Average Error: 6.9 → 0.2
Time: 52.5s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[(\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot \left(-x.im\right)\right))_*\]
\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
↓
(\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot \left(-x.im\right)\right))_*
double f(double x_re, double x_im) {
        double r50438136 = x_re;
        double r50438137 = r50438136 * r50438136;
        double r50438138 = x_im;
        double r50438139 = r50438138 * r50438138;
        double r50438140 = r50438137 - r50438139;
        double r50438141 = r50438140 * r50438136;
        double r50438142 = r50438136 * r50438138;
        double r50438143 = r50438138 * r50438136;
        double r50438144 = r50438142 + r50438143;
        double r50438145 = r50438144 * r50438138;
        double r50438146 = r50438141 - r50438145;
        return r50438146;
}
↓
double f(double x_re, double x_im) {
        double r50438147 = x_im;
        double r50438148 = x_re;
        double r50438149 = r50438147 + r50438148;
        double r50438150 = r50438148 - r50438147;
        double r50438151 = r50438150 * r50438148;
        double r50438152 = r50438148 * r50438147;
        double r50438153 = r50438152 + r50438152;
        double r50438154 = -r50438147;
        double r50438155 = r50438153 * r50438154;
        double r50438156 = fma(r50438149, r50438151, r50438155);
        return r50438156;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.9
Target0.2
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 6.9

    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
  2. Using strategy rm

  3. Applied difference-of-squares6.9

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

  4. Applied associate-*l*0.2

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

  6. Applied fma-neg0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :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)))