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math: Remove slow paths from atan [BZ #15267]

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Remove slow paths from atan. Add ULP annotations.

Reviewed-By: Paul Zimmermann <Paul.Zimmermann@inria.fr>
This commit is contained in:
Wilco Dijkstra 2021-03-10 12:40:56 +00:00 committed by Wilco Dijkstra
parent 476d692e8a
commit e898cd1593
3 changed files with 27 additions and 181 deletions
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4
sysdeps/ieee754/dbl-64/atnat.h
View file

@ -29,7 +29,7 @@
#define M 4
#ifdef BIG_ENDI
static const number
static const mynumber
/* polynomial I */
/**/ d3 = {{0xbfd55555, 0x55555555} }, /* -0.333... */
/**/ d5 = {{0x3fc99999, 0x999997fd} }, /* 0.199... */
@ -79,7 +79,7 @@
#else
#ifdef LITTLE_ENDI
static const number
static const mynumber
/* polynomial I */
/**/ d3 = {{0x55555555, 0xbfd55555} }, /* -0.333... */
/**/ d5 = {{0x999997fd, 0x3fc99999} }, /* 0.199... */

196
sysdeps/ieee754/dbl-64/s_atan.c
View file

@ -20,25 +20,15 @@
/* MODULE_NAME: atnat.c */
/* */
/* FUNCTIONS: uatan */
/* atanMp */
/* signArctan */
/* */
/* */
/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h atnat.h */
/* mpatan.c mpatan2.c mpsqrt.c */
/* FILES NEEDED: dla.h endian.h mydefs.h atnat.h */
/* uatan.tbl */
/* */
/* An ultimate atan() routine. Given an IEEE double machine number x */
/* it computes the correctly rounded (to nearest) value of atan(x). */
/* */
/* Assumption: Machine arithmetic operations are performed in */
/* round to nearest mode of IEEE 754 standard. */
/* */
/************************************************************************/
#include <dla.h>
#include "mpa.h"
#include "MathLib.h"
#include "mydefs.h"
#include "uatan.tbl"
#include "atnat.h"
#include <fenv.h>
@ -47,10 +37,8 @@
#include <math.h>
#include <fenv_private.h>
#include <math-underflow.h>
#include <stap-probe.h>
void __mpatan (mp_no *, mp_no *, int); /* see definition in mpatan.c */
static double atanMp (double, const int[]);
#define TWO52 0x1.0p52
/* Fix the sign of y and return */
static double
@ -59,17 +47,14 @@ __signArctan (double x, double y)
return copysign (y, x);
}
/* An ultimate atan() routine. Given an IEEE double machine number x, */
/* routine computes the correctly rounded (to nearest) value of atan(x). */
/* atan with max ULP of ~0.523 based on random sampling. */
double
__atan (double x)
{
double cor, s1, ss1, s2, ss2, t1, t2, t3, t4, u, u2, u3,
v, vv, w, ww, y, yy, z, zz;
double cor, t1, t2, t3, u,
v, w, ww, y, yy, z;
int i, ux, dx;
static const int pr[M] = { 6, 8, 10, 32 };
number num;
mynumber num;
num.d = x;
ux = num.i[HIGH_HALF];
@ -102,36 +87,14 @@ __atan (double x)
yy = d3.d + v * yy;
yy *= x * v;
if ((y = x + (yy - U1 * x)) == x + (yy + U1 * x))
return y;
EMULV (x, x, v, vv); /* v+vv=x^2 */
s1 = f17.d + v * f19.d;
s1 = f15.d + v * s1;
s1 = f13.d + v * s1;
s1 = f11.d + v * s1;
s1 *= v;
ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
MUL2 (x, 0, s1, ss1, s2, ss2, t1, t2);
ADD2 (x, 0, s2, ss2, s1, ss1, t1, t2);
if ((y = s1 + (ss1 - U5 * s1)) == s1 + (ss1 + U5 * s1))
return y;
return atanMp (x, pr);
y = x + yy;
/* Max ULP is 0.511. */
return y;
}
}
else
{ /* B <= u < C */
i = (TWO52 + TWO8 * u) - TWO52;
i = (TWO52 + 256 * u) - TWO52;
i -= 16;
z = u - cij[i][0].d;
yy = cij[i][5].d + z * cij[i][6].d;
@ -141,44 +104,9 @@ __atan (double x)
yy *= z;
t1 = cij[i][1].d;
if (i < 112)
{
if (i < 48)
u2 = U21; /* u < 1/4 */
else
u2 = U22;
} /* 1/4 <= u < 1/2 */
else
{
if (i < 176)
u2 = U23; /* 1/2 <= u < 3/4 */
else
u2 = U24;
} /* 3/4 <= u <= 1 */
if ((y = t1 + (yy - u2 * t1)) == t1 + (yy + u2 * t1))
return __signArctan (x, y);
z = u - hij[i][0].d;
s1 = hij[i][14].d + z * hij[i][15].d;
s1 = hij[i][13].d + z * s1;
s1 = hij[i][12].d + z * s1;
s1 = hij[i][11].d + z * s1;
s1 *= z;
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
if ((y = s2 + (ss2 - U6 * s2)) == s2 + (ss2 + U6 * s2))
return __signArctan (x, y);
return atanMp (x, pr);
y = t1 + yy;
/* Max ULP is 0.56. */
return __signArctan (x, y);
}
}
else
@ -188,7 +116,7 @@ __atan (double x)
w = 1 / u;
EMULV (w, u, t1, t2);
ww = w * ((1 - t1) - t2);
i = (TWO52 + TWO8 * w) - TWO52;
i = (TWO52 + 256 * w) - TWO52;
i -= 16;
z = (w - cij[i][0].d) + ww;
@ -199,37 +127,9 @@ __atan (double x)
yy = HPI1 - z * yy;
t1 = HPI - cij[i][1].d;
if (i < 112)
u3 = U31; /* w < 1/2 */
else
u3 = U32; /* w >= 1/2 */
if ((y = t1 + (yy - u3)) == t1 + (yy + u3))
return __signArctan (x, y);
DIV2 (1, 0, u, 0, w, ww, t1, t2, t3, t4);
t1 = w - hij[i][0].d;
EADD (t1, ww, z, zz);
s1 = hij[i][14].d + z * hij[i][15].d;
s1 = hij[i][13].d + z * s1;
s1 = hij[i][12].d + z * s1;
s1 = hij[i][11].d + z * s1;
s1 *= z;
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
SUB2 (HPI, HPI1, s2, ss2, s1, ss1, t1, t2);
if ((y = s1 + (ss1 - U7)) == s1 + (ss1 + U7))
return __signArctan (x, y);
return atanMp (x, pr);
y = t1 + yy;
/* Max ULP is 0.503. */
return __signArctan (x, y);
}
else
{
@ -249,34 +149,9 @@ __atan (double x)
ww = w * ((1 - t1) - t2);
ESUB (HPI, w, t3, cor);
yy = ((HPI1 + cor) - ww) - yy;
if ((y = t3 + (yy - U4)) == t3 + (yy + U4))
return __signArctan (x, y);
DIV2 (1, 0, u, 0, w, ww, t1, t2, t3, t4);
MUL2 (w, ww, w, ww, v, vv, t1, t2);
s1 = f17.d + v * f19.d;
s1 = f15.d + v * s1;
s1 = f13.d + v * s1;
s1 = f11.d + v * s1;
s1 *= v;
ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
MUL2 (w, ww, s1, ss1, s2, ss2, t1, t2);
ADD2 (w, ww, s2, ss2, s1, ss1, t1, t2);
SUB2 (HPI, HPI1, s1, ss1, s2, ss2, t1, t2);
if ((y = s2 + (ss2 - U8)) == s2 + (ss2 + U8))
return __signArctan (x, y);
return atanMp (x, pr);
y = t3 + yy;
/* Max ULP is 0.5003. */
return __signArctan (x, y);
}
else
{
@ -290,35 +165,6 @@ __atan (double x)
}
}
/* Final stages. Compute atan(x) by multiple precision arithmetic */
static double
atanMp (double x, const int pr[])
{
mp_no mpx, mpy, mpy2, mperr, mpt1, mpy1;
double y1, y2;
int i, p;
for (i = 0; i < M; i++)
{
p = pr[i];
__dbl_mp (x, &mpx, p);
__mpatan (&mpx, &mpy, p);
__dbl_mp (u9[i].d, &mpt1, p);
__mul (&mpy, &mpt1, &mperr, p);
__add (&mpy, &mperr, &mpy1, p);
__sub (&mpy, &mperr, &mpy2, p);
__mp_dbl (&mpy1, &y1, p);
__mp_dbl (&mpy2, &y2, p);
if (y1 == y2)
{
LIBC_PROBE (slowatan, 3, &p, &x, &y1);
return y1;
}
}
LIBC_PROBE (slowatan_inexact, 3, &p, &x, &y1);
return y1; /*if impossible to do exact computing */
}
#ifndef __atan
libm_alias_double (__atan, atan)
#endif

