Average Error: 43.9 → 43.9

Time: 27.4s

Precision: 64

Internal Precision: 1344

\[\Im(\left(\frac{e^{xre + xim i} + e^{-\left(xre + xim i\right)}}{2 + 0 i}\right))\]

↓

\[\Im(\left(\frac{e^{-\left(xre + xim i\right)} + e^{xre + xim i}}{2 + 0 i}\right))\]

Error

The X axis uses an exponential scale — Bits error versus `xre`

Derivation

Initial program 43.9
\[\Im(\left(\frac{\left(\left(e^{\left(xre + xim i\right)}\right) + \left(e^{\left(-\left(xre + xim i\right)\right)}\right)\right)}{\left(2 + 0 i\right)}\right))\]
Final simplification43.9
\[\leadsto \Im(\left(\frac{e^{-\left(xre + xim i\right)} + e^{xre + xim i}}{2 + 0 i}\right))\]

Reproduce

herbie shell --seed 2019068 +o rules:numerics
(FPCore (xre xim)
  :name "exp with complex power imaginary part (p55)"
  (im (/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0))))