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]

9 files changed, 149 insertions, 406 deletions

diff --git a/src/math/acosh.c b/src/math/acosh.c
index 15f51c6e..4ce9b3d1 100644
--- a/src/math/acosh.c
+++ b/src/math/acosh.c
@@ -1,54 +1,19 @@
-/* origin: FreeBSD /usr/src/lib/msun/src/e_acosh.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.
- * ====================================================
- *
- */
-/* acosh(x)
- * Method :
- *      Based on
- *              acosh(x) = log [ x + sqrt(x*x-1) ]
- *      we have
- *              acosh(x) := log(x)+ln2, if x is large; else
- *              acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
- *              acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
- *
- * Special cases:
- *      acosh(x) is NaN with signal if x<1.
- *      acosh(NaN) is NaN without signal.
- */
-
 #include "libm.h"
 
-static const double
-ln2 = 6.93147180559945286227e-01; /* 0x3FE62E42, 0xFEFA39EF */
-
+/* acosh(x) = log(x + sqrt(x*x-1)) */
 double acosh(double x)
 {
-	double t;
-	int32_t hx;
-	uint32_t lx;
+	union {double f; uint64_t i;} u = {.f = x};
+	unsigned e = u.i >> 52 & 0x7ff;
+
+	/* x < 1 domain error is handled in the called functions */
 
-	EXTRACT_WORDS(hx, lx, x);
-	if (hx < 0x3ff00000) {  /* x < 1 */
-		return (x-x)/(x-x);
-	} else if (hx >= 0x41b00000) {  /* x > 2**28 */
-		if (hx >= 0x7ff00000)  /* x is inf of NaN */
-			return x+x;
-		return log(x) + ln2;   /* acosh(huge) = log(2x) */
-	} else if ((hx-0x3ff00000 | lx) == 0) {
-		return 0.0;            /* acosh(1) = 0 */
-	} else if (hx > 0x40000000) {  /* 2**28 > x > 2 */
-		t = x*x;
-		return log(2.0*x - 1.0/(x+sqrt(t-1.0)));
-	} else {                /* 1 < x < 2 */
-		t = x-1.0;
-		return log1p(t + sqrt(2.0*t+t*t));
-	}
+	if (e < 0x3ff + 1)
+		/* |x| < 2, up to 2ulp error in [1,1.125] */
+		return log1p(x-1 + sqrt((x-1)*(x-1)+2*(x-1)));
+	if (e < 0x3ff + 26)
+		/* |x| < 0x1p26 */
+		return log(2*x - 1/(x+sqrt(x*x-1)));
+	/* |x| >= 0x1p26 or nan */
+	return log(x) + 0.693147180559945309417232121458176568;
 }
diff --git a/src/math/acoshf.c b/src/math/acoshf.c
index 0f7aae2a..4596085e 100644
--- a/src/math/acoshf.c
+++ b/src/math/acoshf.c
@@ -1,42 +1,17 @@
-/* origin: FreeBSD /usr/src/lib/msun/src/e_acoshf.c */
-/*
- * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
- */
-/*
- * ====================================================
- * 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.
- * ====================================================
- */
-
 #include "libm.h"
 
-static const float
-ln2 = 6.9314718246e-01; /* 0x3f317218 */
-
+/* acosh(x) = log(x + sqrt(x*x-1)) */
 float acoshf(float x)
 {
-	float t;
-	int32_t hx;
+	union {float f; int32_t i;} u = {.f = x};
 
-	GET_FLOAT_WORD(hx, x);
-	if (hx < 0x3f800000) {  /* x < 1 */
-		return (x-x)/(x-x);
-	} else if (hx >= 0x4d800000) {  /* x > 2**28 */
-		if (hx >= 0x7f800000)  /* x is inf of NaN */
-			return x + x;
-		return logf(x) + ln2;  /* acosh(huge)=log(2x) */
-	} else if (hx == 0x3f800000) {
-		return 0.0f;  /* acosh(1) = 0 */
-	} else if (hx > 0x40000000) {  /* 2**28 > x > 2 */
-		t = x*x;
-		return logf(2.0f*x - 1.0f/(x+sqrtf(t-1.0f)));
-	} else {                /* 1 < x < 2 */
-		t = x-1.0f;
-		return log1pf(t + sqrtf(2.0f*t+t*t));
-	}
+	if (u.i < 0x3f800000+(1<<23))
+		/* x < 2, invalid if x < 1 or nan */
+		/* up to 2ulp error in [1,1.125] */
+		return log1pf(x-1 + sqrtf((x-1)*(x-1)+2*(x-1)));
+	if (u.i < 0x3f800000+(12<<23))
+		/* x < 0x1p12 */
+		return logf(2*x - 1/(x+sqrtf(x*x-1)));
+	/* x >= 0x1p12 */
+	return logf(x) + 0.693147180559945309417232121458176568f;
 }
diff --git a/src/math/acoshl.c b/src/math/acoshl.c
index a4024516..472c71cb 100644
--- a/src/math/acoshl.c
+++ b/src/math/acoshl.c
@@ -1,28 +1,3 @@
-/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_acoshl.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.
- * ====================================================
- */
-/* acoshl(x)
- * Method :
- *      Based on
- *              acoshl(x) = logl [ x + sqrtl(x*x-1) ]
- *      we have
- *              acoshl(x) := logl(x)+ln2,       if x is large; else
- *              acoshl(x) := logl(2x-1/(sqrtl(x*x-1)+x)) if x>2; else
- *              acoshl(x) := log1pl(t+sqrtl(2.0*t+t*t)); where t=x-1.
- *
- * Special cases:
- *      acoshl(x) is NaN with signal if x<1.
- *      acoshl(NaN) is NaN without signal.
- */
-
 #include "libm.h"
 
