| Partial backports of: |
| |
| |
| commit c5d5d574cbfa96d0f6c1db24d1e072c472627e41 |
| Author: Ondřej Bílka <neleai@seznam.cz> |
| Date: Thu Oct 17 16:03:24 2013 +0200 |
| |
| Format floating routines. |
| |
| commit da08f647d58d674db08cdb3e61c8826c89470e2e |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Fri Dec 21 09:15:10 2012 +0530 |
| |
| Move more constants into static variables |
| |
| Code cleanup. |
| |
| commit f93a8d15699ee699282465dc1e03e033f3fabb52 |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Wed Jan 16 16:06:48 2013 +0530 |
| |
| Consolidate constant defines into mpa.h |
| |
| commit caa99d06e7f1403887294442af520b0f8c6f3de0 |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Fri Jan 18 11:18:13 2013 +0530 |
| |
| Simplify calculation of 2^-m in __mpexp |
| |
| commit 107a5bf085f5c4ef8c28266a34d476724cfc3475 |
| Author: Joseph Myers <joseph@codesourcery.com> |
| Date: Tue Nov 18 15:40:56 2014 +0000 |
| |
| Fix libm mpone, mptwo namespace (bug 17616). |
| |
| To provided __mptwo for __inv. |
| |
| Full backports of the following: |
| |
| |
| commit 44e0d4c20ce5bf3825897e5d4b7caae94016214d |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Wed Jan 2 11:44:13 2013 +0530 |
| |
| Split mantissa calculation loop and add branch prediction |
| |
| commit f8af25d218202ff2f5d167b8e44e4b79f91d147f |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Fri Jan 4 15:09:33 2013 +0530 |
| |
| Remove commented declarations |
| |
| commit a9e48ab40e230c7fe34e4892bec8af4f3f975a20 |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Fri Jan 4 15:42:09 2013 +0530 |
| |
| Clean up comment for MP_NO |
| |
| commit fffb407f4668b40b3df1eb8ee3ae3bc64ee79e20 |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Fri Jan 4 22:52:12 2013 +0530 |
| |
| Remove unused __cr and __cpymn |
| |
| commit 950c99ca9094e7dc6394e90395f51e12093393aa |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Wed Jan 9 19:07:15 2013 +0530 |
| |
| Update comments in mpa.c |
| |
| Fixed comment style and clearer wording in some cases. |
| |
| commit 1066a53440d2744566e97c59bcd0d422186b3e90 |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Mon Jan 14 21:31:25 2013 +0530 |
| |
| Fix code formatting in mpa.c |
| |
| This includes the overridden mpa.c in power4. |
| |
| commit 2a91b5735ac1bc65ce5c2a3646d75ba7208e26e9 |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Mon Jan 14 21:36:58 2013 +0530 |
| |
| Minor tweak to mp multiplication |
| |
| Add a local variable to remove extra copies to/from memory in the Z |
| array. |
| |
| ommit 45f058844c33f670475bd02f266942746bcb332b |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Tue Feb 26 21:28:16 2013 +0530 |
| |
| Another tweak to the multiplication algorithm |
| |
| Reduce the formula to calculate mantissa so that we reduce the net |
| number of multiplications performed. |
| |
| commit bab8a695ee79a5a6e9b2b699938952b006fcbbec |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Thu Feb 21 14:29:18 2013 +0530 |
| |
| Fix whitespace differences between generic and powerpc mpa.c |
| |
| |
| commit 2d0e0f29f85036d1189231cb7c1f19f27ba14a89 |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Fri Feb 15 23:56:20 2013 +0530 |
| |
| Fix determination of lower precision in __mul |
| |
| commit 909279a5cfa938c989c9b01c8f48a0276291ec45 |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Wed Feb 13 14:16:23 2013 +0530 |
| |
| Optimized mp multiplication |
| |
| Don't bother multiplying zeroes since that only wastes cycles. |
| |
| commit bdf028142eb77d6ae59500db875068fa5d7b059d |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Wed Feb 13 13:55:29 2013 +0530 |
| |
| Clean up add_magnitudes and sub_magnitudes |
| |
| commit d6752ccd696c71d23cd3df8fb9cc60b61c32e65a |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Thu Feb 14 10:31:09 2013 +0530 |
| |
| New __sqr function as a faster special case of __mul |
| |
| commit 4709fe7602b56e9f6ee1ab6afb4067409a784f29 |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Sat Feb 16 00:09:29 2013 +0530 |
| |
| Use intermediate variable to compute exponent in __mul |
| |
| commit 20cd7fb3ae63795ae7c9a464abf5ed19b364ade0 |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Wed Feb 20 18:56:20 2013 +0530 |
| |
| Copy comment about inner loop from powerpc mpa.c to the default one |
| |
| commit e69804d14e43f14c3c65dc570afdbfb822c9838b |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Mon Feb 25 16:43:02 2013 +0530 |
| |
| Use long wherever possible in mpa.c |
| |
| Using long throughout like powerpc does is beneficial since it reduces |
| the need to switch to 32-bit instructions. It gives a very minor |
| performance improvement. |
| |
| commit 82a9811d29c00161c7c8ea7f70e2cc30988e192e |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Thu Mar 7 12:23:29 2013 +0530 |
| |
| Use generic mpa.c code for everything except __mul and __sqr |
| |
