cf.data.val, + 307599661); + /* $3\cdot2^{27}\cdot(\sqrt5-1)\approx497706706.78$ and + $3\cdot2^{27}\cdot(3-\sqrt5\,)\approx307599661.22$ */ + if (t.data.val != unity) + num = mp_make_scaled (mp, num, t.data.val); /* |make_scaled(fraction,scaled)=fraction| */ + if (num / 4 >= denom) { + ret->data.val = fraction_four; + } else { + ret->data.val = mp_make_fraction (mp, num, denom); + } +/* |printf ("num,denom=%f,%f -=> %f\n", num/65536.0, denom/65536.0, ret.data.val/65536.0);|*/ +} + + +@ 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 +static 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; + a = a_orig.data.val; + b = b_orig.data.val; + c = c_orig.data.val; + d = d_orig.data.val; + @<Reduce to the case that |a,c>=0|, |b,d>0|@>; + while (1) { + q = a / d; + r = c / b; + if (q != r) { + ret->data.val = (q > r ? 1 : -1); + return; + } + q = a % d; + r = c % b; + if (r == 0) { + ret->data.val = (q ? 1 : 0); + return; + } + if (q == 0) { + ret->data.val = -1; + return; + } + a = b; + b = q; + c = d; + d = r; + } /* now |a>d>0| and |c>b>0| */ +} + + +@ @<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.val = 0; + else + ret->data.val = 1; + return; + } + if (d == 0) { + ret->data.val = (a == 0 ? 0 : -1); + return; + } + q = a; + a = c; + c = q; + q = -b; + b = -d; + d = q; +} else if (b <= 0) { + if (b < 0 && a > 0) { + ret->data.val = -1; + return; + } + ret->data.val = (c == 0 ? 0 : -1); + 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.val = fraction_one + 1; return; } +@d one_crossing { ret->data.val = fraction_one; return; } +@d zero_crossing { ret->data.val = 0; return; } + +@c +static void mp_crossing_point (MP mp, mp_number *ret, mp_number aa, mp_number bb, mp_number cc) { + integer a,b,c; + integer d; /* recursive counter */ + integer x, xx, x0, x1, x2; /* temporary registers for bisection */ + a = aa.data.val; + b = bb.data.val; + c = cc.data.val; + 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 = 1; + x0 = a; + x1 = a - b; + x2 = b - c; + do { + x = (x1 + x2) / 2; + 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 + 1; + } + } + } while (d < fraction_one); + ret->data.val = (d - fraction_one); +} + + +@ 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 = x_orig.data.val; + if (x >= 32768) { + return 1+((x-32768) / 65536); + } else if ( x>=-32768) { + return 0; + } else { + return -(1+((-(x+1)-32768) / 65536)); + } +} + +@ |number_floor| floors a |scaled| + +@c +void mp_number_floor (mp_number *i) { + i->data.val = i->data.val&-65536; +} + +@ |fraction_to_scaled| rounds a |fraction| and converts it to |scaled| +@c +void mp_fraction_to_round_scaled (mp_number *x_orig) { + int x = x_orig->data.val; + x_orig->type = mp_scaled_type; + x_orig->data.val = (x>=2048 ? 1+((x-2048) / 4096) : ( x>=-2048 ? 0 : -(1+((-(x+1)-2048) / 4096)))); +} + + + +@* 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. + +@ To get the square root of a |scaled| number |x|, we want to calculate +$s=\lfloor 2^8\!\sqrt x +{1\over2}\rfloor$. If $x>0$, this is the unique +integer such that $2^{16}x-s\L s^2<2^{16}x+s$. The following subroutine +determines $s$ by an iterative method that maintains the invariant +relations $x=2^{46-2k}x_0\bmod 2^{30}$, $0<y=\lfloor 2^{16-2k}x_0\rfloor +-s^2+s\L q=2s$, where $x_0$ is the initial value of $x$. The value of~$y$ +might, however, be zero at the start of the first iteration. + +@c +void mp_square_rt (MP mp, mp_number *ret, mp_number x_orig) { /* return, x: scaled */ + integer x; + quarterword k; /* iteration control counter */ + integer y; /* register for intermediate calculations */ + integer q; /* register for intermediate calculations */ + x = x_orig.data.val; + if (x <= 0) { + @<Handle square root of zero or negative argument@>; + } else { + k = 23; + q = 2; + while (x < fraction_two) { /* i.e., |while x<@t$2^{29}$@>|\unskip */ + k--; + x = x + x + x + x; + } + if (x < fraction_four) + y = 0; + else { + x = x - fraction_four; + y = 1; + } + do { + @<Decrease |k| by 1, maintaining the invariant + relations between |x|, |y|, and~|q|@>; + } while (k != 0); + ret->data.val = (int) (halfp (q)); + } +} + + +@ @<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 }; + mp_snprintf(msg, 256, "Square root of %s has been replaced by 0", mp_string_scaled (mp, x)); +@.Square root...replaced by 0@>; + mp_error (mp, msg, hlp, true); + } + ret->data.val = 0; + return; +} + + +@ @<Decrease |k| by 1, maintaining...