Average Error: 6.9 → 0.2
Time: 14.1s
Precision: 64
Internal Precision: 384
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right) - {x.im}^{3}\]

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 6.9

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

  2. Applied simplify6.9

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

  3. Taylor expanded around 0 6.9

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

  5. Applied unpow26.9

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

  6. Applied associate-*l*0.2

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

  8. Applied associate-*r*0.2

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

Runtime

Time bar (total: 14.1s)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))