| /* |
| * IBM Accurate Mathematical Library |
| * written by International Business Machines Corp. |
| * Copyright (C) 2001, 2002, 2004 Free Software Foundation |
| * |
| * 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 |
| * the Free Software Foundation; either version 2.1 of the License, or |
| * (at your option) any later version. |
| * |
| * This program is distributed in the hope that it will be useful, |
| * but WITHOUT ANY WARRANTY; without even the implied warranty of |
| * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
| * GNU Lesser General Public License for more details. |
| * |
| * You should have received a copy of the GNU Lesser General Public License |
| * along with this program; if not, write to the Free Software |
| * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. |
| */ |
| /***************************************************************************/ |
| /* MODULE_NAME: upow.c */ |
| /* */ |
| /* FUNCTIONS: upow */ |
| /* power1 */ |
| /* my_log2 */ |
| /* log1 */ |
| /* checkint */ |
| /* FILES NEEDED: dla.h endian.h mpa.h mydefs.h */ |
| /* halfulp.c mpexp.c mplog.c slowexp.c slowpow.c mpa.c */ |
| /* uexp.c upow.c */ |
| /* root.tbl uexp.tbl upow.tbl */ |
| /* An ultimate power routine. Given two IEEE double machine numbers y,x */ |
| /* it computes the correctly rounded (to nearest) value of x^y. */ |
| /* Assumption: Machine arithmetic operations are performed in */ |
| /* round to nearest mode of IEEE 754 standard. */ |
| /* */ |
| /***************************************************************************/ |
| #include "endian.h" |
| #include "upow.h" |
| #include "dla.h" |
| #include "mydefs.h" |
| #include "MathLib.h" |
| #include "upow.tbl" |
| #include "math_private.h" |
| |
| |
| double __exp1(double x, double xx, double error); |
| static double log1(double x, double *delta, double *error); |
| static double my_log2(double x, double *delta, double *error); |
| double __slowpow(double x, double y,double z); |
| static double power1(double x, double y); |
| static int checkint(double x); |
| |
| /***************************************************************************/ |
| /* An ultimate power routine. Given two IEEE double machine numbers y,x */ |
| /* it computes the correctly rounded (to nearest) value of X^y. */ |
| /***************************************************************************/ |
| double __ieee754_pow(double x, double y) { |
| double z,a,aa,error, t,a1,a2,y1,y2; |
| #if 0 |
| double gor=1.0; |
| #endif |
| mynumber u,v; |
| int k; |
| int4 qx,qy; |
| v.x=y; |
| u.x=x; |
| if (v.i[LOW_HALF] == 0) { /* of y */ |
| qx = u.i[HIGH_HALF]&0x7fffffff; |
| /* Checking if x is not too small to compute */ |
| if (((qx==0x7ff00000)&&(u.i[LOW_HALF]!=0))||(qx>0x7ff00000)) return NaNQ.x; |
| if (y == 1.0) return x; |
| if (y == 2.0) return x*x; |
| if (y == -1.0) return 1.0/x; |
| if (y == 0) return 1.0; |
| } |
| /* else */ |
| if(((u.i[HIGH_HALF]>0 && u.i[HIGH_HALF]<0x7ff00000)|| /* x>0 and not x->0 */ |
| (u.i[HIGH_HALF]==0 && u.i[LOW_HALF]!=0)) && |
