Average Error: 6.8 → 0.2
Time: 54.2s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[(\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot \left(-x.im\right)\right))_*\]
double f(double x_re, double x_im) {
        double r29982353 = x_re;
        double r29982354 = r29982353 * r29982353;
        double r29982355 = x_im;
        double r29982356 = r29982355 * r29982355;
        double r29982357 = r29982354 - r29982356;
        double r29982358 = r29982357 * r29982353;
        double r29982359 = r29982353 * r29982355;
        double r29982360 = r29982355 * r29982353;
        double r29982361 = r29982359 + r29982360;
        double r29982362 = r29982361 * r29982355;
        double r29982363 = r29982358 - r29982362;
        return r29982363;
}
↓
double f(double x_re, double x_im) {
        double r29982364 = x_im;
        double r29982365 = x_re;
        double r29982366 = r29982364 + r29982365;
        double r29982367 = r29982365 - r29982364;
        double r29982368 = r29982367 * r29982365;
        double r29982369 = r29982365 * r29982364;
        double r29982370 = r29982369 + r29982369;
        double r29982371 = -r29982364;
        double r29982372 = r29982370 * r29982371;
        double r29982373 = fma(r29982366, r29982368, r29982372);
        return r29982373;
}


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

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.8
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 6.8

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

  3. Applied difference-of-squares6.8

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

  4. Applied associate-*l*0.3

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

  6. Applied fma-neg0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019101 +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)))