math: Use atan2f from CORE-MATH

The CORE-MATH implementation is correctly rounded (for any rounding mode)
and shows slight better performance to the generic atan2f.

The code was adapted to glibc style and to use the definition of
math_config.h (to handle errno, overflow, and underflow).

Benchtest on x64_64 (Ryzen 9 5900X, gcc 14.2.1), aarch64 (Neoverse-N1,
gcc 13.3.1), and powerpc (POWER10, gcc 13.2.1):

Latency                      master        patched   improvement
x86_64                      68.1175        69.2014        -1.59%
x86_64v2                    66.9884        66.0081         1.46%
x86_64v3                    57.7034        61.6407        -6.82%
i686                       189.8690        152.7560       19.55%
aarch64 (Neoverse)          32.6151        24.5382        24.76%
power10                     21.7282        17.1896        20.89%

reciprocal-throughput        master        patched   improvement
x86_64                      34.5202        31.6155         8.41%
x86_64v2                    32.6379        30.3372         7.05%
x86_64v3                    34.3677        23.6455        31.20%
i686                       157.7290        75.8308        51.92%
aarch64 (Neoverse)          27.7788        16.2671        41.44%
power10                     15.5715         8.1588        47.60%

Signed-off-by: Alexei Sibidanov <sibid@uvic.ca>
Signed-off-by: Paul Zimmermann <Paul.Zimmermann@inria.fr>
Signed-off-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
Reviewed-by: DJ Delorie <dj@redhat.com>

SHARED-FILES

@@ -314,3 +314,7 @@ sysdeps/ieee754/flt-32/s_atanf.c:
   (src/binary32/atan/atanf.c in CORE-MATH)
   - The code was adapted to use glibc code style and internal
     functions to handle errno, overflow, and underflow.
+sysdeps/ieee754/flt-32/e_atan2f.c:
+  (src/binary32/atan2/atan2f.c in CORE-MATH)
+  - The code was adapted to use glibc code style and internal
+    functions to handle errno, overflow, and underflow.

sysdeps/aarch64/libm-test-ulps

@@ -142,7 +142,6 @@ double: 1
 ldouble: 1
 
 Function: "atan2":
-float: 1
 ldouble: 2
 
 Function: "atan2_advsimd":
@@ -151,7 +150,6 @@ float: 2
 
 Function: "atan2_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "atan2_sve":
@@ -160,12 +158,10 @@ float: 2
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "atan2_upward":
 double: 1
-float: 1
 ldouble: 2
 
 Function: "atan2pi":
@@ -354,22 +350,18 @@ ldouble: 4
 
 Function: "carg":
 double: 1
-float: 1
 ldouble: 2
 
 Function: "carg_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "carg_upward":
 double: 1
-float: 1
 ldouble: 2
 
 Function: Real part of "casin":

sysdeps/alpha/fpu/libm-test-ulps

@@ -70,22 +70,18 @@ double: 1
 ldouble: 1
 
 Function: "atan2":
-float: 2
 ldouble: 2
 
 Function: "atan2_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "atan2_upward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "atan_downward":
@@ -217,22 +213,18 @@ float: 2
 ldouble: 4
 
 Function: "carg":
-float: 1
 ldouble: 2
 
 Function: "carg_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "carg_upward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: Real part of "casin":

sysdeps/arc/fpu/libm-test-ulps

@@ -54,19 +54,15 @@ double: 1
 
 Function: "atan2":
 double: 7
-float: 2
 
 Function: "atan2_downward":
 double: 5
-float: 2
 
 Function: "atan2_towardzero":
 double: 5
-float: 2
 
 Function: "atan2_upward":
 double: 8
-float: 2
 
 Function: "atan_downward":
 double: 1
@@ -173,19 +169,15 @@ float: 3
 
 Function: "carg":
 double: 7
-float: 2
 
 Function: "carg_downward":
 double: 5
-float: 2
 
 Function: "carg_towardzero":
 double: 5
-float: 2
 
 Function: "carg_upward":
 double: 8
-float: 2
 
 Function: Real part of "casin":
 double: 3

sysdeps/arc/nofpu/libm-test-ulps

@@ -17,7 +17,6 @@ Function: "atan":
 double: 1
 
 Function: "atan2":
-float: 2
 
 Function: "atanh":
 double: 2
@@ -43,7 +42,6 @@ double: 1
 float: 2
 
 Function: "carg":
-float: 1
 
 Function: Real part of "casin":
 double: 1

sysdeps/arm/libm-test-ulps

@@ -85,19 +85,15 @@ Function: "atan":
 double: 1
 
 Function: "atan2":
-float: 2
 
 Function: "atan2_downward":
 double: 1
-float: 2
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 
 Function: "atan2_upward":
 double: 1
-float: 2
 
