2-ancestry mixing, zero discriminant

Percentage Accurate: 76.1% → 98.7%

Time: 2.5s

Alternatives: 8

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: 76.1% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{g}}{\sqrt[3]{a} \cdot {2}^{0.3333333333333333}} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (/ (cbrt g) (* (cbrt a) (pow 2.0 0.3333333333333333))))

C

double code(double g, double a) {
	return cbrt(g) / (cbrt(a) * pow(2.0, 0.3333333333333333));
}

Java

public static double code(double g, double a) {
	return Math.cbrt(g) / (Math.cbrt(a) * Math.pow(2.0, 0.3333333333333333));
}

Julia

function code(g, a)
	return Float64(cbrt(g) / Float64(cbrt(a) * (2.0 ^ 0.3333333333333333)))
end

Wolfram

code[g_, a_] := N[(N[Power[g, 1/3], $MachinePrecision] / N[(N[Power[a, 1/3], $MachinePrecision] * N[Power[2.0, 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.1%

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

   1. lift-+.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. count-2-rev N/A

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

   4. *-commutative N/A

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

   5. cbrt-unprod N/A

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

   6. lower-*.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lower-cbrt.f64 98.2

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

5. Applied rewrites 98.2%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lower-pow.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{a} \cdot \color{blue}{{2}^{0.3333333333333333}}} \]

7. Applied rewrites 98.7%

   \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{a} \cdot \color{blue}{{2}^{0.3333333333333333}}} \]
8. Add Preprocessing

Alternative 2: 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 76.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 3: 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 76.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 4: 80.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{2 \cdot a}}\\ t_1 := \sqrt[3]{\frac{g}{a + a}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-108}:\\ \;\;\;\;e^{\left(\log \left(0.5 \cdot g\right) - \log a\right) \cdot 0.3333333333333333}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log g - \left(\log a + \log 2\right)\right) \cdot 0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (cbrt (/ g (* 2.0 a)))) (t_1 (cbrt (/ g (+ a a)))))
   (if (<= t_0 -4e-106)
     t_1
     (if (<= t_0 5e-108)
       (exp (* (- (log (* 0.5 g)) (log a)) 0.3333333333333333))
       (if (<= t_0 5e+101)
         t_1
         (exp (* (- (log g) (+ (log a) (log 2.0))) 0.3333333333333333)))))))

C

double code(double g, double a) {
	double t_0 = cbrt((g / (2.0 * a)));
	double t_1 = cbrt((g / (a + a)));
	double tmp;
	if (t_0 <= -4e-106) {
		tmp = t_1;
	} else if (t_0 <= 5e-108) {
		tmp = exp(((log((0.5 * g)) - log(a)) * 0.3333333333333333));
	} else if (t_0 <= 5e+101) {
		tmp = t_1;
	} else {
		tmp = exp(((log(g) - (log(a) + log(2.0))) * 0.3333333333333333));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((g / (2.0 * a)));
	double t_1 = Math.cbrt((g / (a + a)));
	double tmp;
	if (t_0 <= -4e-106) {
		tmp = t_1;
	} else if (t_0 <= 5e-108) {
		tmp = Math.exp(((Math.log((0.5 * g)) - Math.log(a)) * 0.3333333333333333));
	} else if (t_0 <= 5e+101) {
		tmp = t_1;
	} else {
		tmp = Math.exp(((Math.log(g) - (Math.log(a) + Math.log(2.0))) * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = cbrt(Float64(g / Float64(2.0 * a)))
	t_1 = cbrt(Float64(g / Float64(a + a)))
	tmp = 0.0
	if (t_0 <= -4e-106)
		tmp = t_1;
	elseif (t_0 <= 5e-108)
		tmp = exp(Float64(Float64(log(Float64(0.5 * g)) - log(a)) * 0.3333333333333333));
	elseif (t_0 <= 5e+101)
		tmp = t_1;
	else
		tmp = exp(Float64(Float64(log(g) - Float64(log(a) + log(2.0))) * 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]}, Block[{t$95$1 = N[Power[N[(g / N[(a + a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t$95$0, -4e-106], t$95$1, If[LessEqual[t$95$0, 5e-108], N[Exp[N[(N[(N[Log[N[(0.5 * g), $MachinePrecision]], $MachinePrecision] - N[Log[a], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 5e+101], t$95$1, N[Exp[N[(N[(N[Log[g], $MachinePrecision] - N[(N[Log[a], $MachinePrecision] + N[Log[2.0], $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 or 5e-108 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 4.99999999999999989e101

