2-ancestry mixing, zero discriminant

Percentage Accurate: 76.2% → 98.7%

Time: 1.8s

Alternatives: 8

Speedup: 0.6×

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: 76.2% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

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

   3. lift-*.f64 N/A

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

   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. mult-flip N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-cbrt.f64 98.7

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

3. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot 0.5}}{\sqrt[3]{a}}} \]
4. 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 76.2%

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

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

   3. cbrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 98.7

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

   7. lift-*.f64 N/A

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

   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: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{2 \cdot a}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-103}:\\ \;\;\;\;e^{\left(\log \left(0.5 \cdot g\right) + -1 \cdot \log a\right) \cdot 0.3333333333333333}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{1}{g \cdot \frac{1}{a + a}}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log g - \log \left(a + a\right)\right) \cdot 0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (cbrt (/ g (* 2.0 a)))))
   (if (<= t_0 2e-103)
     (exp (* (+ (log (* 0.5 g)) (* -1.0 (log a))) 0.3333333333333333))
     (if (<= t_0 5e+102)
       (/ 1.0 (cbrt (/ 1.0 (* g (/ 1.0 (+ a a))))))
       (exp (* (- (log g) (log (+ a a))) 0.3333333333333333))))))

C

double code(double g, double a) {
	double t_0 = cbrt((g / (2.0 * a)));
	double tmp;
	if (t_0 <= 2e-103) {
		tmp = exp(((log((0.5 * g)) + (-1.0 * log(a))) * 0.3333333333333333));
	} else if (t_0 <= 5e+102) {
		tmp = 1.0 / cbrt((1.0 / (g * (1.0 / (a + a)))));
	} else {
		tmp = exp(((log(g) - log((a + a))) * 0.3333333333333333));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((g / (2.0 * a)));
	double tmp;
	if (t_0 <= 2e-103) {
		tmp = Math.exp(((Math.log((0.5 * g)) + (-1.0 * Math.log(a))) * 0.3333333333333333));
	} else if (t_0 <= 5e+102) {
		tmp = 1.0 / Math.cbrt((1.0 / (g * (1.0 / (a + a)))));
	} else {
		tmp = Math.exp(((Math.log(g) - Math.log((a + a))) * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = cbrt(Float64(g / Float64(2.0 * a)))
	tmp = 0.0
	if (t_0 <= 2e-103)
		tmp = exp(Float64(Float64(log(Float64(0.5 * g)) + Float64(-1.0 * log(a))) * 0.3333333333333333));
	elseif (t_0 <= 5e+102)
		tmp = Float64(1.0 / cbrt(Float64(1.0 / Float64(g * Float64(1.0 / Float64(a + a))))));
	else
		tmp = exp(Float64(Float64(log(g) - log(Float64(a + a))) * 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t$95$0, 2e-103], N[Exp[N[(N[(N[Log[N[(0.5 * g), $MachinePrecision]], $MachinePrecision] + N[(-1.0 * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 5e+102], N[(1.0 / N[Power[N[(1.0 / N[(g * N[(1.0 / N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Log[g], $MachinePrecision] - N[Log[N[(a + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 1.99999999999999992e-103

   1. Initial program 76.2%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 36.6

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right)} \cdot 0.3333333333333333} \]

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 36.6

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

   3. Applied rewrites 36.6%

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

   4. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 22.7

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

   6. Applied rewrites 22.7%

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

   if 1.99999999999999992e-103 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 5e102

   1. Initial program 76.2%

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

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

      3. div-flip N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-/.f64 76.2

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 76.2

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

   3. Applied rewrites 76.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 75.5

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

   7. Applied rewrites 75.5%

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

   if 5e102 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 76.2%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 36.6

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right)} \cdot 0.3333333333333333} \]

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 36.6

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

   3. Applied rewrites 36.6%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 22.7

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

   5. Applied rewrites 22.7%

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

Alternative 4: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{2 \cdot a}}\\ t_1 := e^{\left(\log g - \log \left(a + a\right)\right) \cdot 0.3333333333333333}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{1}{g \cdot \frac{1}{a + a}}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (cbrt (/ g (* 2.0 a))))
        (t_1 (exp (* (- (log g) (log (+ a a))) 0.3333333333333333))))
   (if (<= t_0 2e-103)
     t_1
     (if (<= t_0 5e+102) (/ 1.0 (cbrt (/ 1.0 (* g (/ 1.0 (+ a a)))))) t_1))))