@>= +x += x; +y += y; +if (x >= fraction_four) { /* note that |fraction_four=@t$2^{30}$@>| */ + x = x - fraction_four; + y++; +}; +x += x; +y = y + y - q; +q += q; +if (x >= fraction_four) { + x = x - fraction_four; + y++; +}; +if (y > (int) q) { + y -= q; + q += 2; +} else if (y <= 0) { + q -= 2; + y += q; +}; +k-- + +@ Pythagorean addition $\psqrt{a^2+b^2}$ is implemented by an elegant +iterative scheme due to Cleve Moler and Donald Morrison [{\sl IBM Journal +@^Moler, Cleve Barry@> +@^Morrison, Donald Ross@> +of Research and Development\/ \bf27} (1983), 577--581]. It modifies |a| and~|b| +in such a way that their Pythagorean sum remains invariant, while the +smaller argument decreases. + +@c +void mp_pyth_add (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig) { + int a, b; /* a,b : scaled */ + int r; /* register used to transform |a| and |b|, fraction */ + boolean big; /* is the result dangerously near $2^{31}$? */ + a = abs (a_orig.data.val); + b = abs (b_orig.data.val); + if (a < b) { + r = b; + b = a; + a = r; + }; /* now |0<=b<=a| */ + if (b > 0) { + if (a < fraction_two) { + big = false; + } else { + a = a / 4; + b = b / 4; + big = true; + }; /* we reduced the precision to avoid arithmetic overflow */ + @<Replace |a| by an approximation to $\psqrt{a^2+b^2}$@>; + if (big) { + if (a < fraction_two) { + a = a + a + a + a; + } else { + mp->arith_error = true; + a = EL_GORDO; + }; + } + } + ret->data.val = a; +} + + +@ The key idea here is to reflect the vector $(a,b)$ about the +line through $(a,b/2)$. + +@<Replace |a| by an approximation to $\psqrt{a^2+b^2}$@>= +while (1) { + r = mp_make_fraction (mp, b, a); + r = mp_take_fraction (mp, r, r); /* now $r\approx b^2/a^2$ */ + if (r == 0) + break; + r = mp_make_fraction (mp, r, fraction_four + r); + a = a + mp_take_fraction (mp, a + a, r); + b = mp_take_fraction (mp, b, r); +} + + +@ Here is a similar algorithm for $\psqrt{a^2-b^2}$. +It converges slowly when $b$ is near $a$, but otherwise it works fine. + +@c +void mp_pyth_sub (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig) { + int a, b; /* a,b: scaled */ + int r; /* register used to transform |a| and |b|, fraction */ + boolean big; /* is the result dangerously near $2^{31}$? */ + a = abs (a_orig.data.val); + b = abs (b_orig.data.val); + if (a <= b) { + @<Handle erroneous |pyth_sub| and set |a:=0|@>; + } else { + if (a < fraction_four) { + big = false; + } else { + a = (integer) halfp (a); + b = (integer) halfp (b); + big = true; + } + @<Replace |a| by an approximation to $\psqrt{a^2-b^2}$@>; + if (big) + a *= 2; + } + ret->data.val = a; +} + + +@ @<Replace |a| by an approximation to $\psqrt{a^2-b^2}$@>= +while (1) { + r = mp_make_fraction (mp, b, a); + r = mp_take_fraction (mp, r, r); /* now $r\approx b^2/a^2$ */ + if (r == 0) + break; + r = mp_make_fraction (mp, r, fraction_four - r); + a = a - mp_take_fraction (mp, a + a, r); + b = mp_take_fraction (mp, b, r); +} + + +@ @<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 = strdup(mp_string_scaled (mp, a)); + assert (astr); + mp_snprintf (msg, 256, "Pythagorean subtraction %s+-+%s has been replaced by 0", astr, mp_string_scaled (mp, b)); + free(astr); +@.