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math: Remove slow paths from asin and acos [BZ #15267]
This patch series removes all remaining slow paths and related code. First asin/acos, tan, atan, atan2 implementations are updated, and the final patch removes the unused mpa files, headers and probes. Passes buildmanyglibc. Remove slow paths from asin/acos. Add ULP annotations based on previous slow path checks (which are approximate). Update AArch64 and x86_64 libm-test-ulps. Reviewed-By: Paul Zimmermann <Paul.Zimmermann@inria.fr>
This commit is contained in:
Wilco Dijkstra
Wilco Dijkstra
parent
94caafa040
commit
db3f7bb558
@ -57,7 +57,7 @@ float: 1
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ldouble: 1
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Function: "asin_upward":
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double: 1
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double: 2
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float: 1
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ldouble: 2
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@ -21,8 +21,7 @@
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/* */
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/* FUNCTIONS: uasin */
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/* uacos */
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/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h usncs.h */
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/* doasin.c sincos32.c dosincos.c mpa.c */
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/* FILES NEEDED: dla.h endian.h mydefs.h usncs.h */
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/* sincos.tbl asincos.tbl powtwo.tbl root.tbl */
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/* */
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/******************************************************************/
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@ -31,7 +30,6 @@
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#include "asincos.tbl"
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#include "root.tbl"
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#include "powtwo.tbl"
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#include "MathLib.h"
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#include "uasncs.h"
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#include <float.h>
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#include <math.h>
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@ -43,15 +41,11 @@
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# define SECTION
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#endif
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void __doasin(double x, double dx, double w[]);
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void __dubsin(double x, double dx, double v[]);
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void __dubcos(double x, double dx, double v[]);
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void __docos(double x, double dx, double v[]);
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/* asin with max ULP of ~0.516 based on random sampling. */
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double
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SECTION
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__ieee754_asin(double x){
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double x1,x2,xx,s1,s2,res1,p,t,res,r,cor,cc,y,c,z,w[2];
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double x2,xx,res1,p,t,res,r,cor,cc,y,c,z;
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mynumber u,v;
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int4 k,m,n;
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@ -70,27 +64,8 @@ __ieee754_asin(double x){
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x2 = x*x;
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t = (((((f6*x2 + f5)*x2 + f4)*x2 + f3)*x2 + f2)*x2 + f1)*(x2*x);
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res = x+t; /* res=arcsin(x) according to Taylor series */
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cor = (x-res)+t;
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if (res == res+1.025*cor) return res;
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else {
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x1 = x+big;
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xx = x*x;
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x1 -= big;
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x2 = x - x1;
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p = x1*x1*x1;
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s1 = a1.x*p;
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s2 = ((((((c7*xx + c6)*xx + c5)*xx + c4)*xx + c3)*xx + c2)*xx*xx*x +
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((a1.x+a2.x)*x2*x2+ 0.5*x1*x)*x2) + a2.x*p;
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res1 = x+s1;
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s2 = ((x-res1)+s1)+s2;
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res = res1+s2;
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cor = (res1-res)+s2;
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if (res == res+1.00014*cor) return res;
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else {
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__doasin(x,0,w);
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return w[0];
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}
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}
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/* Max ULP is 0.513. */
