Average Error: 6.8 → 0.2
Time: 4.4s
Precision: binary64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[x.re \cdot \left(\left(x.re \cdot 3\right) \cdot x.im\right) - {x.im}^{3}\]
\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
↓
x.re \cdot \left(\left(x.re \cdot 3\right) \cdot x.im\right) - {x.im}^{3}
(FPCore (x.re x.im)
 :precision binary64
 (+
  (* (- (* x.re x.re) (* x.im x.im)) x.im)
  (* (+ (* x.re x.im) (* x.im x.re)) x.re)))
↓
(FPCore (x.re x.im)
 :precision binary64
 (- (* x.re (* (* x.re 3.0) x.im)) (pow x.im 3.0)))
double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}
↓
double code(double x_46_re, double x_46_im) {
	return (x_46_re * ((x_46_re * 3.0) * x_46_im)) - pow(x_46_im, 3.0);
}

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. Simplified0.2

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

  4. Applied associate-*l*_binary64_30880.2

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

  5. Simplified0.2

    \[\leadsto x.re \cdot \left(x.re \cdot \color{blue}{\left(3 \cdot x.im\right)}\right) - {x.im}^{3}\]
  6. Using strategy rm

  7. Applied associate-*r*_binary64_30870.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2021044 
(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)))