Average Error: 6.8 → 0.2
Time: 58.7s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\]
\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
↓
\left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)
double f(double x_re, double x_im) {
        double r15580152 = x_re;
        double r15580153 = r15580152 * r15580152;
        double r15580154 = x_im;
        double r15580155 = r15580154 * r15580154;
        double r15580156 = r15580153 - r15580155;
        double r15580157 = r15580156 * r15580154;
        double r15580158 = r15580152 * r15580154;
        double r15580159 = r15580154 * r15580152;
        double r15580160 = r15580158 + r15580159;
        double r15580161 = r15580160 * r15580152;
        double r15580162 = r15580157 + r15580161;
        return r15580162;
}
↓
double f(double x_re, double x_im) {
        double r15580163 = x_re;
        double r15580164 = x_im;
        double r15580165 = r15580163 - r15580164;
        double r15580166 = r15580165 * r15580164;
        double r15580167 = r15580164 + r15580163;
        double r15580168 = r15580166 * r15580167;
        double r15580169 = r15580163 * r15580164;
        double r15580170 = r15580169 + r15580169;
        double r15580171 = r15580163 * r15580170;
        double r15580172 = r15580168 + r15580171;
        return r15580172;
}

Error

Bits error versus x.re

Bits error versus x.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.8
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 6.8

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

  3. Applied difference-of-squares6.8

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

  4. Applied associate-*l*0.2

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

  5. Taylor expanded around -inf 0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019121 
(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)))