Average Error: 0.4 → 0.4
Time: 8.4s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re + x.re\right)\right)\]
\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)}
↓
x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re + x.re\right)\right)
double f(double x_re, double x_im) {
        double r1251414 = x_re;
        double r1251415 = r1251414 * r1251414;
        double r1251416 = x_im;
        double r1251417 = r1251416 * r1251416;
        double r1251418 = r1251415 - r1251417;
        double r1251419 = r1251418 * r1251416;
        double r1251420 = r1251414 * r1251416;
        double r1251421 = r1251416 * r1251414;
        double r1251422 = r1251420 + r1251421;
        double r1251423 = r1251422 * r1251414;
        double r1251424 = r1251419 + r1251423;
        return r1251424;
}
↓
double f(double x_re, double x_im) {
        double r1251425 = x_im;
        double r1251426 = x_re;
        double r1251427 = r1251425 + r1251426;
        double r1251428 = r1251426 - r1251425;
        double r1251429 = r1251427 * r1251428;
        double r1251430 = r1251426 + r1251426;
        double r1251431 = r1251426 * r1251430;
        double r1251432 = r1251429 + r1251431;
        double r1251433 = r1251425 * r1251432;
        return r1251433;
}

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. Simplified0.4

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

  4. Applied distribute-lft-in0.4

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

  6. Applied p16-distribute-lft-out0.4

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

  7. Final simplification0.4

    \[\leadsto x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re + x.re\right)\right)\]

Reproduce

herbie shell --seed 2019120 
(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)))