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]

1 files changed, 15 insertions, 52 deletions

diff --git a/src/math/atanh.c b/src/math/atanh.c
index dbe241d1..84a84c69 100644
--- a/src/math/atanh.c
+++ b/src/math/atanh.c
@@ -1,58 +1,21 @@
-/* origin: FreeBSD /usr/src/lib/msun/src/e_atanh.c */
-/*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunSoft, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- *
- */
-/* atanh(x)
- * Method :
- *    1.Reduced x to positive by atanh(-x) = -atanh(x)
- *    2.For x>=0.5
- *                  1              2x                          x
- *      atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
- *                  2             1 - x                      1 - x
- *
- *      For x<0.5
- *      atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
- *
- * Special cases:
- *      atanh(x) is NaN if |x| > 1 with signal;
- *      atanh(NaN) is that NaN with no signal;
- *      atanh(+-1) is +-INF with signal.
- *
- */
-
 #include "libm.h"
 
-static const double huge = 1e300;
-
+/* atanh(x) = log((1+x)/(1-x))/2 = log1p(2x/(1-x))/2 ~= x + x^3/3 + o(x^5) */
 double atanh(double x)
 {
-	double t;
-	int32_t hx,ix;
-	uint32_t lx;
+	union {double f; uint64_t i;} u = {.f = x};
+	unsigned e = u.i >> 52 & 0x7ff;
+	unsigned s = u.i >> 63;
+
+	/* |x| */
+	u.i &= (uint64_t)-1/2;
+	x = u.f;
 
-	EXTRACT_WORDS(hx, lx, x);
-	ix = hx & 0x7fffffff;
-	if ((ix | ((lx|-lx)>>31)) > 0x3ff00000)  /* |x| > 1 */
-		return (x-x)/(x-x);
-	if (ix == 0x3ff00000)
-		return x/0.0;
-	if (ix < 0x3e300000 && (huge+x) > 0.0)   /* x < 2**-28 */
-		return x;
-	SET_HIGH_WORD(x, ix);
-	if (ix < 0x3fe00000) {                   /* x < 0.5 */
-		t = x+x;
-		t = 0.5*log1p(t + t*x/(1.0-x));
-	} else
-		t = 0.5*log1p((x+x)/(1.0-x));
-	if (hx >= 0)
-		return t;
-	return -t;
+	if (e < 0x3ff - 1) {
+		/* |x| < 0.5, up to 1.7ulp error */
+		x = 0.5*log1p(2*x + 2*x*x/(1-x));
+	} else {
+		x = 0.5*log1p(2*x/(1-x));
+	}
+	return s ? -x : x;
 }