Average Error: 15.9 → 0.9

Time: 8.8s

Precision: binary64

Cost: 13248

\[\sqrt[3]{\frac{g}{2 \cdot a}}\]

↓

\[\frac{1}{\frac{\sqrt[3]{2 \cdot a}}{\sqrt[3]{g}}}\]

\sqrt[3]{\frac{g}{2 \cdot a}}

↓

\frac{1}{\frac{\sqrt[3]{2 \cdot a}}{\sqrt[3]{g}}}

(FPCore (g a) :precision binary64 (cbrt (/ g (* 2.0 a))))

↓

(FPCore (g a) :precision binary64 (/ 1.0 (/ (cbrt (* 2.0 a)) (cbrt g))))

double code(double g, double a) {
	return cbrt(g / (2.0 * a));
}

↓

double code(double g, double a) {
	return 1.0 / (cbrt(2.0 * a) / cbrt(g));
}

Error

The X axis uses an exponential scale — Bits error versus `g`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	1.8
Cost	77888

\[\frac{\sqrt[3]{\sqrt[3]{g}} \cdot \sqrt[3]{\sqrt[3]{g}}}{\sqrt[3]{\sqrt[3]{2 \cdot a}} \cdot \sqrt[3]{\sqrt[3]{2 \cdot a}}} \cdot \frac{\sqrt[3]{\sqrt[3]{g}}}{\sqrt[3]{\sqrt[3]{2 \cdot a}}}\]

Alternative 2
Error	1.6
Cost	65216

\[\left(\sqrt[3]{g} \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt[3]{2 \cdot a}} \cdot \sqrt[3]{\sqrt[3]{2 \cdot a}}}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt[3]{2 \cdot a}}}\]

Alternative 3
Error	32.6
Cost	64960

\[\frac{\sqrt{\sqrt[3]{g}}}{\sqrt[3]{\sqrt[3]{2 \cdot a}} \cdot \sqrt[3]{\sqrt[3]{2 \cdot a}}} \cdot \frac{\sqrt{\sqrt[3]{g}}}{\sqrt[3]{\sqrt[3]{2 \cdot a}}}\]

Alternative 4
Error	1.6
Cost	58688

\[\sqrt[3]{\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}} \cdot \sqrt[3]{\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}}\right)\]

Alternative 5
Error	1.6
Cost	52160

\[\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt[3]{2 \cdot a}} \cdot \sqrt[3]{\sqrt[3]{2 \cdot a}}} \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{\sqrt[3]{2 \cdot a}}}\]

Alternative 6
Error	1.6
Cost	51776

\[\frac{\sqrt[3]{\sqrt[3]{g}} \cdot \sqrt[3]{\sqrt[3]{g}}}{\sqrt[3]{2}} \cdot \frac{\sqrt[3]{\sqrt[3]{g}}}{\sqrt[3]{a}}\]

Alternative 7
Error	1.6
Cost	46016

\[\sqrt[3]{\frac{1}{\sqrt[3]{2 \cdot a}}} \cdot \left(\sqrt[3]{g} \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{2 \cdot a}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{2 \cdot a}}}\right)\right)\]

Alternative 8
Error	1.6
Cost	45760

\[\frac{\sqrt[3]{g}}{\sqrt[3]{\sqrt[3]{2 \cdot a}}} \cdot \frac{1}{\sqrt[3]{\sqrt[3]{2 \cdot a}} \cdot \sqrt[3]{\sqrt[3]{2 \cdot a}}}\]

Alternative 9
Error	1.6
Cost	45632

\[\frac{\sqrt[3]{g}}{\sqrt[3]{\sqrt[3]{2 \cdot a}} \cdot \left(\sqrt[3]{\sqrt[3]{2 \cdot a}} \cdot \sqrt[3]{\sqrt[3]{2 \cdot a}}\right)}\]

Alternative 10
Error	1.6
Cost	45376

\[\left(\sqrt[3]{\sqrt[3]{g}} \cdot \sqrt[3]{\sqrt[3]{g}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{g}}}{\sqrt[3]{2 \cdot a}}\]

