Average Error: 43.4 → 43.4
Time: 10.4s
Precision: 64
\[\Im(\left(\frac{\left(\left(e^{\left(xre + xim i\right)}\right) + \left(e^{\left(-\left(xre + xim i\right)\right)}\right)\right)}{\left(2 + 0.0 i\right)}\right))\]
↓
\[\Im(\left(\frac{\left(\left(e^{\left(xre + xim i\right)}\right) + \left(e^{\left(-\left(xre + xim i\right)\right)}\right)\right)}{\left(2 + 0.0 i\right)}\right))\]
\Im(\left(\frac{\left(\left(e^{\left(xre + xim i\right)}\right) + \left(e^{\left(-\left(xre + xim i\right)\right)}\right)\right)}{\left(2 + 0.0 i\right)}\right))
↓
\Im(\left(\frac{\left(\left(e^{\left(xre + xim i\right)}\right) + \left(e^{\left(-\left(xre + xim i\right)\right)}\right)\right)}{\left(2 + 0.0 i\right)}\right))
double f(double xre, double xim) {
        double r16138 = xre;
        double r16139 = xim;
        double r16140 = /* ERROR: no complex support in C */;
        double r16141 = exp(r16140);
        double r16142 = -r16140;
        double r16143 = exp(r16142);
        double r16144 = r16141 + r16143;
        double r16145 = 2.0;
        double r16146 = 0.0;
        double r16147 = /* ERROR: no complex support in C */;
        double r16148 = r16144 / r16147;
        double r16149 = /* ERROR: no complex support in C */;
        return r16149;
}
↓
double f(double xre, double xim) {
        double r16150 = xre;
        double r16151 = xim;
        double r16152 = /* ERROR: no complex support in C */;
        double r16153 = exp(r16152);
        double r16154 = -r16152;
        double r16155 = exp(r16154);
        double r16156 = r16153 + r16155;
        double r16157 = 2.0;
        double r16158 = 0.0;
        double r16159 = /* ERROR: no complex support in C */;
        double r16160 = r16156 / r16159;
        double r16161 = /* ERROR: no complex support in C */;
        return r16161;
}

Error

Bits error versus xre

Bits error versus xim

Derivation

  1. Initial program 43.4

    \[\Im(\left(\frac{\left(\left(e^{\left(xre + xim i\right)}\right) + \left(e^{\left(-\left(xre + xim i\right)\right)}\right)\right)}{\left(2 + 0.0 i\right)}\right))\]

  2. Final simplification43.4

    \[\leadsto \Im(\left(\frac{\left(\left(e^{\left(xre + xim i\right)}\right) + \left(e^{\left(-\left(xre + xim i\right)\right)}\right)\right)}{\left(2 + 0.0 i\right)}\right))\]

Reproduce

herbie shell --seed 2020083 
(FPCore (xre xim)
  :name "exp with complex power imaginary part (p55)"
  :precision binary64
  (im (/ (+ (exp (complex xre xim)) (exp (- (complex xre xim)))) (complex 2 0.0))))