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return res;
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}
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/*---------------------0.125 <= |x| < 0.5 -----------------------------*/
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else if (k < 0x3fe00000) {
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@ -103,26 +78,8 @@ __ieee754_asin(double x){
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+xx*asncs.x[n+6]))))+asncs.x[n+7];
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t+=p;
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res =asncs.x[n+8] +t;
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cor = (asncs.x[n+8]-res)+t;
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if (res == res+1.05*cor) return (m>0)?res:-res;
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else {
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r=asncs.x[n+8]+xx*asncs.x[n+9];
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t=((asncs.x[n+8]-r)+xx*asncs.x[n+9])+(p+xx*asncs.x[n+10]);
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res = r+t;
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cor = (r-res)+t;
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if (res == res+1.0005*cor) return (m>0)?res:-res;
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else {
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res1=res+1.1*cor;
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z=0.5*(res1-res);
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__dubsin(res,z,w);
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z=(w[0]-fabs(x))+w[1];
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if (z>1.0e-27) return (m>0)?min(res,res1):-min(res,res1);
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else if (z<-1.0e-27) return (m>0)?max(res,res1):-max(res,res1);
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else {
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return (m>0)?res:-res;
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}
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}
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}
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/* Max ULP is 0.524. */
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return (m>0)?res:-res;
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} /* else if (k < 0x3fe00000) */
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/*-------------------- 0.5 <= |x| < 0.75 -----------------------------*/
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else
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@ -135,26 +92,8 @@ __ieee754_asin(double x){
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+xx*(asncs.x[n+6]+xx*asncs.x[n+7])))))+asncs.x[n+8];
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t+=p;
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res =asncs.x[n+9] +t;
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cor = (asncs.x[n+9]-res)+t;
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if (res == res+1.01*cor) return (m>0)?res:-res;
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else {
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r=asncs.x[n+9]+xx*asncs.x[n+10];
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t=((asncs.x[n+9]-r)+xx*asncs.x[n+10])+(p+xx*asncs.x[n+11]);
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res = r+t;
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cor = (r-res)+t;
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if (res == res+1.0005*cor) return (m>0)?res:-res;
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else {
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res1=res+1.1*cor;
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z=0.5*(res1-res);
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__dubsin(res,z,w);
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z=(w[0]-fabs(x))+w[1];
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if (z>1.0e-27) return (m>0)?min(res,res1):-min(res,res1);
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else if (z<-1.0e-27) return (m>0)?max(res,res1):-max(res,res1);
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else {
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return (m>0)?res:-res;
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}
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}
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}
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/* Max ULP is 0.505. */
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return (m>0)?res:-res;
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} /* else if (k < 0x3fe80000) */
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/*--------------------- 0.75 <= |x|< 0.921875 ----------------------*/
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else
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@ -167,28 +106,8 @@ __ieee754_asin(double x){
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+xx*(asncs.x[n+6]+xx*(asncs.x[n+7]+xx*asncs.x[n+8]))))))+asncs.x[n+9];
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t+=p;
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res =asncs.x[n+10] +t;
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cor = (asncs.x[n+10]-res)+t;
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if (res == res+1.01*cor) return (m>0)?res:-res;
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else {
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r=asncs.x[n+10]+xx*asncs.x[n+11];
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t=((asncs.x[n+10]-r)+xx*asncs.x[n+11])+(p+xx*asncs.x[n+12]);
