Average Error: 7.3 → 0.2
Time: 3.9s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\mathsf{fma}\left(3, \left(x.re \cdot \left(-x.im\right)\right) \cdot x.im, {x.re}^{3}\right)\]
\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im
↓
\mathsf{fma}\left(3, \left(x.re \cdot \left(-x.im\right)\right) \cdot x.im, {x.re}^{3}\right)
double f(double x_re, double x_im) {
        double r272986 = x_re;
        double r272987 = r272986 * r272986;
        double r272988 = x_im;
        double r272989 = r272988 * r272988;
        double r272990 = r272987 - r272989;
        double r272991 = r272990 * r272986;
        double r272992 = r272986 * r272988;
        double r272993 = r272988 * r272986;
        double r272994 = r272992 + r272993;
        double r272995 = r272994 * r272988;
        double r272996 = r272991 - r272995;
        return r272996;
}
↓
double f(double x_re, double x_im) {
        double r272997 = 3.0;
        double r272998 = x_re;
        double r272999 = x_im;
        double r273000 = -r272999;
        double r273001 = r272998 * r273000;
        double r273002 = r273001 * r272999;
        double r273003 = pow(r272998, r272997);
        double r273004 = fma(r272997, r273002, r273003);
        return r273004;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.3
Target0.3
Herbie0.2
\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

  1. Initial program 7.3

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

  2. Simplified7.2

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

  4. Applied distribute-lft-neg-in7.2

    \[\leadsto \mathsf{fma}\left(3, x.re \cdot \color{blue}{\left(\left(-x.im\right) \cdot x.im\right)}, {x.re}^{3}\right)\]

  5. Applied associate-*r*0.2

    \[\leadsto \mathsf{fma}\left(3, \color{blue}{\left(x.re \cdot \left(-x.im\right)\right) \cdot x.im}, {x.re}^{3}\right)\]

  6. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(3, \left(x.re \cdot \left(-x.im\right)\right) \cdot x.im, {x.re}^{3}\right)\]

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))