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 $2^k$. The second involves +a bit more calculation, which the author claims to have done correctly: +|spec_log[k]| is $2^{27}$ times $\ln\bigl(1/(1-2^{-k})\bigr)= +2^{-k}+{1\over2}2^{-2k}+{1\over3}2^{-3k}+\cdots\,$, rounded to the +nearest integer. + +@d two_to_the(A) (1<<(unsigned)(A)) + +@<Declarations@>= +static const integer spec_log[29] = { 0, /* special logarithms */ + 93032640, 38612034, 17922280, 8662214, 4261238, 2113709, + 1052693, 525315, 262400, 131136, 65552, 32772, 16385, + 8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1 +}; + + +@ 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. + +The method is based on exercise 1.2.2--25 in {\sl The Art of Computer +Programming\/}: During the main iteration we have $1\L 2^{-30}x<1/(1-2^{1-k})$, +and the logarithm of $2^{30}x$ remains to be added to an accumulator +register called~$y$. Three auxiliary bits of accuracy are retained in~$y$ +during the calculation, and sixteen auxiliary bits to extend |y| are +kept in~|z| during the initial argument reduction. (We add +$100\cdot2^{16}=6553600$ to~|z| and subtract 100 from~|y| so that |z| will +not become negative; also, the actual amount subtracted from~|y| is~96, +not~100, because we want to add~4 for rounding before the final division by~8.) + +@c +void mp_m_log (MP mp, mp_number *ret, mp_number x_orig) { /* return, x: scaled */ + int x; + integer y, z; /* auxiliary registers */ + integer k; /* iteration counter */ + x = x_orig.data.val; + if (x <= 0) { + @<Handle non-positive logarithm@>; + } else { + y = 1302456956 + 4 - 100; /* $14\times2^{27}\ln2\approx1302456956.421063$ */ + z = 27595 + 6553600; /* and $2^{16}\times .421063\approx 27595$ */ + while (x < fraction_four) { + x = 2*x; + y -= 93032639; + z -= 48782; + } /* $2^{27}\ln2\approx 93032639.74436163$ and $2^{16}\times.74436163\approx 48782$ */ + y = y + (z / unity); + k = 2; + while (x > fraction_four + 4) { + @<Increase |k| until |x| can be multiplied by a + factor of $2^{-k}$, and adjust $y$ accordingly@>; + } + ret->data.val = (y / 8); + } +} + + +@ @<Increase |k| until |x| can...@>= +{ + z = ((x - 1) / two_to_the (k)) + 1; /* $z=\lceil x/2^k\rceil$ */ + while (x < fraction_four + z) { + z = halfp (z + 1); + k++; + }; + y += spec_log[k]; + x -= z; +} + + +@ @<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 }; + mp_snprintf (msg, 256, "Logarithm of %s has been replaced by 0", mp_string_scaled (mp, x)); +@.Logarithm...replaced by 0@>; + mp_error (mp, msg, hlp, true); + ret->data.val = 0; +} + + +@ Conversely, the exponential routine calculates $\exp(x/2^8)$, +when |x| is |scaled|. The result is an integer approximation to +$2^{16}\exp(x/2^{24})$, when |x| is regarded as an integer. + +@c +void mp_m_exp (MP mp, mp_number *ret, mp_number x_orig) { + quarterword k; /* loop control index */ + integer y, z; /* auxiliary registers */ + int x; + x = x_orig.data.val; + if (x > 174436200) { + /* $2^{24}\ln((2^{31}-1)/2^{16})\approx 174436199.51$ */ + mp->arith_error = true; + ret->data.val = EL_GORDO; + } else if (x < -197694359) { + /* $2^{24}\ln(2^{-1}/2^{16})\approx-197694359.45$ */ + ret->data.val = 0; + } else { + if (x <= 0) { + z = -8 * x; + y = 04000000; /* $y=2^{20}$ */ + } else { + if (x <= 127919879) { + z = 1023359037 - 8 * x; + /* $2^{27}\ln((2^{31}-1)/2^{20})\approx 1023359037.125$ */ + } else { + z = 8 * (174436200 - x); /* |z| is always nonnegative */ + } + y = EL_GORDO; + } + @<Multiply |y| by $\exp(-z/2^{27})$@>; + if (x <= 127919879) + ret->data.val = ((y + 8) / 16); + else + ret->data.val = y; + } +} + + +@ The idea here is that subtracting |spec_log[k]| from |z| corresponds +to multiplying |y| by $1-2^{-k}$. + +A subtle point (which had to be checked) was that if $x=127919879$, the +value of~|y| will decrease so that |y+8| doesn't overflow. In fact, +$z$ will be 5 in this case, and |y| will decrease by~64 when |k=25| +and by~16 when |k=27|. + +@<Multiply |y| by...