Alternative 11
Error	1.6
Cost	45376

\[\frac{\sqrt[3]{\sqrt[3]{g}} \cdot \left(\sqrt[3]{\sqrt[3]{g}} \cdot \sqrt[3]{\sqrt[3]{g}}\right)}{\sqrt[3]{2 \cdot a}}\]

Alternative 12
Error	16.6
Cost	39488

\[\sqrt[3]{\sqrt[3]{\frac{g}{2 \cdot a}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{g}{2 \cdot a}}} \cdot \sqrt[3]{\sqrt[3]{\frac{g}{2 \cdot a}}}\right)\]

Alternative 13
Error	32.9
Cost	39104

\[\sqrt{\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}} \cdot \sqrt{\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}}\]

Alternative 14
Error	32.2
Cost	39104

\[\frac{\sqrt[3]{g}}{\sqrt{\sqrt[3]{2 \cdot a}}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\sqrt[3]{2 \cdot a}}}\]

Alternative 15
Error	32.5
Cost	38976

\[\frac{1}{\frac{\sqrt[3]{2}}{\sqrt[3]{\sqrt{g}}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{\sqrt{g}}}}\]

Alternative 16
Error	1.2
Cost	38976

\[\frac{\sqrt[3]{\sqrt[3]{g} \cdot \sqrt[3]{g}}}{\frac{\sqrt[3]{2 \cdot a}}{\sqrt[3]{\sqrt[3]{g}}}}\]

Alternative 17
Error	32.5
Cost	38848

\[\frac{\sqrt[3]{\sqrt{g}}}{\sqrt[3]{2}} \cdot \frac{\sqrt[3]{\sqrt{g}}}{\sqrt[3]{a}}\]

Alternative 18
Error	32.4
Cost	38848

\[\frac{\sqrt{\sqrt[3]{g}}}{\sqrt[3]{2}} \cdot \frac{\sqrt{\sqrt[3]{g}}}{\sqrt[3]{a}}\]

Alternative 19
Error	16.3
Cost	33088

\[\sqrt[3]{\sqrt[3]{\frac{g}{2 \cdot a}} \cdot \sqrt[3]{\frac{g}{2 \cdot a}}} \cdot \sqrt[3]{\sqrt[3]{\frac{g}{2 \cdot a}}}\]

Alternative 20
Error	32.2
Cost	32704

\[\frac{\sqrt[3]{g}}{\sqrt{\sqrt[3]{2 \cdot a}}} \cdot \frac{1}{\sqrt{\sqrt[3]{2 \cdot a}}}\]

Alternative 21
Error	32.2
Cost	32576

\[\frac{\frac{\sqrt[3]{g}}{\sqrt{\sqrt[3]{2 \cdot a}}}}{\sqrt{\sqrt[3]{2 \cdot a}}}\]

Alternative 22
Error	32.2
Cost	32576

\[\frac{\sqrt[3]{g}}{\sqrt{\sqrt[3]{2 \cdot a}} \cdot \sqrt{\sqrt[3]{2 \cdot a}}}\]

Alternative 23
Error	32.4
Cost	32448

\[\sqrt[3]{\sqrt{g}} \cdot \frac{\sqrt[3]{\sqrt{g}}}{\sqrt[3]{2 \cdot a}}\]

Alternative 24
Error	32.3
Cost	32448

\[\frac{\sqrt{\sqrt[3]{g}} \cdot \sqrt{\sqrt[3]{g}}}{\sqrt[3]{2 \cdot a}}\]

Alternative 25
Error	32.4
Cost	32448

\[\frac{\sqrt[3]{\sqrt{g}} \cdot \sqrt[3]{\sqrt{g}}}{\sqrt[3]{2 \cdot a}}\]

Alternative 26
Error	40.0
Cost	26304

\[\sqrt[3]{\sqrt{\frac{g}{2 \cdot a}}} \cdot \sqrt[3]{\sqrt{\frac{g}{2 \cdot a}}}\]

Alternative 27
Error	39.9
Cost	26304

\[\sqrt{\sqrt[3]{\frac{g}{2 \cdot a}}} \cdot \sqrt{\sqrt[3]{\frac{g}{2 \cdot a}}}\]