 Function: "atan2pi":
 double: 1
@@ -233,19 +229,15 @@ double: 3
 float: 2
 
 Function: "carg":
-float: 1
 
 Function: "carg_downward":
 double: 1
-float: 2
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 
 Function: "carg_upward":
 double: 1
-float: 2
 
 Function: Real part of "casin":
 double: 1

sysdeps/csky/fpu/libm-test-ulps

@@ -50,19 +50,15 @@ double: 3
 Function: "atan":
 
 Function: "atan2":
-float: 1
 
 Function: "atan2_downward":
 double: 1
-float: 2
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 
 Function: "atan2_upward":
 double: 1
-float: 2
 
 Function: "atan_downward":
 double: 1
@@ -166,19 +162,15 @@ double: 3
 float: 2
 
 Function: "carg":
-float: 1
 
 Function: "carg_downward":
 double: 1
-float: 2
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 
 Function: "carg_upward":
 double: 1
-float: 2
 
 Function: Real part of "casin":
 double: 1

sysdeps/csky/nofpu/libm-test-ulps

@@ -50,19 +50,15 @@ double: 3
 Function: "atan":
 
 Function: "atan2":
-float: 1
 
 Function: "atan2_downward":
 double: 1
-float: 2
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 
 Function: "atan2_upward":
 double: 1
-float: 2
 
 Function: "atan_downward":
 double: 1
@@ -166,19 +162,15 @@ double: 2
 float: 2
 
 Function: "carg":
-float: 1
 
 Function: "carg_downward":
 double: 1
-float: 2
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 
 Function: "carg_upward":
 double: 1
-float: 2
 
 Function: Real part of "casin":
 double: 1

sysdeps/hppa/fpu/libm-test-ulps

@@ -85,19 +85,15 @@ Function: "atan":
 double: 1
 
 Function: "atan2":
-float: 2
 
 Function: "atan2_downward":
 double: 1
-float: 2
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 
 Function: "atan2_upward":
 double: 1
-float: 2
 
 Function: "atan2pi":
 double: 1
@@ -233,19 +229,15 @@ double: 3
 float: 2
 
 Function: "carg":
-float: 1
 
 Function: "carg_downward":
 double: 1
-float: 2
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 
 Function: "carg_upward":
 double: 1
-float: 2
 
 Function: Real part of "casin":
 double: 1

sysdeps/i386/fpu/e_atan2f.S

@@ -1,30 +0,0 @@
-/*
- * Public domain.
- */
-
-#include <machine/asm.h>
-#include <i386-math-asm.h>
-#include <libm-alias-finite.h>
-
-RCSID("$NetBSD: e_atan2f.S,v 1.1 1995/05/08 23:35:10 jtc Exp $")
-
-DEFINE_FLT_MIN
-
-#ifdef PIC
-# define MO(op) op##@GOTOFF(%ecx)
-#else
-# define MO(op) op
-#endif
-
-	.text
-ENTRY(__ieee754_atan2f)
-#ifdef  PIC
-	LOAD_PIC_REG (cx)
-#endif
-	flds	4(%esp)
-	flds	8(%esp)
-	fpatan
-	FLT_CHECK_FORCE_UFLOW_NARROW
-	ret
-END (__ieee754_atan2f)
-libm_alias_finite (__ieee754_atan2f, __atan2f)

sysdeps/i386/fpu/libm-test-ulps

@@ -139,19 +139,16 @@ ldouble: 1
 
 Function: "atan2_downward":
 double: 1
-float: 1
 float128: 2
 ldouble: 1
 
 Function: "atan2_towardzero":
 double: 1
-float: 1
 float128: 3
 ldouble: 1
 
 Function: "atan2_upward":
 double: 1
-float: 1
 float128: 2
 ldouble: 1
 
@@ -163,19 +160,19 @@ ldouble: 1
 
 Function: "atan2pi_downward":
 double: 1
-float: 1
+float: 2
 float128: 2
 ldouble: 2
 
 Function: "atan2pi_towardzero":
 double: 1
-float: 1
+float: 2
 float128: 2
 ldouble: 2
 
 Function: "atan2pi_upward":
 double: 1
-float: 1
+float: 2
 float128: 2
 ldouble: 2
 
