Average Error: 7.4 → 0.2
Time: 2.7s
Precision: binary64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} + x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\]
\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(x.im \cdot -3\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 (* 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_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 * (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

Original7.4
Target0.2
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. Simplified7.4

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

  4. Applied associate-*r*_binary647.4

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

  5. Simplified7.4

    \[\leadsto {x.re}^{3} + \color{blue}{\left({x.im}^{2} \cdot x.re\right)} \cdot -3\]
  6. Using strategy rm

  7. Applied sqr-pow_binary647.4

    \[\leadsto {x.re}^{3} + \left(\color{blue}{\left({x.im}^{\left(\frac{2}{2}\right)} \cdot {x.im}^{\left(\frac{2}{2}\right)}\right)} \cdot x.re\right) \cdot -3\]

  8. Applied associate-*l*_binary640.2

    \[\leadsto {x.re}^{3} + \color{blue}{\left({x.im}^{\left(\frac{2}{2}\right)} \cdot \left({x.im}^{\left(\frac{2}{2}\right)} \cdot x.re\right)\right)} \cdot -3\]

  9. Simplified0.2

    \[\leadsto {x.re}^{3} + \left({x.im}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right) \cdot -3\]
  10. Using strategy rm

  11. Applied associate-*l*_binary640.2

    \[\leadsto {x.re}^{3} + \color{blue}{{x.im}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)}\]

  12. Simplified0.2

    \[\leadsto {x.re}^{3} + {x.im}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{\left(-3 \cdot \left(x.im \cdot x.re\right)\right)}\]
  13. Using strategy rm

  14. Applied associate-*r*_binary640.2

    \[\leadsto {x.re}^{3} + {x.im}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{\left(\left(-3 \cdot x.im\right) \cdot x.re\right)}\]

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2021173 
(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)))