| /* 2^-1023< x<= 2^-1023 * 0x1.0000ffffffff */ |
| (v.i[HIGH_HALF]&0x7fffffff) < 0x4ff00000) { /* if y<-1 or y>1 */ |
| z = log1(x,&aa,&error); /* x^y =e^(y log (X)) */ |
| t = y*134217729.0; |
| y1 = t - (t-y); |
| y2 = y - y1; |
| t = z*134217729.0; |
| a1 = t - (t-z); |
| a2 = (z - a1)+aa; |
| a = y1*a1; |
| aa = y2*a1 + y*a2; |
| a1 = a+aa; |
| a2 = (a-a1)+aa; |
| error = error*ABS(y); |
| t = __exp1(a1,a2,1.9e16*error); /* return -10 or 0 if wasn't computed exactly */ |
| return (t>0)?t:power1(x,y); |
| } |
| |
| if (x == 0) { |
| if (((v.i[HIGH_HALF] & 0x7fffffff) == 0x7ff00000 && v.i[LOW_HALF] != 0) |
| || (v.i[HIGH_HALF] & 0x7fffffff) > 0x7ff00000) |
| return y; |
| if (ABS(y) > 1.0e20) return (y>0)?0:INF.x; |
| k = checkint(y); |
| if (k == -1) |
| return y < 0 ? 1.0/x : x; |
| else |
| return y < 0 ? 1.0/ABS(x) : 0.0; /* return 0 */ |
| } |
| |
| qx = u.i[HIGH_HALF]&0x7fffffff; /* no sign */ |
| qy = v.i[HIGH_HALF]&0x7fffffff; /* no sign */ |
| |
| if (qx >= 0x7ff00000 && (qx > 0x7ff00000 || u.i[LOW_HALF] != 0)) return NaNQ.x; |
| if (qy >= 0x7ff00000 && (qy > 0x7ff00000 || v.i[LOW_HALF] != 0)) |
| return x == 1.0 ? 1.0 : NaNQ.x; |
| |
| /* if x<0 */ |
| if (u.i[HIGH_HALF] < 0) { |
| k = checkint(y); |
| if (k==0) { |
| if (qy == 0x7ff00000) { |
| if (x == -1.0) return 1.0; |
| else if (x > -1.0) return v.i[HIGH_HALF] < 0 ? INF.x : 0.0; |
| else return v.i[HIGH_HALF] < 0 ? 0.0 : INF.x; |
| } |
| else if (qx == 0x7ff00000) |
| return y < 0 ? 0.0 : INF.x; |
| return NaNQ.x; /* y not integer and x<0 */ |
| } |
| else if (qx == 0x7ff00000) |
| { |
| if (k < 0) |
| return y < 0 ? nZERO.x : nINF.x; |
| else |
| return y < 0 ? 0.0 : INF.x; |
| } |
| return (k==1)?__ieee754_pow(-x,y):-__ieee754_pow(-x,y); /* if y even or odd */ |
| } |
| /* x>0 */ |
| |
| if (qx == 0x7ff00000) /* x= 2^-0x3ff */ |
| {if (y == 0) return NaNQ.x; |
| return (y>0)?x:0; } |
| |
| if (qy > 0x45f00000 && qy < 0x7ff00000) { |
| if (x == 1.0) return 1.0; |
| if (y>0) return (x>1.0)?INF.x:0; |
| if (y<0) return (x<1.0)?INF.x:0; |
| } |
| |
| if (x == 1.0) return 1.0; |
| if (y>0) return (x>1.0)?INF.x:0; |
| if (y<0) return (x<1.0)?INF.x:0; |
| return 0; /* unreachable, to make the compiler happy */ |
| } |
| |
| /**************************************************************************/ |
| /* Computing x^y using more accurate but more slow log routine */ |
| /**************************************************************************/ |
| static double power1(double x, double y) { |
| double z,a,aa,error, t,a1,a2,y1,y2; |
| z = my_log2(x,&aa,&error); |
| t = y*134217729.0; |
| y1 = t - (t-y); |
| y2 = y - y1; |
| t = z*134217729.0; |
| a1 = t - (t-z); |
| a2 = z - a1; |
| a = y*z; |
| aa = ((y1*a1-a)+y1*a2+y2*a1)+y2*a2+aa*y; |
| a1 = a+aa; |
| a2 = (a-a1)+aa; |
| error = error*ABS(y); |
| t = __exp1(a1,a2,1.9e16*error); |
| return (t >= 0)?t:__slowpow(x,y,z); |
| } |
| |
| /****************************************************************************/ |
