math.cube on complex, real part

Average Error: 7.6 → 0.2

Time: 1.9s

Precision: binary64

Math

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

↓

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

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.im (* x.re -3.0)))))

C

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_im * (x_46_re * -3.0)));
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.6
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 7.6

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

2. Simplified 7.5

   \[\leadsto \color{blue}{{x.re}^{3} + \left(x.im \cdot x.im\right) \cdot \left(x.re \cdot -3\right)}\]
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. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020224 
(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)))