@@ -364,19 +361,16 @@ ldouble: 1
 
 Function: "carg_downward":
 double: 1
-float: 1
 float128: 2
 ldouble: 1
 
 Function: "carg_towardzero":
 double: 1
-float: 1
 float128: 3
 ldouble: 1
 
 Function: "carg_upward":
 double: 1
-float: 1
 float128: 2
 ldouble: 1
 

sysdeps/i386/i686/fpu/multiarch/libm-test-ulps

@@ -139,19 +139,16 @@ ldouble: 1
 
 Function: "atan2_downward":
 double: 1
-float: 1
 float128: 2
 ldouble: 1
 
 Function: "atan2_towardzero":
 double: 1
-float: 1
 float128: 3
 ldouble: 1
 
 Function: "atan2_upward":
 double: 1
-float: 1
 float128: 2
 ldouble: 1
 
@@ -163,19 +160,19 @@ ldouble: 1
 
 Function: "atan2pi_downward":
 double: 1
-float: 1
+float: 2
 float128: 2
 ldouble: 2
 
 Function: "atan2pi_towardzero":
 double: 1
-float: 1
+float: 2
 float128: 2
 ldouble: 2
 
 Function: "atan2pi_upward":
 double: 1
-float: 1
+float: 2
 float128: 2
 ldouble: 2
 
@@ -364,19 +361,16 @@ ldouble: 1
 
 Function: "carg_downward":
 double: 1
-float: 1
 float128: 2
 ldouble: 1
 
 Function: "carg_towardzero":
 double: 1
-float: 1
 float128: 3
 ldouble: 1
 
 Function: "carg_upward":
 double: 1
-float: 1
 float128: 2
 ldouble: 1
 

sysdeps/ieee754/flt-32/e_atan2f.c

@@ -1,94 +1,274 @@
-/* e_atan2f.c -- float version of e_atan2.c.
+/* Correctly-rounded arctangent function of two binary32 values.
- */
 
-/*
+Copyright (c) 2022-2024 Alexei Sibidanov and Paul Zimmermann.
- * ====================================================
+
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+The original version of this file was copied from the CORE-MATH
- *
+project (file src/binary32/atan2/atan2f.c, revision 7835c5d).
- * Developed at SunPro, a Sun Microsystems, Inc. business.
+
- * Permission to use, copy, modify, and distribute this
+Permission is hereby granted, free of charge, to any person obtaining a copy
- * software is freely granted, provided that this notice
+of this software and associated documentation files (the "Software"), to deal
- * is preserved.
+in the Software without restriction, including without limitation the rights
- * ====================================================
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
- */
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.
+
+*/
 
 #include <math.h>
-#include <math_private.h>
+#include <stdint.h>
 #include <libm-alias-finite.h>
+#include "math_config.h"
 
-static const float
+static inline double
-tiny  = 1.0e-30,
+muldd (double xh, double xl, double ch, double cl, double *l)
-zero  = 0.0,
+{
-pi_o_4  = 7.8539818525e-01,  /* 0x3f490fdb */
+  double ahlh = ch * xl;
-pi_o_2  = 1.5707963705e+00,  /* 0x3fc90fdb */
+  double alhh = cl * xh;
-pi      = 3.1415927410e+00,  /* 0x40490fdb */
+  double ahhh = ch * xh;
-pi_lo   = -8.7422776573e-08; /* 0xb3bbbd2e */
+  double ahhl = fma (ch, xh, -ahhh);
+  ahhl += alhh + ahlh;
+  ch = ahhh + ahhl;
+  *l = (ahhh - ch) + ahhl;
+  return ch;
+}
+
+static double
+polydd (double xh, double xl, int n, const double c[][2], double *l)
+{
+  int i = n - 1;
+  double ch = c[i][0];
+  double cl = c[i][1];
+  while (--i >= 0)
+    {
+      ch = muldd (xh, xl, ch, cl, &cl);
+      double th = ch + c[i][0];
+      double tl = (c[i][0] - th) + ch;
+      ch = th;
+      cl += tl + c[i][1];
+    }
+  *l = cl;
+  return ch;
+}
+
+/* for y/x tiny, use Taylor approximation z - z^3/3 where z=y/x */
+static float
+cr_atan2f_tiny (float y, float x)
+{
+  double dy = y;
+  double dx = x;
+  double z = dy / dx;
+  double e = fma (-z, x, y);
+  /* z * x + e = y thus y/x = z + e/x */
+  static const double c = -0x1.5555555555555p-2; /* -1/3 rounded to nearest */
+  double zz = z * z;
+  double cz = c * z;
+  e = e / x + cz * zz;
+  uint64_t t = asuint64 (z);
+  if ((t & UINT64_C(0xfffffff)) == 0) /* boundary case */
+    {
+      /* If z and e are of same sign (resp. of different signs), we increase
+	 (resp. decrease) the significant of t by 1 to avoid a double-rounding
+	 issue when rounding t to binary32. */
+      if (z * e > 0)
+	t += 1;
+      else
+	t -= 1;
+    }
+  return asdouble (t);
+}
 