@>= +k = 1; +while (z > 0) { + while (z >= spec_log[k]) { + z -= spec_log[k]; + y = y - 1 - ((y - two_to_the (k - 1)) / two_to_the (k)); + } + k++; +} + +@ The trigonometric subroutines use an auxiliary table such that +|spec_atan[k]| contains an approximation to the |angle| whose tangent +is~$1/2^k$. $\arctan2^{-k}$ times $2^{20}\cdot180/\pi$ + +@<Declarations@>= +static const int spec_atan[27] = { 0, 27855475, 14718068, 7471121, 3750058, + 1876857, 938658, 469357, 234682, 117342, 58671, 29335, 14668, 7334, 3667, + 1833, 917, 458, 229, 115, 57, 29, 14, 7, 4, 2, 1 +}; + + +@ Given integers |x| and |y|, not both zero, the |n_arg| function +returns the |angle| whose tangent points in the direction $(x,y)$. +This subroutine first determines the correct octant, then solves the +problem for |0<=y<=x|, then converts the result appropriately to +return an answer in the range |-one_eighty_deg<=@t$\theta$@><=one_eighty_deg|. +(The answer is |+one_eighty_deg| if |y=0| and |x<0|, but an answer of +|-one_eighty_deg| is possible if, for example, |y=-1| and $x=-2^{30}$.) + +The octants are represented in a ``Gray code,'' since that turns out +to be computationally simplest. + +@d negate_x 1 +@d negate_y 2 +@d switch_x_and_y 4 +@d first_octant 1 +@d second_octant (first_octant+switch_x_and_y) +@d third_octant (first_octant+switch_x_and_y+negate_x) +@d fourth_octant (first_octant+negate_x) +@d fifth_octant (first_octant+negate_x+negate_y) +@d sixth_octant (first_octant+switch_x_and_y+negate_x+negate_y) +@d seventh_octant (first_octant+switch_x_and_y+negate_y) +@d eighth_octant (first_octant+negate_y) + +@c +void mp_n_arg (MP mp, mp_number *ret, mp_number x_orig, mp_number y_orig) { + integer z; /* auxiliary register */ + integer t; /* temporary storage */ + quarterword k; /* loop counter */ + int octant; /* octant code */ + integer x, y; + x = x_orig.data.val; + y = y_orig.data.val; + if (x >= 0) { + octant = first_octant; + } else { + x = -x; + octant = first_octant + negate_x; + } + if (y < 0) { + y = -y; + octant = octant + negate_y; + } + if (x < y) { + t = y; + y = x; + x = t; + octant = octant + switch_x_and_y; + } + if (x == 0) { + @<Handle undefined arg@>; + } else { + ret->type = mp_angle_type; + @<Set variable |z| to the arg of $(x,y)$@>; + @<Return an appropriate answer based on |z| and |octant|@>; + } +} + + +@ @<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.val = 0; +} + + +@ @<Return an appropriate answer...@>= +switch (octant) { +case first_octant: + ret->data.val = z; + break; +case second_octant: + ret->data.val = (ninety_deg - z); + break; +case third_octant: + ret->data.val = (ninety_deg + z); + break; +case fourth_octant: + ret->data.val = (one_eighty_deg - z); + break; +case fifth_octant: + ret->data.val = (z - one_eighty_deg); + break; +case sixth_octant: + ret->data.val = (-z - ninety_deg); + break; +case seventh_octant: + ret->data.val = (z - ninety_deg); + break; +case eighth_octant: + ret->data.val = (-z); + break; +} /* there are no other cases */ + + +@ At this point we have |x>=y>=0|, and |x>0|. The numbers are scaled up +or down until $2^{28}\L x<2^{29}$, so that accurate fixed-point calculations +will be made. + +@<Set variable |z| to the arg...@>= +while (x >= fraction_two) { + x = halfp (x); + y = halfp (y); +} +z = 0; +if (y > 0) { + while (x < fraction_one) { + x += x; + y += y; + }; + @<Increase |z| to the arg of $(x,y)$@>; +} + +@ During the calculations of this section, variables |x| and~|y| +represent actual coordinates $(x,2^{-k}y)$. We will maintain the +condition |x>=y|, so that the tangent will be at most $2^{-k}$. +If $x<2y$, the tangent is greater than $2^{-k-1}$. The transformation +$(a,b)\mapsto(a+b\tan\phi,b-a\tan\phi)$ replaces $(a,b)$ by +coordinates whose angle has decreased by~$\phi$; in the special case +$a=x$, $b=2^{-k}y$, and $\tan\phi=2^{-k-1}$, this operation reduces +to the particularly simple iteration shown here. [Cf.~John E. Meggitt, +@^Meggitt, John E.