text: // The following is adapted from fdlibm (http://www.netlib.org/fdlibm),
  //
  // ====================================================
  // Copyright (C) 1993-2004 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.
  // ====================================================
  //
  // The original source code covered by the above license above has been
  // modified significantly by Google Inc.
  // Copyright 2014 the V8 project authors. All rights reserved.
  //
  // The following is a straightforward translation of fdlibm routines
  // by Raymond Toy (rtoy@google.com).
  
  // Double constants that do not have empty lower 32 bits are found in fdlibm.cc
  // and exposed through kMath as typed array. We assume the compiler to convert
  // from decimal to binary accurately enough to produce the intended values.
  // kMath is initialized to a Float64Array during genesis and not writable.
  var kMath;
  
  const INVPIO2 = kMath[0];
  const PIO2_1  = kMath[1];
  const PIO2_1T = kMath[2];
  const PIO2_2  = kMath[3];
  const PIO2_2T = kMath[4];
  const PIO2_3  = kMath[5];
  const PIO2_3T = kMath[6];
  const PIO4    = kMath[32];
  const PIO4LO  = kMath[33];
  
  // Compute k and r such that x - k*pi/2 = r where |r| < pi/4. For
  // precision, r is returned as two values y0 and y1 such that r = y0 + y1
  // to more than double precision.
  macro REMPIO2(X)
    var n, y0, y1;
    var hx = %_DoubleHi(X);
    var ix = hx & 0x7fffffff;
  
    if (ix < 0x4002d97c) {
      // |X| ~< 3*pi/4, special case with n = +/- 1
      if (hx > 0) {
        var z = X - PIO2_1;
        if (ix != 0x3ff921fb) {
          // 33+53 bit pi is good enough
          y0 = z - PIO2_1T;
          y1 = (z - y0) - PIO2_1T;
        } else {
          // near pi/2, use 33+33+53 bit pi
          z -= PIO2_2;
          y0 = z - PIO2_2T;
          y1 = (z - y0) - PIO2_2T;
        }
        n = 1;
      } else {
        // Negative X
        var z = X + PIO2_1;
        if (ix != 0x3ff921fb) {
          // 33+53 bit pi is good enough
          y0 = z + PIO2_1T;
          y1 = (z - y0) + PIO2_1T;
        } else {
          // near pi/2, use 33+33+53 bit pi
          z += PIO2_2;
          y0 = z + PIO2_2T;
          y1 = (z - y0) + PIO2_2T;
        }
        n = -1;
      }
    } else if (ix <= 0x413921fb) {
      // |X| ~<= 2^19*(pi/2), medium size
      var t = MathAbs(X);
      n = (t * INVPIO2 + 0.5) | 0;
      var r = t - n * PIO2_1;
      var w = n * PIO2_1T;
      // First round good to 85 bit
      y0 = r - w;
      if (ix - (%_DoubleHi(y0) & 0x7ff00000) > 0x1000000) {
        // 2nd iteration needed, good to 118
        t = r;
        w = n * PIO2_2;
        r = t - w;
        w = n * PIO2_2T - ((t - r) - w);
        y0 = r - w;
        if (ix - (%_DoubleHi(y0) & 0x7ff00000) > 0x3100000) {
          // 3rd iteration needed. 151 bits accuracy
          t = r;
          w = n * PIO2_3;
          r = t - w;
          w = n * PIO2_3T - ((t - r) - w);
          y0 = r - w;
        }
      }
      y1 = (r - y0) - w;
      if (hx < 0) {
        n = -n;
        y0 = -y0;
        y1 = -y1;
      }
    } else {
      // Need to do full Payne-Hanek reduction here.
      var r = %RemPiO2(X);
      n = r[0];
      y0 = r[1];
      y1 = r[2];
    }
  endmacro
  
