Average Error: 43.5 → 43.5
Time: 27.4s
Precision: 64
\[\Im(\left(\frac{\left(\left(e^{\left(xre + xim i\right)}\right) + \left(e^{\left(-\left(xre + xim i\right)\right)}\right)\right)}{\left(2 + 0.0 i\right)}\right))\]
↓
\[\Im(\left(\frac{\left(\left(e^{\left(-\left(xre + xim i\right)\right)}\right) + \left(e^{\left(xre + xim i\right)}\right)\right)}{\left(2 + 0.0 i\right)}\right))\]
\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(-\left(xre + xim i\right)\right)}\right) + \left(e^{\left(xre + xim i\right)}\right)\right)}{\left(2 + 0.0 i\right)}\right))
double f(double xre, double xim) {
        double r10283 = xre;
        double r10284 = xim;
        double r10285 = /* ERROR: no complex support in C */;
        double r10286 = exp(r10285);
        double r10287 = -r10285;
        double r10288 = exp(r10287);
        double r10289 = r10286 + r10288;
        double r10290 = 2.0;
        double r10291 = 0.0;
        double r10292 = /* ERROR: no complex support in C */;
        double r10293 = r10289 / r10292;
        double r10294 = /* ERROR: no complex support in C */;
        return r10294;
}
↓
double f(double xre, double xim) {
        double r10295 = xre;
        double r10296 = xim;
        double r10297 = /* ERROR: no complex support in C */;
        double r10298 = -r10297;
        double r10299 = exp(r10298);
        double r10300 = exp(r10297);
        double r10301 = r10299 + r10300;
        double r10302 = 2.0;
        double r10303 = 0.0;
        double r10304 = /* ERROR: no complex support in C */;
        double r10305 = r10301 / r10304;
        double r10306 = /* ERROR: no complex support in C */;
        return r10306;
}

Error

Bits error versus xre

Bits error versus xim

Derivation

  1. Initial program 43.5

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

  2. Final simplification43.5

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (xre xim)
  :name "exp with complex power imaginary part (p55)"
  (im (/ (+ (exp (complex xre xim)) (exp (- (complex xre xim)))) (complex 2.0 0.0))))