math.cube on complex, real part

Average Error: 7.5 → 0.2

Time: 2.2s

Precision: binary64

Math

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

↓

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

FPCore

(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))))

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) (x_46_im * ((double) (((double) (x_46_re * x_46_im)) * -3.0))))));
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.5
Target	0.2
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 7.5

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

2. Simplified 7.4

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

4. Applied associate-*l*_binary64 0.2

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

6. Applied associate-*r*_binary64 0.2

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

7. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020219 
(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)))