2-ancestry mixing, positive discriminant

Percentage Accurate: 43.9% → 95.6%

Time: 24.2s

Alternatives: 5

Speedup: 2.0×

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: 43.9% 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: 95.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{0} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (+ (/ (cbrt (- g)) (cbrt a)) (cbrt 0.0)))

C

double code(double g, double h, double a) {
	return (cbrt(-g) / cbrt(a)) + cbrt(0.0);
}

Java

public static double code(double g, double h, double a) {
	return (Math.cbrt(-g) / Math.cbrt(a)) + Math.cbrt(0.0);
}

Julia

function code(g, h, a)
	return Float64(Float64(cbrt(Float64(-g)) / cbrt(a)) + cbrt(0.0))
end

Wolfram

code[g_, h_, a_] := N[(N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[0.0, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.5%

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

3. Simplified 46.5%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in g around -inf 28.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
6. Step-by-step derivation

   1. *-commutative 28.1%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(g \cdot -2\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

7. Simplified 28.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(g \cdot -2\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

8. Taylor expanded in g around inf 15.7%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{g}\right) \cdot \frac{-0.5}{a}} \]
9. Step-by-step derivation

   1. associate-*l/ 15.7%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5 \cdot \left(g \cdot -2\right)}{a}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   2. cbrt-div 19.4%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}{\sqrt[3]{a}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   3. *-commutative 19.4%

      \[\leadsto \frac{\sqrt[3]{0.5 \cdot \color{blue}{\left(-2 \cdot g\right)}}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   4. associate-*r* 19.4%

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(0.5 \cdot -2\right) \cdot g}}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   5. metadata-eval 19.4%

      \[\leadsto \frac{\sqrt[3]{\color{blue}{-1} \cdot g}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   6. neg-mul-1 19.4%

      \[\leadsto \frac{\sqrt[3]{\color{blue}{-g}}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

10. Applied egg-rr 19.4%

    \[\leadsto \color{blue}{\frac{\sqrt[3]{-g}}{\sqrt[3]{a}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]
11. Step-by-step derivation

    1. add-log-exp 39.4%

       \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\color{blue}{\log \left(e^{\left(g + g\right) \cdot \frac{-0.5}{a}}\right)}} \]

    2. *-commutative 39.4%

       \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\log \left(e^{\color{blue}{\frac{-0.5}{a} \cdot \left(g + g\right)}}\right)} \]

    3. exp-prod 57.7%

       \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\log \color{blue}{\left({\left(e^{\frac{-0.5}{a}}\right)}^{\left(g + g\right)}\right)}} \]

    4. add-sqr-sqrt 56.5%

       \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\log \left({\left(e^{\frac{-0.5}{a}}\right)}^{\left(g + \color{blue}{\sqrt{g} \cdot \sqrt{g}}\right)}\right)} \]

    5. sqrt-prod 67.8%

       \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\log \left({\left(e^{\frac{-0.5}{a}}\right)}^{\left(g + \color{blue}{\sqrt{g \cdot g}}\right)}\right)} \]

    6. sqr-neg 67.8%

       \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\log \left({\left(e^{\frac{-0.5}{a}}\right)}^{\left(g + \sqrt{\color{blue}{\left(-g\right) \cdot \left(-g\right)}}\right)}\right)} \]

    7. sqrt-unprod 65.1%

       \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\log \left({\left(e^{\frac{-0.5}{a}}\right)}^{\left(g + \color{blue}{\sqrt{-g} \cdot \sqrt{-g}}\right)}\right)} \]

    8. add-sqr-sqrt 96.2%

       \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\log \left({\left(e^{\frac{-0.5}{a}}\right)}^{\left(g + \color{blue}{\left(-g\right)}\right)}\right)} \]

    9. sub-neg 96.2%

       \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\log \left({\left(e^{\frac{-0.5}{a}}\right)}^{\color{blue}{\left(g - g\right)}}\right)} \]

    10. +-inverses 96.2%

        \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\log \left({\left(e^{\frac{-0.5}{a}}\right)}^{\color{blue}{0}}\right)} \]

    11. metadata-eval 96.2%

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

    12. metadata-eval 96.2%

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

12. Applied egg-rr 96.2%

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

13. Final simplification 96.2%

    \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{0} \]
14. Add Preprocessing

