From 482ccd2f7497a79ca83e998f54e823e7cedaaa6e Mon Sep 17 00:00:00 2001
From: Szabolcs Nagy <nsz@port70.net>
Date: Tue, 11 Dec 2012 23:06:20 +0100
Subject: math: rewrite inverse hyperbolic functions to be simpler/smaller

modifications:
* avoid unsigned->signed integer conversion
* do not handle special cases when they work correctly anyway
* more strict threshold values (0x1p26 instead of 0x1p28 etc)
* smaller code, cleaner branching logic
* same precision as the old code:
    acosh(x) has up to 2ulp error in [1,1.125]
    asinh(x) has up to 1.6ulp error in [0.125,0.5], [-0.5,-0.125]
    atanh(x) has up to 1.7ulp error in [0.125,0.5], [-0.5,-0.125]
---
 src/math/asinhl.c | 72 +++++++++++++++++++------------------------------------
 1 file changed, 24 insertions(+), 48 deletions(-)

(limited to 'src/math/asinhl.c')

diff --git a/src/math/asinhl.c b/src/math/asinhl.c
index dc5dd71f..db966246 100644
--- a/src/math/asinhl.c
+++ b/src/math/asinhl.c
@@ -1,25 +1,3 @@
-/* origin: OpenBSD /usr/src/lib/libm/src/ld80/s_asinhl.c */
-/*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunPro, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-/* asinhl(x)
- * Method :
- *      Based on
- *              asinhl(x) = signl(x) * logl [ |x| + sqrtl(x*x+1) ]
- *      we have
- *      asinhl(x) := x  if  1+x*x=1,
- *                := signl(x)*(logl(x)+ln2)) for large |x|, else
- *                := signl(x)*logl(2|x|+1/(|x|+sqrtl(x*x+1))) if|x|>2, else
- *                := signl(x)*log1pl(|x| + x^2/(1 + sqrtl(1+x^2)))
- */
-
 #include "libm.h"
 
 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
@@ -28,35 +6,33 @@ long double asinhl(long double x)
 	return asinh(x);
 }
 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
-static const long double
-ln2  = 6.931471805599453094287e-01L, /* 0x3FFE, 0xB17217F7, 0xD1CF79AC */
-huge = 1.000000000000000000e+4900L;
-
+/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */
 long double asinhl(long double x)
 {
-	long double t,w;
-	int32_t hx,ix;
+	union {
+		long double f;
+		struct{uint64_t m; uint16_t se; uint16_t pad;} i;
+	} u = {.f = x};
+	unsigned e = u.i.se & 0x7fff;
+	unsigned s = u.i.se >> 15;
 
-	GET_LDOUBLE_EXP(hx, x);
-	ix = hx & 0x7fff;
-	if (ix == 0x7fff)
-		return x + x;   /* x is inf or NaN */
-	if (ix < 0x3fde) {      /* |x| < 2**-34 */
-		/* return x, raise inexact if x != 0 */
-		if (huge+x > 1.0)
-			return x;
-	}
-	if (ix > 0x4020) {      /* |x| > 2**34 */
-		w = logl(fabsl(x)) + ln2;
-	} else if (ix > 0x4000) { /* 2**34 > |x| > 2.0 */
-		t = fabsl(x);
-		w = logl(2.0*t + 1.0/(sqrtl(x*x + 1.0) + t));
-	} else {                /* 2.0 > |x| > 2**-28 */
-		t = x*x;
-		w =log1pl(fabsl(x) + t/(1.0 + sqrtl(1.0 + t)));
+	/* |x| */
+	u.i.se = e;
+	x = u.f;
+
+	if (e >= 0x3fff + 32) {
+		/* |x| >= 0x1p32 or inf or nan */
+		x = logl(x) + 0.693147180559945309417232121458176568L;
+	} else if (e >= 0x3fff + 1) {
+		/* |x| >= 2 */
+		x = logl(2*x + 1/(sqrtl(x*x+1)+x));
+	} else if (e >= 0x3fff - 32) {
+		/* |x| >= 0x1p-32 */
+		x = log1pl(x + x*x/(sqrtl(x*x+1)+1));
+	} else {
+		/* |x| < 0x1p-32, raise inexact if x!=0 */
+		FORCE_EVAL(x + 0x1p1000);
 	}
-	if (hx & 0x8000)
-		return -w;
-	return w;
+	return s ? -x : x;
 }
 #endif
-- 
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