| /* Computing log(x) (x is left argument). The result is the returned double */ |
| /* + the parameter delta. */ |
| /* The result is bounded by error (rightmost argument) */ |
| /****************************************************************************/ |
| static double log1(double x, double *delta, double *error) { |
| int i,j,m; |
| #if 0 |
| int n; |
| #endif |
| double uu,vv,eps,nx,e,e1,e2,t,t1,t2,res,add=0; |
| #if 0 |
| double cor; |
| #endif |
| mynumber u,v; |
| #ifdef BIG_ENDI |
| mynumber |
| /**/ two52 = {{0x43300000, 0x00000000}}; /* 2**52 */ |
| #else |
| #ifdef LITTLE_ENDI |
| mynumber |
| /**/ two52 = {{0x00000000, 0x43300000}}; /* 2**52 */ |
| #endif |
| #endif |
| |
| u.x = x; |
| m = u.i[HIGH_HALF]; |
| *error = 0; |
| *delta = 0; |
| if (m < 0x00100000) /* 1<x<2^-1007 */ |
| { x = x*t52.x; add = -52.0; u.x = x; m = u.i[HIGH_HALF];} |
| |
| if ((m&0x000fffff) < 0x0006a09e) |
| {u.i[HIGH_HALF] = (m&0x000fffff)|0x3ff00000; two52.i[LOW_HALF]=(m>>20); } |
| else |
| {u.i[HIGH_HALF] = (m&0x000fffff)|0x3fe00000; two52.i[LOW_HALF]=(m>>20)+1; } |
| |
| v.x = u.x + bigu.x; |
| uu = v.x - bigu.x; |
| i = (v.i[LOW_HALF]&0x000003ff)<<2; |
| if (two52.i[LOW_HALF] == 1023) /* nx = 0 */ |
| { |
| if (i > 1192 && i < 1208) /* |x-1| < 1.5*2**-10 */ |
| { |
| t = x - 1.0; |
| t1 = (t+5.0e6)-5.0e6; |
| t2 = t-t1; |
| e1 = t - 0.5*t1*t1; |
| e2 = t*t*t*(r3+t*(r4+t*(r5+t*(r6+t*(r7+t*r8)))))-0.5*t2*(t+t1); |
| res = e1+e2; |
| *error = 1.0e-21*ABS(t); |
| *delta = (e1-res)+e2; |
| return res; |
| } /* |x-1| < 1.5*2**-10 */ |
| else |
| { |
| v.x = u.x*(ui.x[i]+ui.x[i+1])+bigv.x; |
| vv = v.x-bigv.x; |
| j = v.i[LOW_HALF]&0x0007ffff; |
| j = j+j+j; |
| eps = u.x - uu*vv; |
| e1 = eps*ui.x[i]; |
| e2 = eps*(ui.x[i+1]+vj.x[j]*(ui.x[i]+ui.x[i+1])); |
| e = e1+e2; |
| e2 = ((e1-e)+e2); |
| t=ui.x[i+2]+vj.x[j+1]; |
| t1 = t+e; |
| t2 = (((t-t1)+e)+(ui.x[i+3]+vj.x[j+2]))+e2+e*e*(p2+e*(p3+e*p4)); |
| res=t1+t2; |
| *error = 1.0e-24; |
| *delta = (t1-res)+t2; |
| return res; |
| } |
| } /* nx = 0 */ |
| else /* nx != 0 */ |
| { |
| eps = u.x - uu; |
| nx = (two52.x - two52e.x)+add; |
| e1 = eps*ui.x[i]; |
| e2 = eps*ui.x[i+1]; |
| e=e1+e2; |
| e2 = (e1-e)+e2; |
| t=nx*ln2a.x+ui.x[i+2]; |
| t1=t+e; |
| t2=(((t-t1)+e)+nx*ln2b.x+ui.x[i+3]+e2)+e*e*(q2+e*(q3+e*(q4+e*(q5+e*q6)))); |
| res = t1+t2; |
| *error = 1.0e-21; |
| *delta = (t1-res)+t2; |
| return res; |
| } /* nx != 0 */ |
| } |
| |
| /****************************************************************************/ |
| /* More slow but more accurate routine of log */ |
| /* Computing log(x)(x is left argument).The result is return double + delta.*/ |
| /* The result is bounded by error (right argument) */ |
| /****************************************************************************/ |
| static double my_log2(double x, double *delta, double *error) { |
| int i,j,m; |
| #if 0 |
| int n; |
| #endif |
| double uu,vv,eps,nx,e,e1,e2,t,t1,t2,res,add=0; |
| #if 0 |
| double cor; |
| #endif |
| double ou1,ou2,lu1,lu2,ov,lv1,lv2,a,a1,a2; |
| double y,yy,z,zz,j1,j2,j3,j4,j5,j6,j7,j8; |