| commit 6f2e90e78f151bab153c2b38492505ae2012db06 |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Tue Mar 26 19:28:50 2013 +0530 |
| |
| Make mantissa type of mp_no configurable |
| |
| The mantissa of mp_no is intended to take only integral values. This |
| is a relatively good choice for powerpc due to its 4 fpus, but not for |
| other architectures, which suffer due to this choice. This change |
| makes the default mantissa a long integer and allows powerpc to |
| override it. Additionally, some operations have been optimized for |
| integer manipulation, resulting in a significant improvement in |
| performance. |
| |
| commit 5739f705eed5cf58e7b439e5983542e06d7fc2da |
| Author: Siddhesh Poyarekar <siddhesh@redhat.com> |
| Date: Tue Mar 26 20:24:04 2013 +0530 |
| |
| Use integral constants |
| |
| The compiler is smart enough to convert those into double for powerpc, |
| but if we put them as doubles, it adds overhead by performing those |
| operations in floating point mode. |
| |
| commit 89f3b6e18c6e7833438789746fcfc2e7189f7cac |
| Author: Joseph Myers <joseph@codesourcery.com> |
| Date: Thu May 21 23:05:45 2015 +0000 |
| |
| Fix sysdeps/ieee754/dbl-64/mpa.c for -Wuninitialized. |
| |
| If you remove the "override CFLAGS += -Wno-uninitialized" in |
| math/Makefile, one of the errors you get is: |
| |
| ../sysdeps/ieee754/dbl-64/mpa.c: In function '__mp_dbl.part.0': |
| ../sysdeps/ieee754/dbl-64/mpa.c:183:5: error: 'c' may be used uninitialized in this function [-Werror=maybe-uninitialized] |
| c *= X[0]; |
| |
| The problem is that the p < 5 case initializes c if p is 1, 2, 3 or 4 |
| but not otherwise, and in fact p is positive for all calls to this |
| function so the uninitialized case can't actually occur. This patch |
| replaces the "if (p == 4)" last case with a comment so the compiler |
| can see that all paths do initialize c. |
| |
| Tested for x86_64. |
| |
| * sysdeps/ieee754/dbl-64/mpa.c (norm): Remove if condition on |
| (p == 4) case. |
| |
| |
| |
| |
| |
| @@ -1,7 +1,7 @@ |
| /* |
| * IBM Accurate Mathematical Library |
| * written by International Business Machines Corp. |
| - * Copyright (C) 2001, 2011 Free Software Foundation |
| + * Copyright (C) 2001-2017 Free Software Foundation, Inc. |
| * |
| * This program is free software; you can redistribute it and/or modify |
| * it under the terms of the GNU Lesser General Public License as published by |
| @@ -22,9 +22,7 @@ |
| /* FUNCTIONS: */ |
| /* mcr */ |
| /* acr */ |
| -/* cr */ |
| /* cpy */ |
| -/* cpymn */ |
| /* norm */ |
| /* denorm */ |
| /* mp_dbl */ |
| @@ -44,479 +42,868 @@ |
| |
| #include "endian.h" |
| #include "mpa.h" |
| -#include "mpa2.h" |
| -#include <sys/param.h> /* For MIN() */ |
| +#include <sys/param.h> |
| +#include <alloca.h> |
| |
| #ifndef SECTION |
| # define SECTION |
| #endif |
| |
| +#ifndef NO__CONST |
| +/* TODO: With only a partial backport of the constant cleanup from |
| + upstream we limit the definition of these two constants to |
| + this file. */ |
| +static const mp_no __mpone = { 1, { 1.0, 1.0 } }; |
| +static const mp_no __mptwo = { 1, { 1.0, 2.0 } }; |
| +#endif |
| + |
| #ifndef NO___ACR |
| -/* mcr() compares the sizes of the mantissas of two multiple precision */ |
| -/* numbers. Mantissas are compared regardless of the signs of the */ |
| -/* numbers, even if x->d[0] or y->d[0] are zero. Exponents are also */ |
| -/* disregarded. */ |
| +/* Compare mantissa of two multiple precision numbers regardless of the sign |
| + and exponent of the numbers. */ |
| static int |
| -mcr(const mp_no *x, const mp_no *y, int p) { |
| - int i; |
| - for (i=1; i<=p; i++) { |
| - if (X[i] == Y[i]) continue; |
| - else if (X[i] > Y[i]) return 1; |
| - else return -1; } |
| +mcr (const mp_no *x, const mp_no *y, int p) |
| +{ |
| + long i; |
| + long p2 = p; |
| + for (i = 1; i <= p2; i++) |
| + { |
| + if (X[i] == Y[i]) |
| + continue; |
| + else if (X[i] > Y[i]) |
| + return 1; |
| + else |
| + return -1; |
| + } |
| return 0; |
| } |
| |
| - |
| -/* acr() compares the absolute values of two multiple precision numbers */ |
| +/* Compare the absolute values of two multiple precision numbers. */ |
| int |
| -__acr(const mp_no *x, const mp_no *y, int p) { |
| - int i; |
| - |
| - if (X[0] == ZERO) { |
| - if (Y[0] == ZERO) i= 0; |
| - else i=-1; |
| - } |
| - else if (Y[0] == ZERO) i= 1; |
| - else { |
| - if (EX > EY) i= 1; |
| - else if (EX < EY) i=-1; |
| - else i= mcr(x,y,p); |
| - } |
| - |
| - return i; |
| -} |
| -#endif |
| - |
| +__acr (const mp_no *x, const mp_no *y, int p) |
| +{ |
| + long i; |
| |
| -#if 0 |
| -/* cr() compares the values of two multiple precision numbers */ |
| -static int __cr(const mp_no *x, const mp_no *y, int p) { |
| - int i; |
| - |
| - if (X[0] > Y[0]) i= 1; |
| - else if (X[0] < Y[0]) i=-1; |
| - else if (X[0] < ZERO ) i= __acr(y,x,p); |
| - else i= __acr(x,y,p); |
| + if (X[0] == 0) |
| + { |
| + if (Y[0] == 0) |
| + i = 0; |
| + else |
| + i = -1; |
