2-ancestry mixing, zero discriminant

Percentage Accurate: 75.9% → 98.7%

Time: 2.2s

Alternatives: 5

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: 75.9% 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

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

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 75.9%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 2: 75.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{2 \cdot a}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-106}:\\ \;\;\;\;e^{\left(\log g - \log \left(a + a\right)\right) \cdot 0.3333333333333333}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+98}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{a + a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log g - \left(\log 2 + \log a\right)\right) \cdot 0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (cbrt (/ g (* 2.0 a)))))
   (if (<= t_0 4e-106)
     (exp (* (- (log g) (log (+ a a))) 0.3333333333333333))
     (if (<= t_0 5e+98)
       (cbrt (/ 1.0 (/ (+ a a) g)))
       (exp (* (- (log g) (+ (log 2.0) (log a))) 0.3333333333333333))))))

C

double code(double g, double a) {
	double t_0 = cbrt((g / (2.0 * a)));
	double tmp;
	if (t_0 <= 4e-106) {
		tmp = exp(((log(g) - log((a + a))) * 0.3333333333333333));
	} else if (t_0 <= 5e+98) {
		tmp = cbrt((1.0 / ((a + a) / g)));
	} else {
		tmp = exp(((log(g) - (log(2.0) + log(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 <= 4e-106) {
		tmp = Math.exp(((Math.log(g) - Math.log((a + a))) * 0.3333333333333333));
	} else if (t_0 <= 5e+98) {
		tmp = Math.cbrt((1.0 / ((a + a) / g)));
	} else {
		tmp = Math.exp(((Math.log(g) - (Math.log(2.0) + Math.log(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 <= 4e-106)
		tmp = exp(Float64(Float64(log(g) - log(Float64(a + a))) * 0.3333333333333333));
	elseif (t_0 <= 5e+98)
		tmp = cbrt(Float64(1.0 / Float64(Float64(a + a) / g)));
	else
		tmp = exp(Float64(Float64(log(g) - Float64(log(2.0) + log(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, 4e-106], 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, 5e+98], N[Power[N[(1.0 / N[(N[(a + a), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[Exp[N[(N[(N[Log[g], $MachinePrecision] - N[(N[Log[2.0], $MachinePrecision] + N[Log[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))) < 3.99999999999999976e-106

   1. Initial program 75.9%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 36.0

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

      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 36.0

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

   3. Applied rewrites 36.0%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   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 22.8

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

   5. Applied rewrites 22.8%

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

   if 3.99999999999999976e-106 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 4.9999999999999998e98

   1. Initial program 75.9%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 75.2

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

      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 75.2

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

   3. Applied rewrites 75.2%

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

   if 4.9999999999999998e98 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 75.9%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 36.0

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

      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 36.0

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

   3. Applied rewrites 36.0%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   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 22.8

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

   5. Applied rewrites 22.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-log.f64 22.7

         \[\leadsto e^{\left(\log g - \left(\log 2 + \log a\right)\right) \cdot 0.3333333333333333} \]

   8. Applied rewrites 22.7%

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

Alternative 3: 42.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{2 \cdot a}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-106}:\\ \;\;\;\;e^{\left(\log g - \log \left(a + a\right)\right) \cdot 0.3333333333333333}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+98}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{a + a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(0.5 \cdot g\right) + -1 \cdot \log a\right) \cdot 0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (cbrt (/ g (* 2.0 a)))))
   (if (<= t_0 4e-106)
     (exp (* (- (log g) (log (+ a a))) 0.3333333333333333))
     (if (<= t_0 5e+98)
       (cbrt (/ 1.0 (/ (+ a a) g)))
       (exp (* (+ (log (* 0.5 g)) (* -1.0 (log a))) 0.3333333333333333))))))

C

double code(double g, double a) {
	double t_0 = cbrt((g / (2.0 * a)));
	double tmp;
	if (t_0 <= 4e-106) {
		tmp = exp(((log(g) - log((a + a))) * 0.3333333333333333));
	} else if (t_0 <= 5e+98) {
		tmp = cbrt((1.0 / ((a + a) / g)));
	} else {
		tmp = exp(((log((0.5 * g)) + (-1.0 * log(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 <= 4e-106) {
		tmp = Math.exp(((Math.log(g) - Math.log((a + a))) * 0.3333333333333333));
	} else if (t_0 <= 5e+98) {
		tmp = Math.cbrt((1.0 / ((a + a) / g)));
	} else {
		tmp = Math.exp(((Math.log((0.5 * g)) + (-1.0 * Math.log(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 <= 4e-106)
		tmp = exp(Float64(Float64(log(g) - log(Float64(a + a))) * 0.3333333333333333));
	elseif (t_0 <= 5e+98)
		tmp = cbrt(Float64(1.0 / Float64(Float64(a + a) / g)));
	else
		tmp = exp(Float64(Float64(log(Float64(0.5 * g)) + Float64(-1.0 * log(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, 4e-106], 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, 5e+98], N[Power[N[(1.0 / N[(N[(a + a), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 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]]]]

Derivation

1. Split input into 3 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 3.99999999999999976e-106

   1. Initial program 75.9%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 36.0

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

      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 36.0

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

   3. Applied rewrites 36.0%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   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 22.8

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

   5. Applied rewrites 22.8%

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

   if 3.99999999999999976e-106 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 4.9999999999999998e98

   1. Initial program 75.9%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 75.2

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

      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 75.2

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

   3. Applied rewrites 75.2%

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

   if 4.9999999999999998e98 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 75.9%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 36.0

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

      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 36.0

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

   3. Applied rewrites 36.0%

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

   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 22.8

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

   6. Applied rewrites 22.8%

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

Alternative 4: 42.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{2 \cdot a}}\\ t_1 := e^{\left(\log g - \log \left(a + a\right)\right) \cdot 0.3333333333333333}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+98}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{a + a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 4e-106)
     t_1
     (if (<= t_0 5e+98) (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 <= 4e-106) {
		tmp = t_1;
	} else if (t_0 <= 5e+98) {
		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 <= 4e-106) {
		tmp = t_1;
	} else if (t_0 <= 5e+98) {
		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 <= 4e-106)
		tmp = t_1;
	elseif (t_0 <= 5e+98)
		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, 4e-106], t$95$1, If[LessEqual[t$95$0, 5e+98], 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))) < 3.99999999999999976e-106 or 4.9999999999999998e98 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 75.9%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 36.0

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

      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 36.0

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

   3. Applied rewrites 36.0%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   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 22.8

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

   5. Applied rewrites 22.8%

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

   if 3.99999999999999976e-106 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 4.9999999999999998e98

   1. Initial program 75.9%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 75.2

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

      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 75.2

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

   3. Applied rewrites 75.2%

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

Alternative 5: 42.2% 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 75.9%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 75.9

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

3. Applied rewrites 75.9%

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

Reproduce

herbie shell --seed 2025144 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))