Average Error: 6.8 → 0.3
Time: 22.6s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 - \left(x.im \cdot x.im\right) \cdot x.im\]
\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(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 - \left(x.im \cdot x.im\right) \cdot x.im
double f(double x_re, double x_im) {
        double r8695536 = x_re;
        double r8695537 = r8695536 * r8695536;
        double r8695538 = x_im;
        double r8695539 = r8695538 * r8695538;
        double r8695540 = r8695537 - r8695539;
        double r8695541 = r8695540 * r8695538;
        double r8695542 = r8695536 * r8695538;
        double r8695543 = r8695538 * r8695536;
        double r8695544 = r8695542 + r8695543;
        double r8695545 = r8695544 * r8695536;
        double r8695546 = r8695541 + r8695545;
        return r8695546;
}
↓
double f(double x_re, double x_im) {
        double r8695547 = x_re;
        double r8695548 = x_im;
        double r8695549 = r8695547 * r8695548;
        double r8695550 = r8695547 * r8695549;
        double r8695551 = 3.0;
        double r8695552 = r8695550 * r8695551;
        double r8695553 = r8695548 * r8695548;
        double r8695554 = r8695553 * r8695548;
        double r8695555 = r8695552 - r8695554;
        return r8695555;
}

Error

Bits error versus x.re

Bits error versus x.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.8
Target0.2
Herbie0.3
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 6.8

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

  2. Simplified6.8

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

  4. Applied add-cube-cbrt7.5

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

  5. Applied associate-*r*7.5

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

  6. Taylor expanded around -inf 6.7

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

  7. Simplified6.8

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

  9. Applied associate-*r*0.3

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

  10. Final simplification0.3

    \[\leadsto \left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 - \left(x.im \cdot x.im\right) \cdot x.im\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))