 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
@@ -31,29 +6,20 @@ long double acoshl(long double x)
 	return acosh(x);
 }
 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
-static const long double
-ln2 = 6.931471805599453094287e-01L; /* 0x3FFE, 0xB17217F7, 0xD1CF79AC */
-
+/* acosh(x) = log(x + sqrt(x*x-1)) */
 long double acoshl(long double x)
 {
-	long double t;
-	uint32_t se,i0,i1;
+	union {
+		long double f;
+		struct{uint64_t m; int16_t se; uint16_t pad;} i;
+	} u = {.f = x};
 
-	GET_LDOUBLE_WORDS(se, i0, i1, x);
-	if (se < 0x3fff || se & 0x8000) {  /* x < 1 */
-		return (x-x)/(x-x);
-	} else if (se >= 0x401d) {  /* x > 2**30 */
-		if (se >= 0x7fff)  /* x is inf or NaN */
-			return x+x;
-		return logl(x) + ln2;  /* acoshl(huge) = logl(2x) */
-	} else if (((se-0x3fff)|i0|i1) == 0) {
-		return 0.0;            /* acosh(1) = 0 */
-	} else if (se > 0x4000) {  /* x > 2 */
-		t = x*x;
-		return logl(2.0*x - 1.0/(x + sqrtl(t - 1.0)));
-	}
-	/* 1 < x <= 2 */
-	t = x - 1.0;
-	return log1pl(t + sqrtl(2.0*t + t*t));
+	if (u.i.se < 0x3fff + 1)
+		/* x < 2, invalid if x < 1 or nan */
+		return log1pl(x-1 + sqrtl((x-1)*(x-1)+2*(x-1)));
+	if (u.i.se < 0x3fff + 32)
+		/* x < 0x1p32 */
+		return logl(2*x - 1/(x+sqrtl(x*x-1)));
+	return logl(x) + 0.693147180559945309417232121458176568L;
 }
 #endif
diff --git a/src/math/asinh.c b/src/math/asinh.c
index 11bbd71a..4152dc39 100644
--- a/src/math/asinh.c
+++ b/src/math/asinh.c
@@ -1,55 +1,28 @@
-/* origin: FreeBSD /usr/src/lib/msun/src/s_asinh.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.
- * ====================================================
- */
-/* asinh(x)
- * Method :
- *      Based on
- *              asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
- *      we have
- *      asinh(x) := x  if  1+x*x=1,
- *               := sign(x)*(log(x)+ln2)) for large |x|, else
- *               := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
- *               := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
- */
-
 #include "libm.h"
 
-static const double
-ln2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
-huge= 1.00000000000000000000e+300;
-
+/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */
 double asinh(double x)
 {
-	double t,w;
-	int32_t hx,ix;
+	union {double f; uint64_t i;} u = {.f = x};
+	unsigned e = u.i >> 52 & 0x7ff;
+	unsigned s = u.i >> 63;
 
-	GET_HIGH_WORD(hx, x);
-	ix = hx & 0x7fffffff;
-	if (ix >= 0x7ff00000)   /* x is inf or NaN */
-		return x+x;
-	if (ix < 0x3e300000) {  /* |x| < 2**-28 */
-		/* return x inexact except 0 */
-		if (huge+x > 1.0)
-			return x;
-	}
-	if (ix > 0x41b00000) {  /* |x| > 2**28 */
-		w = log(fabs(x)) + ln2;
-	} else if (ix > 0x40000000) {  /* 2**28 > |x| > 2.0 */
-		t = fabs(x);
-		w = log(2.0*t + 1.0/(sqrt(x*x+1.0)+t));
-	} else {                /* 2.0 > |x| > 2**-28 */
-		t = x*x;
-		w =log1p(fabs(x) + t/(1.0+sqrt(1.0+t)));
+	/* |x| */
+	u.i &= (uint64_t)-1/2;
+	x = u.f;
+
+	if (e >= 0x3ff + 26) {
+		/* |x| >= 0x1p26 or inf or nan */
+		x = log(x) + 0.693147180559945309417232121458176568;
+	} else if (e >= 0x3ff + 1) {
+		/* |x| >= 2 */
+		x = log(2*x + 1/(sqrt(x*x+1)+x));
+	} else if (e >= 0x3ff - 26) {
+		/* |x| >= 0x1p-26, up to 1.6ulp error in [0.125,0.5] */
+		x = log1p(x + x*x/(sqrt(x*x+1)+1));
+	} else {
+		/* |x| < 0x1p-26, raise inexact if x != 0 */
+		FORCE_EVAL(x + 0x1p1000);
 	}
-	if (hx > 0)
-		return w;
-	return -w;
+	return s ? -x : x;
 }
diff --git a/src/math/asinhf.c b/src/math/asinhf.c
index efe3af94..fc9f0911 100644
--- a/src/math/asinhf.c
+++ b/src/math/asinhf.c
@@ -1,48 +1,28 @@
-/* origin: FreeBSD /usr/src/lib/msun/src/s_asinhf.c */
-/*
- * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
- */
-/*
- * ====================================================
- * 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.
- * ====================================================
- */
-
 #include "libm.h"
 