  
  // __kernel_sin(X, Y, IY)
  // kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
  // Input X is assumed to be bounded by ~pi/4 in magnitude.
  // Input Y is the tail of X so that x = X + Y.
  //
  // Algorithm
  //  1. Since ieee_sin(-x) = -ieee_sin(x), we need only to consider positive x.
  //  2. ieee_sin(x) is approximated by a polynomial of degree 13 on
  //     [0,pi/4]
  //                           3            13
  //          sin(x) ~ x + S1*x + ... + S6*x
  //     where
  //
  //    |ieee_sin(x)    2     4     6     8     10     12  |     -58
  //    |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x  +S6*x   )| <= 2
  //    |  x                                               |
  //
  //  3. ieee_sin(X+Y) = ieee_sin(X) + sin'(X')*Y
  //              ~ ieee_sin(X) + (1-X*X/2)*Y
  //     For better accuracy, let
  //               3      2      2      2      2
  //          r = X *(S2+X *(S3+X *(S4+X *(S5+X *S6))))
  //     then                   3    2
  //          sin(x) = X + (S1*X + (X *(r-Y/2)+Y))
  //
  macro KSIN(x)
  kMath[7+x]
  endmacro
  
  macro RETURN_KERNELSIN(X, Y, SIGN)
    var z = X * X;
    var v = z * X;
    var r = KSIN(1) + z * (KSIN(2) + z * (KSIN(3) +
                      z * (KSIN(4) + z * KSIN(5))));
    return (X - ((z * (0.5 * Y - v * r) - Y) - v * KSIN(0))) SIGN;
  endmacro
  
  // __kernel_cos(X, Y)
  // kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
  // Input X is assumed to be bounded by ~pi/4 in magnitude.
  // Input Y is the tail of X so that x = X + Y.
  //
  // Algorithm
  //  1. Since ieee_cos(-x) = ieee_cos(x), we need only to consider positive x.
  //  2. ieee_cos(x) is approximated by a polynomial of degree 14 on
  //     [0,pi/4]
  //                                   4            14
  //          cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
  //     where the remez error is
  //
  //  |                   2     4     6     8     10    12     14 |     -58
  //  |ieee_cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
  //  |                                                           |
  //
  //                 4     6     8     10    12     14
  //  3. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
  //         ieee_cos(x) = 1 - x*x/2 + r
  //     since ieee_cos(X+Y) ~ ieee_cos(X) - ieee_sin(X)*Y
  //                    ~ ieee_cos(X) - X*Y,
  //     a correction term is necessary in ieee_cos(x) and hence
  //         cos(X+Y) = 1 - (X*X/2 - (r - X*Y))
  //     For better accuracy when x > 0.3, let qx = |x|/4 with
  //     the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
  //     Then
  //         cos(X+Y) = (1-qx) - ((X*X/2-qx) - (r-X*Y)).
  //     Note that 1-qx and (X*X/2-qx) is EXACT here, and the
  //     magnitude of the latter is at least a quarter of X*X/2,
  //     thus, reducing the rounding error in the subtraction.
  //
  macro KCOS(x)
  kMath[13+x]
  endmacro
  
  macro RETURN_KERNELCOS(X, Y, SIGN)
    var ix = %_DoubleHi(X) & 0x7fffffff;
    var z = X * X;
    var r = z * (KCOS(0) + z * (KCOS(1) + z * (KCOS(2)+
            z * (KCOS(3) + z * (KCOS(4) + z * KCOS(5))))));
    if (ix < 0x3fd33333) {  // |x| ~< 0.3
      return (1 - (0.5 * z - (z * r - X * Y))) SIGN;
    } else {
      var qx;
      if (ix > 0x3fe90000) {  // |x| > 0.78125
        qx = 0.28125;
      } else {
        qx = %_ConstructDouble(%_DoubleHi(0.25 * X), 0);
      }
      var hz = 0.5 * z - qx;
      return (1 - qx - (hz - (z * r - X * Y))) SIGN;
    }
  endmacro
  
