Average Error: 0.4 → 0.3
Time: 1.0m
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.re \cdot x.im + x.re \cdot x.im\right), x.re\right)\right)\]
\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(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\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 r2090941 = x_re;
        double r2090942 = r2090941 * r2090941;
        double r2090943 = x_im;
        double r2090944 = r2090943 * r2090943;
        double r2090945 = r2090942 - r2090944;
        double r2090946 = r2090945 * r2090943;
        double r2090947 = r2090941 * r2090943;
        double r2090948 = r2090943 * r2090941;
        double r2090949 = r2090947 + r2090948;
        double r2090950 = r2090949 * r2090941;
        double r2090951 = r2090946 + r2090950;
        return r2090951;
}
↓
double f(double x_re, double x_im) {
        double r2090952 = x_im;
        double r2090953 = x_re;
        double r2090954 = r2090953 - r2090952;
        double r2090955 = r2090952 * r2090954;
        double r2090956 = r2090952 + r2090953;
        double r2090957 = r2090955 * r2090956;
        double r2090958 = /*Error: no posit support in C */;
        double r2090959 = r2090953 * r2090952;
        double r2090960 = r2090959 + r2090959;
        double r2090961 = /*Error: no posit support in C */;
        double r2090962 = /*Error: no posit support in C */;
        return r2090962;
}

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

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

  7. Applied distribute-lft-in0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019152 +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)))