math.cube on complex, real part

Average Error: 7.2 → 0.2

Time: 4.3s

Precision: binary64

Math

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

↓

\[\left({x.re}^{3} + x.im \cdot {x.re}^{2}\right) - x.im \cdot \left(x.re \cdot \left(x.re + x.im\right) + 2 \cdot \left(x.re \cdot x.im\right)\right)\]

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 (pow x.re 2.0)))
  (* x.im (+ (* x.re (+ x.re x.im)) (* 2.0 (* x.re x.im))))))

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 * pow(x_46_re, 2.0))) - (x_46_im * ((x_46_re * (x_46_re + x_46_im)) + (2.0 * (x_46_re * x_46_im))));
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.2
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 7.2

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

3. Applied difference-of-squares_binary64 7.2

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

4. Applied associate-*l*_binary64 0.2

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

5. Simplified 0.2

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

7. Applied sub-neg_binary64 0.2

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

8. Applied distribute-rgt-in_binary64 0.2

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

9. Applied distribute-rgt-in_binary64 0.2

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

10. Applied associate--l+_binary64 0.2

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

11. Simplified 0.2

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

13. Applied distribute-rgt-in_binary64 0.2

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

14. Simplified 0.2

    \[\leadsto \left(\color{blue}{{x.re}^{3}} + x.im \cdot \left(x.re \cdot x.re\right)\right) + \left(-x.im\right) \cdot \left(x.re \cdot \left(x.im + x.re\right) + 2 \cdot \left(x.im \cdot x.re\right)\right)\]

15. Simplified 0.2

    \[\leadsto \left({x.re}^{3} + \color{blue}{x.im \cdot {x.re}^{2}}\right) + \left(-x.im\right) \cdot \left(x.re \cdot \left(x.im + x.re\right) + 2 \cdot \left(x.im \cdot x.re\right)\right)\]

16. Final simplification 0.2

    \[\leadsto \left({x.re}^{3} + x.im \cdot {x.re}^{2}\right) - x.im \cdot \left(x.re \cdot \left(x.re + x.im\right) + 2 \cdot \left(x.re \cdot x.im\right)\right)\]

Reproduce

herbie shell --seed 2021174 
(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)))