Average Error: 0.4 → 0.4
Time: 23.3s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[x.re \cdot \left(\left(\mathsf{qms}\left(\left(\left(x.re \cdot x.re\right)\right), \left(\frac{\left(\frac{x.im}{x.im}\right)}{x.im}\right), x.im\right)\right)\right)\]
\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)
↓
x.re \cdot \left(\left(\mathsf{qms}\left(\left(\left(x.re \cdot x.re\right)\right), \left(\frac{\left(\frac{x.im}{x.im}\right)}{x.im}\right), x.im\right)\right)\right)
double f(double x_re, double x_im) {
        double r2153763 = x_re;
        double r2153764 = r2153763 * r2153763;
        double r2153765 = x_im;
        double r2153766 = r2153765 * r2153765;
        double r2153767 = r2153764 - r2153766;
        double r2153768 = r2153767 * r2153763;
        double r2153769 = r2153763 * r2153765;
        double r2153770 = r2153765 * r2153763;
        double r2153771 = r2153769 + r2153770;
        double r2153772 = r2153771 * r2153765;
        double r2153773 = r2153768 - r2153772;
        return r2153773;
}
↓
double f(double x_re, double x_im) {
        double r2153774 = x_re;
        double r2153775 = r2153774 * r2153774;
        double r2153776 = /*Error: no posit support in C */;
        double r2153777 = x_im;
        double r2153778 = r2153777 + r2153777;
        double r2153779 = r2153778 + r2153777;
        double r2153780 = /*Error: no posit support in C */;
        double r2153781 = /*Error: no posit support in C */;
        double r2153782 = r2153774 * r2153781;
        return r2153782;
}

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

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

  4. Applied introduce-quire0.4

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

  5. Applied insert-quire-fdp-sub0.4

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

  6. Final simplification0.4

    \[\leadsto x.re \cdot \left(\left(\mathsf{qms}\left(\left(\left(x.re \cdot x.re\right)\right), \left(\frac{\left(\frac{x.im}{x.im}\right)}{x.im}\right), x.im\right)\right)\right)\]

Reproduce

herbie shell --seed 2019163 +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)))