| + } |
| + else if (Y[0] == 0) |
| + i = 1; |
| + else |
| + { |
| + if (EX > EY) |
| + i = 1; |
| + else if (EX < EY) |
| + i = -1; |
| + else |
| + i = mcr (x, y, p); |
| + } |
| |
| return i; |
| } |
| #endif |
| |
| - |
| #ifndef NO___CPY |
| -/* Copy a multiple precision number. Set *y=*x. x=y is permissible. */ |
| -void __cpy(const mp_no *x, mp_no *y, int p) { |
| +/* Copy multiple precision number X into Y. They could be the same |
| + number. */ |
| +void |
| +__cpy (const mp_no *x, mp_no *y, int p) |
| +{ |
| + long i; |
| + |
| EY = EX; |
| - for (int i=0; i <= p; i++) Y[i] = X[i]; |
| + for (i = 0; i <= p; i++) |
| + Y[i] = X[i]; |
| } |
| #endif |
| |
| +#ifndef NO___MP_DBL |
| +/* Convert a multiple precision number *X into a double precision |
| + number *Y, normalized case (|x| >= 2**(-1022))). X has precision |
| + P, which is positive. */ |
| +static void |
| +norm (const mp_no *x, double *y, int p) |
| +{ |
| +# define R RADIXI |
| + long i; |
| + double c; |
| + mantissa_t a, u, v, z[5]; |
| + if (p < 5) |
| + { |
| + if (p == 1) |
| + c = X[1]; |
| + else if (p == 2) |
| + c = X[1] + R * X[2]; |
| + else if (p == 3) |
| + c = X[1] + R * (X[2] + R * X[3]); |
| + else /* p == 4. */ |
| + c = (X[1] + R * X[2]) + R * R * (X[3] + R * X[4]); |
| + } |
| + else |
| + { |
| + for (a = 1, z[1] = X[1]; z[1] < TWO23; ) |
| + { |
| + a *= 2; |
| + z[1] *= 2; |
| + } |
| |
| -#if 0 |
| -/* Copy a multiple precision number x of precision m into a */ |
| -/* multiple precision number y of precision n. In case n>m, */ |
| -/* the digits of y beyond the m'th are set to zero. In case */ |
| -/* n<m, the digits of x beyond the n'th are ignored. */ |
| -/* x=y is permissible. */ |
| - |
| -static void __cpymn(const mp_no *x, int m, mp_no *y, int n) { |
| - |
| - int i,k; |
| - |
| - EY = EX; k=MIN(m,n); |
| - for (i=0; i <= k; i++) Y[i] = X[i]; |
| - for ( ; i <= n; i++) Y[i] = ZERO; |
| -} |
| -#endif |
| + for (i = 2; i < 5; i++) |
| + { |
| + mantissa_store_t d, r; |
| + d = X[i] * (mantissa_store_t) a; |
| + DIV_RADIX (d, r); |
| + z[i] = r; |
| + z[i - 1] += d; |
| + } |
| |
| + u = ALIGN_DOWN_TO (z[3], TWO19); |
| + v = z[3] - u; |
| |
| -#ifndef NO___MP_DBL |
| -/* Convert a multiple precision number *x into a double precision */ |
| -/* number *y, normalized case (|x| >= 2**(-1022))) */ |
| -static void norm(const mp_no *x, double *y, int p) |
| -{ |
| - #define R radixi.d |
| - int i; |
| -#if 0 |
| - int k; |
| -#endif |
| - double a,c,u,v,z[5]; |
| - if (p<5) { |
| - if (p==1) c = X[1]; |
| - else if (p==2) c = X[1] + R* X[2]; |
| - else if (p==3) c = X[1] + R*(X[2] + R* X[3]); |
| - else if (p==4) c =(X[1] + R* X[2]) + R*R*(X[3] + R*X[4]); |
| - } |
| - else { |
| - for (a=ONE, z[1]=X[1]; z[1] < TWO23; ) |
| - {a *= TWO; z[1] *= TWO; } |
| - |
| - for (i=2; i<5; i++) { |
| - z[i] = X[i]*a; |
| - u = (z[i] + CUTTER)-CUTTER; |
| - if (u > z[i]) u -= RADIX; |
| - z[i] -= u; |
| - z[i-1] += u*RADIXI; |
| - } |
| - |
| - u = (z[3] + TWO71) - TWO71; |
| - if (u > z[3]) u -= TWO19; |
| - v = z[3]-u; |
| - |
| - if (v == TWO18) { |
| - if (z[4] == ZERO) { |
| - for (i=5; i <= p; i++) { |
| - if (X[i] == ZERO) continue; |
| - else {z[3] += ONE; break; } |
| + if (v == TWO18) |
| + { |
| + if (z[4] == 0) |
| + { |
| + for (i = 5; i <= p; i++) |
| + { |
| + if (X[i] == 0) |
| + continue; |
| + else |
| + { |
| + z[3] += 1; |
| + break; |
| + } |
| + } |
| + } |
| + else |
| + z[3] += 1; |
| } |
| - } |
| - else z[3] += ONE; |
| - } |
| |
| - c = (z[1] + R *(z[2] + R * z[3]))/a; |
| - } |
| + c = (z[1] + R * (z[2] + R * z[3])) / a; |
| + } |
| |
| c *= X[0]; |
| |
| - for (i=1; i<EX; i++) c *= RADIX; |
| - for (i=1; i>EX; i--) c *= RADIXI; |
| + for (i = 1; i < EX; i++) |
| + c *= RADIX; |
| + for (i = 1; i > EX; i--) |
| + c *= RADIXI; |
| |
| *y = c; |
| -#undef R |
| +# undef R |
| } |
| |
| -/* Convert a multiple precision number *x into a double precision */ |
| -/* number *y, denormalized case (|x| < 2**(-1022))) */ |
| -static void denorm(const mp_no *x, double *y, int p) |
| +/* Convert a multiple precision number *X into a double precision |
| + number *Y, Denormal case (|x| < 2**(-1022))). */ |
| +static void |
| +denorm (const mp_no *x, double *y, int p) |
| { |
| - int i,k; |
| - double c,u,z[5]; |
| -#if 0 |
| - double a,v; |
| -#endif |
| + long i, k; |
| + long p2 = p; |
| + double c; |
| + mantissa_t u, z[5]; |
| + |
| +# define R RADIXI |
| + if (EX < -44 || (EX == -44 && X[1] < TWO5)) |
| + { |
| + *y = 0; |
| + return; |
| + } |
| |
| -#define R radixi.d |
| - if (EX<-44 || (EX==-44 && X[1]<TWO5)) |
| - { *y=ZERO; return; } |
| - |
| - if (p==1) { |
| - if (EX==-42) {z[1]=X[1]+TWO10; z[2]=ZERO; z[3]=ZERO; k=3;} |
| - else if (EX==-43) {z[1]= TWO10; z[2]=X[1]; z[3]=ZERO; k=2;} |
| - else {z[1]= TWO10; z[2]=ZERO; z[3]=X[1]; k=1;} |
| - } |
| - else if (p==2) { |
| - if (EX==-42) {z[1]=X[1]+TWO10; z[2]=X[2]; z[3]=ZERO; k=3;} |
