Average Error: 0.4 → 0.4
Time: 13.3s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re + x.re\right)\right)\]
\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)}
↓
x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re + x.re\right)\right)
double f(double x_re, double x_im) {
        double r2675957 = x_re;
        double r2675958 = r2675957 * r2675957;
        double r2675959 = x_im;
        double r2675960 = r2675959 * r2675959;
        double r2675961 = r2675958 - r2675960;
        double r2675962 = r2675961 * r2675959;
        double r2675963 = r2675957 * r2675959;
        double r2675964 = r2675959 * r2675957;
        double r2675965 = r2675963 + r2675964;
        double r2675966 = r2675965 * r2675957;
        double r2675967 = r2675962 + r2675966;
        return r2675967;
}
↓
double f(double x_re, double x_im) {
        double r2675968 = x_im;
        double r2675969 = x_re;
        double r2675970 = r2675968 + r2675969;
        double r2675971 = r2675969 - r2675968;
        double r2675972 = r2675970 * r2675971;
        double r2675973 = r2675969 + r2675969;
        double r2675974 = r2675969 * r2675973;
        double r2675975 = r2675972 + r2675974;
        double r2675976 = r2675968 * r2675975;
        return r2675976;
}

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. Simplified0.4

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

  4. Applied sub-neg0.4

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

  5. Applied distribute-rgt-in0.4

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

  7. Applied distribute-rgt-out0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019128 +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)))