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, 18 insertions, 51 deletions

diff --git a/src/math/atanhl.c b/src/math/atanhl.c
index 931bae32..b4c5e58b 100644
--- a/src/math/atanhl.c
+++ b/src/math/atanhl.c
@@ -1,31 +1,3 @@
-/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_atanh.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.
- * ====================================================
- */
-/* atanhl(x)
- * Method :
- *    1.Reduced x to positive by atanh(-x) = -atanh(x)
- *    2.For x>=0.5
- *                   1              2x                          x
- *      atanhl(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
- *                   2             1 - x                      1 - x
- *
- *      For x<0.5
- *      atanhl(x) = 0.5*log1pl(2x+2x*x/(1-x))
- *
- * Special cases:
- *      atanhl(x) is NaN if |x| > 1 with signal;
- *      atanhl(NaN) is that NaN with no signal;
- *      atanhl(+-1) is +-INF with signal.
- */
-
 #include "libm.h"
 
 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
@@ -34,31 +6,26 @@ long double atanhl(long double x)
 	return atanh(x);
 }
 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
-static const long double huge = 1e4900L;
-
+/* atanh(x) = log((1+x)/(1-x))/2 = log1p(2x/(1-x))/2 ~= x + x^3/3 + o(x^5) */
 long double atanhl(long double x)
 {
-	long double t;
-	int32_t ix;
-	uint32_t se,i0,i1;
+	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;
+
+	/* |x| */
+	u.i.se = e;
+	x = u.f;
 
-	GET_LDOUBLE_WORDS(se, i0, i1, x);
-	ix = se & 0x7fff;
-	if ((ix+((((i0&0x7fffffff)|i1)|(-((i0&0x7fffffff)|i1)))>>31)) > 0x3fff)
-		/* |x| > 1 */
-		return (x-x)/(x-x);
-	if (ix == 0x3fff)
-		return x/0.0;
-	if (ix < 0x3fe3 && huge+x > 0.0)  /* x < 2**-28 */
-		return x;
-	SET_LDOUBLE_EXP(x, ix);
-	if (ix < 0x3ffe) {  /* x < 0.5 */
-		t = x + x;
-		t = 0.5*log1pl(t + t*x/(1.0 - x));
-	} else
-		t = 0.5*log1pl((x + x)/(1.0 - x));
-	if (se <= 0x7fff)
-		return t;
-	return -t;
+	if (e < 0x3fff - 1) {
+		/* |x| < 0.5, up to 1.7ulp error */
+		x = 0.5*log1pl(2*x + 2*x*x/(1-x));
+	} else {
+		x = 0.5*log1pl(2*x/(1-x));
+	}
+	return s ? -x : x;
 }
 #endif