Alternative 2: 73.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

3. Simplified 46.5%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in g around -inf 28.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
6. Step-by-step derivation

   1. *-commutative 28.1%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(g \cdot -2\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

7. Simplified 28.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(g \cdot -2\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

8. Taylor expanded in g around -inf 76.5%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{-1 \cdot g}\right) \cdot \frac{-0.5}{a}} \]
9. Step-by-step derivation

   1. neg-mul-1 76.5%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{\left(-g\right)}\right) \cdot \frac{-0.5}{a}} \]

10. Simplified 76.5%

    \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{\left(-g\right)}\right) \cdot \frac{-0.5}{a}} \]
11. Step-by-step derivation

    1. associate-*l/ 76.5%

       \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5 \cdot \left(g \cdot -2\right)}{a}}} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

    2. *-commutative 76.5%

       \[\leadsto \sqrt[3]{\frac{0.5 \cdot \color{blue}{\left(-2 \cdot g\right)}}{a}} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

    3. associate-*r* 76.8%

       \[\leadsto \sqrt[3]{\frac{\color{blue}{\left(0.5 \cdot -2\right) \cdot g}}{a}} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

    4. metadata-eval 76.8%

       \[\leadsto \sqrt[3]{\frac{\color{blue}{-1} \cdot g}{a}} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

    5. neg-mul-1 76.8%

       \[\leadsto \sqrt[3]{\frac{\color{blue}{-g}}{a}} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

12. Applied egg-rr 76.8%

    \[\leadsto \sqrt[3]{\color{blue}{\frac{-g}{a}}} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

13. Final simplification 76.8%

    \[\leadsto \sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} \]
14. Add Preprocessing

Alternative 3: 7.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{-2} + \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-2} + \frac{\sqrt[3]{g}}{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= a -5e-310)
   (+ (cbrt -2.0) (cbrt g))
   (+ (cbrt -2.0) (/ (cbrt g) -2.0))))

C

double code(double g, double h, double a) {
	double tmp;
	if (a <= -5e-310) {
		tmp = cbrt(-2.0) + cbrt(g);
	} else {
		tmp = cbrt(-2.0) + (cbrt(g) / -2.0);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (a <= -5e-310) {
		tmp = Math.cbrt(-2.0) + Math.cbrt(g);
	} else {
		tmp = Math.cbrt(-2.0) + (Math.cbrt(g) / -2.0);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (a <= -5e-310)
		tmp = Float64(cbrt(-2.0) + cbrt(g));
	else
		tmp = Float64(cbrt(-2.0) + Float64(cbrt(g) / -2.0));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[a, -5e-310], N[(N[Power[-2.0, 1/3], $MachinePrecision] + N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[-2.0, 1/3], $MachinePrecision] + N[(N[Power[g, 1/3], $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.999999999999985e-310

   1. Initial program 44.7%

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

   3. Simplified 44.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in g around -inf 26.1%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 26.1%

         \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(g \cdot -2\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

   7. Simplified 26.1%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(g \cdot -2\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

   8. Taylor expanded in g around inf 15.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{g}\right) \cdot \frac{-0.5}{a}} \]
   9. Step-by-step derivation

      1. associate-*l/ 15.8%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5 \cdot \left(g \cdot -2\right)}{a}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      2. associate-/l* 15.8%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{\frac{a}{g \cdot -2}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      3. add-sqr-sqrt 8.4%

         \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\sqrt{g \cdot -2} \cdot \sqrt{g \cdot -2}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      4. sqrt-unprod 9.4%

         \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\sqrt{\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      5. *-commutative 9.4%

         \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(-2 \cdot g\right)} \cdot \left(g \cdot -2\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      6. *-commutative 9.4%

         \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\left(-2 \cdot g\right) \cdot \color{blue}{\left(-2 \cdot g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      7. swap-sqr 9.4%

         \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(-2 \cdot -2\right) \cdot \left(g \cdot g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      8. metadata-eval 9.4%

         \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{4} \cdot \left(g \cdot g\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      9. metadata-eval 9.4%