| mynumber u,v; |
| #ifdef BIG_ENDI |
| mynumber |
| /**/ two52 = {{0x43300000, 0x00000000}}; /* 2**52 */ |
| #else |
| #ifdef LITTLE_ENDI |
| mynumber |
| /**/ two52 = {{0x00000000, 0x43300000}}; /* 2**52 */ |
| #endif |
| #endif |
| |
| u.x = x; |
| m = u.i[HIGH_HALF]; |
| *error = 0; |
| *delta = 0; |
| add=0; |
| if (m<0x00100000) { /* x < 2^-1022 */ |
| x = x*t52.x; add = -52.0; u.x = x; m = u.i[HIGH_HALF]; } |
| |
| if ((m&0x000fffff) < 0x0006a09e) |
| {u.i[HIGH_HALF] = (m&0x000fffff)|0x3ff00000; two52.i[LOW_HALF]=(m>>20); } |
| else |
| {u.i[HIGH_HALF] = (m&0x000fffff)|0x3fe00000; two52.i[LOW_HALF]=(m>>20)+1; } |
| |
| v.x = u.x + bigu.x; |
| uu = v.x - bigu.x; |
| i = (v.i[LOW_HALF]&0x000003ff)<<2; |
| /*------------------------------------- |x-1| < 2**-11------------------------------- */ |
| if ((two52.i[LOW_HALF] == 1023) && (i == 1200)) |
| { |
| t = x - 1.0; |
| EMULV(t,s3,y,yy,j1,j2,j3,j4,j5); |
| ADD2(-0.5,0,y,yy,z,zz,j1,j2); |
| MUL2(t,0,z,zz,y,yy,j1,j2,j3,j4,j5,j6,j7,j8); |
| MUL2(t,0,y,yy,z,zz,j1,j2,j3,j4,j5,j6,j7,j8); |
| |
| e1 = t+z; |
| e2 = (((t-e1)+z)+zz)+t*t*t*(ss3+t*(s4+t*(s5+t*(s6+t*(s7+t*s8))))); |
| res = e1+e2; |
| *error = 1.0e-25*ABS(t); |
| *delta = (e1-res)+e2; |
| return res; |
| } |
| /*----------------------------- |x-1| > 2**-11 -------------------------- */ |
| else |
| { /*Computing log(x) according to log table */ |
| nx = (two52.x - two52e.x)+add; |
| ou1 = ui.x[i]; |
| ou2 = ui.x[i+1]; |
| lu1 = ui.x[i+2]; |
| lu2 = ui.x[i+3]; |
| v.x = u.x*(ou1+ou2)+bigv.x; |
| vv = v.x-bigv.x; |
| j = v.i[LOW_HALF]&0x0007ffff; |
| j = j+j+j; |
| eps = u.x - uu*vv; |
| ov = vj.x[j]; |
| lv1 = vj.x[j+1]; |
| lv2 = vj.x[j+2]; |
| a = (ou1+ou2)*(1.0+ov); |
| a1 = (a+1.0e10)-1.0e10; |
| a2 = a*(1.0-a1*uu*vv); |
| e1 = eps*a1; |
| e2 = eps*a2; |
| e = e1+e2; |
| e2 = (e1-e)+e2; |
| t=nx*ln2a.x+lu1+lv1; |
| t1 = t+e; |
| t2 = (((t-t1)+e)+(lu2+lv2+nx*ln2b.x+e2))+e*e*(p2+e*(p3+e*p4)); |
| res=t1+t2; |
| *error = 1.0e-27; |
| *delta = (t1-res)+t2; |
| return res; |
| } |
| } |
| |
| /**********************************************************************/ |
| /* Routine receives a double x and checks if it is an integer. If not */ |
| /* it returns 0, else it returns 1 if even or -1 if odd. */ |
| /**********************************************************************/ |
| static int checkint(double x) { |
| union {int4 i[2]; double x;} u; |
| int k,m,n; |
| #if 0 |
| int l; |
| #endif |
| u.x = x; |
| m = u.i[HIGH_HALF]&0x7fffffff; /* no sign */ |
| if (m >= 0x7ff00000) return 0; /* x is +/-inf or NaN */ |
| if (m >= 0x43400000) return 1; /* |x| >= 2**53 */ |
| if (m < 0x40000000) return 0; /* |x| < 2, can not be 0 or 1 */ |
| n = u.i[LOW_HALF]; |
| k = (m>>20)-1023; /* 1 <= k <= 52 */ |
| if (k == 52) return (n&1)? -1:1; /* odd or even*/ |
| if (k>20) { |
| if (n<<(k-20)) return 0; /* if not integer */ |
| return (n<<(k-21))?-1:1; |
| } |
| if (n) return 0; /*if not integer*/ |
| if (k == 20) return (m&1)? -1:1; |
| if (m<<(k+12)) return 0; |
| return (m<<(k+11))?-1:1; |
| } |