C

double code(double g, double a) {
	double t_0 = cbrt((g / (2.0 * a)));
	double t_1 = exp(((log(g) - log((a + a))) * 0.3333333333333333));
	double tmp;
	if (t_0 <= 2e-103) {
		tmp = t_1;
	} else if (t_0 <= 5e+102) {
		tmp = 1.0 / cbrt((1.0 / (g * (1.0 / (a + a)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((g / (2.0 * a)));
	double t_1 = Math.exp(((Math.log(g) - Math.log((a + a))) * 0.3333333333333333));
	double tmp;
	if (t_0 <= 2e-103) {
		tmp = t_1;
	} else if (t_0 <= 5e+102) {
		tmp = 1.0 / Math.cbrt((1.0 / (g * (1.0 / (a + a)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = cbrt(Float64(g / Float64(2.0 * a)))
	t_1 = exp(Float64(Float64(log(g) - log(Float64(a + a))) * 0.3333333333333333))
	tmp = 0.0
	if (t_0 <= 2e-103)
		tmp = t_1;
	elseif (t_0 <= 5e+102)
		tmp = Float64(1.0 / cbrt(Float64(1.0 / Float64(g * Float64(1.0 / Float64(a + a))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[Log[g], $MachinePrecision] - N[Log[N[(a + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 2e-103], t$95$1, If[LessEqual[t$95$0, 5e+102], N[(1.0 / N[Power[N[(1.0 / N[(g * N[(1.0 / N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 1.99999999999999992e-103 or 5e102 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 76.2%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 36.6

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right)} \cdot 0.3333333333333333} \]

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 36.6

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

   3. Applied rewrites 36.6%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 22.7

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

   5. Applied rewrites 22.7%

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

   if 1.99999999999999992e-103 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 5e102

   1. Initial program 76.2%

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

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

      3. div-flip N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-/.f64 76.2

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 76.2

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

   3. Applied rewrites 76.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 75.5

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

   7. Applied rewrites 75.5%

      \[\leadsto \frac{1}{\sqrt[3]{\frac{1}{\color{blue}{g \cdot \frac{1}{a + a}}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{2 \cdot a}}\\ t_1 := e^{\left(\log g - \log \left(a + a\right)\right) \cdot 0.3333333333333333}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{a + a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (cbrt (/ g (* 2.0 a))))
        (t_1 (exp (* (- (log g) (log (+ a a))) 0.3333333333333333))))
   (if (<= t_0 2e-103)
     t_1
     (if (<= t_0 5e+102) (cbrt (/ 1.0 (/ (+ a a) g))) t_1))))

C

double code(double g, double a) {
	double t_0 = cbrt((g / (2.0 * a)));
	double t_1 = exp(((log(g) - log((a + a))) * 0.3333333333333333));
	double tmp;
	if (t_0 <= 2e-103) {
		tmp = t_1;
	} else if (t_0 <= 5e+102) {
		tmp = cbrt((1.0 / ((a + a) / g)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((g / (2.0 * a)));
	double t_1 = Math.exp(((Math.log(g) - Math.log((a + a))) * 0.3333333333333333));
	double tmp;
	if (t_0 <= 2e-103) {
		tmp = t_1;
	} else if (t_0 <= 5e+102) {
		tmp = Math.cbrt((1.0 / ((a + a) / g)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = cbrt(Float64(g / Float64(2.0 * a)))
	t_1 = exp(Float64(Float64(log(g) - log(Float64(a + a))) * 0.3333333333333333))
	tmp = 0.0
	if (t_0 <= 2e-103)
		tmp = t_1;
	elseif (t_0 <= 5e+102)
		tmp = cbrt(Float64(1.0 / Float64(Float64(a + a) / g)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[Log[g], $MachinePrecision] - N[Log[N[(a + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 2e-103], t$95$1, If[LessEqual[t$95$0, 5e+102], N[Power[N[(1.0 / N[(N[(a + a), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 1.99999999999999992e-103 or 5e102 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 76.2%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 36.6

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right)} \cdot 0.3333333333333333} \]

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 36.6

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

   3. Applied rewrites 36.6%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 22.7

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

   5. Applied rewrites 22.7%

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

   if 1.99999999999999992e-103 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 5e102

   1. Initial program 76.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 75.6

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 75.6

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

   3. Applied rewrites 75.6%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{a + a}{g}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 43.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt[3]{\frac{a + a}{g}}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ 1.0 (cbrt (/ (+ a a) g))))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

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

   3. div-flip N/A

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

   4. cbrt-div N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-/.f64 76.2

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

   9. lift-*.f64 N/A

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

   10. count-2-rev N/A

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

   11. lower-+.f64 76.2

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

3. Applied rewrites 76.2%

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{a + a}{g}}}} \]
4. Add Preprocessing

Alternative 7: 43.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval 76.2

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

3. Applied rewrites 76.2%

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

Alternative 8: 43.6% 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 76.2%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 76.2

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

3. Applied rewrites 76.2%

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

Reproduce

herbie shell --seed 2025148 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))