 float
 __ieee754_atan2f (float y, float x)
 {
-	float z;
+  static const double cn[] =
-	int32_t k,m,hx,hy,ix,iy;
+    {
-
+      0x1p+0,               0x1.40e0698f94c35p+1, 0x1.248c5da347f0dp+1,
-	GET_FLOAT_WORD(hx,x);
+      0x1.d873386572976p-1, 0x1.46fa40b20f1dp-3,  0x1.33f5e041eed0fp-7,
-	ix = hx&0x7fffffff;
+      0x1.546bbf28667c5p-14
-	GET_FLOAT_WORD(hy,y);
+    };
-	iy = hy&0x7fffffff;
+  static const double cd[] =
-	if((ix>0x7f800000)||
+    {
-	   (iy>0x7f800000))	/* x or y is NaN */
+      0x1p+0,               0x1.6b8b143a3f6dap+1, 0x1.8421201d18ed5p+1,
-	   return x+y;
+      0x1.8221d086914ebp+0, 0x1.670657e3a07bap-2, 0x1.0f4951fd1e72dp-5,
-	if(hx==0x3f800000) return __atanf(y);   /* x=1.0 */
+      0x1.b3874b8798286p-11
-	m = ((hy>>31)&1)|((hx>>30)&2);	/* 2*sign(x)+sign(y) */
+    };
-
+  static const double m[] = { 0, 1 };
-    /* when y = 0 */
+#define pi 0x1.921fb54442d18p+1
-	if(iy==0) {
+#define pi2 0x1.921fb54442d18p+0
-	    switch(m) {
+#define pi2l 0x1.1a62633145c07p-54
-		case 0:
+  static const double off[] = { 0.0f, pi2, pi, pi2, -0.0f, -pi2, -pi, -pi2 };
-		case 1: return y;	/* atan(+-0,+anything)=+-0 */
+  static const double offl[] =
-		case 2: return  pi+tiny;/* atan(+0,-anything) = pi */
+    {
-		case 3: return -pi-tiny;/* atan(-0,-anything) =-pi */
+      0.0f, pi2l, 2 * pi2l, pi2l, -0.0f, -pi2l, -2 * pi2l, -pi2l
-	    }
+    };
+  static const double sgn[] = { 1, -1 };
+  uint32_t ux = asuint (x);
+  uint32_t uy = asuint (y);
+  uint32_t ax = ux & (~0u >> 1);
+  uint32_t ay = uy & (~0u >> 1);
+  if (__glibc_unlikely (ay >= (0xff << 23) || ax >= (0xff << 23)))
+    {
+      /* we use x+y below so that the invalid exception is set
+	 for (x,y) = (qnan,snan) or (snan,qnan) */
+      if (ay > (0xff << 23))
+	return x + y; /* nan */
+      if (ax > (0xff << 23))
+	return x + y; /* nan */
+      bool yinf = ay == (0xff << 23);
+      bool xinf = ax == (0xff << 23);
+      if (yinf & xinf)
+	{
+	  if (ux >> 31)
+	    return 0x1.2d97c7f3321d2p+1 * sgn[uy >> 31];
+	  else
+	    return 0x1.921fb54442d18p-1 * sgn[uy >> 31];
 	}
-    /* when x = 0 */
+      if (xinf)
-	if(ix==0) return (hy<0)?  -pi_o_2-tiny: pi_o_2+tiny;
+	{
-
+	  if (ux >> 31)
-    /* when x is INF */
+	    return pi * sgn[uy >> 31];
-	if(ix==0x7f800000) {
+	  else
-	    if(iy==0x7f800000) {
+	    return 0.0f * sgn[uy >> 31];
-		switch(m) {
-		    case 0: return  pi_o_4+tiny;/* atan(+INF,+INF) */
-		    case 1: return -pi_o_4-tiny;/* atan(-INF,+INF) */
-		    case 2: return  (float)3.0*pi_o_4+tiny;/*atan(+INF,-INF)*/
-		    case 3: return (float)-3.0*pi_o_4-tiny;/*atan(-INF,-INF)*/
-		}
-	    } else {
-		switch(m) {
-		    case 0: return  zero  ;	/* atan(+...,+INF) */
-		    case 1: return -zero  ;	/* atan(-...,+INF) */
-		    case 2: return  pi+tiny  ;	/* atan(+...,-INF) */
-		    case 3: return -pi-tiny  ;	/* atan(-...,-INF) */
-		}
-	    }
 	}
-    /* when y is INF */
+      if (yinf)
-	if(iy==0x7f800000) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny;
+	return pi2 * sgn[uy >> 31];
-
+    }
-    /* compute y/x */
+  if (__glibc_unlikely (ay == 0))
-	k = (iy-ix)>>23;
+    {
-	if(k > 60) z=pi_o_2+(float)0.5*pi_lo;	/* |y/x| >  2**60 */
+      if (__glibc_unlikely (!(ay | ax)))
-	else if(hx<0&&k<-60) z=0.0;	/* |y|/x < -2**60 */
+	{
-	else z=__atanf(fabsf(y/x));	/* safe to do y/x */
+	  uint32_t i = (uy >> 31) * 4 + (ux >> 31) * 2;
-	switch (m) {
+	  if (ux >> 31)
-	    case 0: return       z  ;	/* atan(+,+) */
+	    return off[i] + offl[i];
-	    case 1: {
+	  else
-		      uint32_t zh;
+	    return off[i];
-		      GET_FLOAT_WORD(zh,z);
-		      SET_FLOAT_WORD(z,zh ^ 0x80000000);
-		    }
-		    return       z  ;	/* atan(-,+) */
-	    case 2: return  pi-(z-pi_lo);/* atan(+,-) */
-	    default: /* case 3 */
-		    return  (z-pi_lo)-pi;/* atan(-,-) */
 	}
+      if (!(ux >> 31))
+	return 0.0f * sgn[uy >> 31];
+    }
+  uint32_t gt = ay > ax;
+  uint32_t i = (uy >> 31) * 4 + (ux >> 31) * 2 + gt;
+
+  double zx = x;