  
  // kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
  // Input x is assumed to be bounded by ~pi/4 in magnitude.
  // Input y is the tail of x.
  // Input k indicates whether ieee_tan (if k = 1) or -1/tan (if k = -1)
  // is returned.
  //
  // Algorithm
  //  1. Since ieee_tan(-x) = -ieee_tan(x), we need only to consider positive x.
  //  2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
  //  3. ieee_tan(x) is approximated by a odd polynomial of degree 27 on
  //     [0,0.67434]
  //                           3             27
  //          tan(x) ~ x + T1*x + ... + T13*x
  //     where
  //
  //     |ieee_tan(x)    2     4            26   |     -59.2
  //     |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
  //     |  x                                    |
  //
  //     Note: ieee_tan(x+y) = ieee_tan(x) + tan'(x)*y
  //                    ~ ieee_tan(x) + (1+x*x)*y
  //     Therefore, for better accuracy in computing ieee_tan(x+y), let
  //               3      2      2       2       2
  //          r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
  //     then
  //                              3    2
  //          tan(x+y) = x + (T1*x + (x *(r+y)+y))
  //
  //  4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
  //          tan(x) = ieee_tan(pi/4-y) = (1-ieee_tan(y))/(1+ieee_tan(y))
  //                 = 1 - 2*(ieee_tan(y) - (ieee_tan(y)^2)/(1+ieee_tan(y)))
  //
  // Set returnTan to 1 for tan; -1 for cot.  Anything else is illegal
  // and will cause incorrect results.
  //
  macro KTAN(x)
  kMath[19+x]
  endmacro
  
  function KernelTan(x, y, returnTan) {
    var z;
    var w;
    var hx = %_DoubleHi(x);
    var ix = hx & 0x7fffffff;
  
    if (ix < 0x3e300000) {  // |x| < 2^-28
      if (((ix | %_DoubleLo(x)) | (returnTan + 1)) == 0) {
        // x == 0 && returnTan = -1
        return 1 / MathAbs(x);
      } else {
        if (returnTan == 1) {
          return x;
        } else {
          // Compute -1/(x + y) carefully
          var w = x + y;
          var z = %_ConstructDouble(%_DoubleHi(w), 0);
          var v = y - (z - x);
          var a = -1 / w;
          var t = %_ConstructDouble(%_DoubleHi(a), 0);
          var s = 1 + t * z;
          return t + a * (s + t * v);
        }
      }
    }
    if (ix >= 0x3fe59428) {  // |x| > .6744
      if (x < 0) {
        x = -x;
        y = -y;
      }
      z = PIO4 - x;
      w = PIO4LO - y;
      x = z + w;
      y = 0;
    }
    z = x * x;
    w = z * z;
  
    // Break x^5 * (T1 + x^2*T2 + ...) into
    // x^5 * (T1 + x^4*T3 + ... + x^20*T11) +
    // x^5 * (x^2 * (T2 + x^4*T4 + ... + x^22*T12))
    var r = KTAN(1) + w * (KTAN(3) + w * (KTAN(5) +
                      w * (KTAN(7) + w * (KTAN(9) + w * KTAN(11)))));
    var v = z * (KTAN(2) + w * (KTAN(4) + w * (KTAN(6) +
                           w * (KTAN(8) + w * (KTAN(10) + w * KTAN(12))))));
    var s = z * x;
    r = y + z * (s * (r + v) + y);
    r = r + KTAN(0) * s;
    w = x + r;
    if (ix >= 0x3fe59428) {
      return (1 - ((hx >> 30) & 2)) *
        (returnTan - 2.0 * (x - (w * w / (w + returnTan) - r)));
    }
    if (returnTan == 1) {
      return w;
    } else {
      z = %_ConstructDouble(%_DoubleHi(w), 0);
      v = r - (z - x);
      var a = -1 / w;
      var t = %_ConstructDouble(%_DoubleHi(a), 0);
      s = 1 + t * z;
      return t + a * (s + t * v);
    }
  }
  
  function MathSinSlow(x) {
    REMPIO2(x);
    var sign = 1 - (n & 2);
    if (n & 1) {
      RETURN_KERNELCOS(y0, y1, * sign);
    } else {
      RETURN_KERNELSIN(y0, y1, * sign);
    }
  }
  
  function MathCosSlow(x) {
    REMPIO2(x);
    if (n & 1) {
      var sign = (n & 2) - 1;
      RETURN_KERNELSIN(y0, y1, * sign);
    } else {
      var sign = 1 - (n & 2);
      RETURN_KERNELCOS(y0, y1, * sign);
    }
  }
  