| - else if (EX==-43) {z[1]= TWO10; z[2]=X[1]; z[3]=X[2]; k=2;} |
| - else {z[1]= TWO10; z[2]=ZERO; z[3]=X[1]; k=1;} |
| - } |
| - else { |
| - if (EX==-42) {z[1]=X[1]+TWO10; z[2]=X[2]; k=3;} |
| - else if (EX==-43) {z[1]= TWO10; z[2]=X[1]; k=2;} |
| - else {z[1]= TWO10; z[2]=ZERO; k=1;} |
| - z[3] = X[k]; |
| - } |
| - |
| - u = (z[3] + TWO57) - TWO57; |
| - if (u > z[3]) u -= TWO5; |
| - |
| - if (u==z[3]) { |
| - for (i=k+1; i <= p; i++) { |
| - if (X[i] == ZERO) continue; |
| - else {z[3] += ONE; break; } |
| - } |
| - } |
| - |
| - c = X[0]*((z[1] + R*(z[2] + R*z[3])) - TWO10); |
| - |
| - *y = c*TWOM1032; |
| -#undef R |
| -} |
| - |
| -/* Convert a multiple precision number *x into a double precision number *y. */ |
| -/* The result is correctly rounded to the nearest/even. *x is left unchanged */ |
| - |
| -void __mp_dbl(const mp_no *x, double *y, int p) { |
| -#if 0 |
| - int i,k; |
| - double a,c,u,v,z[5]; |
| -#endif |
| + if (p2 == 1) |
| + { |
| + if (EX == -42) |
| + { |
| + z[1] = X[1] + TWO10; |
| + z[2] = 0; |
| + z[3] = 0; |
| + k = 3; |
| + } |
| + else if (EX == -43) |
| + { |
| + z[1] = TWO10; |
| + z[2] = X[1]; |
| + z[3] = 0; |
| + k = 2; |
| + } |
| + else |
| + { |
| + z[1] = TWO10; |
| + z[2] = 0; |
| + z[3] = X[1]; |
| + k = 1; |
| + } |
| + } |
| + else if (p2 == 2) |
| + { |
| + if (EX == -42) |
| + { |
| + z[1] = X[1] + TWO10; |
| + z[2] = X[2]; |
| + z[3] = 0; |
| + k = 3; |
| + } |
| + else if (EX == -43) |
| + { |
| + z[1] = TWO10; |
| + z[2] = X[1]; |
| + z[3] = X[2]; |
| + k = 2; |
| + } |
| + else |
| + { |
| + z[1] = TWO10; |
| + z[2] = 0; |
| + z[3] = X[1]; |
| + k = 1; |
| + } |
| + } |
| + else |
| + { |
| + if (EX == -42) |
| + { |
| + z[1] = X[1] + TWO10; |
| + z[2] = X[2]; |
| + k = 3; |
| + } |
| + else if (EX == -43) |
| + { |
| + z[1] = TWO10; |
| + z[2] = X[1]; |
| + k = 2; |
| + } |
| + else |
| + { |
| + z[1] = TWO10; |
| + z[2] = 0; |
| + k = 1; |
| + } |
| + z[3] = X[k]; |
| + } |
| |
| - if (X[0] == ZERO) {*y = ZERO; return; } |
| + u = ALIGN_DOWN_TO (z[3], TWO5); |
| |
| - if (EX> -42) norm(x,y,p); |
| - else if (EX==-42 && X[1]>=TWO10) norm(x,y,p); |
| - else denorm(x,y,p); |
| + if (u == z[3]) |
| + { |
| + for (i = k + 1; i <= p2; i++) |
| + { |
| + if (X[i] == 0) |
| + continue; |
| + else |
| + { |
| + z[3] += 1; |
| + break; |
| + } |
| + } |
| + } |
| + |
| + c = X[0] * ((z[1] + R * (z[2] + R * z[3])) - TWO10); |
| + |
| + *y = c * TWOM1032; |
| +# undef R |
| } |
| -#endif |
| |
| +/* Convert multiple precision number *X into double precision number *Y. The |
| + result is correctly rounded to the nearest/even. */ |
| +void |
| +__mp_dbl (const mp_no *x, double *y, int p) |
| +{ |
| + if (X[0] == 0) |
| + { |
| + *y = 0; |
| + return; |
| + } |
| |
| -/* dbl_mp() converts a double precision number x into a multiple precision */ |
| -/* number *y. If the precision p is too small the result is truncated. x is */ |
| -/* left unchanged. */ |
| + if (__glibc_likely (EX > -42 || (EX == -42 && X[1] >= TWO10))) |
| + norm (x, y, p); |
| + else |
| + denorm (x, y, p); |
| +} |
| +#endif |
| |
| +/* Get the multiple precision equivalent of X into *Y. If the precision is too |
| + small, the result is truncated. */ |
| void |
| SECTION |
| -__dbl_mp(double x, mp_no *y, int p) { |
| +__dbl_mp (double x, mp_no *y, int p) |
| +{ |
| + long i, n; |
| + long p2 = p; |
| |
| - int i,n; |
| - double u; |
| + /* Sign. */ |
| + if (x == 0) |
| + { |
| + Y[0] = 0; |
| + return; |
| + } |
| + else if (x > 0) |
| + Y[0] = 1; |
| + else |
| + { |
| + Y[0] = -1; |
| + x = -x; |
| + } |
| |
| - /* Sign */ |
| - if (x == ZERO) {Y[0] = ZERO; return; } |
| - else if (x > ZERO) Y[0] = ONE; |
| - else {Y[0] = MONE; x=-x; } |
| - |
| - /* Exponent */ |
| - for (EY=ONE; x >= RADIX; EY += ONE) x *= RADIXI; |
| - for ( ; x < ONE; EY -= ONE) x *= RADIX; |
| - |
| - /* Digits */ |
| - n=MIN(p,4); |
| - for (i=1; i<=n; i++) { |
| - u = (x + TWO52) - TWO52; |
| - if (u>x) u -= ONE; |
| - Y[i] = u; x -= u; x *= RADIX; } |
| - for ( ; i<=p; i++) Y[i] = ZERO; |
| + /* Exponent. */ |
| + for (EY = 1; x >= RADIX; EY += 1) |
| + x *= RADIXI; |
| + for (; x < 1; EY -= 1) |
| + x *= RADIX; |
| + |
| + /* Digits. */ |
| + n = MIN (p2, 4); |
| + for (i = 1; i <= n; i++) |
| + { |
| + INTEGER_OF (x, Y[i]); |
| + x *= RADIX; |
| + } |
| + for (; i <= p2; i++) |
| + Y[i] = 0; |
| } |
| |
| - |
| -/* add_magnitudes() adds the magnitudes of *x & *y assuming that */ |
| -/* abs(*x) >= abs(*y) > 0. */ |
| -/* The sign of the sum *z is undefined. x&y may overlap but not x&z or y&z. */ |
| -/* No guard digit is used. The result equals the exact sum, truncated. */ |
| -/* *x & *y are left unchanged. */ |
| - |
| +/* Add magnitudes of *X and *Y assuming that abs (*X) >= abs (*Y) > 0. The |
| + sign of the sum *Z is not changed. X and Y may overlap but not X and Z or |
| + Y and Z. No guard digit is used. The result equals the exact sum, |
| + truncated. */ |
