2-ancestry mixing, zero discriminant

Average Error: 15.7 → 0.9

Time: 6.7s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 (1.0 / Math.cbrt(a)) * (1.0 / Math.cbrt((2.0 / g)));
}

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 15.7

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

2. Applied egg-rr 48.5

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

3. Simplified 15.7

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

   [Start] 48.5	\[ e^{\mathsf{log1p}\left(\sqrt[3]{g \cdot \frac{0.5}{a}}\right)} - 1 \]
   expm1-def [=>] 28.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{g \cdot \frac{0.5}{a}}\right)\right)} \]
   expm1-log1p [=>] 15.7	\[ \color{blue}{\sqrt[3]{g \cdot \frac{0.5}{a}}} \]

4. Applied egg-rr 15.8

   \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g \cdot 0.5}{{\left(\sqrt[3]{a}\right)}^{2}}}{\sqrt[3]{a}}}} \]

5. Applied egg-rr 15.7

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

6. Simplified 15.7

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

   [Start] 15.7	\[ \frac{1}{\sqrt[3]{2 \cdot \frac{a}{g}}} \]
   associate-*r/ [=>] 15.7	\[ \frac{1}{\sqrt[3]{\color{blue}{\frac{2 \cdot a}{g}}}} \]
   associate-/l* [=>] 16.2	\[ \frac{1}{\sqrt[3]{\color{blue}{\frac{2}{\frac{g}{a}}}}} \]
   associate-/r/ [=>] 15.7	\[ \frac{1}{\sqrt[3]{\color{blue}{\frac{2}{g} \cdot a}}} \]

7. Applied egg-rr 0.9

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{2}{g}}\right)}^{-1} \cdot {\left(\sqrt[3]{a}\right)}^{-1}} \]

8. Simplified 0.9

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

   [Start] 0.9	\[ {\left(\sqrt[3]{\frac{2}{g}}\right)}^{-1} \cdot {\left(\sqrt[3]{a}\right)}^{-1} \]
   *-commutative [=>] 0.9	\[ \color{blue}{{\left(\sqrt[3]{a}\right)}^{-1} \cdot {\left(\sqrt[3]{\frac{2}{g}}\right)}^{-1}} \]
   unpow-1 [=>] 0.9	\[ \color{blue}{\frac{1}{\sqrt[3]{a}}} \cdot {\left(\sqrt[3]{\frac{2}{g}}\right)}^{-1} \]
   unpow-1 [=>] 0.9	\[ \frac{1}{\sqrt[3]{a}} \cdot \color{blue}{\frac{1}{\sqrt[3]{\frac{2}{g}}}} \]

9. Final simplification 0.9

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

Alternatives

Alternative 1
Error	0.8
Cost	13120

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

Alternative 2
Error	0.8
Cost	13120

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

Alternative 3
Error	0.8
Cost	13120

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

Alternative 4
Error	15.7
Cost	6848

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

Alternative 5
Error	15.7
Cost	6720

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

Reproduce

herbie shell --seed 2023054 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))