math.cube on complex, real part

Average Error: 7.8 → 0.3

Time: 4.7s

Precision: binary64

Math

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

↓

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

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
 (- (* (* x.re x.re) (+ x.re x.im)) (* x.im (* x.re (+ x.re (* x.im 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 ((x_46_re * x_46_re) * (x_46_re + x_46_im)) - (x_46_im * (x_46_re * (x_46_re + (x_46_im * 3.0))));
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.8
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 7.8

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

3. Applied difference-of-squares_binary64_2775 7.8

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

4. Applied associate-*l*_binary64_2747 0.3

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

5. Simplified 0.3

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

7. Applied sub-neg_binary64_2799 0.3

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

8. Applied distribute-rgt-in_binary64_2756 0.3

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

9. Applied distribute-rgt-in_binary64_2756 0.3

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

10. Applied associate--l+_binary64_2743 0.3

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

11. Simplified 0.2

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

13. Applied distribute-lft-out_binary64_2757 0.3

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

14. Simplified 0.3

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

15. Final simplification 0.3

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

Reproduce

herbie shell --seed 2021009 
(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)))