Average Error: 0.4 → 0.3
Time: 24.1s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(\frac{x.im}{x.re}\right)\right)\right), \left(x.re \cdot \left(\frac{x.im}{x.im}\right)\right), x.re\right)\right)\]
\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(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(\frac{x.im}{x.re}\right)\right)\right), \left(x.re \cdot \left(\frac{x.im}{x.im}\right)\right), x.re\right)\right)
double f(double x_re, double x_im) {
        double r1979493 = x_re;
        double r1979494 = r1979493 * r1979493;
        double r1979495 = x_im;
        double r1979496 = r1979495 * r1979495;
        double r1979497 = r1979494 - r1979496;
        double r1979498 = r1979497 * r1979495;
        double r1979499 = r1979493 * r1979495;
        double r1979500 = r1979495 * r1979493;
        double r1979501 = r1979499 + r1979500;
        double r1979502 = r1979501 * r1979493;
        double r1979503 = r1979498 + r1979502;
        return r1979503;
}
↓
double f(double x_re, double x_im) {
        double r1979504 = x_im;
        double r1979505 = x_re;
        double r1979506 = r1979505 - r1979504;
        double r1979507 = r1979504 * r1979506;
        double r1979508 = r1979504 + r1979505;
        double r1979509 = r1979507 * r1979508;
        double r1979510 = /*Error: no posit support in C */;
        double r1979511 = r1979504 + r1979504;
        double r1979512 = r1979505 * r1979511;
        double r1979513 = /*Error: no posit support in C */;
        double r1979514 = /*Error: no posit support in C */;
        return r1979514;
}

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 introduce-quire0.4

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

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

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

  5. Simplified0.3

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

  7. Applied distribute-lft-in0.3

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

  9. Applied distribute-lft-out0.3

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

  10. Applied associate-*r*0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(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)))