   1. Initial program 91.5%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 91.5

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

   3. Applied rewrites 91.5%

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

   if -3.99999999999999976e-106 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 5e-108

   1. Initial program 5.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 4.5

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

   3. Applied rewrites 4.5%

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

   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 22.4

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

   6. Applied rewrites 22.4%

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

   7. Taylor expanded in g around inf

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

      1. diff-log N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 22.0

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

   9. Applied rewrites 22.0%

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

   11. Applied rewrites 22.0%

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

   if 4.99999999999999989e101 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 91.5%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 42.6

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

   3. Applied rewrites 42.6%

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

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

      3. lift-/.f64 N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lift-+.f64 21.0

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

   5. Applied rewrites 21.0%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. count-2-rev N/A

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

      4. sum-log N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-log.f64 21.0

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

   7. Applied rewrites 21.0%

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

Alternative 5: 80.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{2 \cdot a}}\\ t_1 := \sqrt[3]{\frac{g}{a + a}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-108}:\\ \;\;\;\;e^{\left(\log \left(0.5 \cdot g\right) - \log a\right) \cdot 0.3333333333333333}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log g - \log \left(a + a\right)\right) \cdot 0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (cbrt (/ g (* 2.0 a)))) (t_1 (cbrt (/ g (+ a a)))))
   (if (<= t_0 -4e-106)
     t_1
     (if (<= t_0 5e-108)
       (exp (* (- (log (* 0.5 g)) (log a)) 0.3333333333333333))
       (if (<= t_0 5e+101)
         t_1
         (exp (* (- (log g) (log (+ a a))) 0.3333333333333333)))))))

C

double code(double g, double a) {
	double t_0 = cbrt((g / (2.0 * a)));
	double t_1 = cbrt((g / (a + a)));
	double tmp;
	if (t_0 <= -4e-106) {
		tmp = t_1;
	} else if (t_0 <= 5e-108) {
		tmp = exp(((log((0.5 * g)) - log(a)) * 0.3333333333333333));
	} else if (t_0 <= 5e+101) {
		tmp = t_1;
	} 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 t_1 = Math.cbrt((g / (a + a)));
	double tmp;
	if (t_0 <= -4e-106) {
		tmp = t_1;
	} else if (t_0 <= 5e-108) {
		tmp = Math.exp(((Math.log((0.5 * g)) - Math.log(a)) * 0.3333333333333333));
	} else if (t_0 <= 5e+101) {
		tmp = t_1;
	} 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)))
	t_1 = cbrt(Float64(g / Float64(a + a)))
	tmp = 0.0
	if (t_0 <= -4e-106)
		tmp = t_1;
	elseif (t_0 <= 5e-108)
		tmp = exp(Float64(Float64(log(Float64(0.5 * g)) - log(a)) * 0.3333333333333333));
	elseif (t_0 <= 5e+101)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[Power[N[(g / N[(a + a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t$95$0, -4e-106], t$95$1, If[LessEqual[t$95$0, 5e-108], N[Exp[N[(N[(N[Log[N[(0.5 * g), $MachinePrecision]], $MachinePrecision] - N[Log[a], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 5e+101], t$95$1, 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))) < -3.99999999999999976e-106 or 5e-108 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 4.99999999999999989e101

   1. Initial program 91.5%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 91.5

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

   3. Applied rewrites 91.5%

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

   if -3.99999999999999976e-106 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 5e-108

   1. Initial program 5.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 4.5

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

   3. Applied rewrites 4.5%

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

   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 22.4

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

   6. Applied rewrites 22.4%

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

   7. Taylor expanded in g around inf

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

      1. diff-log N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 22.0

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

   9. Applied rewrites 22.0%

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

   11. Applied rewrites 22.0%

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

   if 4.99999999999999989e101 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 91.5%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 42.6

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

   3. Applied rewrites 42.6%

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

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

      3. lift-/.f64 N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lift-+.f64 21.0

