Average Error: 0.4 → 0.3
Time: 2.1m
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\right)\right), \left(x.re \cdot x.im + x.re \cdot x.im\right), x.re\right)\right)\]
\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(x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\right)\right), \left(x.re \cdot x.im + x.re \cdot x.im\right), x.re\right)\right)
double f(double x_re, double x_im) {
        double r2838105 = x_re;
        double r2838106 = r2838105 * r2838105;
        double r2838107 = x_im;
        double r2838108 = r2838107 * r2838107;
        double r2838109 = r2838106 - r2838108;
        double r2838110 = r2838109 * r2838107;
        double r2838111 = r2838105 * r2838107;
        double r2838112 = r2838107 * r2838105;
        double r2838113 = r2838111 + r2838112;
        double r2838114 = r2838113 * r2838105;
        double r2838115 = r2838110 + r2838114;
        return r2838115;
}
↓
double f(double x_re, double x_im) {
        double r2838116 = x_im;
        double r2838117 = x_re;
        double r2838118 = r2838116 + r2838117;
        double r2838119 = r2838117 - r2838116;
        double r2838120 = r2838118 * r2838119;
        double r2838121 = r2838116 * r2838120;
        double r2838122 = /*Error: no posit support in C */;
        double r2838123 = r2838117 * r2838116;
        double r2838124 = r2838123 + r2838123;
        double r2838125 = /*Error: no posit support in C */;
        double r2838126 = /*Error: no posit support in C */;
        return r2838126;
}

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

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

  7. Applied distribute-lft-in0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019144 
(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)))