| static void |
| SECTION |
| -add_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) { |
| - |
| - int i,j,k; |
| +add_magnitudes (const mp_no *x, const mp_no *y, mp_no *z, int p) |
| +{ |
| + long i, j, k; |
| + long p2 = p; |
| + mantissa_t zk; |
| |
| EZ = EX; |
| |
| - i=p; j=p+ EY - EX; k=p+1; |
| + i = p2; |
| + j = p2 + EY - EX; |
| + k = p2 + 1; |
| + |
| + if (__glibc_unlikely (j < 1)) |
| + { |
| + __cpy (x, z, p); |
| + return; |
| + } |
| |
| - if (j<1) |
| - {__cpy(x,z,p); return; } |
| - else Z[k] = ZERO; |
| - |
| - for (; j>0; i--,j--) { |
| - Z[k] += X[i] + Y[j]; |
| - if (Z[k] >= RADIX) { |
| - Z[k] -= RADIX; |
| - Z[--k] = ONE; } |
| - else |
| - Z[--k] = ZERO; |
| - } |
| - |
| - for (; i>0; i--) { |
| - Z[k] += X[i]; |
| - if (Z[k] >= RADIX) { |
| - Z[k] -= RADIX; |
| - Z[--k] = ONE; } |
| - else |
| - Z[--k] = ZERO; |
| - } |
| - |
| - if (Z[1] == ZERO) { |
| - for (i=1; i<=p; i++) Z[i] = Z[i+1]; } |
| - else EZ += ONE; |
| -} |
| + zk = 0; |
| |
| + for (; j > 0; i--, j--) |
| + { |
| + zk += X[i] + Y[j]; |
| + if (zk >= RADIX) |
| + { |
| + Z[k--] = zk - RADIX; |
| + zk = 1; |
| + } |
| + else |
| + { |
| + Z[k--] = zk; |
| + zk = 0; |
| + } |
| + } |
| |
| -/* sub_magnitudes() subtracts the magnitudes of *x & *y assuming that */ |
| -/* abs(*x) > abs(*y) > 0. */ |
| -/* The sign of the difference *z is undefined. x&y may overlap but not x&z */ |
| -/* or y&z. One guard digit is used. The error is less than one ulp. */ |
| -/* *x & *y are left unchanged. */ |
| + for (; i > 0; i--) |
| + { |
| + zk += X[i]; |
| + if (zk >= RADIX) |
| + { |
| + Z[k--] = zk - RADIX; |
| + zk = 1; |
| + } |
| + else |
| + { |
| + Z[k--] = zk; |
| + zk = 0; |
| + } |
| + } |
| |
| + if (zk == 0) |
| + { |
| + for (i = 1; i <= p2; i++) |
| + Z[i] = Z[i + 1]; |
| + } |
| + else |
| + { |
| + Z[1] = zk; |
| + EZ += 1; |
| + } |
| +} |
| + |
| +/* Subtract the magnitudes of *X and *Y assuming that abs (*x) > abs (*y) > 0. |
| + The sign of the difference *Z is not changed. X and Y may overlap but not X |
| + and Z or Y and Z. One guard digit is used. The error is less than one |
| + ULP. */ |
| static void |
| SECTION |
| -sub_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) { |
| - |
| - int i,j,k; |
| +sub_magnitudes (const mp_no *x, const mp_no *y, mp_no *z, int p) |
| +{ |
| + long i, j, k; |
| + long p2 = p; |
| + mantissa_t zk; |
| |
| EZ = EX; |
| + i = p2; |
| + j = p2 + EY - EX; |
| + k = p2; |
| + |
| + /* Y is too small compared to X, copy X over to the result. */ |
| + if (__glibc_unlikely (j < 1)) |
| + { |
| + __cpy (x, z, p); |
| + return; |
| + } |
| + |
| + /* The relevant least significant digit in Y is non-zero, so we factor it in |
| + to enhance accuracy. */ |
| + if (j < p2 && Y[j + 1] > 0) |
| + { |
| + Z[k + 1] = RADIX - Y[j + 1]; |
| + zk = -1; |
| + } |
| + else |
| + zk = Z[k + 1] = 0; |
| |
| - if (EX == EY) { |
| - i=j=k=p; |
| - Z[k] = Z[k+1] = ZERO; } |
| - else { |
| - j= EX - EY; |
| - if (j > p) {__cpy(x,z,p); return; } |
| - else { |
| - i=p; j=p+1-j; k=p; |
| - if (Y[j] > ZERO) { |
| - Z[k+1] = RADIX - Y[j--]; |
| - Z[k] = MONE; } |
| - else { |
| - Z[k+1] = ZERO; |
| - Z[k] = ZERO; j--;} |
| - } |
| - } |
| - |
| - for (; j>0; i--,j--) { |
| - Z[k] += (X[i] - Y[j]); |
| - if (Z[k] < ZERO) { |
| - Z[k] += RADIX; |
| - Z[--k] = MONE; } |
| - else |
| - Z[--k] = ZERO; |
| - } |
| - |
| - for (; i>0; i--) { |
| - Z[k] += X[i]; |
| - if (Z[k] < ZERO) { |
| - Z[k] += RADIX; |
| - Z[--k] = MONE; } |
| - else |
| - Z[--k] = ZERO; |
| - } |
| + /* Subtract and borrow. */ |
| + for (; j > 0; i--, j--) |
| + { |
| + zk += (X[i] - Y[j]); |
| + if (zk < 0) |
| + { |
| + Z[k--] = zk + RADIX; |
| + zk = -1; |
| + } |
| + else |
| + { |
| + Z[k--] = zk; |
| + zk = 0; |
| + } |
| + } |
| |
| - for (i=1; Z[i] == ZERO; i++) ; |
| + /* We're done with digits from Y, so it's just digits in X. */ |
| + for (; i > 0; i--) |
| + { |
| + zk += X[i]; |
| + if (zk < 0) |
| + { |
| + Z[k--] = zk + RADIX; |
| + zk = -1; |
| + } |
| + else |
| + { |
| + Z[k--] = zk; |
| + zk = 0; |
| + } |
| + } |
| + |
| + /* Normalize. */ |
| + for (i = 1; Z[i] == 0; i++) |
| + ; |
| EZ = EZ - i + 1; |
| - for (k=1; i <= p+1; ) |
| + for (k = 1; i <= p2 + 1; ) |
| Z[k++] = Z[i++]; |
| - for (; k <= p; ) |
| - Z[k++] = ZERO; |
| + for (; k <= p2; ) |
| + Z[k++] = 0; |
| } |
| |
| - |
| -/* Add two multiple precision numbers. Set *z = *x + *y. x&y may overlap */ |
| -/* but not x&z or y&z. One guard digit is used. The error is less than */ |
| -/* one ulp. *x & *y are left unchanged. */ |
| - |
| +/* Add *X and *Y and store the result in *Z. X and Y may overlap, but not X |
| + and Z or Y and Z. One guard digit is used. The error is less than one |
| + ULP. */ |
| void |
| SECTION |
| -__add(const mp_no *x, const mp_no *y, mp_no *z, int p) { |
| - |
| +__add (const mp_no *x, const mp_no *y, mp_no *z, int p) |
| +{ |
| int n; |
| |
