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math: Remove slow paths from atan2 [BZ #15267]

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Remove slow paths from atan2. Add ULP annotations.

Reviewed-By: Paul Zimmermann <Paul.Zimmermann@inria.fr>
This commit is contained in:
Wilco Dijkstra 2021-03-10 12:41:20 +00:00 committed by Wilco Dijkstra
parent e898cd1593
commit 4e1a870b9a
2 changed files with 40 additions and 294 deletions
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4
sysdeps/ieee754/dbl-64/atnat2.h
View file

@ -34,7 +34,7 @@
#define MM 5
#ifdef BIG_ENDI
static const number
static const mynumber
/* polynomial I */
/**/ d3 = {{0xbfd55555, 0x55555555} }, /* -0.333... */
/**/ d5 = {{0x3fc99999, 0x999997fd} }, /* 0.199... */
@ -96,7 +96,7 @@
#else
#ifdef LITTLE_ENDI
static const number
static const mynumber
/* polynomial I */
/**/ d3 = {{0x55555555, 0xbfd55555} }, /* -0.333... */
/**/ d5 = {{0x999997fd, 0x3fc99999} }, /* 0.199... */

330
sysdeps/ieee754/dbl-64/e_atan2.c
View file

@ -20,25 +20,14 @@
/* MODULE_NAME: atnat2.c */
/* */
/* FUNCTIONS: uatan2 */
/* atan2Mp */
/* signArctan2 */
/* normalized */
/* */
/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h atnat2.h */
/* mpatan.c mpatan2.c mpsqrt.c */
/* FILES NEEDED: dla.h endian.h mydefs.h atnat2.h */
/* uatan.tbl */
/* */
/* An ultimate atan2() routine. Given two IEEE double machine numbers y,*/
/* x it computes the correctly rounded (to nearest) value of atan2(y,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 "atnat2.h"
@ -48,20 +37,15 @@
#include <math-barriers.h>
#include <math_private.h>
#include <fenv_private.h>
#include <stap-probe.h>
#include <libm-alias-finite.h>
#ifndef SECTION
# define SECTION
#endif
/************************************************************************/
/* An ultimate atan2 routine. Given two IEEE double machine numbers y,x */
/* it computes the correctly rounded (to nearest) value of atan2(y,x). */
/* Assumption: Machine arithmetic operations are performed in */
/* round to nearest mode of IEEE 754 standard. */
/************************************************************************/
static double atan2Mp (double, double, const int[]);
#define TWO52 0x1.0p52
#define TWOM1022 0x1.0p-1022
/* Fix the sign and return after stage 1 or stage 2 */
static double
signArctan2 (double y, double z)
@ -69,18 +53,15 @@ signArctan2 (double y, double z)
return copysign (z, y);
}
static double normalized (double, double, double, double);
void __mpatan2 (mp_no *, mp_no *, mp_no *, int);
/* atan2 with max ULP of ~0.524 based on random sampling. */
double
SECTION
__ieee754_atan2 (double y, double x)
{
int i, de, ux, dx, uy, dy;
static const int pr[MM] = { 6, 8, 10, 20, 32 };
double ax, ay, u, du, u9, ua, v, vv, dv, t1, t2, t3,
z, zz, cor, s1, ss1, s2, ss2;
number num;
double ax, ay, u, du, v, vv, dv, t1, t2, t3,
z, zz, cor;
mynumber num;
static const int ep = 59768832, /* 57*16**5 */
em = -59768832; /* -57*16**5 */
@ -208,10 +189,8 @@ __ieee754_atan2 (double y, double x)
if (x > 0)
{
double ret;
if ((z = ay / ax) < TWOM1022)
ret = normalized (ax, ay, y, z);
else
ret = signArctan2 (y, z);
z = ay / ax;
ret = signArctan2 (y, z);
if (fabs (ret) < DBL_MIN)
{
double vret = ret ? ret : DBL_MIN;
@ -270,30 +249,12 @@ __ieee754_atan2 (double y, double x)
+ v * (d11.d
+ v * d13.d)))));
if ((z = u + (zz - u1.d * u)) == u + (zz + u1.d * u))
return signArctan2 (y, z);
MUL2 (u, du, u, du, v, vv, t1, t2);
s1 = v * (f11.d + v * (f13.d
+ v * (f15.d + v * (f17.d + v * f19.d))));
