Average Error: 7.9 → 0.2
Time: 4.7s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\mathsf{fma}\left(3, \left(x.re \cdot x.im\right) \cdot \left(-x.im\right), {x.re}^{3}\right)\]
\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 x.im\right) \cdot \left(-x.im\right), {x.re}^{3}\right)
double f(double x_re, double x_im) {
        double r181179 = x_re;
        double r181180 = r181179 * r181179;
        double r181181 = x_im;
        double r181182 = r181181 * r181181;
        double r181183 = r181180 - r181182;
        double r181184 = r181183 * r181179;
        double r181185 = r181179 * r181181;
        double r181186 = r181181 * r181179;
        double r181187 = r181185 + r181186;
        double r181188 = r181187 * r181181;
        double r181189 = r181184 - r181188;
        return r181189;
}
↓
double f(double x_re, double x_im) {
        double r181190 = 3.0;
        double r181191 = x_re;
        double r181192 = x_im;
        double r181193 = r181191 * r181192;
        double r181194 = -r181192;
        double r181195 = r181193 * r181194;
        double r181196 = pow(r181191, r181190);
        double r181197 = fma(r181190, r181195, r181196);
        return r181197;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.9
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 7.9

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

  2. Simplified7.8

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

  4. Applied distribute-rgt-neg-in7.8

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

  5. Applied associate-*r*0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020046 +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)))