Average Error: 0.4 → 0.4
Time: 9.2s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(x.re \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(\left(x.im + x.im\right) \cdot x.re\right)\]
\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 \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(\left(x.im + x.im\right) \cdot x.re\right)
double f(double x_re, double x_im) {
        double r1271063 = x_re;
        double r1271064 = r1271063 * r1271063;
        double r1271065 = x_im;
        double r1271066 = r1271065 * r1271065;
        double r1271067 = r1271064 - r1271066;
        double r1271068 = r1271067 * r1271063;
        double r1271069 = r1271063 * r1271065;
        double r1271070 = r1271065 * r1271063;
        double r1271071 = r1271069 + r1271070;
        double r1271072 = r1271071 * r1271065;
        double r1271073 = r1271068 - r1271072;
        return r1271073;
}
↓
double f(double x_re, double x_im) {
        double r1271074 = x_re;
        double r1271075 = x_im;
        double r1271076 = r1271075 + r1271074;
        double r1271077 = r1271074 * r1271076;
        double r1271078 = r1271074 - r1271075;
        double r1271079 = r1271077 * r1271078;
        double r1271080 = r1271075 + r1271075;
        double r1271081 = r1271080 * r1271074;
        double r1271082 = r1271075 * r1271081;
        double r1271083 = r1271079 - r1271082;
        return r1271083;
}

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. Simplified0.4

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

  4. Applied associate-*r*0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019124 
(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)))