1 /* @(#)s_atan.c 5.1 93/09/24 */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunPro, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 */ 12 13 #include <sys/cdefs.h> 14 //__FBSDID("$FreeBSD: src/lib/msun/src/s_atan.c,v 1.13 2011/02/10 07:37:50 das Exp $"); 15 16 /* atan(x) 17 * Method 18 * 1. Reduce x to positive by atan(x) = -atan(-x). 19 * 2. According to the integer k=4t+0.25 chopped, t=x, the argument 20 * is further reduced to one of the following intervals and the 21 * arctangent of t is evaluated by the corresponding formula: 22 * 23 * [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...) 24 * [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) ) 25 * [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) ) 26 * [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) ) 27 * [39/16,INF] atan(x) = atan(INF) + atan( -1/t ) 28 * 29 * Constants: 30 * The hexadecimal values are the intended ones for the following 31 * constants. The decimal values may be used, provided that the 32 * compiler will convert from decimal to binary accurately enough 33 * to produce the hexadecimal values shown. 34 */ 35 36 #include <float.h> 37 38 #include "math.h" 39 #include "math_private.h" 40 41 static const double atanhi[] = { 42 4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */ 43 7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */ 44 9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */ 45 1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */ 46 }; 47 48 static const double atanlo[] = { 49 2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */ 50 3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */ 51 1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */ 52 6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */ 53 }; 54 55 static const double aT[] = { 56 3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */ 57 -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */ 58 1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */ 59 -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */ 60 9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */ 61 -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */ 62 6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */ 63 -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */ 64 4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */ 65 -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */ 66 1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */ 67 }; 68 69 static const double 70 one = 1.0, 71 huge = 1.0e300; 72 73 double 74 atan(double x) 75 { 76 double w,s1,s2,z; 77 int32_t ix,hx,id; 78 79 GET_HIGH_WORD(hx,x); 80 ix = hx&0x7fffffff; 81 if(ix>=0x44100000) { /* if |x| >= 2^66 */ 82 u_int32_t low; 83 GET_LOW_WORD(low,x); 84 if(ix>0x7ff00000|| 85 (ix==0x7ff00000&&(low!=0))) 86 return x+x; /* NaN */ 87 if(hx>0) return atanhi[3]+*(volatile double *)&atanlo[3]; 88 else return -atanhi[3]-*(volatile double *)&atanlo[3]; 89 } if (ix < 0x3fdc0000) { /* |x| < 0.4375 */ 90 if (ix < 0x3e400000) { /* |x| < 2^-27 */ 91 if(huge+x>one) return x; /* raise inexact */ 92 } 93 id = -1; 94 } else { 95 x = fabs(x); 96 if (ix < 0x3ff30000) { /* |x| < 1.1875 */ 97 if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */ 98 id = 0; x = (2.0*x-one)/(2.0+x); 99 } else { /* 11/16<=|x|< 19/16 */ 100 id = 1; x = (x-one)/(x+one); 101 } 102 } else { 103 if (ix < 0x40038000) { /* |x| < 2.4375 */ 104 id = 2; x = (x-1.5)/(one+1.5*x); 105 } else { /* 2.4375 <= |x| < 2^66 */ 106 id = 3; x = -1.0/x; 107 } 108 }} 109 /* end of argument reduction */ 110 z = x*x; 111 w = z*z; 112 /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */ 113 s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10]))))); 114 s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9])))); 115 if (id<0) return x - x*(s1+s2); 116 else { 117 z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x); 118 return (hx<0)? -z:z; 119 } 120 } 121 122 #if LDBL_MANT_DIG == 53 123 __weak_reference(atan, atanl); 124 #endif 125