math.cube on complex, imaginary part

Average Error: 7.3 → 0.2

Time: 4.7s

Precision: binary64

Cost: 7040

Math

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

↓

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

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+
  (* (- (* x.re x.re) (* x.im x.im)) x.im)
  (* (+ (* x.re x.im) (* x.im x.re)) x.re)))

↓

(FPCore (x.re x.im)
 :precision binary64
 (- (* x.re (* x.re (* x.im 3.0))) (pow 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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

↓

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

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.3
Target	0.2
Herbie	0.2

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

Alternatives

Alternative 1
Error	23.1
Cost	33024

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

Alternative 2
Error	0.6
Cost	27008

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

Alternative 3
Error	0.6
Cost	26496

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

Alternative 4
Error	0.2
Cost	26496

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

Alternative 5
Error	0.6
Cost	26496

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

Alternative 6
Error	26.7
Cost	20224

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

Alternative 7
Error	0.3
Cost	19968

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

Alternative 8
Error	32.3
Cost	19968

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

Alternative 9
Error	13.5
Cost	19904

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

Alternative 10
Error	33.9
Cost	19904

\[\sqrt{x.re} \cdot \left(\left(x.im \cdot 3\right) \cdot {x.re}^{1.5}\right) - {x.im}^{3}\]

Alternative 11
Error	7.2
Cost	13376

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

Alternative 12
Error	7.3
Cost	13376

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

Alternative 13
Error	0.2
Cost	7040

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

Alternative 14
Error	26.3
Cost	6784

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

Alternative 15
Error	28.2
Cost	6592

\[-{x.im}^{3}\]

Alternative 16
Error	7.3
Cost	1216

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

Alternative 17
Error	61.9
Cost	64

\[1\]

Alternative 18
Error	46.7
Cost	64

\[0\]

Alternative 19
Error	61.8
Cost	64

\[-1\]

Derivation

1. Initial program 7.3

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

2. Simplified 0.2

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

4. Applied associate-*l*_binary64_2676 0.2

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

5. Simplified 0.2

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

6. Final simplification 0.2

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

Reproduce

herbie shell --seed 2021042 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

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