math.cube on complex, real part

Average Error: 7.3 → 0.3

Time: 9.6s

Precision: binary64

Cost: 1216

Math

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

↓

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

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

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.3
Target	0.3
Herbie	0.3

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

Alternatives

Alternative 1
Error	7.7
Cost	59712

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

Alternative 2
Error	27.9
Cost	40384

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

Alternative 3
Error	7.8
Cost	27776

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

Alternative 4
Error	7.8
Cost	27008

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

Alternative 5
Error	32.1
Cost	26752

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

Alternative 6
Error	7.6
Cost	21312

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

Alternative 7
Error	0.6
Cost	21312

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

Alternative 8
Error	0.8
Cost	21184

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

Alternative 9
Error	0.7
Cost	20928

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

Alternative 10
Error	0.7
Cost	20672

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

Alternative 11
Error	0.6
Cost	20544

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

Alternative 12
Error	29.7
Cost	14400

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

Alternative 13
Error	32.1
Cost	14400

\[\frac{x.re \cdot \left({x.re}^{4} - {x.im}^{4}\right)}{x.re \cdot x.re + x.im \cdot x.im} - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\]

Alternative 14
Error	26.0
Cost	14400

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

Alternative 15
Error	20.5
Cost	14400

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

Alternative 16
Error	29.7
Cost	14272

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

Alternative 17
Error	31.9
Cost	14272

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

Alternative 18
Error	32.5
Cost	14144

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

Alternative 19
Error	32.8
Cost	14144

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

Alternative 20
Error	21.3
Cost	14080

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

Alternative 21
Error	31.8
Cost	14016

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

Alternative 22
Error	19.1
Cost	13952

\[x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \sqrt[3]{{\left(x.re \cdot x.im\right)}^{3} \cdot 8}\]

Alternative 23
Error	20.6
Cost	13952

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

Alternative 24
Error	45.9
Cost	9088

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

Alternative 25
Error	41.5
Cost	9088

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

Alternative 26
Error	23.9
Cost	8704

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

Alternative 27
Error	7.3
Cost	7040

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

Alternative 28
Error	25.8
Cost	6784

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

Alternative 29
Error	28.7
Cost	6528

\[{x.re}^{3}\]

Alternative 30
Error	0.2
Cost	1344

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

Alternative 31
Error	0.2
Cost	1216

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

Alternative 32
Error	7.3
Cost	1216

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

Alternative 33
Error	18.7
Cost	1152

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

Alternative 34
Error	7.3
Cost	1088

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

Alternative 35
Error	7.4
Cost	1088

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

Alternative 36
Error	0.3
Cost	1088

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

Alternative 37
Error	0.2
Cost	1088

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

Alternative 38
Error	1.0
Cost	960

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

Alternative 39
Error	1.0
Cost	960

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

Alternative 40
Error	28.7
Cost	832

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

Alternative 41
Error	61.8
Cost	64

\[1\]

Alternative 42
Error	46.7
Cost	64

\[0\]

Alternative 43
Error	61.9
Cost	64

\[-1\]

Derivation

1. Initial program 7.3

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

3. Applied difference-of-squares_binary64_3116 7.3

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

4. Applied associate-*l*_binary64_3088 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. Simplified 0.3

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

7. Final simplification 0.3

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

Reproduce

herbie shell --seed 2021022 
(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)))