+  double zy = y;
+  double z = (m[gt] * zx + m[1 - gt] * zy) / (m[gt] * zy + m[1 - gt] * zx);
+  /* z = x/y if |y| > |x|, and z = y/x otherwise */
+  double r;
+  int d = (int) ax - (int) ay;
+  if (__glibc_likely (d < (27 << 23) && d > (-(27 << 23))))
+    {
+      double z2 = z * z, z4 = z2 * z2, z8 = z4 * z4;
+      /* z2 cannot underflow, since for |y|=0x1p-149 and |x|=0x1.fffffep+127
+	 we get |z| > 2^-277 thus z2 > 2^-554, but z4 and z8 might underflow,
+	 which might give spurious underflow exceptions. */
+      double cn0 = cn[0] + z2 * cn[1];
+      double cn2 = cn[2] + z2 * cn[3];
+      double cn4 = cn[4] + z2 * cn[5];
+      double cn6 = cn[6];
+      cn0 += z4 * cn2;
+      cn4 += z4 * cn6;
+      cn0 += z8 * cn4;
+      double cd0 = cd[0] + z2 * cd[1];
+      double cd2 = cd[2] + z2 * cd[3];
+      double cd4 = cd[4] + z2 * cd[5];
+      double cd6 = cd[6];
+      cd0 += z4 * cd2;
+      cd4 += z4 * cd6;
+      cd0 += z8 * cd4;
+      r = cn0 / cd0;
+    }
+  else
+    r = 1;
+  z *= sgn[gt];
+  r = z * r + off[i];
+  if (__glibc_unlikely (((asuint64 (r) + 8) & 0xfffffff) <= 16))
+    {
+      /* check tiny y/x */
+      if (ay < ax && ((ax - ay) >> 23 >= 25))
+	return cr_atan2f_tiny (y, x);
+      double zh;
+      double zl;
+      if (gt == 0)
+	{
+	  zh = zy / zx;
+	  zl = fma (zh, -zx, zy) / zx;
+	}
+      else
+	{
+	  zh = zx / zy;
+	  zl = fma (zh, -zy, zx) / zy;
+	}
+      double z2l;
+      double z2h = muldd (zh, zl, zh, zl, &z2l);
+      static const double c[32][2] =
+	{
+	  {  0x1p+0,                -0x1.8c1dac5492248p-87 },
+	  { -0x1.5555555555555p-2,  -0x1.55553bf3a2abep-56 },
+	  {  0x1.999999999999ap-3,  -0x1.99deed1ec9071p-57 },
+	  { -0x1.2492492492492p-3,  -0x1.fd99c8d18269ap-58 },
+	  {  0x1.c71c71c71c717p-4,  -0x1.651eee4c4d9dp-61 },
+	  { -0x1.745d1745d1649p-4,  -0x1.632683d6c44a6p-58 },
+	  {  0x1.3b13b13b11c63p-4,   0x1.bf69c1f8af41dp-58 },
+	  { -0x1.11111110e6338p-4,   0x1.3c3e431e8bb68p-61 },
+	  {  0x1.e1e1e1dc45c4ap-5,  -0x1.be2db05c77bbfp-59 },
+	  { -0x1.af286b8164b4fp-5,   0x1.a4673491f0942p-61 },
+	  {  0x1.86185e9ad4846p-5,   0x1.e12e32d79fceep-59 },
+	  { -0x1.642c6d5161faep-5,   0x1.3ce76c1ca03fp-59 },
+	  {  0x1.47ad6f277e5bfp-5,  -0x1.abd8d85bdb714p-60 },
+	  { -0x1.2f64a2ee8896dp-5,   0x1.ef87d4b615323p-61 },
+	  {  0x1.1a6a2b31741b5p-5,   0x1.a5d9d973547eep-62 },
+	  { -0x1.07fbdad65e0a6p-5,  -0x1.65ac07f5d35f4p-61 },
+	  {  0x1.ee9932a9a5f8bp-6,   0x1.f8b9623f6f55ap-61 },
+	  { -0x1.ce8b5b9584dc6p-6,   0x1.fe5af96e8ea2dp-61 },
+	  {  0x1.ac9cb288087b7p-6,  -0x1.450cdfceaf5cap-60 },
+	  { -0x1.84b025351f3e6p-6,   0x1.579561b0d73dap-61 },
+	  {  0x1.52f5b8ecdd52bp-6,   0x1.036bd2c6fba47p-60 },
+	  { -0x1.163a8c44909dcp-6,   0x1.18f735ffb9f16p-60 },
+	  {  0x1.a400dce3eea6fp-7,  -0x1.c90569c0c1b5cp-61 },
+	  { -0x1.1caa78ae6db3ap-7,  -0x1.4c60f8161ea09p-61 },
+	  {  0x1.52672453c0731p-8,   0x1.834efb598c338p-62 },
+	  { -0x1.5850c5be137cfp-9,  -0x1.445fc150ca7f5p-63 },
+	  {  0x1.23eb98d22e1cap-10, -0x1.388fbaf1d783p-64 },
+	  { -0x1.8f4e974a40741p-12,  0x1.271198a97da34p-66 },
+	  {  0x1.a5cf2e9cf76e5p-14, -0x1.887eb4a63b665p-68 },
+	  { -0x1.420c270719e32p-16,  0x1.efd595b27888bp-71 },
+	  {  0x1.3ba2d69b51677p-19, -0x1.4fb06829cdfc7p-73 },
+	  { -0x1.29b7e6f676385p-23, -0x1.a783b6de718fbp-77 }
+	};
+      double pl;
+      double ph = polydd (z2h, z2l, 32, c, &pl);
+      zh *= sgn[gt];
+      zl *= sgn[gt];
+      ph = muldd (zh, zl, ph, pl, &pl);
+      double sh = ph + off[i];
+      double sl = ((off[i] - sh) + ph) + pl + offl[i];
+      float rf = sh;
+      double th = rf;
+      double dh = sh - th;
+      double tm = dh + sl;
+      uint64_t tth = asuint64 (th);
+      if (th + th * 0x1p-60 == th - th * 0x1p-60)
+	{
+	  tth &= UINT64_C(0x7ff) << 52;
+	  tth -= UINT64_C(24) << 52;
+	  if (fabs (tm) > asdouble (tth))
+	    tm *= 1.25;
+	  else
+	    tm *= 0.75;
+	}
+      r = th + tm;
+    }
+  return r;
 }
 libm_alias_finite (__ieee754_atan2f, __atan2f)

