Average Error: 7.4 → 0.2
Time: 3.6s
Precision: binary64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - x.im \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)\]
\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
↓
{x.re}^{3} - x.im \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)
(FPCore (x.re x.im)
 :precision binary64
 (-
  (* (- (* x.re x.re) (* x.im x.im)) x.re)
  (* (+ (* x.re x.im) (* x.im x.re)) x.im)))
↓
(FPCore (x.re x.im)
 :precision binary64
 (- (pow x.re 3.0) (* x.im (* x.re (* 3.0 x.im)))))
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}
↓
double code(double x_46_re, double x_46_im) {
	return pow(x_46_re, 3.0) - (x_46_im * (x_46_re * (3.0 * 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

Original7.4
Target0.3
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 7.4

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

  3. Applied difference-of-squares_binary64_31167.4

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

  4. Applied associate-*l*_binary64_30880.2

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

  5. Simplified0.2

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

  7. Applied sub-neg_binary64_31400.2

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

  8. Applied distribute-rgt-in_binary64_30970.2

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

  9. Applied distribute-rgt-in_binary64_30970.2

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

  10. Applied associate--l+_binary64_30840.2

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

  11. Simplified0.2

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

  13. Applied distribute-lft-neg-out_binary64_31060.2

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

  14. Simplified0.2

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

  16. Applied distribute-rgt-in_binary64_30970.2

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

  17. Applied distribute-rgt-in_binary64_30970.2

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

  18. Applied distribute-neg-in_binary64_31080.2

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

  19. Applied associate-+r+_binary64_30790.2

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

  20. Simplified0.2

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

  21. Final simplification0.2

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

Reproduce

herbie shell --seed 2020356 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))