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 (u, du, s1, ss1, s2, ss2, t1, t2);
ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
if ((z = s1 + (ss1 - u5.d * s1)) == s1 + (ss1 + u5.d * s1))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
z = u + zz;
/* Max ULP is 0.504. */
return signArctan2 (y, z);
}
i = (TWO52 + TWO8 * u) - TWO52;
i = (TWO52 + 256 * u) - TWO52;
i -= 16;
t3 = u - cij[i][0].d;
EADD (t3, du, v, dv);
@ -304,43 +265,9 @@ __ieee754_atan2 (double y, double x)
+ v * (cij[i][4].d
+ v * (cij[i][5].d
+ v * cij[i][6].d))));
if (i < 112)
{
if (i < 48)
u9 = u91.d; /* u < 1/4 */
else
u9 = u92.d;
} /* 1/4 <= u < 1/2 */
else
{
if (i < 176)
u9 = u93.d; /* 1/2 <= u < 3/4 */
else
u9 = u94.d;
} /* 3/4 <= u <= 1 */
if ((z = t1 + (zz - u9 * t1)) == t1 + (zz + u9 * t1))
return signArctan2 (y, z);
t1 = u - hij[i][0].d;
EADD (t1, du, v, vv);
s1 = v * (hij[i][11].d
+ v * (hij[i][12].d
+ v * (hij[i][13].d
+ v * (hij[i][14].d
+ v * hij[i][15].d))));
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
if ((z = s2 + (ss2 - ub.d * s2)) == s2 + (ss2 + ub.d * s2))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
z = t1 + zz;
/* Max ULP is 0.56. */
return signArctan2 (y, z);
}
/* (ii) x>0, abs(x)<=abs(y): pi/2-atan(ax/ay) */
@ -355,31 +282,12 @@ __ieee754_atan2 (double y, double x)
+ v * d13.d)))));
ESUB (hpi.d, u, t2, cor);
t3 = ((hpi1.d + cor) - du) - zz;
if ((z = t2 + (t3 - u2.d)) == t2 + (t3 + u2.d))
return signArctan2 (y, z);
MUL2 (u, du, u, du, v, vv, t1, t2);
s1 = v * (f11.d
+ v * (f13.d
+ v * (f15.d + v * (f17.d + v * f19.d))));
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 (u, du, s1, ss1, s2, ss2, t1, t2);
ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
SUB2 (hpi.d, hpi1.d, s1, ss1, s2, ss2, t1, t2);
if ((z = s2 + (ss2 - u6.d)) == s2 + (ss2 + u6.d))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
z = t2 + t3;
/* Max ULP is 0.501. */
return signArctan2 (y, z);
}
i = (TWO52 + TWO8 * u) - TWO52;
i = (TWO52 + 256 * u) - TWO52;
i -= 16;
v = (u - cij[i][0].d) + du;
@ -389,36 +297,9 @@ __ieee754_atan2 (double y, double x)
+ v * (cij[i][5].d
+ v * cij[i][6].d))));
t1 = hpi.d - cij[i][1].d;
if (i < 112)
ua = ua1.d; /* w < 1/2 */
else
ua = ua2.d; /* w >= 1/2 */
if ((z = t1 + (zz - ua)) == t1 + (zz + ua))
return signArctan2 (y, z);
t1 = u - hij[i][0].d;
EADD (t1, du, v, vv);
s1 = v * (hij[i][11].d
+ v * (hij[i][12].d
+ v * (hij[i][13].d
+ v * (hij[i][14].d
+ v * hij[i][15].d))));
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
SUB2 (hpi.d, hpi1.d, s2, ss2, s1, ss1, t1, t2);
if ((z = s1 + (ss1 - uc.d)) == s1 + (ss1 + uc.d))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
z = t1 + zz;
/* Max ULP is 0.503. */
return signArctan2 (y, z);
}
/* (iii) x<0, abs(x)< abs(y): pi/2+atan(ax/ay) */
@ -434,30 +315,12 @@ __ieee754_atan2 (double y, double x)
+ v * (d11.d + v * d13.d)))));
EADD (hpi.d, u, t2, cor);
t3 = ((hpi1.d + cor) + du) + zz;
if ((z = t2 + (t3 - u3.d)) == t2 + (t3 + u3.d))
return signArctan2 (y, z);
MUL2 (u, du, u, du, v, vv, t1, t2);
s1 = v * (f11.d
+ v * (f13.d + v * (f15.d + v * (f17.d + v * f19.d))));
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 (u, du, s1, ss1, s2, ss2, t1, t2);
ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
ADD2 (hpi.d, hpi1.d, s1, ss1, s2, ss2, t1, t2);
if ((z = s2 + (ss2 - u7.d)) == s2 + (ss2 + u7.d))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
z = t2 + t3;
/* Max ULP is 0.501. */
return signArctan2 (y, z);
}
i = (TWO52 + TWO8 * u) - TWO52;