sysdeps/loongarch/lp64/libm-test-ulps

@@ -70,22 +70,18 @@ double: 1
 ldouble: 1
 
 Function: "atan2":
-float: 2
 ldouble: 2
 
 Function: "atan2_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "atan2_upward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "atan_downward":
@@ -217,22 +213,18 @@ float: 2
 ldouble: 4
 
 Function: "carg":
-float: 1
 ldouble: 2
 
 Function: "carg_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "carg_upward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: Real part of "casin":

sysdeps/m68k/coldfire/fpu/libm-test-ulps

@@ -2,7 +2,6 @@
 
 # Maximal error of functions:
 Function: "atan2":
-float: 1
 
 Function: "atanh":
 float: 1

sysdeps/m68k/m680x0/fpu/libm-test-ulps

@@ -50,22 +50,18 @@ ldouble: 3
 
 Function: "atan2":
 double: 1
-float: 1
 ldouble: 1
 
 Function: "atan2_downward":
 double: 1
-float: 1
 ldouble: 2
 
 Function: "atan2_towardzero":
 double: 1
-float: 1
 ldouble: 2
 
 Function: "atan2_upward":
 double: 1
-float: 1
 ldouble: 2
 
 Function: "atan_downward":
@@ -191,22 +187,18 @@ float: 2
 ldouble: 3
 
 Function: "carg":
-float: 1
 ldouble: 1
 
 Function: "carg_downward":
 double: 1
-float: 1
 ldouble: 1
 
 Function: "carg_towardzero":
 double: 1
-float: 1
 ldouble: 1
 
 Function: "carg_upward":
 double: 1
-float: 1
 ldouble: 1
 
 Function: Real part of "casin":