  // ECMA 262 - 15.8.2.16
  function MathSin(x) {
    x = x * 1;  // Convert to number.
    if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) {
      // |x| < pi/4, approximately.  No reduction needed.
      RETURN_KERNELSIN(x, 0, /* empty */);
    }
    return MathSinSlow(x);
  }
  
  // ECMA 262 - 15.8.2.7
  function MathCos(x) {
    x = x * 1;  // Convert to number.
    if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) {
      // |x| < pi/4, approximately.  No reduction needed.
      RETURN_KERNELCOS(x, 0, /* empty */);
    }
    return MathCosSlow(x);
  }
  
  // ECMA 262 - 15.8.2.18
  function MathTan(x) {
    x = x * 1;  // Convert to number.
    if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) {
      // |x| < pi/4, approximately.  No reduction needed.
      return KernelTan(x, 0, 1);
    }
    REMPIO2(x);
    return KernelTan(y0, y1, (n & 1) ? -1 : 1);
  }
  
  // ES6 draft 09-27-13, section 20.2.2.20.
  // Math.log1p
  //
  // Method :                  
  //   1. Argument Reduction: find k and f such that 
  //                      1+x = 2^k * (1+f), 
  //         where  sqrt(2)/2 < 1+f < sqrt(2) .
  //
  //      Note. If k=0, then f=x is exact. However, if k!=0, then f
  //      may not be representable exactly. In that case, a correction
  //      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
  //      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
  //      and add back the correction term c/u.
  //      (Note: when x > 2**53, one can simply return log(x))
  //
  //   2. Approximation of log1p(f).
  //      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  //            = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  //            = 2s + s*R
  //      We use a special Reme algorithm on [0,0.1716] to generate 
  //      a polynomial of degree 14 to approximate R The maximum error 
  //      of this polynomial approximation is bounded by 2**-58.45. In
  //      other words,
  //                      2      4      6      8      10      12      14
  //          R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
  //      (the values of Lp1 to Lp7 are listed in the program)
  //      and
  //          |      2          14          |     -58.45
  //          | Lp1*s +...+Lp7*s    -  R(z) | <= 2 
  //          |                             |
  //      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  //      In order to guarantee error in log below 1ulp, we compute log
  //      by
  //              log1p(f) = f - (hfsq - s*(hfsq+R)).
  //
  //      3. Finally, log1p(x) = k*ln2 + log1p(f).  
  //                           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  //         Here ln2 is split into two floating point number: 
  //                      ln2_hi + ln2_lo,
  //         where n*ln2_hi is always exact for |n| < 2000.
  //
  // Special cases:
  //      log1p(x) is NaN with signal if x < -1 (including -INF) ; 
  //      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
  //      log1p(NaN) is that NaN with no signal.
  //
  // Accuracy:
  //      according to an error analysis, the error is always less than
  //      1 ulp (unit in the last place).
  //
  // Constants:
  //      Constants are found in fdlibm.cc. We assume the C++ compiler to convert
  //      from decimal to binary accurately enough to produce the intended values.
  //
  // Note: Assuming log() return accurate answer, the following
  //       algorithm can be used to compute log1p(x) to within a few ULP:
  //
  //              u = 1+x;
  //              if (u==1.0) return x ; else
  //                          return log(u)*(x/(u-1.0));
  //
  //       See HP-15C Advanced Functions Handbook, p.193.
  //
  const LN2_HI    = kMath[34];
  const LN2_LO    = kMath[35];
  const TWO54     = kMath[36];
  const TWO_THIRD = kMath[37];
  macro KLOG1P(x)
  (kMath[38+x])
  endmacro
  
  function MathLog1p(x) {
    x = x * 1;  // Convert to number.
    var hx = %_DoubleHi(x);
    var ax = hx & 0x7fffffff;
    var k = 1;
    var f = x;
    var hu = 1;
    var c = 0;
    var u = x;
  
