Average Error: 0.4 → 0.4
Time: 9.4s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\left(x.re \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(\left(x.im + x.im\right) \cdot x.re\right)\]
double f(double x_re, double x_im) {
        double r1099007 = x_re;
        double r1099008 = r1099007 * r1099007;
        double r1099009 = x_im;
        double r1099010 = r1099009 * r1099009;
        double r1099011 = r1099008 - r1099010;
        double r1099012 = r1099011 * r1099007;
        double r1099013 = r1099007 * r1099009;
        double r1099014 = r1099009 * r1099007;
        double r1099015 = r1099013 + r1099014;
        double r1099016 = r1099015 * r1099009;
        double r1099017 = r1099012 - r1099016;
        return r1099017;
}
↓
double f(double x_re, double x_im) {
        double r1099018 = x_re;
        double r1099019 = x_im;
        double r1099020 = r1099019 + r1099018;
        double r1099021 = r1099018 * r1099020;
        double r1099022 = r1099018 - r1099019;
        double r1099023 = r1099021 * r1099022;
        double r1099024 = r1099019 + r1099019;
        double r1099025 = r1099024 * r1099018;
        double r1099026 = r1099019 * r1099025;
        double r1099027 = r1099023 - r1099026;
        return r1099027;
}


\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.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)

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 2019101 +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)))