Average Error: 7.3 → 0.2

Time: 5.3s

Precision: binary64

Cost: 7040

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

↓

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

\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

↓

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

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

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 * (3.0 * (x_46_re * x_46_im))) - pow(x_46_im, 3.0);
}

Error

The X axis uses an exponential scale — Bits error versus `x.re`

Try it out

In
Out

Enter valid numbers for all inputs

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	32.6
Cost	45824

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

Alternative 2
Error	23.4
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 3
Error	0.7
Cost	27008

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

Alternative 4
Error	0.6
Cost	26496

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

Alternative 5
Error	0.3
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 6
Error	7.5
Cost	26432

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

Alternative 7
Error	26.1
Cost	20224

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

Alternative 8
Error	0.3
Cost	19968

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

Alternative 9
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 10
Error	34.1
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	13.5
Cost	19904

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

Alternative 12
Error	7.3
Cost	13376

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

Alternative 13
Error	7.3
Cost	13376

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

Alternative 14
Error	0.2
Cost	7040

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

Alternative 15
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 16
Error	61.8
Cost	64

\[1\]

Alternative 17
Error	46.9
Cost	64

\[0\]

Alternative 18
Error	61.8
Cost	64

\[-1\]

Error

Derivation

Initial program 7.3
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
Simplified0.2
\[\leadsto \color{blue}{x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right) - {x.im}^{3}}\]
Simplified0.2
\[\leadsto \color{blue}{x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right) - {x.im}^{3}}\]
Final simplification0.2
\[\leadsto x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right) - {x.im}^{3}\]

Reproduce

herbie shell --seed 2021022 
(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)))