From 359d84fb9e0a22ae9d6def2bc1a002542354c50c Mon Sep 17 00:00:00 2001
From: Raymond Toy <toy.raymond@gmail.com>
Date: Sat, 2 Aug 2014 08:35:39 -0700
Subject: [PATCH] Import inverse trig functions from fdlibm, as is.

---
 src/lisp/e_acos.c | 105 ++++++++++++++++++++++++++++++++++++
 src/lisp/e_asin.c | 114 +++++++++++++++++++++++++++++++++++++++
 src/lisp/s_atan.c | 134 ++++++++++++++++++++++++++++++++++++++++++++++
 3 files changed, 353 insertions(+)
 create mode 100644 src/lisp/e_acos.c
 create mode 100644 src/lisp/e_asin.c
 create mode 100644 src/lisp/s_atan.c

diff --git a/src/lisp/e_acos.c b/src/lisp/e_acos.c
new file mode 100644
index 000000000..d7c9ed225
--- /dev/null
+++ b/src/lisp/e_acos.c
@@ -0,0 +1,105 @@
+
+/* @(#)e_acos.c 1.3 95/01/18 */
+/*
+ * ====================================================
+ * 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.
+ * ====================================================
+ */
+
+/* __ieee754_acos(x)
+ * Method :                  
+ *	acos(x)  = pi/2 - asin(x)
+ *	acos(-x) = pi/2 + asin(x)
+ * For |x|<=0.5
+ *	acos(x) = pi/2 - (x + x*x^2*R(x^2))	(see asin.c)
+ * For x>0.5
+ * 	acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
+ *		= 2asin(sqrt((1-x)/2))  
+ *		= 2s + 2s*z*R(z) 	...z=(1-x)/2, s=sqrt(z)
+ *		= 2f + (2c + 2s*z*R(z))
+ *     where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
+ *     for f so that f+c ~ sqrt(z).
+ * For x<-0.5
+ *	acos(x) = pi - 2asin(sqrt((1-|x|)/2))
+ *		= pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
+ *
+ * Special cases:
+ *	if x is NaN, return x itself;
+ *	if |x|>1, return NaN with invalid signal.
+ *
+ * Function needed: sqrt
+ */
+
+#include "fdlibm.h"
+
+#ifdef __STDC__
+static const double 
+#else
+static double 
+#endif
+one=  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
+pi =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
+pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
+pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
+pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
+pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
+pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
+pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
+pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
+pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
+qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
+qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
+qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
+qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
+
+#ifdef __STDC__
+	double __ieee754_acos(double x)
+#else
+	double __ieee754_acos(x)
+	double x;
+#endif
+{
+	double z,p,q,r,w,s,c,df;
+	int hx,ix;
+	hx = __HI(x);
+	ix = hx&0x7fffffff;
+	if(ix>=0x3ff00000) {	/* |x| >= 1 */
+	    if(((ix-0x3ff00000)|__LO(x))==0) {	/* |x|==1 */
+		if(hx>0) return 0.0;		/* acos(1) = 0  */
+		else return pi+2.0*pio2_lo;	/* acos(-1)= pi */
+	    }
+	    return (x-x)/(x-x);		/* acos(|x|>1) is NaN */
+	}
+	if(ix<0x3fe00000) {	/* |x| < 0.5 */
+	    if(ix<=0x3c600000) return pio2_hi+pio2_lo;/*if|x|<2**-57*/
+	    z = x*x;
+	    p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
+	    q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
+	    r = p/q;
+	    return pio2_hi - (x - (pio2_lo-x*r));
+	} else  if (hx<0) {		/* x < -0.5 */
+	    z = (one+x)*0.5;
+	    p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
+	    q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
+	    s = sqrt(z);
+	    r = p/q;
+	    w = r*s-pio2_lo;
+	    return pi - 2.0*(s+w);
+	} else {			/* x > 0.5 */
+	    z = (one-x)*0.5;
+	    s = sqrt(z);
+	    df = s;
+	    __LO(df) = 0;
+	    c  = (z-df*df)/(s+df);
+	    p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
+	    q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
+	    r = p/q;
+	    w = r*s+c;
+	    return 2.0*(df+w);
+	}
+}
diff --git a/src/lisp/e_asin.c b/src/lisp/e_asin.c
new file mode 100644
index 000000000..8e37e22ad
--- /dev/null
+++ b/src/lisp/e_asin.c
@@ -0,0 +1,114 @@
+
+/* @(#)e_asin.c 1.3 95/01/18 */
+/*
+ * ====================================================
+ * 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.
