2-ancestry mixing, zero discriminant

Average Error: 15.2 → 0.9

Time: 6.1s

Precision: binary64

Cost: 13248

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Java

public static double code(double g, double a) {
	return Math.cbrt((g / (2.0 * a)));
}

↓

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

Julia

function code(g, a)
	return cbrt(Float64(g / Float64(2.0 * a)))
end

↓

function code(g, a)
	return Float64(cbrt(Float64(g / 2.0)) * cbrt(Float64(1.0 / a)))
end

Wolfram

code[g_, a_] := N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

↓

code[g_, a_] := N[(N[Power[N[(g / 2.0), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.2

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

2. Applied egg-rr 48.7

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{g \cdot \frac{0.5}{a}}\right)} - 1} \]

3. Simplified 15.2

   \[\leadsto \color{blue}{\sqrt[3]{g \cdot \frac{0.5}{a}}} \]
   Proof

   [Start] 48.7	\[ e^{\mathsf{log1p}\left(\sqrt[3]{g \cdot \frac{0.5}{a}}\right)} - 1 \]
   expm1-def [=>] 27.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{g \cdot \frac{0.5}{a}}\right)\right)} \]
   expm1-log1p [=>] 15.2	\[ \color{blue}{\sqrt[3]{g \cdot \frac{0.5}{a}}} \]

4. Applied egg-rr 15.6

   \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{a}{g \cdot 0.5}}}} \]

5. Applied egg-rr 0.9

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

6. Final simplification 0.9

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

Alternatives

Alternative 1
Error	0.8
Cost	13120

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

Alternative 2
Error	0.9
Cost	13120

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

Alternative 3
Error	0.8
Cost	13120

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

Alternative 4
Error	15.2
Cost	6848

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

Alternative 5
Error	15.2
Cost	6848

\[\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}} \]

Alternative 6
Error	15.2
Cost	6720

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

Alternative 7
Error	15.2
Cost	6720

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

Reproduce

herbie shell --seed 2023066 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))