fix inaccuracy of j0f for x >= 2^127 when sin(x)+cos(x) is tiny

[Paul Zimmermann]

       Dear Andreas,

yes thanks. Sorry my english is not perfect.

Paul

>From 6b731f36b1a5badf4704645d0dda40957cedd0db Mon Sep 17 00:00:00 2001
From: Paul Zimmermann <Paul.Zimmermann@inria.fr>
Date: Mon, 27 Jul 2020 19:01:18 +0200
Subject: [PATCH 1/3] fix inaccuracy of j0f for x >= 2^127 when sin(x)+cos(x)
 is tiny

---
 sysdeps/ieee754/flt-32/e_j0f.c | 16 ++++++++++++++++
 1 file changed, 16 insertions(+)

diff --git a/sysdeps/ieee754/flt-32/e_j0f.c b/sysdeps/ieee754/flt-32/e_j0f.c
index c89b9f2688..f85d8a59e0 100644
--- a/sysdeps/ieee754/flt-32/e_j0f.c
+++ b/sysdeps/ieee754/flt-32/e_j0f.c
@@ -56,6 +56,22 @@ __ieee754_j0f(float x)
 		    if ((s*c)<zero) cc = z/ss;
 		    else	    ss = z/cc;
 		}
+                else {
+                  /* we subtract (exactly) a value x0 such that cos(x0)+sin(x0)
+                     is very near from 0, and use the identity
+                     sin(x-x0) = sin(x)*cos(x0)-cos(x)*sin(x0) to get
+                     sin(x) + cos(x) with extra accuracy */
+                   float x0 = 3.153646966e+38f;
+                   float y = x - x0; /* exact */
+                   /* sin(y) = sin(x)*cos(x0)-cos(x)*sin(x0) */
+                   z = __sinf (y);
+                   float eps = 8.17583368e-8f;
+                   /* cos(x0) ~ -sin(x0) + eps */
+                   z += eps * __cosf (x);
+                   /* now z ~ (sin(x)-cos(x))*cos(x0) */
+                   float cosx0 = -0.707106740f;
+                   cc = z / cosx0;
+                }
 	/*
 	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
-- 
2.27.0

>From 44124c42fe519c7dcac829160181ba0bb6c8751c Mon Sep 17 00:00:00 2001
From: Paul Zimmermann <Paul.Zimmermann@inria.fr>
Date: Tue, 28 Jul 2020 10:05:38 +0200
Subject: [PATCH 2/3] use hex float constants as advised by Joseph Myers

---
 sysdeps/ieee754/flt-32/e_j0f.c | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/sysdeps/ieee754/flt-32/e_j0f.c b/sysdeps/ieee754/flt-32/e_j0f.c
index f85d8a59e0..883a1a4d13 100644
--- a/sysdeps/ieee754/flt-32/e_j0f.c
+++ b/sysdeps/ieee754/flt-32/e_j0f.c
@@ -61,15 +61,15 @@ __ieee754_j0f(float x)
                      is very near from 0, and use the identity
                      sin(x-x0) = sin(x)*cos(x0)-cos(x)*sin(x0) to get
                      sin(x) + cos(x) with extra accuracy */
-                   float x0 = 3.153646966e+38f;
+                   float x0 = 0xe.d4108p+124f;
                    float y = x - x0; /* exact */
                    /* sin(y) = sin(x)*cos(x0)-cos(x)*sin(x0) */
                    z = __sinf (y);
-                   float eps = 8.17583368e-8f;
+                   float eps = 0x1.5f263ep-24f;
                    /* cos(x0) ~ -sin(x0) + eps */
                    z += eps * __cosf (x);
                    /* now z ~ (sin(x)-cos(x))*cos(x0) */
-                   float cosx0 = -0.707106740f;
+                   float cosx0 = -0xb.504f3p-4f;
                    cc = z / cosx0;
                 }
 	/*
-- 
2.27.0

>From 409781259c5d150ec62079e608ea94b520eafa58 Mon Sep 17 00:00:00 2001
From: Paul Zimmermann <Paul.Zimmermann@inria.fr>
Date: Tue, 28 Jul 2020 12:47:31 +0200
Subject: [PATCH 3/3] fixed typo (thanks Andreas Schwab)

---
 sysdeps/ieee754/flt-32/e_j0f.c | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/sysdeps/ieee754/flt-32/e_j0f.c b/sysdeps/ieee754/flt-32/e_j0f.c
index 883a1a4d13..66249d2812 100644
--- a/sysdeps/ieee754/flt-32/e_j0f.c
+++ b/sysdeps/ieee754/flt-32/e_j0f.c
@@ -58,7 +58,7 @@ __ieee754_j0f(float x)
 		}
                 else {
                   /* we subtract (exactly) a value x0 such that cos(x0)+sin(x0)
-                     is very near from 0, and use the identity
+                     is very near to 0, and use the identity
                      sin(x-x0) = sin(x)*cos(x0)-cos(x)*sin(x0) to get
                      sin(x) + cos(x) with extra accuracy */
                    float x0 = 0xe.d4108p+124f;
-- 
2.27.0

> From: Andreas Schwab <schwab@linux-m68k.org>
> Date: Tue, 28 Jul 2020 11:19:28 +0200
> 
> On Jul 28 2020, Paul Zimmermann wrote:
> 
> > +                  /* we subtract (exactly) a value x0 such that cos(x0)+sin(x0)
> > +                     is very near from 0, and use the identity
> 
> Did you mean "near to"?
> 
> Andreas.
> 
> -- 
> Andreas Schwab, schwab@linux-m68k.org
> GPG Key fingerprint = 7578 EB47 D4E5 4D69 2510  2552 DF73 E780 A9DA AEC1
> "And now for something completely different."