Average Error: 0.4 → 0.4
Time: 46.1s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)
↓
\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
double f(double x_re, double x_im) {
        double r5173024 = x_re;
        double r5173025 = r5173024 * r5173024;
        double r5173026 = x_im;
        double r5173027 = r5173026 * r5173026;
        double r5173028 = r5173025 - r5173027;
        double r5173029 = r5173028 * r5173024;
        double r5173030 = r5173024 * r5173026;
        double r5173031 = r5173026 * r5173024;
        double r5173032 = r5173030 + r5173031;
        double r5173033 = r5173032 * r5173026;
        double r5173034 = r5173029 - r5173033;
        return r5173034;
}
↓
double f(double x_re, double x_im) {
        double r5173035 = x_re;
        double r5173036 = x_im;
        double r5173037 = r5173035 + r5173036;
        double r5173038 = r5173035 - r5173036;
        double r5173039 = r5173038 * r5173035;
        double r5173040 = r5173037 * r5173039;
        double r5173041 = r5173035 * r5173036;
        double r5173042 = r5173036 * r5173035;
        double r5173043 = r5173041 + r5173042;
        double r5173044 = r5173043 * r5173036;
        double r5173045 = r5173040 - r5173044;
        return r5173045;
}

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 0.4

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

  3. Applied difference-of-squares0.4

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

  5. Applied associate-*l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  (-.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.re) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.im)))