Average Error: 6.9 → 7.0

Time: 26.0s

Precision: 64

Internal Precision: 128

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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `x.re`

Target

Original	6.9
Target	0.2
Herbie	7.0

\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

Initial program 6.9
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
Simplified7.0
\[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re + \left(x.re + x.re\right)\right) - x.im \cdot x.im\right)}\]
Final simplification7.0
\[\leadsto x.im \cdot \left(\left(x.re + \left(x.re + x.re\right)\right) \cdot x.re - x.im \cdot x.im\right)\]

Reproduce

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

  :herbie-target
  (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))