e_asin.c 3.7 KB

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  1. /* @(#)e_asin.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. #if defined(LIBM_SCCS) && !defined(lint)
  13. static char rcsid[] = "$NetBSD: e_asin.c,v 1.9 1995/05/12 04:57:22 jtc Exp $";
  14. #endif
  15. /* __ieee754_asin(x)
  16. * Method :
  17. * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
  18. * we approximate asin(x) on [0,0.5] by
  19. * asin(x) = x + x*x^2*R(x^2)
  20. * where
  21. * R(x^2) is a rational approximation of (asin(x)-x)/x^3
  22. * and its remez error is bounded by
  23. * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
  24. *
  25. * For x in [0.5,1]
  26. * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
  27. * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
  28. * then for x>0.98
  29. * asin(x) = pi/2 - 2*(s+s*z*R(z))
  30. * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
  31. * For x<=0.98, let pio4_hi = pio2_hi/2, then
  32. * f = hi part of s;
  33. * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
  34. * and
  35. * asin(x) = pi/2 - 2*(s+s*z*R(z))
  36. * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
  37. * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
  38. *
  39. * Special cases:
  40. * if x is NaN, return x itself;
  41. * if |x|>1, return NaN with invalid signal.
  42. *
  43. */
  44. #include "math.h"
  45. #include "math_private.h"
  46. #ifdef __STDC__
  47. static const double
  48. #else
  49. static double
  50. #endif
  51. one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  52. huge = 1.000e+300,
  53. pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
  54. pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
  55. pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
  56. /* coefficient for R(x^2) */
  57. pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
  58. pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
  59. pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
  60. pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
  61. pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
  62. pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
  63. qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
  64. qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
  65. qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
  66. qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
  67. #ifdef __STDC__
  68. double attribute_hidden __ieee754_asin(double x)
  69. #else
  70. double attribute_hidden __ieee754_asin(x)
  71. double x;
  72. #endif
  73. {
  74. double t=0.0,w,p,q,c,r,s;
  75. int32_t hx,ix;
  76. GET_HIGH_WORD(hx,x);
  77. ix = hx&0x7fffffff;
  78. if(ix>= 0x3ff00000) { /* |x|>= 1 */
  79. u_int32_t lx;
  80. GET_LOW_WORD(lx,x);
  81. if(((ix-0x3ff00000)|lx)==0)
  82. /* asin(1)=+-pi/2 with inexact */
  83. return x*pio2_hi+x*pio2_lo;
  84. return (x-x)/(x-x); /* asin(|x|>1) is NaN */
  85. } else if (ix<0x3fe00000) { /* |x|<0.5 */
  86. if(ix<0x3e400000) { /* if |x| < 2**-27 */
  87. if(huge+x>one) return x;/* return x with inexact if x!=0*/
  88. } else {
  89. t = x*x;
  90. p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
  91. q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
  92. w = p/q;
  93. return x+x*w;
  94. }
  95. }
  96. /* 1> |x|>= 0.5 */
  97. w = one-fabs(x);
  98. t = w*0.5;
  99. p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
  100. q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
  101. s = __ieee754_sqrt(t);
  102. if(ix>=0x3FEF3333) { /* if |x| > 0.975 */
  103. w = p/q;
  104. t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
  105. } else {
  106. w = s;
  107. SET_LOW_WORD(w,0);
  108. c = (t-w*w)/(s+w);
  109. r = p/q;
  110. p = 2.0*s*r-(pio2_lo-2.0*c);
  111. q = pio4_hi-2.0*w;
  112. t = pio4_hi-(p-q);
  113. }
  114. if(hx>0) return t; else return -t;
  115. }