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

   5. Applied rewrites 21.0%

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

Alternative 6: 80.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(\log g - \log \left(a + a\right)\right) \cdot 0.3333333333333333}\\ t_1 := \sqrt[3]{\frac{g}{2 \cdot a}}\\ t_2 := \sqrt[3]{\frac{g}{a + a}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-106}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+101}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (exp (* (- (log g) (log (+ a a))) 0.3333333333333333)))
        (t_1 (cbrt (/ g (* 2.0 a))))
        (t_2 (cbrt (/ g (+ a a)))))
   (if (<= t_1 -4e-106)
     t_2
     (if (<= t_1 5e-108) t_0 (if (<= t_1 5e+101) t_2 t_0)))))

C

double code(double g, double a) {
	double t_0 = exp(((log(g) - log((a + a))) * 0.3333333333333333));
	double t_1 = cbrt((g / (2.0 * a)));
	double t_2 = cbrt((g / (a + a)));
	double tmp;
	if (t_1 <= -4e-106) {
		tmp = t_2;
	} else if (t_1 <= 5e-108) {
		tmp = t_0;
	} else if (t_1 <= 5e+101) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.exp(((Math.log(g) - Math.log((a + a))) * 0.3333333333333333));
	double t_1 = Math.cbrt((g / (2.0 * a)));
	double t_2 = Math.cbrt((g / (a + a)));
	double tmp;
	if (t_1 <= -4e-106) {
		tmp = t_2;
	} else if (t_1 <= 5e-108) {
		tmp = t_0;
	} else if (t_1 <= 5e+101) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = exp(Float64(Float64(log(g) - log(Float64(a + a))) * 0.3333333333333333))
	t_1 = cbrt(Float64(g / Float64(2.0 * a)))
	t_2 = cbrt(Float64(g / Float64(a + a)))
	tmp = 0.0
	if (t_1 <= -4e-106)
		tmp = t_2;
	elseif (t_1 <= 5e-108)
		tmp = t_0;
	elseif (t_1 <= 5e+101)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Exp[N[(N[(N[Log[g], $MachinePrecision] - N[Log[N[(a + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(g / N[(a + a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t$95$1, -4e-106], t$95$2, If[LessEqual[t$95$1, 5e-108], t$95$0, If[LessEqual[t$95$1, 5e+101], t$95$2, t$95$0]]]]]]

Derivation

1. Split input into 2 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < -3.99999999999999976e-106 or 5e-108 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 4.99999999999999989e101

   1. Initial program 91.5%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 91.5

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

   3. Applied rewrites 91.5%

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

   if -3.99999999999999976e-106 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 5e-108 or 4.99999999999999989e101 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 5.7%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 4.9

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

   3. Applied rewrites 4.9%

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

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

      3. lift-/.f64 N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lift-+.f64 30.1

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

   5. Applied rewrites 30.1%

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

Alternative 7: 76.1% 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 76.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 76.1

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

3. Applied rewrites 76.1%

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

Alternative 8: 4.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

   1. lift-+.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. count-2-rev N/A

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

   4. *-commutative N/A

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

   5. cbrt-unprod N/A

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

   6. lower-*.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lower-cbrt.f64 98.2

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

5. Applied rewrites 98.2%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lower-pow.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{a} \cdot \color{blue}{{2}^{0.3333333333333333}}} \]

7. Applied rewrites 98.7%

   \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{a} \cdot \color{blue}{{2}^{0.3333333333333333}}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow1/3 N/A

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

   7. cbrt-unprod N/A

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

   8. *-commutative N/A

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

   9. count-2-rev N/A

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

   10. cbrt-undiv N/A

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

   11. rem-exp-log N/A

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

   12. diff-log N/A

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

   13. lower-cbrt.f64 N/A

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

   14. diff-log N/A

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

   15. rem-exp-log N/A

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

   16. flip-+ N/A

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

   17. +-inverses N/A

       \[\leadsto \sqrt[3]{\frac{g}{\frac{\color{blue}{0}}{a - a}}} \]

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. +-inverses N/A

       \[\leadsto \sqrt[3]{\frac{g}{\frac{1 \cdot 1 - 1 \cdot 1}{\color{blue}{0}}}} \]

   22. metadata-eval N/A

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

   23. flip-+ N/A

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

   24. metadata-eval N/A

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

   25. lower-/.f64 4.8

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

9. Applied rewrites 4.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025120 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))