| - if (X[0] == ZERO) {__cpy(y,z,p); return; } |
| - else if (Y[0] == ZERO) {__cpy(x,z,p); return; } |
| + if (X[0] == 0) |
| + { |
| + __cpy (y, z, p); |
| + return; |
| + } |
| + else if (Y[0] == 0) |
| + { |
| + __cpy (x, z, p); |
| + return; |
| + } |
| |
| - if (X[0] == Y[0]) { |
| - if (__acr(x,y,p) > 0) {add_magnitudes(x,y,z,p); Z[0] = X[0]; } |
| - else {add_magnitudes(y,x,z,p); Z[0] = Y[0]; } |
| - } |
| - else { |
| - if ((n=__acr(x,y,p)) == 1) {sub_magnitudes(x,y,z,p); Z[0] = X[0]; } |
| - else if (n == -1) {sub_magnitudes(y,x,z,p); Z[0] = Y[0]; } |
| - else Z[0] = ZERO; |
| - } |
| + if (X[0] == Y[0]) |
| + { |
| + if (__acr (x, y, p) > 0) |
| + { |
| + add_magnitudes (x, y, z, p); |
| + Z[0] = X[0]; |
| + } |
| + else |
| + { |
| + add_magnitudes (y, x, z, p); |
| + Z[0] = Y[0]; |
| + } |
| + } |
| + else |
| + { |
| + if ((n = __acr (x, y, p)) == 1) |
| + { |
| + sub_magnitudes (x, y, z, p); |
| + Z[0] = X[0]; |
| + } |
| + else if (n == -1) |
| + { |
| + sub_magnitudes (y, x, z, p); |
| + Z[0] = Y[0]; |
| + } |
| + else |
| + Z[0] = 0; |
| + } |
| } |
| |
| +/* Subtract *Y from *X and return the result in *Z. X and Y may overlap but |
| + not X and Z or Y and Z. One guard digit is used. The error is less than |
| + one ULP. */ |
| +void |
| +SECTION |
| +__sub (const mp_no *x, const mp_no *y, mp_no *z, int p) |
| +{ |
| + int n; |
| + |
| + if (X[0] == 0) |
| + { |
| + __cpy (y, z, p); |
| + Z[0] = -Z[0]; |
| + return; |
| + } |
| + else if (Y[0] == 0) |
| + { |
| + __cpy (x, z, p); |
| + return; |
| + } |
| |
| -/* Subtract two multiple precision numbers. *z is set to *x - *y. x&y may */ |
| -/* overlap but not x&z or y&z. One guard digit is used. The error is */ |
| -/* less than one ulp. *x & *y are left unchanged. */ |
| + if (X[0] != Y[0]) |
| + { |
| + if (__acr (x, y, p) > 0) |
| + { |
| + add_magnitudes (x, y, z, p); |
| + Z[0] = X[0]; |
| + } |
| + else |
| + { |
| + add_magnitudes (y, x, z, p); |
| + Z[0] = -Y[0]; |
| + } |
| + } |
| + else |
| + { |
| + if ((n = __acr (x, y, p)) == 1) |
| + { |
| + sub_magnitudes (x, y, z, p); |
| + Z[0] = X[0]; |
| + } |
| + else if (n == -1) |
| + { |
| + sub_magnitudes (y, x, z, p); |
| + Z[0] = -Y[0]; |
| + } |
| + else |
| + Z[0] = 0; |
| + } |
| +} |
| |
| +#ifndef NO__MUL |
| +/* Multiply *X and *Y and store result in *Z. X and Y may overlap but not X |
| + and Z or Y and Z. For P in [1, 2, 3], the exact result is truncated to P |
| + digits. In case P > 3 the error is bounded by 1.001 ULP. */ |
| void |
| SECTION |
| -__sub(const mp_no *x, const mp_no *y, mp_no *z, int p) { |
| +__mul (const mp_no *x, const mp_no *y, mp_no *z, int p) |
| +{ |
| + long i, j, k, ip, ip2; |
| + long p2 = p; |
| + mantissa_store_t zk; |
| + const mp_no *a; |
| + mantissa_store_t *diag; |
| + |
| + /* Is z=0? */ |
| + if (__glibc_unlikely (X[0] * Y[0] == 0)) |
| + { |
| + Z[0] = 0; |
| + return; |
| + } |
| |
| - int n; |
| + /* We need not iterate through all X's and Y's since it's pointless to |
| + multiply zeroes. Here, both are zero... */ |
| + for (ip2 = p2; ip2 > 0; ip2--) |
| + if (X[ip2] != 0 || Y[ip2] != 0) |
| + break; |
| + |
| + a = X[ip2] != 0 ? y : x; |
| + |
| + /* ... and here, at least one of them is still zero. */ |
| + for (ip = ip2; ip > 0; ip--) |
| + if (a->d[ip] != 0) |
| + break; |
| + |
| + /* The product looks like this for p = 3 (as an example): |
| + |
| + |
| + a1 a2 a3 |
| + x b1 b2 b3 |
| + ----------------------------- |
| + a1*b3 a2*b3 a3*b3 |
| + a1*b2 a2*b2 a3*b2 |
| + a1*b1 a2*b1 a3*b1 |
| + |
| + So our K needs to ideally be P*2, but we're limiting ourselves to P + 3 |
| + for P >= 3. We compute the above digits in two parts; the last P-1 |
| + digits and then the first P digits. The last P-1 digits are a sum of |
| + products of the input digits from P to P-k where K is 0 for the least |
| + significant digit and increases as we go towards the left. The product |
| + term is of the form X[k]*X[P-k] as can be seen in the above example. |
| + |
| + The first P digits are also a sum of products with the same product term, |
| + except that the sum is from 1 to k. This is also evident from the above |
| + example. |
| + |
| + Another thing that becomes evident is that only the most significant |
| + ip+ip2 digits of the result are non-zero, where ip and ip2 are the |
| + 'internal precision' of the input numbers, i.e. digits after ip and ip2 |
| + are all 0. */ |
| + |
| + k = (__glibc_unlikely (p2 < 3)) ? p2 + p2 : p2 + 3; |
| + |
| + while (k > ip + ip2 + 1) |
| + Z[k--] = 0; |
| + |
| + zk = 0; |
| + |
| + /* Precompute sums of diagonal elements so that we can directly use them |
| + later. See the next comment to know we why need them. */ |
| + diag = alloca (k * sizeof (mantissa_store_t)); |
| + mantissa_store_t d = 0; |
| + for (i = 1; i <= ip; i++) |
| + { |
| + d += X[i] * (mantissa_store_t) Y[i]; |
| + diag[i] = d; |
| + } |
| + while (i < k) |
| + diag[i++] = d; |
| |
