Average Error: 6.9 → 0.2
Time: 15.0s
Precision: 64
Internal Precision: 320
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[(3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right) + \left(-{x.im}^{3}\right))_*\]

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.9
Target0.2
Herbie0.2
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 6.9

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

  2. Initial simplification6.9

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

  3. Taylor expanded around 0 6.9

    \[\leadsto \color{blue}{3 \cdot \left(x.im \cdot {x.re}^{2}\right) - {x.im}^{3}}\]
  4. Using strategy rm

  5. Applied unpow26.9

    \[\leadsto 3 \cdot \left(x.im \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) - {x.im}^{3}\]

  6. Applied associate-*r*0.2

    \[\leadsto 3 \cdot \color{blue}{\left(\left(x.im \cdot x.re\right) \cdot x.re\right)} - {x.im}^{3}\]
  7. Using strategy rm

  8. Applied fma-neg0.2

    \[\leadsto \color{blue}{(3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right) + \left(-{x.im}^{3}\right))_*}\]

  9. Final simplification0.2

    \[\leadsto (3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right) + \left(-{x.im}^{3}\right))_*\]

Runtime

Time bar (total: 15.0s)Debug logProfile

herbie shell --seed 2018227 +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)))