Average Error: 6.7 → 0.2
Time: 1.6m
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right) - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im\]
double f(double x_re, double x_im) {
        double r67114719 = x_re;
        double r67114720 = r67114719 * r67114719;
        double r67114721 = x_im;
        double r67114722 = r67114721 * r67114721;
        double r67114723 = r67114720 - r67114722;
        double r67114724 = r67114723 * r67114719;
        double r67114725 = r67114719 * r67114721;
        double r67114726 = r67114721 * r67114719;
        double r67114727 = r67114725 + r67114726;
        double r67114728 = r67114727 * r67114721;
        double r67114729 = r67114724 - r67114728;
        return r67114729;
}
↓
double f(double x_re, double x_im) {
        double r67114730 = x_re;
        double r67114731 = x_im;
        double r67114732 = r67114730 - r67114731;
        double r67114733 = r67114732 * r67114730;
        double r67114734 = r67114731 + r67114730;
        double r67114735 = r67114733 * r67114734;
        double r67114736 = r67114730 * r67114731;
        double r67114737 = r67114736 + r67114736;
        double r67114738 = r67114737 * r67114731;
        double r67114739 = r67114735 - r67114738;
        return r67114739;
}


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

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.7
Target0.3
Herbie0.2
\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

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. Using strategy rm

  3. Applied difference-of-squares6.7

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

  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im))))

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