    if (hx < 0x3fda827a) {
      // x < 0.41422
      if (ax >= 0x3ff00000) {  // |x| >= 1
        if (x === -1) {
          return -INFINITY;  // log1p(-1) = -inf
        } else {
          return NAN;  // log1p(x<-1) = NaN
        }
      } else if (ax < 0x3c900000)  {
        // For |x| < 2^-54 we can return x.
        return x;
      } else if (ax < 0x3e200000) {
        // For |x| < 2^-29 we can use a simple two-term Taylor series.
        return x - x * x * 0.5;
      }
  
      if ((hx > 0) || (hx <= -0x402D413D)) {  // (int) 0xbfd2bec3 = -0x402d413d
        // -.2929 < x < 0.41422
        k = 0;
      }
    }
  
    // Handle Infinity and NAN
    if (hx >= 0x7ff00000) return x;
  
    if (k !== 0) {
      if (hx < 0x43400000) {
        // x < 2^53
        u = 1 + x;
        hu = %_DoubleHi(u);
        k = (hu >> 20) - 1023;
        c = (k > 0) ? 1 - (u - x) : x - (u - 1);
        c = c / u;
      } else {
        hu = %_DoubleHi(u);
        k = (hu >> 20) - 1023;
      }
      hu = hu & 0xfffff;
      if (hu < 0x6a09e) {
        u = %_ConstructDouble(hu | 0x3ff00000, %_DoubleLo(u));  // Normalize u.
      } else {
        ++k;
        u = %_ConstructDouble(hu | 0x3fe00000, %_DoubleLo(u));  // Normalize u/2.
        hu = (0x00100000 - hu) >> 2;
      }
      f = u - 1;
    }
  
    var hfsq = 0.5 * f * f;
    if (hu === 0) {
      // |f| < 2^-20;
      if (f === 0) {
        if (k === 0) {
          return 0.0;
        } else {
          return k * LN2_HI + (c + k * LN2_LO);
        }
      }
      var R = hfsq * (1 - TWO_THIRD * f);
      if (k === 0) {
        return f - R;
      } else {
        return k * LN2_HI - ((R - (k * LN2_LO + c)) - f);
      }
    }
  
    var s = f / (2 + f); 
    var z = s * s;
    var R = z * (KLOG1P(0) + z * (KLOG1P(1) + z *
                (KLOG1P(2) + z * (KLOG1P(3) + z *
                (KLOG1P(4) + z * (KLOG1P(5) + z * KLOG1P(6)))))));
    if (k === 0) {
      return f - (hfsq - s * (hfsq + R));
    } else {
      return k * LN2_HI - ((hfsq - (s * (hfsq + R) + (k * LN2_LO + c))) - f);
    }
  }
  