         \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(2 \cdot 2\right)} \cdot \left(g \cdot g\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      10. swap-sqr 9.4%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(2 \cdot g\right) \cdot \left(2 \cdot g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      11. count-2 9.4%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(g + g\right)} \cdot \left(2 \cdot g\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      12. count-2 9.4%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\left(g + g\right) \cdot \color{blue}{\left(g + g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      13. sqrt-unprod 2.1%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\sqrt{g + g} \cdot \sqrt{g + g}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      14. add-sqr-sqrt 3.5%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{g + g}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      15. flip-+ 0.0%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\frac{g \cdot g - g \cdot g}{g - g}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      16. +-inverses 0.0%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{\color{blue}{0}}{g - g}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      17. +-inverses 0.0%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{0}{\color{blue}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   10. Applied egg-rr 0.0%

       \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{\frac{a}{\frac{0}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   12. Simplified 43.1%

       \[\leadsto \sqrt[3]{\color{blue}{-2}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]
   13. Step-by-step derivation

       1. expm1-log1p-u 18.2%

          \[\leadsto \sqrt[3]{-2} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}}\right)\right)} \]

       2. expm1-udef 18.2%

          \[\leadsto \sqrt[3]{-2} + \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}}\right)} - 1\right)} \]

   14. Applied egg-rr 2.2%

       \[\leadsto \sqrt[3]{-2} + \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{g}{a}}\right)} - 1\right)} \]

   16. Simplified 6.9%

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

   if -4.999999999999985e-310 < a

   1. Initial program 48.3%

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

   3. Simplified 48.3%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in g around -inf 30.3%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 30.3%

         \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(g \cdot -2\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

   7. Simplified 30.3%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(g \cdot -2\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

   8. Taylor expanded in g around inf 15.6%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{g}\right) \cdot \frac{-0.5}{a}} \]
   9. Step-by-step derivation

      1. associate-*l/ 15.6%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5 \cdot \left(g \cdot -2\right)}{a}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      2. associate-/l* 15.6%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{\frac{a}{g \cdot -2}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      3. add-sqr-sqrt 7.3%

         \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\sqrt{g \cdot -2} \cdot \sqrt{g \cdot -2}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      4. sqrt-unprod 10.5%

         \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\sqrt{\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      5. *-commutative 10.5%

         \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(-2 \cdot g\right)} \cdot \left(g \cdot -2\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      6. *-commutative 10.5%

         \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\left(-2 \cdot g\right) \cdot \color{blue}{\left(-2 \cdot g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      7. swap-sqr 10.4%

         \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(-2 \cdot -2\right) \cdot \left(g \cdot g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      8. metadata-eval 10.4%

         \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{4} \cdot \left(g \cdot g\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      9. metadata-eval 10.4%

         \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(2 \cdot 2\right)} \cdot \left(g \cdot g\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      10. swap-sqr 10.5%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(2 \cdot g\right) \cdot \left(2 \cdot g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      11. count-2 10.5%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(g + g\right)} \cdot \left(2 \cdot g\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      12. count-2 10.5%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\left(g + g\right) \cdot \color{blue}{\left(g + g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      13. sqrt-unprod 1.6%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\sqrt{g + g} \cdot \sqrt{g + g}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      14. add-sqr-sqrt 3.1%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{g + g}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      15. flip-+ 0.0%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\frac{g \cdot g - g \cdot g}{g - g}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      16. +-inverses 0.0%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{\color{blue}{0}}{g - g}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      17. +-inverses 0.0%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{0}{\color{blue}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   10. Applied egg-rr 0.0%

       \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{\frac{a}{\frac{0}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   12. Simplified 37.3%

       \[\leadsto \sqrt[3]{\color{blue}{-2}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]
   13. Step-by-step derivation

       1. frac-2neg 37.3%

          \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\left(g + g\right) \cdot \color{blue}{\frac{--0.5}{-a}}} \]

       2. metadata-eval 37.3%

          \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\left(g + g\right) \cdot \frac{\color{blue}{0.5}}{-a}} \]

       3. associate-*r/ 37.3%

          \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\color{blue}{\frac{\left(g + g\right) \cdot 0.5}{-a}}} \]