| - if (X[0] == ZERO) {__cpy(y,z,p); Z[0] = -Z[0]; return; } |
| - else if (Y[0] == ZERO) {__cpy(x,z,p); return; } |
| + while (k > p2) |
| + { |
| + long lim = k / 2; |
| |
| - if (X[0] != Y[0]) { |
| - if (__acr(x,y,p) > 0) {add_magnitudes(x,y,z,p); Z[0] = X[0]; } |
| - else {add_magnitudes(y,x,z,p); Z[0] = -Y[0]; } |
| - } |
| - else { |
| - if ((n=__acr(x,y,p)) == 1) {sub_magnitudes(x,y,z,p); Z[0] = X[0]; } |
| - else if (n == -1) {sub_magnitudes(y,x,z,p); Z[0] = -Y[0]; } |
| - else Z[0] = ZERO; |
| - } |
| -} |
| + if (k % 2 == 0) |
| + /* We want to add this only once, but since we subtract it in the sum |
| + of products above, we add twice. */ |
| + zk += 2 * X[lim] * (mantissa_store_t) Y[lim]; |
| |
| + for (i = k - p2, j = p2; i < j; i++, j--) |
| + zk += (X[i] + X[j]) * (mantissa_store_t) (Y[i] + Y[j]); |
| |
| -/* Multiply two multiple precision numbers. *z is set to *x * *y. x&y */ |
| -/* may overlap but not x&z or y&z. In case p=1,2,3 the exact result is */ |
| -/* truncated to p digits. In case p>3 the error is bounded by 1.001 ulp. */ |
| -/* *x & *y are left unchanged. */ |
| + zk -= diag[k - 1]; |
| |
| -void |
| -SECTION |
| -__mul(const mp_no *x, const mp_no *y, mp_no *z, int p) { |
| + DIV_RADIX (zk, Z[k]); |
| + k--; |
| + } |
| |
| - int i, i1, i2, j, k, k2; |
| - double u; |
| + /* The real deal. Mantissa digit Z[k] is the sum of all X[i] * Y[j] where i |
| + goes from 1 -> k - 1 and j goes the same range in reverse. To reduce the |
| + number of multiplications, we halve the range and if k is an even number, |
| + add the diagonal element X[k/2]Y[k/2]. Through the half range, we compute |
| + X[i] * Y[j] as (X[i] + X[j]) * (Y[i] + Y[j]) - X[i] * Y[i] - X[j] * Y[j]. |
| + |
| + This reduction tells us that we're summing two things, the first term |
| + through the half range and the negative of the sum of the product of all |
| + terms of X and Y in the full range. i.e. |
| + |
| + SUM(X[i] * Y[i]) for k terms. This is precalculated above for each k in |
| + a single loop so that it completes in O(n) time and can hence be directly |
| + used in the loop below. */ |
| + while (k > 1) |
| + { |
| + long lim = k / 2; |
| + |
| + if (k % 2 == 0) |
| + /* We want to add this only once, but since we subtract it in the sum |
| + of products above, we add twice. */ |
| + zk += 2 * X[lim] * (mantissa_store_t) Y[lim]; |
| |
| - /* Is z=0? */ |
| - if (X[0]*Y[0]==ZERO) |
| - { Z[0]=ZERO; return; } |
| - |
| - /* Multiply, add and carry */ |
| - k2 = (p<3) ? p+p : p+3; |
| - Z[k2]=ZERO; |
| - for (k=k2; k>1; ) { |
| - if (k > p) {i1=k-p; i2=p+1; } |
| - else {i1=1; i2=k; } |
| - for (i=i1,j=i2-1; i<i2; i++,j--) Z[k] += X[i]*Y[j]; |
| - |
| - u = (Z[k] + CUTTER)-CUTTER; |
| - if (u > Z[k]) u -= RADIX; |
| - Z[k] -= u; |
| - Z[--k] = u*RADIXI; |
| - } |
| - |
| - /* Is there a carry beyond the most significant digit? */ |
| - if (Z[1] == ZERO) { |
| - for (i=1; i<=p; i++) Z[i]=Z[i+1]; |
| - EZ = EX + EY - 1; } |
| - else |
| - EZ = EX + EY; |
| + for (i = 1, j = k - 1; i < j; i++, j--) |
| + zk += (X[i] + X[j]) * (mantissa_store_t) (Y[i] + Y[j]); |
| + |
| + zk -= diag[k - 1]; |
| + |
| + DIV_RADIX (zk, Z[k]); |
| + k--; |
| + } |
| + Z[k] = zk; |
| + |
| + /* Get the exponent sum into an intermediate variable. This is a subtle |
| + optimization, where given enough registers, all operations on the exponent |
| + happen in registers and the result is written out only once into EZ. */ |
| + int e = EX + EY; |
| + |
| + /* Is there a carry beyond the most significant digit? */ |
| + if (__glibc_unlikely (Z[1] == 0)) |
| + { |
| + for (i = 1; i <= p2; i++) |
| + Z[i] = Z[i + 1]; |
| + e--; |
| + } |
| |
| + EZ = e; |
| Z[0] = X[0] * Y[0]; |
| } |
| +#endif |
| + |
| +#ifndef NO__SQR |
| +/* Square *X and store result in *Y. X and Y may not overlap. For P in |
| + [1, 2, 3], the exact result is truncated to P digits. In case P > 3 the |
| + error is bounded by 1.001 ULP. This is a faster special case of |
| + multiplication. */ |
| +void |
| +SECTION |
| +__sqr (const mp_no *x, mp_no *y, int p) |
| +{ |
| + long i, j, k, ip; |
| + mantissa_store_t yk; |
| |
| + /* Is z=0? */ |
| + if (__glibc_unlikely (X[0] == 0)) |
| + { |
| + Y[0] = 0; |
| + return; |
| + } |
| |
| -/* Invert a multiple precision number. Set *y = 1 / *x. */ |
| -/* Relative error bound = 1.001*r**(1-p) for p=2, 1.063*r**(1-p) for p=3, */ |
| -/* 2.001*r**(1-p) for p>3. */ |
| -/* *x=0 is not permissible. *x is left unchanged. */ |
| + /* We need not iterate through all X's since it's pointless to |
| + multiply zeroes. */ |
| + for (ip = p; ip > 0; ip--) |
| + if (X[ip] != 0) |
| + break; |
| |
| -static |
| -SECTION |
| -void __inv(const mp_no *x, mp_no *y, int p) { |
| - int i; |
| -#if 0 |
| - int l; |
| + k = (__glibc_unlikely (p < 3)) ? p + p : p + 3; |
| + |
| + while (k > 2 * ip + 1) |
| + Y[k--] = 0; |
| + |
| + yk = 0; |
| + |
| + while (k > p) |
| + { |
| + mantissa_store_t yk2 = 0; |
