e_exp.c 5.0 KB

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  1. /*
  2. * ====================================================
  3. * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  4. *
  5. * Developed at SunPro, a Sun Microsystems, Inc. business.
  6. * Permission to use, copy, modify, and distribute this
  7. * software is freely granted, provided that this notice
  8. * is preserved.
  9. * ====================================================
  10. */
  11. /* __ieee754_exp(x)
  12. * Returns the exponential of x.
  13. *
  14. * Method
  15. * 1. Argument reduction:
  16. * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
  17. * Given x, find r and integer k such that
  18. *
  19. * x = k*ln2 + r, |r| <= 0.5*ln2.
  20. *
  21. * Here r will be represented as r = hi-lo for better
  22. * accuracy.
  23. *
  24. * 2. Approximation of exp(r) by a special rational function on
  25. * the interval [0,0.34658]:
  26. * Write
  27. * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
  28. * We use a special Reme algorithm on [0,0.34658] to generate
  29. * a polynomial of degree 5 to approximate R. The maximum error
  30. * of this polynomial approximation is bounded by 2**-59. In
  31. * other words,
  32. * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
  33. * (where z=r*r, and the values of P1 to P5 are listed below)
  34. * and
  35. * | 5 | -59
  36. * | 2.0+P1*z+...+P5*z - R(z) | <= 2
  37. * | |
  38. * The computation of exp(r) thus becomes
  39. * 2*r
  40. * exp(r) = 1 + -------
  41. * R - r
  42. * r*R1(r)
  43. * = 1 + r + ----------- (for better accuracy)
  44. * 2 - R1(r)
  45. * where
  46. * 2 4 10
  47. * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
  48. *
  49. * 3. Scale back to obtain exp(x):
  50. * From step 1, we have
  51. * exp(x) = 2^k * exp(r)
  52. *
  53. * Special cases:
  54. * exp(INF) is INF, exp(NaN) is NaN;
  55. * exp(-INF) is 0, and
  56. * for finite argument, only exp(0)=1 is exact.
  57. *
  58. * Accuracy:
  59. * according to an error analysis, the error is always less than
  60. * 1 ulp (unit in the last place).
  61. *
  62. * Misc. info.
  63. * For IEEE double
  64. * if x > 7.09782712893383973096e+02 then exp(x) overflow
  65. * if x < -7.45133219101941108420e+02 then exp(x) underflow
  66. *
  67. * Constants:
  68. * The hexadecimal values are the intended ones for the following
  69. * constants. The decimal values may be used, provided that the
  70. * compiler will convert from decimal to binary accurately enough
  71. * to produce the hexadecimal values shown.
  72. */
  73. #include "math.h"
  74. #include "math_private.h"
  75. static const double
  76. one = 1.0,
  77. halF[2] = {0.5,-0.5,},
  78. huge = 1.0e+300,
  79. twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
  80. o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
  81. u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
  82. ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
  83. -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
  84. ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
  85. -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
  86. invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
  87. P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
  88. P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
  89. P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
  90. P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
  91. P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
  92. double __ieee754_exp(double x) /* default IEEE double exp */
  93. {
  94. double y;
  95. double hi = 0.0;
  96. double lo = 0.0;
  97. double c;
  98. double t;
  99. int32_t k=0;
  100. int32_t xsb;
  101. u_int32_t hx;
  102. GET_HIGH_WORD(hx,x);
  103. xsb = (hx>>31)&1; /* sign bit of x */
  104. hx &= 0x7fffffff; /* high word of |x| */
  105. /* filter out non-finite argument */
  106. if(hx >= 0x40862E42) { /* if |x|>=709.78... */
  107. if(hx>=0x7ff00000) {
  108. u_int32_t lx;
  109. GET_LOW_WORD(lx,x);
  110. if(((hx&0xfffff)|lx)!=0)
  111. return x+x; /* NaN */
  112. else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
  113. }
  114. if(x > o_threshold) return huge*huge; /* overflow */
  115. if(x < u_threshold) return twom1000*twom1000; /* underflow */
  116. }
  117. /* argument reduction */
  118. if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
  119. if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
  120. hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
  121. } else {
  122. k = (int32_t)(invln2*x+halF[xsb]);
  123. t = k;
  124. hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
  125. lo = t*ln2LO[0];
  126. }
  127. x = hi - lo;
  128. }
  129. else if(hx < 0x3e300000) { /* when |x|<2**-28 */
  130. if(huge+x>one) return one+x;/* trigger inexact */
  131. }
  132. else k = 0;
  133. /* x is now in primary range */
  134. t = x*x;
  135. c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
  136. if(k==0) return one-((x*c)/(c-2.0)-x);
  137. else y = one-((lo-(x*c)/(2.0-c))-hi);
  138. if(k >= -1021) {
  139. u_int32_t hy;
  140. GET_HIGH_WORD(hy,y);
  141. SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */
  142. return y;
  143. } else {
  144. u_int32_t hy;
  145. GET_HIGH_WORD(hy,y);
  146. SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
  147. return y*twom1000;
  148. }
  149. }