+ * ====================================================
+ */
+
+/* __ieee754_asin(x)
+ * Method :                  
+ *	Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
+ *	we approximate asin(x) on [0,0.5] by
+ *		asin(x) = x + x*x^2*R(x^2)
+ *	where
+ *		R(x^2) is a rational approximation of (asin(x)-x)/x^3 
+ *	and its remez error is bounded by
+ *		|(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
+ *
+ *	For x in [0.5,1]
+ *		asin(x) = pi/2-2*asin(sqrt((1-x)/2))
+ *	Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
+ *	then for x>0.98
+ *		asin(x) = pi/2 - 2*(s+s*z*R(z))
+ *			= pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
+ *	For x<=0.98, let pio4_hi = pio2_hi/2, then
+ *		f = hi part of s;
+ *		c = sqrt(z) - f = (z-f*f)/(s+f) 	...f+c=sqrt(z)
+ *	and
+ *		asin(x) = pi/2 - 2*(s+s*z*R(z))
+ *			= pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
+ *			= pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
+ *
+ * Special cases:
+ *	if x is NaN, return x itself;
+ *	if |x|>1, return NaN with invalid signal.
+ *
+ */
+
+
+#include "fdlibm.h"
+
+#ifdef __STDC__
+static const double 
+#else
+static double 
+#endif
+one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
+huge =  1.000e+300,
+pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
+pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
+pio4_hi =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
+	/* coefficient for R(x^2) */
+pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
+pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
+pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
+pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
+pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
+pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
+qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
+qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
+qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
+qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
+
+#ifdef __STDC__
+	double __ieee754_asin(double x)
+#else
+	double __ieee754_asin(x)
+	double x;
+#endif
+{
+	double t,w,p,q,c,r,s;
+	int hx,ix;
+	hx = __HI(x);
+	ix = hx&0x7fffffff;
+	if(ix>= 0x3ff00000) {		/* |x|>= 1 */
+	    if(((ix-0x3ff00000)|__LO(x))==0)
+		    /* asin(1)=+-pi/2 with inexact */
+		return x*pio2_hi+x*pio2_lo;	
+	    return (x-x)/(x-x);		/* asin(|x|>1) is NaN */   
+	} else if (ix<0x3fe00000) {	/* |x|<0.5 */
+	    if(ix<0x3e400000) {		/* if |x| < 2**-27 */
+		if(huge+x>one) return x;/* return x with inexact if x!=0*/
+	    } else 
+		t = x*x;
+		p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
+		q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
+		w = p/q;
+		return x+x*w;
+	}
+	/* 1> |x|>= 0.5 */
+	w = one-fabs(x);
+	t = w*0.5;
+	p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
+	q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
+	s = sqrt(t);
+	if(ix>=0x3FEF3333) { 	/* if |x| > 0.975 */
+	    w = p/q;
+	    t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
+	} else {
+	    w  = s;
+	    __LO(w) = 0;
+	    c  = (t-w*w)/(s+w);
+	    r  = p/q;
+	    p  = 2.0*s*r-(pio2_lo-2.0*c);
+	    q  = pio4_hi-2.0*w;
+	    t  = pio4_hi-(p-q);
+	}    
+	if(hx>0) return t; else return -t;    
+}
diff --git a/src/lisp/s_atan.c b/src/lisp/s_atan.c
new file mode 100644
index 000000000..0093eafdf
--- /dev/null
+++ b/src/lisp/s_atan.c
@@ -0,0 +1,134 @@
+
+/* @(#)s_atan.c 1.3 95/01/18 */
+/*
+ * ====================================================
+ * 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.