       4. add-sqr-sqrt 22.1%

          \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{\color{blue}{\left(\sqrt{g + g} \cdot \sqrt{g + g}\right)} \cdot 0.5}{-a}} \]

       5. sqrt-unprod 11.9%

          \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{\color{blue}{\sqrt{\left(g + g\right) \cdot \left(g + g\right)}} \cdot 0.5}{-a}} \]

       6. count-2 11.9%

          \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{\sqrt{\color{blue}{\left(2 \cdot g\right)} \cdot \left(g + g\right)} \cdot 0.5}{-a}} \]

       7. count-2 11.9%

          \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{\sqrt{\left(2 \cdot g\right) \cdot \color{blue}{\left(2 \cdot g\right)}} \cdot 0.5}{-a}} \]

       8. swap-sqr 11.9%

          \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{\sqrt{\color{blue}{\left(2 \cdot 2\right) \cdot \left(g \cdot g\right)}} \cdot 0.5}{-a}} \]

       9. metadata-eval 11.9%

          \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{\sqrt{\color{blue}{4} \cdot \left(g \cdot g\right)} \cdot 0.5}{-a}} \]

       10. metadata-eval 11.9%

           \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{\sqrt{\color{blue}{\left(-2 \cdot -2\right)} \cdot \left(g \cdot g\right)} \cdot 0.5}{-a}} \]

       11. swap-sqr 11.9%

           \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{\sqrt{\color{blue}{\left(-2 \cdot g\right) \cdot \left(-2 \cdot g\right)}} \cdot 0.5}{-a}} \]

       12. *-commutative 11.9%

           \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{\sqrt{\color{blue}{\left(g \cdot -2\right)} \cdot \left(-2 \cdot g\right)} \cdot 0.5}{-a}} \]

       13. *-commutative 11.9%

           \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{\sqrt{\left(g \cdot -2\right) \cdot \color{blue}{\left(g \cdot -2\right)}} \cdot 0.5}{-a}} \]

       14. sqrt-unprod 0.6%

           \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{\color{blue}{\left(\sqrt{g \cdot -2} \cdot \sqrt{g \cdot -2}\right)} \cdot 0.5}{-a}} \]

       15. add-sqr-sqrt 2.4%

           \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{\color{blue}{\left(g \cdot -2\right)} \cdot 0.5}{-a}} \]

       16. *-commutative 2.4%

           \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{\color{blue}{0.5 \cdot \left(g \cdot -2\right)}}{-a}} \]

       17. *-commutative 2.4%

           \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{0.5 \cdot \color{blue}{\left(-2 \cdot g\right)}}{-a}} \]

       18. associate-*r* 2.4%

           \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{\color{blue}{\left(0.5 \cdot -2\right) \cdot g}}{-a}} \]

       19. metadata-eval 2.4%

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

       20. neg-mul-1 2.4%

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

   14. Applied egg-rr 2.5%

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

   16. Simplified 6.8%

       \[\leadsto \sqrt[3]{-2} + \color{blue}{\frac{\sqrt[3]{g}}{-2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 6.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{-2} + \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-2} + \frac{\sqrt[3]{g}}{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.8% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (g h a) :precision binary64 (+ (cbrt (/ (- g) a)) (cbrt -2.0)))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

3. Simplified 46.5%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in g around -inf 28.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
6. Step-by-step derivation

   1. *-commutative 28.1%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(g \cdot -2\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

7. Simplified 28.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(g \cdot -2\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

8. Taylor expanded in g around inf 15.7%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{g}\right) \cdot \frac{-0.5}{a}} \]
9. Step-by-step derivation

