Average Error: 6.6 → 0.7
Time: 5.0m
Precision: 64
Internal Precision: 384
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\left(\sqrt[3]{x.re + x.im} \cdot \sqrt[3]{x.re + x.im}\right) \cdot \left(\sqrt[3]{x.re + x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]

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. Using strategy rm

  3. Applied difference-of-squares6.6

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

  4. Applied associate-*l*0.2

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

  6. Applied add-cube-cbrt0.7

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

  7. Applied associate-*l*0.7

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

Runtime

Time bar (total: 5.0m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(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)))