Average Error: 6.7 → 0.2

Time: 23.0s

Precision: 64

Internal Precision: 576

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

↓

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

Error

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

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Enter valid numbers for all inputs

Target

Original	6.7
Target	0.2
Herbie	0.2

\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

Initial program 6.7
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
Initial simplification6.7
\[\leadsto (x.re \cdot \left((\left(-x.im\right) \cdot \left(x.im + x.im\right) + \left(x.re \cdot x.re\right))_*\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(-x.re\right)\right))_*\]
Taylor expanded around 0 6.7
\[\leadsto \color{blue}{{x.re}^{3} - 3 \cdot \left({x.im}^{2} \cdot x.re\right)}\]
Taylor expanded around -inf 62.6
\[\leadsto {x.re}^{3} - \color{blue}{3 \cdot \left(e^{2 \cdot \left(\log -1 - \log \left(\frac{-1}{x.im}\right)\right)} \cdot x.re\right)}\]
Simplified0.2
\[\leadsto {x.re}^{3} - \color{blue}{\left(x.im \cdot 3\right) \cdot \left(x.im \cdot x.re\right)}\]
Final simplification0.2
\[\leadsto {x.re}^{3} - \left(x.im \cdot 3\right) \cdot \left(x.im \cdot x.re\right)\]

Runtime

herbie shell --seed 2018216 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im))))

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