Average Error: 6.8 → 0.2

Time: 15.5s

Precision: 64

Internal Precision: 576

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

↓

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

Error

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

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In
Out

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Target

Original	6.8
Target	0.2
Herbie	0.2

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

Derivation

Initial program 6.8
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
Initial simplification6.8
\[\leadsto (x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(\left(x.re \cdot x.re\right) \cdot \left(x.im + x.im\right)\right))_*\]
Taylor expanded around 0 6.8
\[\leadsto \color{blue}{3 \cdot \left(x.im \cdot {x.re}^{2}\right) - {x.im}^{3}}\]
Using strategy rm
Applied unpow26.8
\[\leadsto 3 \cdot \left(x.im \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) - {x.im}^{3}\]
Applied associate-*r*0.2
\[\leadsto 3 \cdot \color{blue}{\left(\left(x.im \cdot x.re\right) \cdot x.re\right)} - {x.im}^{3}\]
Final simplification0.2
\[\leadsto \left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 - {x.im}^{3}\]

Runtime

herbie shell --seed 2018216 +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)))