2-ancestry mixing, zero discriminant

Percentage Accurate: 76.3% → 98.7%

Time: 2.0s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{2 \cdot a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{2 \cdot a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{0.5 \cdot g}}{\sqrt[3]{a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ (cbrt (* 0.5 g)) (cbrt a)))

C

double code(double g, double a) {
	return cbrt((0.5 * g)) / cbrt(a);
}

Java

public static double code(double g, double a) {
	return Math.cbrt((0.5 * g)) / Math.cbrt(a);
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(0.5 * g)) / cbrt(a))
end

Wolfram

code[g_, a_] := N[(N[Power[N[(0.5 * g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Step-by-step derivation

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cbrt.f64 98.7

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

3. Applied rewrites 98.7%

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

4. Taylor expanded in g around 0

   \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{1}{2} \cdot g}}}{\sqrt[3]{a}} \]
5. Step-by-step derivation

   1. lower-*.f64 98.7

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

6. Applied rewrites 98.7%

   \[\leadsto \frac{\sqrt[3]{\color{blue}{0.5 \cdot g}}}{\sqrt[3]{a}} \]
7. Add Preprocessing

Alternative 2: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{g}}{\sqrt[3]{a + a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ (cbrt g) (cbrt (+ a a))))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.3%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Step-by-step derivation

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. cbrt-div N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. count-2-rev N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{a + a}}} \]

   9. lower-+.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{a + a}}} \]

3. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{a + a}}} \]
4. Add Preprocessing

Alternative 3: 78.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{g}{2 \cdot a}} \leq 4 \cdot 10^{+101}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a + a}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(-\log \left(\frac{-1}{g}\right)\right) + \log \left(\frac{-0.5}{a}\right)\right) \cdot 0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (cbrt (/ g (* 2.0 a))) 4e+101)
   (cbrt (/ g (+ a a)))
   (exp (* (+ (- (log (/ -1.0 g))) (log (/ -0.5 a))) 0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if (cbrt((g / (2.0 * a))) <= 4e+101) {
		tmp = cbrt((g / (a + a)));
	} else {
		tmp = exp(((-log((-1.0 / g)) + log((-0.5 / a))) * 0.3333333333333333));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if (Math.cbrt((g / (2.0 * a))) <= 4e+101) {
		tmp = Math.cbrt((g / (a + a)));
	} else {
		tmp = Math.exp(((-Math.log((-1.0 / g)) + Math.log((-0.5 / a))) * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (cbrt(Float64(g / Float64(2.0 * a))) <= 4e+101)
		tmp = cbrt(Float64(g / Float64(a + a)));
	else
		tmp = exp(Float64(Float64(Float64(-log(Float64(-1.0 / g))) + log(Float64(-0.5 / a))) * 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 4e+101], N[Power[N[(g / N[(a + a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[Exp[N[(N[((-N[Log[N[(-1.0 / g), $MachinePrecision]], $MachinePrecision]) + N[Log[N[(-0.5 / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 3.9999999999999999e101

   1. Initial program 81.0%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

         \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a + a}}} \]

      3. lower-+.f64 81.0

         \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a + a}}} \]

   3. Applied rewrites 81.0%

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a + a}}} \]

   if 3.9999999999999999e101 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 6.5%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Step-by-step derivation

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow1/3 N/A

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

      5. pow-to-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. count-2-rev N/A

          \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot \frac{1}{3}} \]

      11. lower-+.f64 6.4

          \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot 0.3333333333333333} \]

   3. Applied rewrites 6.4%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]

   4. Taylor expanded in g around -inf

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{-1}{2}}{a}\right) + -1 \cdot \log \left(\frac{-1}{g}\right)\right)} \cdot \frac{1}{3}} \]
   5. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto e^{\left(\left(\mathsf{neg}\left(\log \left(\frac{-1}{g}\right)\right)\right) + \log \color{blue}{\left(\frac{\frac{-1}{2}}{a}\right)}\right) \cdot \frac{1}{3}} \]

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

         \[\leadsto e^{\left(\left(-\log \left(\frac{-1}{g}\right)\right) + \log \left(\frac{\frac{-1}{2}}{a}\right)\right) \cdot \frac{1}{3}} \]

      7. lower-log.f64 N/A

         \[\leadsto e^{\left(\left(-\log \left(\frac{-1}{g}\right)\right) + \log \left(\frac{\frac{-1}{2}}{a}\right)\right) \cdot \frac{1}{3}} \]

      8. lower-/.f64 45.1

         \[\leadsto e^{\left(\left(-\log \left(\frac{-1}{g}\right)\right) + \log \left(\frac{-0.5}{a}\right)\right) \cdot 0.3333333333333333} \]

   6. Applied rewrites 45.1%

      \[\leadsto e^{\color{blue}{\left(\left(-\log \left(\frac{-1}{g}\right)\right) + \log \left(\frac{-0.5}{a}\right)\right)} \cdot 0.3333333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{a + a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt (/ g (+ a a))))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.3%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a + a}}} \]

   3. lower-+.f64 76.3

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a + a}}} \]

3. Applied rewrites 76.3%

   \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a + a}}} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2025107 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))