  // ES6 draft 09-27-13, section 20.2.2.14.
  // Math.expm1
  // Returns exp(x)-1, the exponential of x minus 1.
  //
  // Method
  //   1. Argument reduction:
  //      Given x, find r and integer k such that
  //
  //               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658  
  //
  //      Here a correction term c will be computed to compensate 
  //      the error in r when rounded to a floating-point number.
  //
  //   2. Approximating expm1(r) by a special rational function on
  //      the interval [0,0.34658]:
  //      Since
  //          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
  //      we define R1(r*r) by
  //          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
  //      That is,
  //          R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
  //                   = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
  //                   = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
  //      We use a special Remes algorithm on [0,0.347] to generate 
  //      a polynomial of degree 5 in r*r to approximate R1. The 
  //      maximum error of this polynomial approximation is bounded 
  //      by 2**-61. In other words,
  //          R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
  //      where   Q1  =  -1.6666666666666567384E-2,
  //              Q2  =   3.9682539681370365873E-4,
  //              Q3  =  -9.9206344733435987357E-6,
  //              Q4  =   2.5051361420808517002E-7,
  //              Q5  =  -6.2843505682382617102E-9;
  //      (where z=r*r, and the values of Q1 to Q5 are listed below)
  //      with error bounded by
  //          |                  5           |     -61
  //          | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2 
  //          |                              |
  //
  //      expm1(r) = exp(r)-1 is then computed by the following 
  //      specific way which minimize the accumulation rounding error: 
  //                             2     3
  //                            r     r    [ 3 - (R1 + R1*r/2)  ]
  //            expm1(r) = r + --- + --- * [--------------------]
  //                            2     2    [ 6 - r*(3 - R1*r/2) ]
  //
  //      To compensate the error in the argument reduction, we use
  //              expm1(r+c) = expm1(r) + c + expm1(r)*c 
  //                         ~ expm1(r) + c + r*c 
  //      Thus c+r*c will be added in as the correction terms for
  //      expm1(r+c). Now rearrange the term to avoid optimization 
  //      screw up:
  //                      (      2                                    2 )
  //                      ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
  //       expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
  //                      ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
  //                      (                                             )
  //
  //                 = r - E
  //   3. Scale back to obtain expm1(x):
  //      From step 1, we have
  //         expm1(x) = either 2^k*[expm1(r)+1] - 1
  //                  = or     2^k*[expm1(r) + (1-2^-k)]
  //   4. Implementation notes:
  //      (A). To save one multiplication, we scale the coefficient Qi
  //           to Qi*2^i, and replace z by (x^2)/2.
  //      (B). To achieve maximum accuracy, we compute expm1(x) by
  //        (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
  //        (ii)  if k=0, return r-E
  //        (iii) if k=-1, return 0.5*(r-E)-0.5
  //        (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
  //                     else          return  1.0+2.0*(r-E);
  //        (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
  //        (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
  //        (vii) return 2^k(1-((E+2^-k)-r)) 
  //
  // Special cases:
  //      expm1(INF) is INF, expm1(NaN) is NaN;
  //      expm1(-INF) is -1, and
  //      for finite argument, only expm1(0)=0 is exact.
  //
  // Accuracy:
  //      according to an error analysis, the error is always less than
  //      1 ulp (unit in the last place).
  //
  // Misc. info.
  //      For IEEE double 
  //          if x > 7.09782712893383973096e+02 then expm1(x) overflow
  //
  const KEXPM1_OVERFLOW = kMath[45];
  const INVLN2          = kMath[46];
  macro KEXPM1(x)
  (kMath[47+x])
  endmacro
  
  function MathExpm1(x) {
    x = x * 1;  // Convert to number.
    var y;
    var hi;
    var lo;
    var k;
    var t;
    var c;
      
    var hx = %_DoubleHi(x);
    var xsb = hx & 0x80000000;     // Sign bit of x
    var y = (xsb === 0) ? x : -x;  // y = |x|
    hx &= 0x7fffffff;              // High word of |x|
  
    // Filter out huge and non-finite argument
    if (hx >= 0x4043687a) {     // if |x| ~=> 56 * ln2
      if (hx >= 0x40862e42) {   // if |x| >= 709.78
        if (hx >= 0x7ff00000) {
          // expm1(inf) = inf; expm1(-inf) = -1; expm1(nan) = nan;
          return (x === -INFINITY) ? -1 : x;
        }
        if (x > KEXPM1_OVERFLOW) return INFINITY;  // Overflow
      }
      if (xsb != 0) return -1;  // x < -56 * ln2, return -1.
    }
  
    // Argument reduction
    if (hx > 0x3fd62e42) {    // if |x| > 0.5 * ln2
      if (hx < 0x3ff0a2b2) {  // and |x| < 1.5 * ln2
        if (xsb === 0) {
          hi = x - LN2_HI;
          lo = LN2_LO;
          k = 1;
        } else {
          hi = x + LN2_HI;
          lo = -LN2_LO;
          k = -1;
        }
      } else {
        k = (INVLN2 * x + ((xsb === 0) ? 0.5 : -0.5)) | 0;
        t = k;
        // t * ln2_hi is exact here.
        hi = x - t * LN2_HI;
        lo = t * LN2_LO;
      }
      x = hi - lo;
      c = (hi - x) - lo;
    } else if (hx < 0x3c900000)	{
      // When |x| < 2^-54, we can return x.
      return x;
    } else {
      // Fall through.
      k = 0;
    }
  