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res = r+t;
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cor = (r-res)+t;
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if (res == res+1.0008*cor) return (m>0)?res:-res;
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else {
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res1=res+1.1*cor;
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z=0.5*(res1-res);
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y=hp0.x-res;
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z=((hp0.x-y)-res)+(hp1.x-z);
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__dubcos(y,z,w);
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z=(w[0]-fabs(x))+w[1];
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if (z>1.0e-27) return (m>0)?min(res,res1):-min(res,res1);
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else if (z<-1.0e-27) return (m>0)?max(res,res1):-max(res,res1);
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else {
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return (m>0)?res:-res;
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}
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}
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}
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/* Max ULP is 0.505. */
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return (m>0)?res:-res;
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} /* else if (k < 0x3fed8000) */
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/*-------------------0.921875 <= |x| < 0.953125 ------------------------*/
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else
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@ -203,29 +122,8 @@ __ieee754_asin(double x){
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xx*asncs.x[n+9])))))))+asncs.x[n+10];
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t+=p;
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res =asncs.x[n+11] +t;
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cor = (asncs.x[n+11]-res)+t;
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if (res == res+1.01*cor) return (m>0)?res:-res;
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else {
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r=asncs.x[n+11]+xx*asncs.x[n+12];
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t=((asncs.x[n+11]-r)+xx*asncs.x[n+12])+(p+xx*asncs.x[n+13]);
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res = r+t;
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cor = (r-res)+t;
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if (res == res+1.0007*cor) return (m>0)?res:-res;
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else {
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res1=res+1.1*cor;
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z=0.5*(res1-res);
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y=(hp0.x-res)-z;
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z=y+hp1.x;
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y=(y-z)+hp1.x;
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__dubcos(z,y,w);
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z=(w[0]-fabs(x))+w[1];
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if (z>1.0e-27) return (m>0)?min(res,res1):-min(res,res1);
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else if (z<-1.0e-27) return (m>0)?max(res,res1):-max(res,res1);
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else {
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return (m>0)?res:-res;
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}
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}
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}
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/* Max ULP is 0.505. */
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return (m>0)?res:-res;
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} /* else if (k < 0x3fee8000) */
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/*--------------------0.953125 <= |x| < 0.96875 ------------------------*/
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@ -241,29 +139,8 @@ __ieee754_asin(double x){
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xx*(asncs.x[n+9]+xx*asncs.x[n+10]))))))))+asncs.x[n+11];
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t+=p;
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res =asncs.x[n+12] +t;
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cor = (asncs.x[n+12]-res)+t;
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if (res == res+1.01*cor) return (m>0)?res:-res;
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else {
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r=asncs.x[n+12]+xx*asncs.x[n+13];
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t=((asncs.x[n+12]-r)+xx*asncs.x[n+13])+(p+xx*asncs.x[n+14]);
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res = r+t;
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cor = (r-res)+t;
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if (res == res+1.0007*cor) return (m>0)?res:-res;
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else {
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res1=res+1.1*cor;
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z=0.5*(res1-res);
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y=(hp0.x-res)-z;
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z=y+hp1.x;
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y=(y-z)+hp1.x;
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__dubcos(z,y,w);
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z=(w[0]-fabs(x))+w[1];