| + long lim = k / 2; |
| + |
| + if (k % 2 == 0) |
| + yk += X[lim] * (mantissa_store_t) X[lim]; |
| + |
| + /* In __mul, this loop (and the one within the next while loop) run |
| + between a range to calculate the mantissa as follows: |
| + |
| + Z[k] = X[k] * Y[n] + X[k+1] * Y[n-1] ... + X[n-1] * Y[k+1] |
| + + X[n] * Y[k] |
| + |
| + For X == Y, we can get away with summing halfway and doubling the |
| + result. For cases where the range size is even, the mid-point needs |
| + to be added separately (above). */ |
| + for (i = k - p, j = p; i < j; i++, j--) |
| + yk2 += X[i] * (mantissa_store_t) X[j]; |
| + |
| + yk += 2 * yk2; |
| + |
| + DIV_RADIX (yk, Y[k]); |
| + k--; |
| + } |
| + |
| + while (k > 1) |
| + { |
| + mantissa_store_t yk2 = 0; |
| + long lim = k / 2; |
| + |
| + if (k % 2 == 0) |
| + yk += X[lim] * (mantissa_store_t) X[lim]; |
| + |
| + /* Likewise for this loop. */ |
| + for (i = 1, j = k - 1; i < j; i++, j--) |
| + yk2 += X[i] * (mantissa_store_t) X[j]; |
| + |
| + yk += 2 * yk2; |
| + |
| + DIV_RADIX (yk, Y[k]); |
| + k--; |
| + } |
| + Y[k] = yk; |
| + |
| + /* Squares are always positive. */ |
| + Y[0] = 1; |
| + |
| + /* Get the exponent sum into an intermediate variable. This is a subtle |
| + optimization, where given enough registers, all operations on the exponent |
| + happen in registers and the result is written out only once into EZ. */ |
| + int e = EX * 2; |
| + |
| + /* Is there a carry beyond the most significant digit? */ |
| + if (__glibc_unlikely (Y[1] == 0)) |
| + { |
| + for (i = 1; i <= p; i++) |
| + Y[i] = Y[i + 1]; |
| + e--; |
| + } |
| + |
| + EY = e; |
| +} |
| #endif |
| + |
| +/* Invert *X and store in *Y. Relative error bound: |
| + - For P = 2: 1.001 * R ^ (1 - P) |
| + - For P = 3: 1.063 * R ^ (1 - P) |
| + - For P > 3: 2.001 * R ^ (1 - P) |
| + |
| + *X = 0 is not permissible. */ |
| +static void |
| +SECTION |
| +__inv (const mp_no *x, mp_no *y, int p) |
| +{ |
| + long i; |
| double t; |
| - mp_no z,w; |
| - static const int np1[] = {0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3, |
| - 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}; |
| - const mp_no mptwo = {1,{1.0,2.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0, |
| - 0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0, |
| - 0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0, |
| - 0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}}; |
| - |
| - __cpy(x,&z,p); z.e=0; __mp_dbl(&z,&t,p); |
| - t=ONE/t; __dbl_mp(t,y,p); EY -= EX; |
| - |
| - for (i=0; i<np1[p]; i++) { |
| - __cpy(y,&w,p); |
| - __mul(x,&w,y,p); |
| - __sub(&mptwo,y,&z,p); |
| - __mul(&w,&z,y,p); |
| - } |
| + mp_no z, w; |
| + static const int np1[] = |
| + { 0, 0, 0, 0, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, |
| + 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 |
| + }; |
| + |
| + __cpy (x, &z, p); |
| + z.e = 0; |
| + __mp_dbl (&z, &t, p); |
| + t = 1 / t; |
| + __dbl_mp (t, y, p); |
| + EY -= EX; |
| + |
| + for (i = 0; i < np1[p]; i++) |
| + { |
| + __cpy (y, &w, p); |
| + __mul (x, &w, y, p); |
| + __sub (&__mptwo, y, &z, p); |
| + __mul (&w, &z, y, p); |
| + } |
| } |
| |
| +/* Divide *X by *Y and store result in *Z. X and Y may overlap but not X and Z |
| + or Y and Z. Relative error bound: |
| + - For P = 2: 2.001 * R ^ (1 - P) |
| + - For P = 3: 2.063 * R ^ (1 - P) |
| + - For P > 3: 3.001 * R ^ (1 - P) |
| |
| -/* Divide one multiple precision number by another.Set *z = *x / *y. *x & *y */ |
| -/* are left unchanged. x&y may overlap but not x&z or y&z. */ |
| -/* Relative error bound = 2.001*r**(1-p) for p=2, 2.063*r**(1-p) for p=3 */ |
| -/* and 3.001*r**(1-p) for p>3. *y=0 is not permissible. */ |
| - |
| + *X = 0 is not permissible. */ |
| void |
| SECTION |
| -__dvd(const mp_no *x, const mp_no *y, mp_no *z, int p) { |
| - |
| +__dvd (const mp_no *x, const mp_no *y, mp_no *z, int p) |
| +{ |
| mp_no w; |
| |
| - if (X[0] == ZERO) Z[0] = ZERO; |
| - else {__inv(y,&w,p); __mul(x,&w,z,p);} |
| + if (X[0] == 0) |
| + Z[0] = 0; |
| + else |
| + { |
| + __inv (y, &w, p); |
| + __mul (x, &w, z, p); |
| + } |
| } |
| |
| |
| |
| |
| @@ -1,7 +1,7 @@ |
| /* |
| * IBM Accurate Mathematical Library |
| * Written by International Business Machines Corp. |
| - * Copyright (C) 2001, 2011 Free Software Foundation, Inc. |
| + * Copyright (C) 2001-2017 Free Software Foundation, Inc. |
| * |
| * This program is free software; you can redistribute it and/or modify |
| * it under the terms of the GNU Lesser General Public License as published by |
| @@ -23,36 +23,58 @@ |
| /* FUNCTIONS: */ |
| /* mcr */ |
| /* acr */ |
| -/* cr */ |
| /* cpy */ |
| -/* cpymn */ |
| /* mp_dbl */ |
| /* dbl_mp */ |
| /* add */ |
| /* sub */ |
| /* mul */ |
| -/* inv */ |
| /* dvd */ |
| /* */ |
| /* Arithmetic functions for multiple precision numbers. */ |
| /* Common types and definition */ |
| / |
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