Average Error: 43.6 → 43.6
Time: 10.6s
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 r20049 = xre;
        double r20050 = xim;
        double r20051 = /* ERROR: no complex support in C */;
        double r20052 = exp(r20051);
        double r20053 = -r20051;
        double r20054 = exp(r20053);
        double r20055 = r20052 + r20054;
        double r20056 = 2.0;
        double r20057 = 0.0;
        double r20058 = /* ERROR: no complex support in C */;
        double r20059 = r20055 / r20058;
        double r20060 = /* ERROR: no complex support in C */;
        return r20060;
}
↓
double f(double xre, double xim) {
        double r20061 = xre;
        double r20062 = xim;
        double r20063 = /* ERROR: no complex support in C */;
        double r20064 = exp(r20063);
        double r20065 = -r20063;
        double r20066 = exp(r20065);
        double r20067 = r20064 + r20066;
        double r20068 = 2.0;
        double r20069 = 0.0;
        double r20070 = /* ERROR: no complex support in C */;
        double r20071 = r20067 / r20070;
        double r20072 = /* ERROR: no complex support in C */;
        return r20072;
}

Error

Bits error versus xre

Bits error versus xim

Derivation

  1. Initial program 43.6

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

  2. Final simplification43.6

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(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))))