Alternative 28
Error	36.1
Cost	26048

\[\sqrt[3]{\frac{\sqrt{g}}{2}} \cdot \sqrt[3]{\frac{\sqrt{g}}{a}}\]

Alternative 29
Error	34.0
Cost	25920

\[\frac{e^{\log \left(\sqrt[3]{g}\right)}}{\sqrt[3]{2 \cdot a}}\]

Alternative 30
Error	33.9
Cost	25920

\[\frac{\sqrt[3]{g}}{e^{\log \left(\sqrt[3]{2 \cdot a}\right)}}\]

Alternative 31
Error	16.2
Cost	19840

\[\frac{\sqrt[3]{-1}}{\sqrt[3]{2}} \cdot \frac{1}{\sqrt[3]{\frac{-a}{g}}}\]

Alternative 32
Error	16.2
Cost	19776

\[\frac{1}{\sqrt[3]{\frac{a}{g}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{1}}}\]

Alternative 33
Error	1.1
Cost	19648

\[\frac{1}{\frac{\sqrt[3]{2} \cdot \sqrt[3]{a}}{\sqrt[3]{g}}}\]

Alternative 34
Error	16.4
Cost	19648

\[\sqrt[3]{\frac{g}{a}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{2}}\]

Alternative 35
Error	1.1
Cost	19648

\[\sqrt[3]{g} \cdot \frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{a}}\]

Alternative 36
Error	1.1
Cost	19520

\[\frac{\frac{\sqrt[3]{g}}{\sqrt[3]{2}}}{\sqrt[3]{a}}\]

Alternative 37
Error	1.1
Cost	19520

\[\frac{\sqrt[3]{g}}{\sqrt[3]{2} \cdot \sqrt[3]{a}}\]

Alternative 38
Error	41.2
Cost	19520

\[e^{\log \left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)}\]

Alternative 39
Error	0.9
Cost	13376

\[\frac{\frac{1}{\sqrt[3]{2 \cdot a}}}{\frac{1}{\sqrt[3]{g}}}\]

Alternative 40
Error	16.4
Cost	13248

\[\frac{1}{\sqrt[3]{2}} \cdot \sqrt[3]{\frac{g}{a}}\]

Alternative 41
Error	16.4
Cost	13248

\[\frac{1}{\frac{\sqrt[3]{2}}{\sqrt[3]{\frac{g}{a}}}}\]

Alternative 42
Error	16.2
Cost	13248

\[\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{\frac{a}{g}}}\]

Alternative 43
Error	0.9
Cost	13248

\[\sqrt[3]{g} \cdot \frac{1}{\sqrt[3]{2 \cdot a}}\]

Alternative 44
Error	0.9
Cost	13120

\[\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}\]

Alternative 45
Error	16.4
Cost	13120

\[\frac{\sqrt[3]{\frac{g}{a}}}{\sqrt[3]{2}}\]

Alternative 46
Error	0.8
Cost	13120

\[\sqrt[3]{g} \cdot \sqrt[3]{\frac{0.5}{a}}\]

Alternative 47
Error	16.0
Cost	13120

\[\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{0.5}\]

Alternative 48
Error	15.9
Cost	6848

\[\frac{1}{\sqrt[3]{\frac{2 \cdot a}{g}}}\]

Alternative 49
Error	15.9
Cost	6720

\[\sqrt[3]{\frac{g}{2 \cdot a}}\]

Alternative 50
Error	61.1
Cost	64

\[1\]

Alternative 51
Error	61.9
Cost	64

\[0\]

Alternative 52
Error	61.1
Cost	64

\[-1\]

Error

Derivation

Initial program 15.9
\[\sqrt[3]{\frac{g}{2 \cdot a}}\]
Using strategy rm
Applied cbrt-div_binary64_35200.9
\[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}}\]
Using strategy rm
Applied clear-num_binary64_34870.9
\[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{2 \cdot a}}{\sqrt[3]{g}}}}\]
Simplified0.9
\[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{2 \cdot a}}{\sqrt[3]{g}}}}\]
Final simplification0.9
\[\leadsto \frac{1}{\frac{\sqrt[3]{2 \cdot a}}{\sqrt[3]{g}}}\]

Reproduce

herbie shell --seed 2021022 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))