2-ancestry mixing, positive discriminant

Percentage Accurate: 44.1% → 95.7%

Time: 15.5s

Alternatives: 8

Speedup: 2.6×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{t\_0 \cdot \left(\left(-g\right) - t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
   (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[((-g) - t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Initial Program: 44.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{t\_0 \cdot \left(\left(-g\right) - t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
   (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[((-g) - t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 95.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.0%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

4. Applied rewrites 55.2%

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

5. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. metadata-eval N/A

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

   7. rem-square-sqrt N/A

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

   8. unpow2 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. unpow2 N/A

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

   11. rem-square-sqrt N/A

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

   12. metadata-eval 72.4

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

7. Applied rewrites 72.4%

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

9. Applied rewrites 95.6%

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

11. Applied rewrites 95.6%

    \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\frac{-1}{a}} \]
12. Add Preprocessing

Alternative 2: 88.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{2 \cdot a} \leq -5 \cdot 10^{-294}:\\ \;\;\;\;\sqrt[3]{g} \cdot {\left(-a\right)}^{-0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (/ 1.0 (* 2.0 a)) -5e-294)
   (* (cbrt g) (pow (- a) -0.3333333333333333))
   (* (cbrt (- g)) (pow a -0.3333333333333333))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (2.0 * a)) <= -5e-294) {
		tmp = cbrt(g) * pow(-a, -0.3333333333333333);
	} else {
		tmp = cbrt(-g) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (2.0 * a)) <= -5e-294) {
		tmp = Math.cbrt(g) * Math.pow(-a, -0.3333333333333333);
	} else {
		tmp = Math.cbrt(-g) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(1.0 / Float64(2.0 * a)) <= -5e-294)
		tmp = Float64(cbrt(g) * (Float64(-a) ^ -0.3333333333333333));
	else
		tmp = Float64(cbrt(Float64(-g)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -5e-294], N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[(-a), -0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(N[Power[(-g), 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < -5.0000000000000003e-294

   1. Initial program 56.4%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. metadata-eval N/A

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

      7. rem-square-sqrt N/A

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

      8. unpow2 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. unpow2 N/A

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

      11. rem-square-sqrt N/A

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

      12. metadata-eval 74.9

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

   7. Applied rewrites 74.9%

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

   9. Applied rewrites 96.2%

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

   11. Applied rewrites 90.1%

       \[\leadsto {\left(-a\right)}^{-0.3333333333333333} \cdot \color{blue}{\sqrt[3]{g}} \]

   if -5.0000000000000003e-294 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 44.7%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. metadata-eval N/A

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

      7. rem-square-sqrt N/A

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

      8. unpow2 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. unpow2 N/A

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

      11. rem-square-sqrt N/A

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

      12. metadata-eval 69.5

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

   7. Applied rewrites 69.5%

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

   9. Applied rewrites 88.7%

      \[\leadsto \sqrt[3]{-g} \cdot \color{blue}{{a}^{-0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{2 \cdot a} \leq -5 \cdot 10^{-294}:\\ \;\;\;\;\sqrt[3]{g} \cdot {\left(-a\right)}^{-0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 4 \cdot 10^{-307}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{-g}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (* 2.0 a) 4e-307)
   (/ 1.0 (cbrt (/ a (- g))))
   (* (cbrt (- g)) (pow a -0.3333333333333333))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((2.0 * a) <= 4e-307) {
		tmp = 1.0 / cbrt((a / -g));
	} else {
		tmp = cbrt(-g) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((2.0 * a) <= 4e-307) {
		tmp = 1.0 / Math.cbrt((a / -g));
	} else {
		tmp = Math.cbrt(-g) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= 4e-307)
		tmp = Float64(1.0 / cbrt(Float64(a / Float64(-g))));
	else
		tmp = Float64(cbrt(Float64(-g)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], 4e-307], N[(1.0 / N[Power[N[(a / (-g)), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[(-g), 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < 3.99999999999999964e-307

   1. Initial program 56.1%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 60.2%

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

   5. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. metadata-eval N/A

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

      7. rem-square-sqrt N/A

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

      8. unpow2 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. unpow2 N/A

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

      11. rem-square-sqrt N/A

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

      12. metadata-eval 75.1

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

   7. Applied rewrites 75.1%

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

   9. Applied rewrites 75.2%

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

   if 3.99999999999999964e-307 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 45.0%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 49.3%

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

   5. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. metadata-eval N/A

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

      7. rem-square-sqrt N/A

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

      8. unpow2 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. unpow2 N/A

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

      11. rem-square-sqrt N/A

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

      12. metadata-eval 69.2

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

   7. Applied rewrites 69.2%

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

   9. Applied rewrites 88.7%

      \[\leadsto \sqrt[3]{-g} \cdot \color{blue}{{a}^{-0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 4 \cdot 10^{-307}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{-g}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (g h a) :precision binary64 (/ (cbrt (- g)) (cbrt a)))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.0%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

4. Applied rewrites 55.2%

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

5. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. metadata-eval N/A

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

   7. rem-square-sqrt N/A

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

   8. unpow2 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. unpow2 N/A

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

   11. rem-square-sqrt N/A

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

   12. metadata-eval 72.4

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

7. Applied rewrites 72.4%

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

9. Applied rewrites 95.6%

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

Alternative 5: 73.7% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (g h a) :precision binary64 (/ 1.0 (cbrt (/ a (- g)))))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.0%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

4. Applied rewrites 55.2%

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

5. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. metadata-eval N/A

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

   7. rem-square-sqrt N/A

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

   8. unpow2 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. unpow2 N/A

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

   11. rem-square-sqrt N/A

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

   12. metadata-eval 72.4

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

7. Applied rewrites 72.4%

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

9. Applied rewrites 73.2%

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

10. Final simplification 73.2%

    \[\leadsto \frac{1}{\sqrt[3]{\frac{a}{-g}}} \]
11. Add Preprocessing

Alternative 6: 73.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.0%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

4. Applied rewrites 55.2%

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

5. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. metadata-eval N/A

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

   7. rem-square-sqrt N/A

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

   8. unpow2 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. unpow2 N/A

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

   11. rem-square-sqrt N/A

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

   12. metadata-eval 72.4

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

7. Applied rewrites 72.4%

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

9. Applied rewrites 72.4%

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

11. Applied rewrites 72.4%

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

12. Final simplification 72.4%

    \[\leadsto \sqrt[3]{g \cdot \frac{-1}{a}} \]
13. Add Preprocessing

Alternative 7: 73.2% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (g h a) :precision binary64 (cbrt (- (/ g a))))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.0%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

4. Applied rewrites 55.2%

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

5. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. metadata-eval N/A

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

   7. rem-square-sqrt N/A

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

   8. unpow2 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. unpow2 N/A

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

   11. rem-square-sqrt N/A

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

   12. metadata-eval 72.4

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

7. Applied rewrites 72.4%

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

9. Applied rewrites 72.4%

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

10. Final simplification 72.4%

    \[\leadsto \sqrt[3]{-\frac{g}{a}} \]
11. Add Preprocessing

Alternative 8: 1.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.0%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

4. Applied rewrites 55.2%

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

5. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. metadata-eval N/A

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

   7. rem-square-sqrt N/A

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

   8. unpow2 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. unpow2 N/A

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

   11. rem-square-sqrt N/A

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

   12. metadata-eval 72.4

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

7. Applied rewrites 72.4%

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

9. Applied rewrites 95.6%

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

11. Applied rewrites 1.4%

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

Reproduce

herbie shell --seed 2024237 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  :precision binary64
  (+ (cbrt (* (/ 1.0 (* 2.0 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))