Average Error: 44.1 → 44.1
Time: 11.2s
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 r18281 = xre;
        double r18282 = xim;
        double r18283 = /* ERROR: no complex support in C */;
        double r18284 = exp(r18283);
        double r18285 = -r18283;
        double r18286 = exp(r18285);
        double r18287 = r18284 + r18286;
        double r18288 = 2.0;
        double r18289 = 0.0;
        double r18290 = /* ERROR: no complex support in C */;
        double r18291 = r18287 / r18290;
        double r18292 = /* ERROR: no complex support in C */;
        return r18292;
}
↓
double f(double xre, double xim) {
        double r18293 = xre;
        double r18294 = xim;
        double r18295 = /* ERROR: no complex support in C */;
        double r18296 = exp(r18295);
        double r18297 = -r18295;
        double r18298 = exp(r18297);
        double r18299 = r18296 + r18298;
        double r18300 = 2.0;
        double r18301 = 0.0;
        double r18302 = /* ERROR: no complex support in C */;
        double r18303 = r18299 / r18302;
        double r18304 = /* ERROR: no complex support in C */;
        return r18304;
}

Error

Bits error versus xre

Bits error versus xim

Derivation

  1. Initial program 44.1

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

  2. Final simplification44.1

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

Reproduce

herbie shell --seed 2019356 +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))))