Average Error: 0.4 → 0.4
Time: 12.6s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}
↓
\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re
double f(double x_re, double x_im) {
        double r1283774 = x_re;
        double r1283775 = r1283774 * r1283774;
        double r1283776 = x_im;
        double r1283777 = r1283776 * r1283776;
        double r1283778 = r1283775 - r1283777;
        double r1283779 = r1283778 * r1283776;
        double r1283780 = r1283774 * r1283776;
        double r1283781 = r1283776 * r1283774;
        double r1283782 = r1283780 + r1283781;
        double r1283783 = r1283782 * r1283774;
        double r1283784 = r1283779 + r1283783;
        return r1283784;
}
↓
double f(double x_re, double x_im) {
        double r1283785 = x_re;
        double r1283786 = x_im;
        double r1283787 = r1283785 + r1283786;
        double r1283788 = r1283785 - r1283786;
        double r1283789 = r1283788 * r1283786;
        double r1283790 = r1283787 * r1283789;
        double r1283791 = r1283785 * r1283786;
        double r1283792 = r1283786 * r1283785;
        double r1283793 = r1283791 + r1283792;
        double r1283794 = r1283793 * r1283785;
        double r1283795 = r1283790 + r1283794;
        return r1283795;
}

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 0.4

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

  3. Applied difference-of-squares0.4

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

  4. Applied associate-*l*0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  (+.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.im) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.re)))