i = (TWO52 + 256 * u) - TWO52;
i -= 16;
v = (u - cij[i][0].d) + du;
zz = hpi1.d + v * (cij[i][2].d
@ -466,34 +329,9 @@ __ieee754_atan2 (double y, double x)
+ v * (cij[i][5].d
+ v * cij[i][6].d))));
t1 = hpi.d + cij[i][1].d;
if (i < 112)
ua = ua1.d; /* w < 1/2 */
else
ua = ua2.d; /* w >= 1/2 */
if ((z = t1 + (zz - ua)) == t1 + (zz + ua))
return signArctan2 (y, z);
t1 = u - hij[i][0].d;
EADD (t1, du, v, vv);
s1 = v * (hij[i][11].d
+ v * (hij[i][12].d
+ v * (hij[i][13].d
+ v * (hij[i][14].d
+ v * hij[i][15].d))));
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
ADD2 (hpi.d, hpi1.d, s2, ss2, s1, ss1, t1, t2);
if ((z = s1 + (ss1 - uc.d)) == s1 + (ss1 + uc.d))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
z = t1 + zz;
/* Max ULP is 0.503. */
return signArctan2 (y, z);
}
/* (iv) x<0, abs(y)<=abs(x): pi-atan(ax/ay) */
@ -506,29 +344,12 @@ __ieee754_atan2 (double y, double x)
+ v * (d9.d + v * (d11.d + v * d13.d)))));
ESUB (opi.d, u, t2, cor);
t3 = ((opi1.d + cor) - du) - zz;
if ((z = t2 + (t3 - u4.d)) == t2 + (t3 + u4.d))
return signArctan2 (y, z);
MUL2 (u, du, u, du, v, vv, t1, t2);
s1 = v * (f11.d + v * (f13.d + v * (f15.d + v * (f17.d + v * f19.d))));
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 (u, du, s1, ss1, s2, ss2, t1, t2);
ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
SUB2 (opi.d, opi1.d, s1, ss1, s2, ss2, t1, t2);
if ((z = s2 + (ss2 - u8.d)) == s2 + (ss2 + u8.d))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
z = t2 + t3;
/* Max ULP is 0.501. */
return signArctan2 (y, z);
}
i = (TWO52 + TWO8 * u) - TWO52;
i = (TWO52 + 256 * u) - TWO52;
i -= 16;
v = (u - cij[i][0].d) + du;
zz = opi1.d - v * (cij[i][2].d
@ -536,86 +357,11 @@ __ieee754_atan2 (double y, double x)
+ v * (cij[i][4].d
+ v * (cij[i][5].d + v * cij[i][6].d))));
t1 = opi.d - cij[i][1].d;
if (i < 112)
ua = ua1.d; /* w < 1/2 */
else
ua = ua2.d; /* w >= 1/2 */
if ((z = t1 + (zz - ua)) == t1 + (zz + ua))
return signArctan2 (y, z);
t1 = u - hij[i][0].d;
EADD (t1, du, v, vv);
s1 = v * (hij[i][11].d
+ v * (hij[i][12].d
+ v * (hij[i][13].d
+ v * (hij[i][14].d + v * hij[i][15].d))));
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
SUB2 (opi.d, opi1.d, s2, ss2, s1, ss1, t1, t2);
if ((z = s1 + (ss1 - uc.d)) == s1 + (ss1 + uc.d))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
z = t1 + zz;
/* Max ULP is 0.502. */
return signArctan2 (y, z);
}
#ifndef __ieee754_atan2
libm_alias_finite (__ieee754_atan2, __atan2)
#endif
/* Treat the Denormalized case */
static double
SECTION
normalized (double ax, double ay, double y, double z)
{
int p;
mp_no mpx, mpy, mpz, mperr, mpz2, mpt1;
p = 6;
__dbl_mp (ax, &mpx, p);
__dbl_mp (ay, &mpy, p);
__dvd (&mpy, &mpx, &mpz, p);
__dbl_mp (ue.d, &mpt1, p);
__mul (&mpz, &mpt1, &mperr, p);
__sub (&mpz, &mperr, &mpz2, p);
__mp_dbl (&mpz2, &z, p);
return signArctan2 (y, z);
}
/* Stage 3: Perform a multi-Precision computation */
static double
SECTION
atan2Mp (double x, double y, const int pr[])
{
double z1, z2;
int i, p;
mp_no mpx, mpy, mpz, mpz1, mpz2, mperr, mpt1;
for (i = 0; i < MM; i++)
{
p = pr[i];
__dbl_mp (x, &mpx, p);
__dbl_mp (y, &mpy, p);
__mpatan2 (&mpy, &mpx, &mpz, p);
__dbl_mp (ud[i].d, &mpt1, p);
__mul (&mpz, &mpt1, &mperr, p);
__add (&mpz, &mperr, &mpz1, p);
__sub (&mpz, &mperr, &mpz2, p);
__mp_dbl (&mpz1, &z1, p);
__mp_dbl (&mpz2, &z2, p);
if (z1 == z2)
{
LIBC_PROBE (slowatan2, 4, &p, &x, &y, &z1);
return z1;
}
}
LIBC_PROBE (slowatan2_inexact, 4, &p, &x, &y, &z1);
return z1; /*if impossible to do exact computing */
}