-static const float
-ln2 = 6.9314718246e-01, /* 0x3f317218 */
-huge= 1.0000000000e+30;
-
+/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */
 float asinhf(float x)
 {
-	float t,w;
-	int32_t hx,ix;
+	union {float f; uint32_t i;} u = {.f = x};
+	uint32_t i = u.i & 0x7fffffff;
+	unsigned s = u.i >> 31;
 
-	GET_FLOAT_WORD(hx, x);
-	ix = hx & 0x7fffffff;
-	if (ix >= 0x7f800000)   /* x is inf or NaN */
-		return x+x;
-	if (ix < 0x31800000) {  /* |x| < 2**-28 */
-		/* return x inexact except 0 */
-		if (huge+x > 1.0f)
-			return x;
-	}
-	if (ix > 0x4d800000) {  /* |x| > 2**28 */
-		w = logf(fabsf(x)) + ln2;
-	} else if (ix > 0x40000000) {  /* 2**28 > |x| > 2.0 */
-		t = fabsf(x);
-		w = logf(2.0f*t + 1.0f/(sqrtf(x*x+1.0f)+t));
-	} else {                /* 2.0 > |x| > 2**-28 */
-		t = x*x;
-		w =log1pf(fabsf(x) + t/(1.0f+sqrtf(1.0f+t)));
+	/* |x| */
+	u.i = i;
+	x = u.f;
+
+	if (i >= 0x3f800000 + (12<<23)) {
+		/* |x| >= 0x1p12 or inf or nan */
+		x = logf(x) + 0.693147180559945309417232121458176568f;
+	} else if (i >= 0x3f800000 + (1<<23)) {
+		/* |x| >= 2 */
+		x = logf(2*x + 1/(sqrtf(x*x+1)+x));
+	} else if (i >= 0x3f800000 - (12<<23)) {
+		/* |x| >= 0x1p-12, up to 1.6ulp error in [0.125,0.5] */
+		x = log1pf(x + x*x/(sqrtf(x*x+1)+1));
+	} else {
+		/* |x| < 0x1p-12, raise inexact if x!=0 */
+		FORCE_EVAL(x + 0x1p120f);
 	}
-	if (hx > 0)
-		return w;
-	return -w;
+	return s ? -x : x;
 }
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
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;
 }
diff --git a/src/math/atanhf.c b/src/math/atanhf.c
index 2be780bb..ca106efc 100644
--- a/src/math/atanhf.c
+++ b/src/math/atanhf.c
@@ -1,42 +1,20 @@
-/* origin: FreeBSD /usr/src/lib/msun/src/e_atanhf.c */
-/*
- * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
- */
-/*
- * ====================================================
- * 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.
- * ====================================================
- */
-
 #include "libm.h"
 
-static const float huge = 1e30;
-
+/* atanh(x) = log((1+x)/(1-x))/2 = log1p(2x/(1-x))/2 ~= x + x^3/3 + o(x^5) */
 float atanhf(float x)
 {
-	float t;
-	int32_t hx,ix;
+	union {float f; uint32_t i;} u = {.f = x};
+	unsigned s = u.i >> 31;
+
+	/* |x| */
+	u.i &= 0x7fffffff;
+	x = u.f;
 
-	GET_FLOAT_WORD(hx, x);
-	ix = hx & 0x7fffffff;
-	if (ix > 0x3f800000)                   /* |x| > 1 */
-		return (x-x)/(x-x);
-	if (ix == 0x3f800000)
-		return x/0.0f;
-	if (ix < 0x31800000 && huge+x > 0.0f)  /* x < 2**-28 */
-		return x;
-	SET_FLOAT_WORD(x, ix);
-	if (ix < 0x3f000000) {                 /* x < 0.5 */
-		t = x+x;
-		t = 0.5f*log1pf(t + t*x/(1.0f-x));
-	} else
-		t = 0.5f*log1pf((x+x)/(1.0f-x));
-	if (hx >= 0)
-		return t;
-	return -t;
+	if (u.i < 0x3f800000 - (1<<23)) {
+		/* |x| < 0.5, up to 1.7ulp error */
+		x = 0.5f*log1pf(2*x + 2*x*x/(1-x));
+	} else {
+		x = 0.5f*log1pf(2*x/(1-x));
+	}
+	return s ? -x : x;
 }
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