Average Error: 7.3 → 0.3
Time: 7.2s
Precision: binary64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) - x.im \cdot \left(2 \cdot \left(x.re \cdot x.im\right)\right)\]
\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
↓
\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) - x.im \cdot \left(2 \cdot \left(x.re \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
 (- (* (+ x.re x.im) (* x.re (- x.re x.im))) (* x.im (* 2.0 (* x.re 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 ((x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))) - (x_46_im * (2.0 * (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

Original7.3
Target0.3
Herbie0.3
\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

  1. Initial program 7.3

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

  3. Applied difference-of-squares_binary64_31167.3

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

  4. Applied associate-*l*_binary64_30880.3

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

  5. Simplified0.3

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

  7. Applied *-un-lft-identity_binary64_31470.3

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

  8. Applied associate-*r*_binary64_30870.3

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2021022 
(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)))