Average Error: 7.1 → 0.2
Time: 4.4m
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\left(x.im + x.re\right) \cdot x.im\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\]
\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.im + x.re\right) \cdot x.im\right) \cdot \left(x.re - x.im\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 r82919255 = x_re;
        double r82919256 = r82919255 * r82919255;
        double r82919257 = x_im;
        double r82919258 = r82919257 * r82919257;
        double r82919259 = r82919256 - r82919258;
        double r82919260 = r82919259 * r82919257;
        double r82919261 = r82919255 * r82919257;
        double r82919262 = r82919257 * r82919255;
        double r82919263 = r82919261 + r82919262;
        double r82919264 = r82919263 * r82919255;
        double r82919265 = r82919260 + r82919264;
        return r82919265;
}
↓
double f(double x_re, double x_im) {
        double r82919266 = x_im;
        double r82919267 = x_re;
        double r82919268 = r82919266 + r82919267;
        double r82919269 = r82919268 * r82919266;
        double r82919270 = r82919267 - r82919266;
        double r82919271 = r82919269 * r82919270;
        double r82919272 = r82919267 * r82919266;
        double r82919273 = r82919272 + r82919272;
        double r82919274 = r82919267 * r82919273;
        double r82919275 = r82919271 + r82919274;
        return r82919275;
}

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

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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019120 
(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)))