e_exp.c 5.3 KB

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  1. /* @(#)e_exp.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_exp.c,v 1.8 1995/05/10 20:45:03 jtc Exp $";
  14. #endif
  15. /* __ieee754_exp(x)
  16. * Returns the exponential of x.
  17. *
  18. * Method
  19. * 1. Argument reduction:
  20. * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
  21. * Given x, find r and integer k such that
  22. *
  23. * x = k*ln2 + r, |r| <= 0.5*ln2.
  24. *
  25. * Here r will be represented as r = hi-lo for better
  26. * accuracy.
  27. *
  28. * 2. Approximation of exp(r) by a special rational function on
  29. * the interval [0,0.34658]:
  30. * Write
  31. * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
  32. * We use a special Reme algorithm on [0,0.34658] to generate
  33. * a polynomial of degree 5 to approximate R. The maximum error
  34. * of this polynomial approximation is bounded by 2**-59. In
  35. * other words,
  36. * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
  37. * (where z=r*r, and the values of P1 to P5 are listed below)
  38. * and
  39. * | 5 | -59
  40. * | 2.0+P1*z+...+P5*z - R(z) | <= 2
  41. * | |
  42. * The computation of exp(r) thus becomes
  43. * 2*r
  44. * exp(r) = 1 + -------
  45. * R - r
  46. * r*R1(r)
  47. * = 1 + r + ----------- (for better accuracy)
  48. * 2 - R1(r)
  49. * where
  50. * 2 4 10
  51. * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
  52. *
  53. * 3. Scale back to obtain exp(x):
  54. * From step 1, we have
  55. * exp(x) = 2^k * exp(r)
  56. *
  57. * Special cases:
  58. * exp(INF) is INF, exp(NaN) is NaN;
  59. * exp(-INF) is 0, and
  60. * for finite argument, only exp(0)=1 is exact.
  61. *
  62. * Accuracy:
  63. * according to an error analysis, the error is always less than
  64. * 1 ulp (unit in the last place).
  65. *
  66. * Misc. info.
  67. * For IEEE double
  68. * if x > 7.09782712893383973096e+02 then exp(x) overflow
  69. * if x < -7.45133219101941108420e+02 then exp(x) underflow
  70. *
  71. * Constants:
  72. * The hexadecimal values are the intended ones for the following
  73. * constants. The decimal values may be used, provided that the
  74. * compiler will convert from decimal to binary accurately enough
  75. * to produce the hexadecimal values shown.
  76. */
  77. #include "math.h"
  78. #include "math_private.h"
  79. #ifdef __STDC__
  80. static const double
  81. #else
  82. static double
  83. #endif
  84. one = 1.0,
  85. halF[2] = {0.5,-0.5,},
  86. huge = 1.0e+300,
  87. twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
  88. o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
  89. u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
  90. ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
  91. -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
  92. ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
  93. -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
  94. invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
  95. P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
  96. P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
  97. P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
  98. P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
  99. P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
  100. #ifdef __STDC__
  101. double __ieee754_exp(double x) /* default IEEE double exp */
  102. #else
  103. double __ieee754_exp(x) /* default IEEE double exp */
  104. double x;
  105. #endif
  106. {
  107. double y;
  108. double hi = 0;
  109. double lo = 0;
  110. double c;
  111. double t;
  112. int32_t k=0;
  113. int32_t xsb;
  114. u_int32_t hx;
  115. GET_HIGH_WORD(hx,x);
  116. xsb = (hx>>31)&1; /* sign bit of x */
  117. hx &= 0x7fffffff; /* high word of |x| */
  118. /* filter out non-finite argument */
  119. if(hx >= 0x40862E42) { /* if |x|>=709.78... */
  120. if(hx>=0x7ff00000) {
  121. u_int32_t lx;
  122. GET_LOW_WORD(lx,x);
  123. if(((hx&0xfffff)|lx)!=0)
  124. return x+x; /* NaN */
  125. else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
  126. }
  127. if(x > o_threshold) return huge*huge; /* overflow */
  128. if(x < u_threshold) return twom1000*twom1000; /* underflow */
  129. }
  130. /* argument reduction */
  131. if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
  132. if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
  133. hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
  134. } else {
  135. k = invln2*x+halF[xsb];
  136. t = k;
  137. hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
  138. lo = t*ln2LO[0];
  139. }
  140. x = hi - lo;
  141. }
  142. else if(hx < 0x3e300000) { /* when |x|<2**-28 */
  143. if(huge+x>one) return one+x;/* trigger inexact */
  144. }
  145. else k = 0;
  146. /* x is now in primary range */
  147. t = x*x;
  148. c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
  149. if(k==0) return one-((x*c)/(c-2.0)-x);
  150. else y = one-((lo-(x*c)/(2.0-c))-hi);
  151. if(k >= -1021) {
  152. u_int32_t hy;
  153. GET_HIGH_WORD(hy,y);
  154. SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */
  155. return y;
  156. } else {
  157. u_int32_t hy;
  158. GET_HIGH_WORD(hy,y);
  159. SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
  160. return y*twom1000;
  161. }
  162. }