    // x is now in primary range
    var hfx = 0.5 * x;
    var hxs = x * hfx;
    var r1 = 1 + hxs * (KEXPM1(0) + hxs * (KEXPM1(1) + hxs *
                       (KEXPM1(2) + hxs * (KEXPM1(3) + hxs * KEXPM1(4)))));
    t = 3 - r1 * hfx;
    var e = hxs * ((r1 - t) / (6 - x * t));
    if (k === 0) {  // c is 0
      return x - (x*e - hxs);
    } else {
      e = (x * (e - c) - c);
      e -= hxs;
      if (k === -1) return 0.5 * (x - e) - 0.5;
      if (k === 1) {
        if (x < -0.25) return -2 * (e - (x + 0.5));
        return 1 + 2 * (x - e);
      }
  
      if (k <= -2 || k > 56) {
        // suffice to return exp(x) + 1
        y = 1 - (e - x);
        // Add k to y's exponent
        y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y));
        return y - 1;
      }
      if (k < 20) {
        // t = 1 - 2^k
        t = %_ConstructDouble(0x3ff00000 - (0x200000 >> k), 0);
        y = t - (e - x);
        // Add k to y's exponent
        y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y));
      } else {
        // t = 2^-k
        t = %_ConstructDouble((0x3ff - k) << 20, 0);
        y = x - (e + t);
        y += 1;
        // Add k to y's exponent
        y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y));
      }
    }
    return y;
  }
  
  
  // ES6 draft 09-27-13, section 20.2.2.30.
  // Math.sinh
  // Method :
  // mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2
  //      1. Replace x by |x| (sinh(-x) = -sinh(x)).
  //      2.
  //                                                  E + E/(E+1)
  //          0        <= x <= 22     :  sinh(x) := --------------, E=expm1(x)
  //                                                      2
  //
  //          22       <= x <= lnovft :  sinh(x) := exp(x)/2 
  //          lnovft   <= x <= ln2ovft:  sinh(x) := exp(x/2)/2 * exp(x/2)
  //          ln2ovft  <  x           :  sinh(x) := x*shuge (overflow)
  //
  // Special cases:
  //      sinh(x) is |x| if x is +Infinity, -Infinity, or NaN.
  //      only sinh(0)=0 is exact for finite x.
  //
  const KSINH_OVERFLOW = kMath[52];
  const TWO_M28 = 3.725290298461914e-9;  // 2^-28, empty lower half
  const LOG_MAXD = 709.7822265625;  // 0x40862e42 00000000, empty lower half
  
  function MathSinh(x) {
    x = x * 1;  // Convert to number.
    var h = (x < 0) ? -0.5 : 0.5;
    // |x| in [0, 22]. return sign(x)*0.5*(E+E/(E+1))
    var ax = MathAbs(x);
    if (ax < 22) {
      // For |x| < 2^-28, sinh(x) = x
      if (ax < TWO_M28) return x;
      var t = MathExpm1(ax);
      if (ax < 1) return h * (2 * t - t * t / (t + 1));
      return h * (t + t / (t + 1));
    }
    // |x| in [22, log(maxdouble)], return 0.5 * exp(|x|)
    if (ax < LOG_MAXD) return h * MathExp(ax);
    // |x| in [log(maxdouble), overflowthreshold]
    // overflowthreshold = 710.4758600739426
    if (ax <= KSINH_OVERFLOW) {
      var w = MathExp(0.5 * ax);
      var t = h * w;
      return t * w;
    }
    // |x| > overflowthreshold or is NaN.
    // Return Infinity of the appropriate sign or NaN.
    return x * INFINITY;
  }
  