+ * ====================================================
+ *
+ */
+
+/* atan(x)
+ * Method
+ *   1. Reduce x to positive by atan(x) = -atan(-x).
+ *   2. According to the integer k=4t+0.25 chopped, t=x, the argument
+ *      is further reduced to one of the following intervals and the
+ *      arctangent of t is evaluated by the corresponding formula:
+ *
+ *      [0,7/16]      atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
+ *      [7/16,11/16]  atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
+ *      [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
+ *      [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
+ *      [39/16,INF]   atan(x) = atan(INF) + atan( -1/t )
+ *
+ * Constants:
+ * The hexadecimal values are the intended ones for the following 
+ * constants. The decimal values may be used, provided that the 
+ * compiler will convert from decimal to binary accurately enough 
+ * to produce the hexadecimal values shown.
+ */
+
+#include "fdlibm.h"
+
+#ifdef __STDC__
+static const double atanhi[] = {
+#else
+static double atanhi[] = {
+#endif
+  4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
+  7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
+  9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
+  1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
+};
+
+#ifdef __STDC__
+static const double atanlo[] = {
+#else
+static double atanlo[] = {
+#endif
+  2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
+  3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
+  1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
+  6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
+};
+
+#ifdef __STDC__
+static const double aT[] = {
+#else
+static double aT[] = {
+#endif
+  3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
+ -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
+  1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
+ -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
+  9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
+ -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
+  6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
+ -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
+  4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
+ -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
+  1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
+};
+
+#ifdef __STDC__
+	static const double 
+#else
+	static double 
+#endif
+one   = 1.0,
+huge   = 1.0e300;
+
+#ifdef __STDC__
+	double atan(double x)
+#else
+	double atan(x)
+	double x;
+#endif
+{
+	double w,s1,s2,z;
+	int ix,hx,id;
+
+	hx = __HI(x);
+	ix = hx&0x7fffffff;
+	if(ix>=0x44100000) {	/* if |x| >= 2^66 */
+	    if(ix>0x7ff00000||
+		(ix==0x7ff00000&&(__LO(x)!=0)))
+		return x+x;		/* NaN */
+	    if(hx>0) return  atanhi[3]+atanlo[3];
+	    else     return -atanhi[3]-atanlo[3];
+	} if (ix < 0x3fdc0000) {	/* |x| < 0.4375 */
+	    if (ix < 0x3e200000) {	/* |x| < 2^-29 */
+		if(huge+x>one) return x;	/* raise inexact */
+	    }
+	    id = -1;
+	} else {
+	x = fabs(x);
+	if (ix < 0x3ff30000) {		/* |x| < 1.1875 */
+	    if (ix < 0x3fe60000) {	/* 7/16 <=|x|<11/16 */
+		id = 0; x = (2.0*x-one)/(2.0+x); 
+	    } else {			/* 11/16<=|x|< 19/16 */
+		id = 1; x  = (x-one)/(x+one); 
+	    }
+	} else {
+	    if (ix < 0x40038000) {	/* |x| < 2.4375 */
+		id = 2; x  = (x-1.5)/(one+1.5*x);
+	    } else {			/* 2.4375 <= |x| < 2^66 */
+		id = 3; x  = -1.0/x;
+	    }
+	}}
+    /* end of argument reduction */
+	z = x*x;
+	w = z*z;
+    /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
+	s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10])))));
+	s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9]))));
+	if (id<0) return x - x*(s1+s2);
+	else {
+	    z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
+	    return (hx<0)? -z:z;
+	}
+}
-- 
GitLab