Average Error: 0.4 → 0.4
Time: 10.8s
Precision: 64
Internal Precision: 320
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[x.re \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot \left(\left(x.re + x.re\right) \cdot x.im\right)\]

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 0.4

    \[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
  2. Using strategy rm

  3. Applied difference-of-squares0.4

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

  4. Applied associate-*l*0.4

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

  6. Applied p16-flip--0.4

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

  7. Applied associate-*l/0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  (-.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.re) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.im)))