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if (z>1.0e-27) return (m>0)?min(res,res1):-min(res,res1);
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else if (z<-1.0e-27) return (m>0)?max(res,res1):-max(res,res1);
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else {
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return (m>0)?res:-res;
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}
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}
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}
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/* Max ULP is 0.505. */
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return (m>0)?res:-res;
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} /* else if (k < 0x3fef0000) */
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/*--------------------0.96875 <= |x| < 1 --------------------------------*/
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else
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@ -282,16 +159,8 @@ __ieee754_asin(double x){
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cor = (hp1.x - 2.0*cc)-2.0*(y+cc)*p;
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res1 = hp0.x - 2.0*y;
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res =res1 + cor;
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if (res == res+1.003*((res1-res)+cor)) return (m>0)?res:-res;
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else {
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c=y+cc;
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cc=(y-c)+cc;
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__doasin(c,cc,w);
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res1=hp0.x-2.0*w[0];
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cor=((hp0.x-res1)-2.0*w[0])+(hp1.x-2.0*w[1]);
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res = res1+cor;
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return (m>0)?res:-res;
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}
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/* Max ULP is 0.5015. */
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return (m>0)?res:-res;
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} /* else if (k < 0x3ff00000) */
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/*---------------------------- |x|>=1 -------------------------------*/
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else if (k==0x3ff00000 && u.i[LOW_HALF]==0) return (m>0)?hp0.x:-hp0.x;
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@ -315,11 +184,12 @@ libm_alias_finite (__ieee754_asin, __asin)
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/* */
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/*******************************************************************/
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/* acos with max ULP of ~0.523 based on random sampling. */
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double
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SECTION
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__ieee754_acos(double x)
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{
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double x1,x2,xx,s1,s2,res1,p,t,res,r,cor,cc,y,c,z,w[2],eps;
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double x2,xx,res1,p,t,res,r,cor,cc,y,c,z;
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mynumber u,v;
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int4 k,m,n;
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u.x = x;
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@ -336,32 +206,8 @@ __ieee754_acos(double x)
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r=hp0.x-x;
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cor=(((hp0.x-r)-x)+hp1.x)-t;
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res = r+cor;
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cor = (r-res)+cor;
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if (res == res+1.004*cor) return res;
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else {
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x1 = x+big;
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xx = x*x;
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x1 -= big;
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x2 = x - x1;
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p = x1*x1*x1;
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s1 = a1.x*p;
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s2 = ((((((c7*xx + c6)*xx + c5)*xx + c4)*xx + c3)*xx + c2)*xx*xx*x +
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((a1.x+a2.x)*x2*x2+ 0.5*x1*x)*x2) + a2.x*p;
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res1 = x+s1;
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s2 = ((x-res1)+s1)+s2;
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r=hp0.x-res1;
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cor=(((hp0.x-r)-res1)+hp1.x)-s2;
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res = r+cor;
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cor = (r-res)+cor;
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if (res == res+1.00004*cor) return res;
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else {
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__doasin(x,0,w);
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r=hp0.x-w[0];
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cor=((hp0.x-r)-w[0])+(hp1.x-w[1]);
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res=r+cor;
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return res;
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}
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}
