Average Error: 6.7 → 0.2
Time: 28.0s
Precision: 64
Internal Precision: 384
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - 3 \cdot \left(\left(x.re \cdot x.im\right) \cdot x.im\right)\]

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 6.7

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

  2. Applied simplify6.7

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

  3. Taylor expanded around 0 6.7

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

  5. Applied unpow26.7

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

  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: 28.0s)Debug log

herbie shell --seed '#(2151728608 3767143998 4065594768 704903143 4284770345 1704475780)' 
(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)))