math.cube on complex, real part

Average Error: 7.8 → 0.2

Time: 9.3s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

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_re)) - ((double) (((double) (((double) (x_46_re * x_46_im)) + ((double) (x_46_im * x_46_re)))) * x_46_im))));
}

↓

double code(double x_46_re, double x_46_im) {
	return ((double) (((double) pow(x_46_re, 3.0)) - ((double) (((double) (x_46_im * ((double) (3.0 * x_46_re)))) * x_46_im))));
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.8
Target	0.3
Herbie	0.2

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

Derivation

1. Initial program 7.8

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

2. Simplified 0.2

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

4. Applied associate-*r* 0.2

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

5. Simplified 0.2

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

6. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020182 
(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)))