   1. associate-*l/ 15.7%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5 \cdot \left(g \cdot -2\right)}{a}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   2. associate-/l* 15.7%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{\frac{a}{g \cdot -2}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   3. add-sqr-sqrt 7.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\sqrt{g \cdot -2} \cdot \sqrt{g \cdot -2}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   4. sqrt-unprod 9.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\sqrt{\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   5. *-commutative 9.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(-2 \cdot g\right)} \cdot \left(g \cdot -2\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   6. *-commutative 9.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\left(-2 \cdot g\right) \cdot \color{blue}{\left(-2 \cdot g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   7. swap-sqr 9.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(-2 \cdot -2\right) \cdot \left(g \cdot g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   8. metadata-eval 9.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{4} \cdot \left(g \cdot g\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   9. metadata-eval 9.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(2 \cdot 2\right)} \cdot \left(g \cdot g\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   10. swap-sqr 9.9%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(2 \cdot g\right) \cdot \left(2 \cdot g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   11. count-2 9.9%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(g + g\right)} \cdot \left(2 \cdot g\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   12. count-2 9.9%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\left(g + g\right) \cdot \color{blue}{\left(g + g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   13. sqrt-unprod 1.8%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\sqrt{g + g} \cdot \sqrt{g + g}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   14. add-sqr-sqrt 3.3%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{g + g}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   15. flip-+ 0.0%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\frac{g \cdot g - g \cdot g}{g - g}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   16. +-inverses 0.0%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{\color{blue}{0}}{g - g}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   17. +-inverses 0.0%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{0}{\color{blue}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

10. Applied egg-rr 0.0%

    \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{\frac{a}{\frac{0}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

12. Simplified 40.3%

    \[\leadsto \sqrt[3]{\color{blue}{-2}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

13. Taylor expanded in g around 0 40.6%

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

    1. associate-*r/ 40.6%

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

    2. mul-1-neg 40.6%

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

15. Simplified 40.6%

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

16. Final simplification 40.6%

    \[\leadsto \sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-2} \]
17. Add Preprocessing

Alternative 5: 4.4% accurate, 4.3× speedup

Math

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

FPCore

(FPCore (g h a) :precision binary64 (cbrt -2.0))

C

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

Java

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

Julia

function code(g, h, a)
	return cbrt(-2.0)
end

Wolfram

code[g_, h_, a_] := N[Power[-2.0, 1/3], $MachinePrecision]

Derivation

1. Initial program 46.5%

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

3. Simplified 46.5%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in g around -inf 28.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
6. Step-by-step derivation

   1. *-commutative 28.1%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(g \cdot -2\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

7. Simplified 28.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(g \cdot -2\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

8. Taylor expanded in g around inf 15.7%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{g}\right) \cdot \frac{-0.5}{a}} \]
9. Step-by-step derivation

   1. associate-*l/ 15.7%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5 \cdot \left(g \cdot -2\right)}{a}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   2. associate-/l* 15.7%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{\frac{a}{g \cdot -2}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   3. add-sqr-sqrt 7.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\sqrt{g \cdot -2} \cdot \sqrt{g \cdot -2}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   4. sqrt-unprod 9.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\sqrt{\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   5. *-commutative 9.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(-2 \cdot g\right)} \cdot \left(g \cdot -2\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   6. *-commutative 9.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\left(-2 \cdot g\right) \cdot \color{blue}{\left(-2 \cdot g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   7. swap-sqr 9.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(-2 \cdot -2\right) \cdot \left(g \cdot g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   8. metadata-eval 9.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{4} \cdot \left(g \cdot g\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   9. metadata-eval 9.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(2 \cdot 2\right)} \cdot \left(g \cdot g\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   10. swap-sqr 9.9%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(2 \cdot g\right) \cdot \left(2 \cdot g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   11. count-2 9.9%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\color{blue}{\left(g + g\right)} \cdot \left(2 \cdot g\right)}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   12. count-2 9.9%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\sqrt{\left(g + g\right) \cdot \color{blue}{\left(g + g\right)}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   13. sqrt-unprod 1.8%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\sqrt{g + g} \cdot \sqrt{g + g}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   14. add-sqr-sqrt 3.3%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{g + g}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   15. flip-+ 0.0%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\color{blue}{\frac{g \cdot g - g \cdot g}{g - g}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   16. +-inverses 0.0%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{\color{blue}{0}}{g - g}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   17. +-inverses 0.0%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{0}{\color{blue}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

10. Applied egg-rr 0.0%

    \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{\frac{a}{\frac{0}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

12. Simplified 40.3%

    \[\leadsto \sqrt[3]{\color{blue}{-2}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

13. Taylor expanded in g around 0 4.7%

    \[\leadsto \color{blue}{\sqrt[3]{-2}} \]

14. Final simplification 4.7%

    \[\leadsto \sqrt[3]{-2} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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))))))))