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/* Max ULP is 0.502. */
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return res;
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} /* else if (k < 0x3fc00000) */
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/*---------------------- 0.125 <= |x| < 0.5 --------------------*/
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else
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@ -377,35 +223,16 @@ __ieee754_acos(double x)
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y = (m>0)?(hp0.x-asncs.x[n+8]):(hp0.x+asncs.x[n+8]);
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t = (m>0)?(hp1.x-t):(hp1.x+t);
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res = y+t;
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if (res == res+1.02*((y-res)+t)) return res;
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else {
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r=asncs.x[n+8]+xx*asncs.x[n+9];
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t=((asncs.x[n+8]-r)+xx*asncs.x[n+9])+(p+xx*asncs.x[n+10]);
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if (m>0)
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{p = hp0.x-r; t = (((hp0.x-p)-r)-t)+hp1.x; }
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else
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{p = hp0.x+r; t = ((hp0.x-p)+r)+(hp1.x+t); }
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res = p+t;
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cor = (p-res)+t;
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if (res == (res+1.0002*cor)) return res;
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else {
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res1=res+1.1*cor;
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z=0.5*(res1-res);
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__docos(res,z,w);
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z=(w[0]-x)+w[1];
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if (z>1.0e-27) return max(res,res1);
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else if (z<-1.0e-27) return min(res,res1);
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else return res;
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}
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}
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/* Max ULP is 0.51. */
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return res;
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} /* else if (k < 0x3fe00000) */
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/*--------------------------- 0.5 <= |x| < 0.75 ---------------------*/
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else
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if (k < 0x3fe80000) {
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n = 1056+((k&0x000fe000)>>11)*3;
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if (m>0) {xx = x - asncs.x[n]; eps=1.04; }
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else {xx = -x - asncs.x[n]; eps=1.02; }
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if (m>0) {xx = x - asncs.x[n]; }
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else {xx = -x - asncs.x[n]; }
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t = asncs.x[n+1]*xx;
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p=xx*xx*(asncs.x[n+2]+xx*(asncs.x[n+3]+xx*(asncs.x[n+4]+
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xx*(asncs.x[n+5]+xx*(asncs.x[n+6]+
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@ -414,33 +241,16 @@ __ieee754_acos(double x)
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y = (m>0)?(hp0.x-asncs.x[n+9]):(hp0.x+asncs.x[n+9]);
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t = (m>0)?(hp1.x-t):(hp1.x+t);
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res = y+t;
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if (res == res+eps*((y-res)+t)) return res;
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else {
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r=asncs.x[n+9]+xx*asncs.x[n+10];
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t=((asncs.x[n+9]-r)+xx*asncs.x[n+10])+(p+xx*asncs.x[n+11]);
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if (m>0) {p = hp0.x-r; t = (((hp0.x-p)-r)-t)+hp1.x; eps=1.0004; }
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else {p = hp0.x+r; t = ((hp0.x-p)+r)+(hp1.x+t); eps=1.0002; }
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res = p+t;
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cor = (p-res)+t;
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if (res == (res+eps*cor)) return res;
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else {
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res1=res+1.1*cor;
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z=0.5*(res1-res);
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__docos(res,z,w);
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z=(w[0]-x)+w[1];
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if (z>1.0e-27) return max(res,res1);
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else if (z<-1.0e-27) return min(res,res1);
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else return res;
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}
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}
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/* Max ULP is 0.523 based on random sampling. */
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return res;
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} /* else if (k < 0x3fe80000) */
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|