sysdeps/microblaze/libm-test-ulps

@@ -14,7 +14,6 @@ double: 1
 Function: "atan":
 
 Function: "atan2":
-float: 1
 
 Function: "atanh":
 double: 2
@@ -40,7 +39,6 @@ double: 1
 float: 2
 
 Function: "carg":
-float: 1
 
 Function: Real part of "casin":
 double: 1

sysdeps/mips/mips32/libm-test-ulps

@@ -53,19 +53,15 @@ Function: "atan":
 double: 1
 
 Function: "atan2":
-float: 2
 
 Function: "atan2_downward":
 double: 1
-float: 2
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 
 Function: "atan2_upward":
 double: 1
-float: 2
 
 Function: "atan_downward":
 double: 1
@@ -169,19 +165,15 @@ double: 3
 float: 2
 
 Function: "carg":
-float: 1
 
 Function: "carg_downward":
 double: 1
-float: 2
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 
 Function: "carg_upward":
 double: 1
-float: 2
 
 Function: Real part of "casin":
 double: 1

sysdeps/mips/mips64/libm-test-ulps

@@ -70,22 +70,18 @@ double: 1
 ldouble: 1
 
 Function: "atan2":
-float: 2
 ldouble: 2
 
 Function: "atan2_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "atan2_upward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "atan_downward":
@@ -217,22 +213,18 @@ float: 2
 ldouble: 4
 
 Function: "carg":
-float: 1
 ldouble: 2
 
 Function: "carg_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "carg_upward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: Real part of "casin":

sysdeps/or1k/fpu/libm-test-ulps

@@ -53,19 +53,15 @@ Function: "atan":
 double: 1
 
 Function: "atan2":
-float: 2
 
 Function: "atan2_downward":
 double: 5
-float: 2
 
 Function: "atan2_towardzero":
 double: 5
-float: 2
 
 Function: "atan2_upward":
 double: 8
-float: 2
 
 Function: "atan_downward":
 double: 1
@@ -169,19 +165,15 @@ double: 3
 float: 2
 
 Function: "carg":
-float: 1
 
 Function: "carg_downward":
 double: 5
-float: 2
 
 Function: "carg_towardzero":
 double: 5
-float: 2
 
 Function: "carg_upward":
 double: 8
-float: 2
 
 Function: Real part of "casin":
 double: 1

sysdeps/or1k/nofpu/libm-test-ulps

@@ -61,19 +61,15 @@ Function: "atan":
 double: 1
 
 Function: "atan2":
-float: 2
 
 Function: "atan2_downward":
 double: 5
-float: 2
 
 Function: "atan2_towardzero":
 double: 5
-float: 2
 
 Function: "atan2_upward":
 double: 8
-float: 2
 
 Function: "atan2pi":
 double: 1
@@ -185,19 +181,15 @@ double: 3
 float: 2
 
 Function: "carg":
-float: 1
 
 Function: "carg_downward":
 double: 5
-float: 2
 
 Function: "carg_towardzero":
 double: 5
-float: 2
 
 Function: "carg_upward":
 double: 8
-float: 2
 
 Function: Real part of "casin":
 double: 1

sysdeps/powerpc/fpu/libm-test-ulps

@@ -139,25 +139,21 @@ float128: 1
 ldouble: 1
 
 Function: "atan2":
-float: 1
 float128: 2
 ldouble: 2
 
 Function: "atan2_downward":
 double: 1
-float: 2
 float128: 2
 ldouble: 5
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 float128: 3
 ldouble: 6
 
 Function: "atan2_upward":
 double: 1
-float: 1
 float128: 2
 ldouble: 3
 
@@ -366,25 +362,21 @@ ldouble: 8
 
 Function: "carg":
 double: 1
-float: 1
 float128: 2
 ldouble: 2
 
 Function: "carg_downward":
 double: 1
-float: 2
 float128: 2
 ldouble: 5
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 float128: 3
 ldouble: 6
 
