Average Error: 7.1 → 0.2
Time: 45.3s
Precision: 64
Internal Precision: 576
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.im\]

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 7.1

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

  2. Applied simplify7.1

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

  3. Taylor expanded around 0 7.1

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

  5. Applied unpow27.1

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

  6. Applied associate-*r*0.2

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

  8. Applied associate-*r*0.2

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

  10. Applied associate-*r*0.2

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

Runtime

Time bar (total: 45.3s)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))