Average Error: 0.4 → 0.4
Time: 14.1s
Precision: 64
Internal Precision: 320
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right) - x.im \cdot x.im\right) \cdot x.im\]

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 0.4

    \[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]

  2. Simplified0.4

    \[\leadsto \color{blue}{x.im \cdot \left(\frac{\left(\left(\frac{x.im}{x.re}\right) \cdot \left(x.re - x.im\right)\right)}{\left(x.re \cdot \left(\frac{x.re}{x.re}\right)\right)}\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{\left(\left(x.re \cdot \left(\frac{\left(\frac{x.re}{x.re}\right)}{x.re}\right)\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im}\]

  4. Final simplification0.4

    \[\leadsto \left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right) - x.im \cdot x.im\right) \cdot x.im\]

Reproduce

herbie shell --seed 2019091 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  (+.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.im) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.re)))