Average Error: 0.4 → 0.3
Time: 55.5s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(\left(x.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.im\right)\right)\right), \left(\left(x.im + x.im\right) \cdot x.re\right), x.re\right)\right)\]
\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.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.im\right)\right)\right), \left(\left(x.im + x.im\right) \cdot x.re\right), x.re\right)\right)
double f(double x_re, double x_im) {
        double r3217127 = x_re;
        double r3217128 = r3217127 * r3217127;
        double r3217129 = x_im;
        double r3217130 = r3217129 * r3217129;
        double r3217131 = r3217128 - r3217130;
        double r3217132 = r3217131 * r3217129;
        double r3217133 = r3217127 * r3217129;
        double r3217134 = r3217129 * r3217127;
        double r3217135 = r3217133 + r3217134;
        double r3217136 = r3217135 * r3217127;
        double r3217137 = r3217132 + r3217136;
        return r3217137;
}
↓
double f(double x_re, double x_im) {
        double r3217138 = x_re;
        double r3217139 = x_im;
        double r3217140 = r3217138 - r3217139;
        double r3217141 = r3217139 + r3217138;
        double r3217142 = r3217141 * r3217139;
        double r3217143 = r3217140 * r3217142;
        double r3217144 = /*Error: no posit support in C */;
        double r3217145 = r3217139 + r3217139;
        double r3217146 = r3217145 * r3217138;
        double r3217147 = /*Error: no posit support in C */;
        double r3217148 = /*Error: no posit support in C */;
        return r3217148;
}

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

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019151 
(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)))