Average Error: 0.4 → 0.3
Time: 23.8s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(\mathsf{qms}\left(\left(\left(\left(x.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.re\right)\right)\right), x.im, \left(x.re \cdot \left(x.im + x.im\right)\right)\right)\right)\]
\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)
↓
\left(\mathsf{qms}\left(\left(\left(\left(x.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.re\right)\right)\right), x.im, \left(x.re \cdot \left(x.im + x.im\right)\right)\right)\right)
double f(double x_re, double x_im) {
        double r2252937 = x_re;
        double r2252938 = r2252937 * r2252937;
        double r2252939 = x_im;
        double r2252940 = r2252939 * r2252939;
        double r2252941 = r2252938 - r2252940;
        double r2252942 = r2252941 * r2252937;
        double r2252943 = r2252937 * r2252939;
        double r2252944 = r2252939 * r2252937;
        double r2252945 = r2252943 + r2252944;
        double r2252946 = r2252945 * r2252939;
        double r2252947 = r2252942 - r2252946;
        return r2252947;
}
↓
double f(double x_re, double x_im) {
        double r2252948 = x_re;
        double r2252949 = x_im;
        double r2252950 = r2252948 - r2252949;
        double r2252951 = r2252949 + r2252948;
        double r2252952 = r2252951 * r2252948;
        double r2252953 = r2252950 * r2252952;
        double r2252954 = /*Error: no posit support in C */;
        double r2252955 = r2252949 + r2252949;
        double r2252956 = r2252948 * r2252955;
        double r2252957 = /*Error: no posit support in C */;
        double r2252958 = /*Error: no posit support in C */;
        return r2252958;
}

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. Using strategy rm

  3. Applied *-commutative0.4

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

  4. Applied distribute-lft-out0.4

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

  5. Applied associate-*l*0.4

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

  6. Applied introduce-quire0.4

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

  7. Applied insert-quire-fdp-sub0.3

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

  8. Simplified0.3

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

  10. Applied associate-*l*0.3

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

  12. Applied *-commutative0.3

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

  13. Applied *-commutative0.3

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

  14. Applied distribute-lft-out0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019158 
(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)))