Average Error: 6.9 → 0.2
Time: 3.2s
Precision: binary64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(x.re \cdot \left(x.re \cdot x.im\right) - {x.im}^{3}\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\]
\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(x.re \cdot \left(x.re \cdot x.im\right) - {x.im}^{3}\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)
double code(double x_46_re, double x_46_im) {
	return ((double) (((double) (((double) (((double) (x_46_re * x_46_re)) - ((double) (x_46_im * x_46_im)))) * x_46_im)) + ((double) (((double) (((double) (x_46_re * x_46_im)) + ((double) (x_46_im * x_46_re)))) * x_46_re))));
}
↓
double code(double x_46_re, double x_46_im) {
	return ((double) (((double) (((double) (x_46_re * ((double) (x_46_re * x_46_im)))) - ((double) pow(x_46_im, 3.0)))) + ((double) (x_46_re * ((double) (((double) (x_46_re * x_46_im)) + ((double) (x_46_re * x_46_im))))))));
}

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

Original6.9
Target0.3
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 6.9

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

  3. Applied pow16.9

    \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot \color{blue}{{x.im}^{1}} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

  4. Applied pow16.9

    \[\leadsto \color{blue}{{\left(x.re \cdot x.re - x.im \cdot x.im\right)}^{1}} \cdot {x.im}^{1} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

  5. Applied pow-prod-down6.9

    \[\leadsto \color{blue}{{\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im\right)}^{1}} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

  6. Simplified0.2

    \[\leadsto {\color{blue}{\left(x.re \cdot \left(x.re \cdot x.im\right) - {x.im}^{3}\right)}}^{1} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2020196 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 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)))