Average Error: 6.6 → 0.2
Time: 44.1s
Precision: 64
Internal Precision: 576
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - 3 \cdot \left(\left(x.re \cdot x.im\right) \cdot x.im\right)\]

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 6.6

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

  2. Applied simplify6.6

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

  3. Taylor expanded around 0 6.6

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

  5. Applied unpow26.6

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

  6. Applied associate-*r*0.2

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

Runtime

Time bar (total: 44.1s)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' +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)))