@> +{\sl IBM Journal of Research and Development\/ \bf6} (1962), 210--226.] + +The initial value of |x| will be multiplied by at most +$(1+{1\over2})(1+{1\over8})(1+{1\over32})\cdots\approx 1.7584$; hence +there is no chance of integer overflow. + +@<Increase |z|...@>= +k = 0; +do { + y += y; + k++; + if (y > x) { + z = z + spec_atan[k]; + t = x; + x = x + (y / two_to_the (k + k)); + y = y - t; + }; +} while (k != 15); +do { + y += y; + k++; + if (y > x) { + z = z + spec_atan[k]; + y = y - x; + }; +} while (k != 26) + +@ 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 forty_five_deg 0264000000 /* $45\cdot2^{20}$, represents $45^\circ$ */ +@d ninety_deg 0550000000 /* $90\cdot2^{20}$, represents $90^\circ$ */ +@d one_eighty_deg 01320000000 /* $180\cdot2^{20}$, represents $180^\circ$ */ +@d three_sixty_deg 02640000000 /* $360\cdot2^{20}$, represents $360^\circ$ */ + +@d odd(A) (abs(A)%2==1) + +@ Compute a multiple of the sine and cosine + +@c +void mp_n_sin_cos (MP mp, mp_number z_orig, mp_number *n_cos, mp_number *n_sin) { + quarterword k; /* loop control variable */ + int q; /* specifies the quadrant */ + integer x, y, t; /* temporary registers */ + int z; /* scaled */ + mp_number x_n, y_n, ret; + new_number (ret); + new_number (x_n); + new_number (y_n); + z = z_orig.data.val; + while (z < 0) + z = z + three_sixty_deg; + z = z % three_sixty_deg; /* now |0<=z<three_sixty_deg| */ + q = z / forty_five_deg; + z = z % forty_five_deg; + x = fraction_one; + y = x; + if (!odd (q)) + z = forty_five_deg - z; + @<Subtract angle |z| from |(x,y)|@>; + @<Convert |(x,y)| to the octant determined by~|q|@>; + x_n.data.val = x; + y_n.data.val = y; + mp_pyth_add (mp, &ret, x_n, y_n); + n_cos->data.val = mp_make_fraction (mp, x, ret.data.val); + n_sin->data.val = mp_make_fraction (mp, y, ret.data.val); + free_number(ret); + free_number(x_n); + free_number(y_n); +} + + +@ In this case the octants are numbered sequentially. + +@<Convert |(x,...@>= +switch (q) { +case 0: + break; +case 1: + t = x; + x = y; + y = t; + break; +case 2: + t = x; + x = -y; + y = t; + break; +case 3: + x = -x; + break; +case 4: + x = -x; + y = -y; + break; +case 5: + t = x; + x = -y; + y = -t; + break; +case 6: + t = x; + x = y; + y = -t; + break; +case 7: + y = -y; + break; +} /* there are no other cases */ + + +@ The main iteration of |n_sin_cos| is similar to that of |n_arg| but +applied in reverse. The values of |spec_atan[k]| decrease slowly enough +that this loop is guaranteed to terminate before the (nonexistent) value +|spec_atan[27]| would be required. + +@<Subtract angle |z|...@>= +k = 1; +while (z > 0) { + if (z >= spec_atan[k]) { + z = z - spec_atan[k]; + t = x; + x = t + y / two_to_the (k); + y = y - t / two_to_the (k); + } + k++; +} +if (y < 0) + y = 0 /* this precaution may never be needed */ + + +@ 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.val = j; + } + mp_new_randoms (mp); + mp_new_randoms (mp); + mp_new_randoms (mp); /* ``warm up'' the array */ +} + + +@ @c +void mp_print_number (MP mp, mp_number n) { + mp_print_scaled (mp, n.data.val); +} + + +@ @c +char * mp_number_tostring (MP mp, mp_number n) { + return mp_string_scaled(mp, n.data.val); +} + +@ @c +void mp_number_modulo (mp_number *a, mp_number b) { + a->data.val = a->data.val % b.data.val; +} + + + + + +@ 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_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_number_abs (&abs_x); + mp_next_random(mp, &u); + /*|take_fraction (y, abs_x, u);|*/ + mp_number_take_fraction (mp,&y, abs_x,u); + free_number (u); + if (mp_number_equal(y, abs_x)) { + /*|set_number_to_zero(*ret);|*/ + 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_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_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_number_abs (&abs_x); + } while (!mp_number_less(abs_x, u)); + mp_number_make_fraction (mp, &r, xa, u); + mp_number_clone (&xa, r); + mp_m_log (mp,&la, u); + mp_set_number_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); +} |