  
  // ES6 draft 09-27-13, section 20.2.2.12.
  // Math.cosh
  // Method : 
  // mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2
  //      1. Replace x by |x| (cosh(x) = cosh(-x)). 
  //      2.
  //                                                      [ exp(x) - 1 ]^2 
  //          0        <= x <= ln2/2  :  cosh(x) := 1 + -------------------
  //                                                         2*exp(x)
  //
  //                                                 exp(x) + 1/exp(x)
  //          ln2/2    <= x <= 22     :  cosh(x) := -------------------
  //                                                        2
  //          22       <= x <= lnovft :  cosh(x) := exp(x)/2 
  //          lnovft   <= x <= ln2ovft:  cosh(x) := exp(x/2)/2 * exp(x/2)
  //          ln2ovft  <  x           :  cosh(x) := huge*huge (overflow)
  //
  // Special cases:
  //      cosh(x) is |x| if x is +INF, -INF, or NaN.
  //      only cosh(0)=1 is exact for finite x.
  //
  const KCOSH_OVERFLOW = kMath[52];
  
  function MathCosh(x) {
    x = x * 1;  // Convert to number.
    var ix = %_DoubleHi(x) & 0x7fffffff;
    // |x| in [0,0.5*log2], return 1+expm1(|x|)^2/(2*exp(|x|))
    if (ix < 0x3fd62e43) {
      var t = MathExpm1(MathAbs(x));
      var w = 1 + t;
      // For |x| < 2^-55, cosh(x) = 1
      if (ix < 0x3c800000) return w;
      return 1 + (t * t) / (w + w);
    }
    // |x| in [0.5*log2, 22], return (exp(|x|)+1/exp(|x|)/2
    if (ix < 0x40360000) {
      var t = MathExp(MathAbs(x));
      return 0.5 * t + 0.5 / t;
    }
    // |x| in [22, log(maxdouble)], return half*exp(|x|)
    if (ix < 0x40862e42) return 0.5 * MathExp(MathAbs(x));
    // |x| in [log(maxdouble), overflowthreshold]
    if (MathAbs(x) <= KCOSH_OVERFLOW) {
      var w = MathExp(0.5 * MathAbs(x));
      var t = 0.5 * w;
      return t * w;
    }
    if (NUMBER_IS_NAN(x)) return x;
    // |x| > overflowthreshold.
    return INFINITY;
  }
  
  // ES6 draft 09-27-13, section 20.2.2.21.
  // Return the base 10 logarithm of x
  //
  // Method :
  //      Let log10_2hi = leading 40 bits of log10(2) and
  //          log10_2lo = log10(2) - log10_2hi,
  //          ivln10   = 1/log(10) rounded.
  //      Then
  //              n = ilogb(x),
  //              if(n<0)  n = n+1;
  //              x = scalbn(x,-n);
  //              log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
  //
  // Note 1:
  //      To guarantee log10(10**n)=n, where 10**n is normal, the rounding
  //      mode must set to Round-to-Nearest.
  // Note 2:
  //      [1/log(10)] rounded to 53 bits has error .198 ulps;
  //      log10 is monotonic at all binary break points.
  //
  // Special cases:
  //      log10(x) is NaN if x < 0;
  //      log10(+INF) is +INF; log10(0) is -INF;
  //      log10(NaN) is that NaN;
  //      log10(10**N) = N  for N=0,1,...,22.
  //
  
  const IVLN10 = kMath[53];
  const LOG10_2HI = kMath[54];
  const LOG10_2LO = kMath[55];
  
  function MathLog10(x) {
    x = x * 1;  // Convert to number.
    var hx = %_DoubleHi(x);
    var lx = %_DoubleLo(x);
    var k = 0;
  
    if (hx < 0x00100000) {
      // x < 2^-1022
      // log10(+/- 0) = -Infinity.
      if (((hx & 0x7fffffff) | lx) === 0) return -INFINITY;
      // log10 of negative number is NaN.
      if (hx < 0) return NAN;
      // Subnormal number. Scale up x.
      k -= 54;
      x *= TWO54;
      hx = %_DoubleHi(x);
      lx = %_DoubleLo(x);
    }
  
    // Infinity or NaN.
    if (hx >= 0x7ff00000) return x;
  
    k += (hx >> 20) - 1023;
    i = (k & 0x80000000) >> 31;
    hx = (hx & 0x000fffff) | ((0x3ff - i) << 20);
    y = k + i;
    x = %_ConstructDouble(hx, lx);
  
    z = y * LOG10_2LO + IVLN10 * MathLog(x);
    return z + y * LOG10_2HI;
  }
  