||||
/*------------------------- 0.75 <= |x| < 0.921875 -------------*/
|
||||
else
|
||||
if (k < 0x3fed8000) {
|
||||
n = 992+((k&0x000fe000)>>13)*13;
|
||||
if (m>0) {xx = x - asncs.x[n]; eps = 1.04; }
|
||||
else {xx = -x - asncs.x[n]; eps = 1.01; }
|
||||
if (m>0) {xx = x - asncs.x[n]; }
|
||||
else {xx = -x - asncs.x[n]; }
|
||||
t = asncs.x[n+1]*xx;
|
||||
p=xx*xx*(asncs.x[n+2]+xx*(asncs.x[n+3]+xx*(asncs.x[n+4]+
|
||||
xx*(asncs.x[n+5]+xx*(asncs.x[n+6]+xx*(asncs.x[n+7]+
|
||||
@ -449,33 +259,16 @@ __ieee754_acos(double x)
|
||||
y = (m>0)?(hp0.x-asncs.x[n+10]):(hp0.x+asncs.x[n+10]);
|
||||
t = (m>0)?(hp1.x-t):(hp1.x+t);
|
||||
res = y+t;
|
||||
if (res == res+eps*((y-res)+t)) return res;
|
||||
else {
|
||||
r=asncs.x[n+10]+xx*asncs.x[n+11];
|
||||
t=((asncs.x[n+10]-r)+xx*asncs.x[n+11])+(p+xx*asncs.x[n+12]);
|
||||
if (m>0) {p = hp0.x-r; t = (((hp0.x-p)-r)-t)+hp1.x; eps=1.0032; }
|
||||
else {p = hp0.x+r; t = ((hp0.x-p)+r)+(hp1.x+t); eps=1.0008; }
|
||||
res = p+t;
|
||||
cor = (p-res)+t;
|
||||
if (res == (res+eps*cor)) return res;
|
||||
else {
|
||||
res1=res+1.1*cor;
|
||||
z=0.5*(res1-res);
|
||||
__docos(res,z,w);
|
||||
z=(w[0]-x)+w[1];
|
||||
if (z>1.0e-27) return max(res,res1);
|
||||
else if (z<-1.0e-27) return min(res,res1);
|
||||
else return res;
|
||||
}
|
||||
}
|
||||
/* Max ULP is 0.523 based on random sampling. */
|
||||
return res;
|
||||
} /* else if (k < 0x3fed8000) */
|
||||
|
||||
/*-------------------0.921875 <= |x| < 0.953125 ------------------*/
|
||||
else
|
||||
if (k < 0x3fee8000) {
|
||||
n = 884+((k&0x000fe000)>>13)*14;
|
||||
if (m>0) {xx = x - asncs.x[n]; eps=1.04; }
|
||||
else {xx = -x - asncs.x[n]; eps =1.005; }
|
||||
if (m>0) {xx = x - asncs.x[n]; }
|
||||
else {xx = -x - asncs.x[n]; }
|
||||
t = asncs.x[n+1]*xx;
|
||||
p=xx*xx*(asncs.x[n+2]+xx*(asncs.x[n+3]+xx*(asncs.x[n+4]+
|
||||
xx*(asncs.x[n+5]+xx*(asncs.x[n+6]
|
||||
@ -485,33 +278,16 @@ __ieee754_acos(double x)
|
||||
y = (m>0)?(hp0.x-asncs.x[n+11]):(hp0.x+asncs.x[n+11]);
|
||||
t = (m>0)?(hp1.x-t):(hp1.x+t);
|
||||
res = y+t;
|
||||
if (res == res+eps*((y-res)+t)) return res;
|
||||
else {
|
||||
r=asncs.x[n+11]+xx*asncs.x[n+12];
|
||||
t=((asncs.x[n+11]-r)+xx*asncs.x[n+12])+(p+xx*asncs.x[n+13]);
|
||||
if (m>0) {p = hp0.x-r; t = (((hp0.x-p)-r)-t)+hp1.x; eps=1.0030; }
|
||||
else {p = hp0.x+r; t = ((hp0.x-p)+r)+(hp1.x+t); eps=1.0005; }
|
||||
res = p+t;
|
||||
cor = (p-res)+t;
|
||||
if (res == (res+eps*cor)) return res;
|
||||
else {
|
||||
res1=res+1.1*cor;
|
||||
z=0.5*(res1-res);
|
||||
__docos(res,z,w);
|
||||
z=(w[0]-x)+w[1];
|
||||
if (z>1.0e-27) return max(res,res1);
|
||||
else if (z<-1.0e-27) return min(res,res1);
|
||||
else return res;
|
||||
}
|
||||
}
|
||||
/* Max ULP is 0.523 based on random sampling. */
|
||||
return res;
|
||||
} /* else if (k < 0x3fee8000) */
|
||||
|
||||
/*--------------------0.953125 <= |x| < 0.96875 ----------------*/
|
||||
else
|
||||
if (k < 0x3fef0000) {
|
||||
n = 768+((k&0x000fe000)>>13)*15;
|
||||
if (m>0) {xx = x - asncs.x[n]; eps=1.04; }
|
||||
else {xx = -x - asncs.x[n]; eps=1.005;}
|
||||
if (m>0) {xx = x - asncs.x[n]; }
|
||||
else {xx = -x - asncs.x[n]; }
|
||||
t = asncs.x[n+1]*xx;
|
||||
p=xx*xx*(asncs.x[n+2]+xx*(asncs.x[n+3]+xx*(asncs.x[n+4]+
|
||||
xx*(asncs.x[n+5]+xx*(asncs.x[n+6]
|
||||
@ -521,25 +297,8 @@ __ieee754_acos(double x)
|
||||
y = (m>0)?(hp0.x-asncs.x[n+12]):(hp0.x+asncs.x[n+12]);
|
||||
t = (m>0)?(hp1.x-t):(hp1.x+t);
|
||||
res = y+t;
|
||||
if (res == res+eps*((y-res)+t)) return res;
|
||||
else {
|
||||
r=asncs.x[n+12]+xx*asncs.x[n+13];
|
||||
t=((asncs.x[n+12]-r)+xx*asncs.x[n+13])+(p+xx*asncs.x[n+14]);
|
||||
if (m>0) {p = hp0.x-r; t = (((hp0.x-p)-r)-t)+hp1.x; eps=1.0030; }
|
||||
else {p = hp0.x+r; t = ((hp0.x-p)+r)+(hp1.x+t); eps=1.0005; }
|
||||
res = p+t;
|
||||
cor = (p-res)+t;
|
||||
if (res == (res+eps*cor)) return res;
|
||||
else {
|
||||
res1=res+1.1*cor;
|
||||
z=0.5*(res1-res);
|
||||
__docos(res,z,w);
|
||||
z=(w[0]-x)+w[1];
|
||||
if (z>1.0e-27) return max(res,res1);
|
||||
else if (z<-1.0e-27) return min(res,res1);
|
||||
else return res;
|
||||
}
|
||||
}
|
||||
/* Max ULP is 0.523 based on random sampling. */
|
||||
return res;
|
||||
} /* else if (k < 0x3fef0000) */
|
||||
/*-----------------0.96875 <= |x| < 1 ---------------------------*/
|
||||
|
||||
@ -560,28 +319,14 @@ __ieee754_acos(double x)
|
||||
cor = (hp1.x - cc)-(y+cc)*p;
|
||||
res1 = hp0.x - y;
|
||||
res =res1 + cor;
|
||||
if (res == res+1.002*((res1-res)+cor)) return (res+res);
|
||||
else {
|
||||
c=y+cc;
|
||||
cc=(y-c)+cc;
|
||||
__doasin(c,cc,w);
|
||||
res1=hp0.x-w[0];
|
||||
cor=((hp0.x-res1)-w[0])+(hp1.x-w[1]);
|
||||
res = res1+cor;
|
||||
return (res+res);
|
||||
}
|
||||
/* Max ULP is 0.501. */
|
||||
return (res+res);
|
||||
}
|
||||
else {
|
||||
cor = cc+p*(y+cc);
|
||||
res = y + cor;
|
||||
if (res == res+1.03*((y-res)+cor)) return (res+res);
|
||||
else {
|
||||
c=y+cc;
|
||||
cc=(y-c)+cc;
|
||||
__doasin(c,cc,w);
|
||||
res = w[0];
|
||||
return (res+res);
|
||||
}
|
||||
/* Max ULP is 0.515. */
|
||||
return (res+res);
|
||||
}
|
||||
} /* else if (k < 0x3ff00000) */
|
||||
|
||||
|
||||
@ -68,7 +68,7 @@ float128: 1
|
||||
ldouble: 1
|
||||
|
||||
Function: "asin_upward":
|
||||
double: 1
|
||||
double: 2
|
||||
float: 1
|
||||
float128: 2
|
||||
ldouble: 1
|
||||
@ -98,6 +98,7 @@ float128: 4
|
||||
ldouble: 5
|
||||
|
||||
Function: "atan":
|
||||
double: 1
|
||||
float: 1
|
||||
float128: 1
|
||||
ldouble: 1
|
||||
@ -673,6 +674,7 @@ float128: 2
|
||||
ldouble: 3
|
||||
|
||||
Function: Imaginary part of "clog":
|
||||
double: 1
|
||||
float: 1
|
||||
float128: 1
|
||||
ldouble: 1
|
||||
|
||||
Loading…
Reference in New Issue
Block a user