Average Error: 0.4 → 0.4
Time: 22.6s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[x.re \cdot \left(\mathsf{qms}\left(\left(\left(x.re \cdot x.re\right)\right), \left(\left(x.im + x.im\right) + x.im\right), x.im\right)\right)\]
\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(\mathsf{qms}\left(\left(\left(x.re \cdot x.re\right)\right), \left(\left(x.im + x.im\right) + x.im\right), x.im\right)\right)
double f(double x_re, double x_im) {
        double r1840185 = x_re;
        double r1840186 = r1840185 * r1840185;
        double r1840187 = x_im;
        double r1840188 = r1840187 * r1840187;
        double r1840189 = r1840186 - r1840188;
        double r1840190 = r1840189 * r1840185;
        double r1840191 = r1840185 * r1840187;
        double r1840192 = r1840187 * r1840185;
        double r1840193 = r1840191 + r1840192;
        double r1840194 = r1840193 * r1840187;
        double r1840195 = r1840190 - r1840194;
        return r1840195;
}
↓
double f(double x_re, double x_im) {
        double r1840196 = x_re;
        double r1840197 = r1840196 * r1840196;
        double r1840198 = /*Error: no posit support in C */;
        double r1840199 = x_im;
        double r1840200 = r1840199 + r1840199;
        double r1840201 = r1840200 + r1840199;
        double r1840202 = /*Error: no posit support in C */;
        double r1840203 = /*Error: no posit support in C */;
        double r1840204 = r1840196 * r1840203;
        return r1840204;
}

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

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

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)))