Average Error: 0.4 → 0.3
Time: 24.9s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(\left(\left(x.im + x.re\right) \cdot x.im\right) \cdot \left(x.re - x.im\right)\right)\right), x.re, \left(x.im \cdot x.re + x.im \cdot x.re\right)\right)\right)\]
\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}
↓
\left(\mathsf{qma}\left(\left(\left(\left(\left(x.im + x.re\right) \cdot x.im\right) \cdot \left(x.re - x.im\right)\right)\right), x.re, \left(x.im \cdot x.re + x.im \cdot x.re\right)\right)\right)
double f(double x_re, double x_im) {
        double r1727524 = x_re;
        double r1727525 = r1727524 * r1727524;
        double r1727526 = x_im;
        double r1727527 = r1727526 * r1727526;
        double r1727528 = r1727525 - r1727527;
        double r1727529 = r1727528 * r1727526;
        double r1727530 = r1727524 * r1727526;
        double r1727531 = r1727526 * r1727524;
        double r1727532 = r1727530 + r1727531;
        double r1727533 = r1727532 * r1727524;
        double r1727534 = r1727529 + r1727533;
        return r1727534;
}
↓
double f(double x_re, double x_im) {
        double r1727535 = x_im;
        double r1727536 = x_re;
        double r1727537 = r1727535 + r1727536;
        double r1727538 = r1727537 * r1727535;
        double r1727539 = r1727536 - r1727535;
        double r1727540 = r1727538 * r1727539;
        double r1727541 = /*Error: no posit support in C */;
        double r1727542 = r1727535 * r1727536;
        double r1727543 = r1727542 + r1727542;
        double r1727544 = /*Error: no posit support in C */;
        double r1727545 = /*Error: no posit support in C */;
        return r1727545;
}

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 0.4

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

  3. Applied *-commutative0.4

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

  4. Applied distribute-lft-out0.4

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

  5. Applied associate-*l*0.4

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

  6. Applied introduce-quire0.4

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

  7. Applied insert-quire-fdp-add0.3

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019158 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  (+.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.im) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.re)))