 Function: "carg_upward":
 double: 1
-float: 1
 float128: 2
 ldouble: 3
 

sysdeps/powerpc/nofpu/libm-test-ulps

@@ -74,22 +74,18 @@ double: 1
 ldouble: 1
 
 Function: "atan2":
-float: 2
 ldouble: 2
 
 Function: "atan2_downward":
 double: 1
-float: 2
 ldouble: 5
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 ldouble: 6
 
 Function: "atan2_upward":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "atan_downward":
@@ -221,22 +217,18 @@ float: 2
 ldouble: 5
 
 Function: "carg":
-float: 1
 ldouble: 2
 
 Function: "carg_downward":
 double: 1
-float: 2
 ldouble: 5
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 ldouble: 6
 
 Function: "carg_upward":
 double: 1
-float: 2
 ldouble: 3
 
 Function: Real part of "casin":

sysdeps/riscv/nofpu/libm-test-ulps

@@ -70,22 +70,18 @@ double: 1
 ldouble: 1
 
 Function: "atan2":
-float: 2
 ldouble: 2
 
 Function: "atan2_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "atan2_upward":
 double: 1
-float: 1
 ldouble: 2
 
 Function: "atan_downward":
@@ -217,22 +213,18 @@ float: 2
 ldouble: 4
 
 Function: "carg":
-float: 1
 ldouble: 2
 
 Function: "carg_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "carg_upward":
 double: 1
-float: 1
 ldouble: 2
 
 Function: Real part of "casin":

sysdeps/riscv/rvd/libm-test-ulps

@@ -70,22 +70,18 @@ double: 1
 ldouble: 1
 
 Function: "atan2":
-float: 1
 ldouble: 2
 
 Function: "atan2_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "atan2_upward":
 double: 1
-float: 1
 ldouble: 2
 
 Function: "atan_downward":
@@ -217,22 +213,18 @@ float: 2
 ldouble: 4
 
 Function: "carg":
-float: 1
 ldouble: 2
 
 Function: "carg_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "carg_upward":
 double: 1
-float: 1
 ldouble: 2
 
 Function: Real part of "casin":

sysdeps/s390/fpu/libm-test-ulps

@@ -70,22 +70,18 @@ double: 1
 ldouble: 1
 
 Function: "atan2":
-float: 1
 ldouble: 2
 
 Function: "atan2_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "atan2_upward":
 double: 1
-float: 1
 ldouble: 2
 
 Function: "atan_downward":
@@ -217,22 +213,18 @@ float: 2
 ldouble: 4
 
 Function: "carg":
-float: 1
 ldouble: 2
 
 Function: "carg_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "carg_upward":
 double: 1
-float: 1
 ldouble: 2
 
 Function: Real part of "casin":

sysdeps/sh/libm-test-ulps

@@ -26,11 +26,9 @@ double: 2
 Function: "atan":
 
 Function: "atan2":
-float: 1
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 
 Function: "atan_towardzero":
 double: 1
@@ -82,11 +80,9 @@ double: 3
 float: 2
 
 Function: "carg":
-float: 1
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 
 Function: Real part of "casin":
 double: 1

sysdeps/sparc/fpu/libm-test-ulps

@@ -70,22 +70,18 @@ double: 1
 ldouble: 1
 
 Function: "atan2":
-float: 2
 ldouble: 2
 
 Function: "atan2_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "atan2_upward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "atan_downward":
@@ -217,22 +213,18 @@ float: 2
 ldouble: 4
 
 Function: "carg":
-float: 1
 ldouble: 2
 
 Function: "carg_downward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 ldouble: 3
 
 Function: "carg_upward":
 double: 1
-float: 2
 ldouble: 2
 
 Function: Real part of "casin":

sysdeps/x86_64/fpu/libm-test-ulps

@@ -212,25 +212,21 @@ float128: 1
 ldouble: 1
 
 Function: "atan2":
-float: 2
 float128: 2
 ldouble: 1
 
 Function: "atan2_downward":
 double: 1
-float: 2
 float128: 2
 ldouble: 1
 
 Function: "atan2_towardzero":
 double: 1
-float: 2
 float128: 3
 ldouble: 1
 
 Function: "atan2_upward":
 double: 1
-float: 2
 float128: 2
 ldouble: 1
 
@@ -498,25 +494,21 @@ float128: 4
 ldouble: 3
 
 Function: "carg":
-float: 1
 float128: 2
 ldouble: 1
 
 Function: "carg_downward":
 double: 1
-float: 2
 float128: 2
 ldouble: 1
 
 Function: "carg_towardzero":
 double: 1
-float: 2
 float128: 3
 ldouble: 1
 
 Function: "carg_upward":
 double: 1
-float: 2
 float128: 2
 ldouble: 1