Average Error: 7.1 → 0.2
Time: 51.9s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\mathsf{fma}\left(\left(x.im \cdot \left(x.re + x.im\right)\right), \left(x.re - x.im\right), \left(x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\right)\right)\]
\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re
↓
\mathsf{fma}\left(\left(x.im \cdot \left(x.re + x.im\right)\right), \left(x.re - x.im\right), \left(x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\right)\right)
double f(double x_re, double x_im) {
        double r41421124 = x_re;
        double r41421125 = r41421124 * r41421124;
        double r41421126 = x_im;
        double r41421127 = r41421126 * r41421126;
        double r41421128 = r41421125 - r41421127;
        double r41421129 = r41421128 * r41421126;
        double r41421130 = r41421124 * r41421126;
        double r41421131 = r41421126 * r41421124;
        double r41421132 = r41421130 + r41421131;
        double r41421133 = r41421132 * r41421124;
        double r41421134 = r41421129 + r41421133;
        return r41421134;
}
↓
double f(double x_re, double x_im) {
        double r41421135 = x_im;
        double r41421136 = x_re;
        double r41421137 = r41421136 + r41421135;
        double r41421138 = r41421135 * r41421137;
        double r41421139 = r41421136 - r41421135;
        double r41421140 = r41421135 * r41421136;
        double r41421141 = r41421140 + r41421140;
        double r41421142 = r41421136 * r41421141;
        double r41421143 = fma(r41421138, r41421139, r41421142);
        return r41421143;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.1
Target0.2
Herbie0.2
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 7.1

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

  2. Taylor expanded around -inf 7.0

    \[\leadsto \color{blue}{\left(x.im \cdot {x.re}^{2} - {x.im}^{3}\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

  3. Simplified0.2

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

  5. Applied fma-def0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

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