8
sysdeps/ieee754/dbl-64/uatan.tbl
View file

@ -25,7 +25,7 @@
#ifdef BIG_ENDI
static const number
static const mynumber
cij[241][7] = { /* x0,cij for (1/16,1) */
/**/ {{{0X3FB04006, 0X65E0244E} },
/**/ {{0X3FB03A73, 0X7B53DD20} },
@ -1716,7 +1716,7 @@
/**/ {{0XBF99B9A7, 0X18A3BA58} } },
};
static const number
static const mynumber
hij[241][16] = { /* x0,hij for (1/16,1) */
/**/ {{{0x3fb04000, 0x00000000} },
/**/ {{0x3fb03a6d, 0x1c06693d} },
@ -5579,7 +5579,7 @@
#else
#ifdef LITTLE_ENDI
static const number
static const mynumber
cij[241][7] = { /* x0,cij for (1/16,1) */
/**/ {{{0X65E0244E, 0X3FB04006} },
/**/ {{0X7B53DD20, 0X3FB03A73} },
@ -7270,7 +7270,7 @@
/**/ {{0X18A3BA58, 0XBF99B9A7} } },
};
static const number
static const mynumber
hij[241][16] = { /* x0,hij for (1/16,1) */
/**/ {{{0x00000000, 0x3fb04000} },
/**/ {{0x1c06693d, 0x3fb03a6d} },