2-ancestry mixing, positive discriminant

Percentage Accurate: 44.1% → 95.8%

Time: 26.4s

Alternatives: 9

Speedup: 2.1×

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: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Simplified 37.8%

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

5. Taylor expanded in g around -inf 24.0%

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

   1. *-commutative 24.0%

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

7. Simplified 24.0%

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

8. Taylor expanded in g around -inf 69.8%

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

   1. neg-mul-1 69.8%

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

10. Simplified 69.8%

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

    1. associate-*l/ 69.8%

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

    2. cbrt-div 95.7%

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

    3. *-commutative 95.7%

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

    4. associate-*r* 96.0%

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

    5. metadata-eval 96.0%

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

    6. neg-mul-1 96.0%

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

    7. add-cube-cbrt 95.8%

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

    8. unpow2 95.8%

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

    9. distribute-rgt-neg-in 95.8%

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

    10. unpow2 95.8%

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

    11. sqr-neg 95.8%

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

    12. add-cbrt-cube 96.0%

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

    13. add-sqr-sqrt 45.9%

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

    14. sqrt-unprod 46.6%

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

    15. sqr-neg 46.6%

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

    16. sqrt-prod 0.7%

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

    17. add-sqr-sqrt 1.3%

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

    18. frac-2neg 1.3%

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

12. Applied egg-rr 96.0%

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

13. Final simplification 96.0%

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

Alternative 2: 89.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \mathbf{if}\;a \leq -5.2 \cdot 10^{-81}:\\ \;\;\;\;t\_0 + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-50}:\\ \;\;\;\;\sqrt[3]{-1} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \sqrt[3]{\frac{g}{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (- g g) (/ -0.5 a)))))
   (if (<= a -5.2e-81)
     (+ t_0 (cbrt (* (/ 0.5 a) (* g -2.0))))
     (if (<= a 2.3e-50)
       (+ (cbrt -1.0) (/ (cbrt (- g)) (cbrt a)))
       (+ t_0 (cbrt (/ g (- a))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((g - g) * (-0.5 / a)));
	double tmp;
	if (a <= -5.2e-81) {
		tmp = t_0 + cbrt(((0.5 / a) * (g * -2.0)));
	} else if (a <= 2.3e-50) {
		tmp = cbrt(-1.0) + (cbrt(-g) / cbrt(a));
	} else {
		tmp = t_0 + cbrt((g / -a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((g - g) * (-0.5 / a)));
	double tmp;
	if (a <= -5.2e-81) {
		tmp = t_0 + Math.cbrt(((0.5 / a) * (g * -2.0)));
	} else if (a <= 2.3e-50) {
		tmp = Math.cbrt(-1.0) + (Math.cbrt(-g) / Math.cbrt(a));
	} else {
		tmp = t_0 + Math.cbrt((g / -a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(g - g) * Float64(-0.5 / a)))
	tmp = 0.0
	if (a <= -5.2e-81)
		tmp = Float64(t_0 + cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))));
	elseif (a <= 2.3e-50)
		tmp = Float64(cbrt(-1.0) + Float64(cbrt(Float64(-g)) / cbrt(a)));
	else
		tmp = Float64(t_0 + cbrt(Float64(g / Float64(-a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[a, -5.2e-81], N[(t$95$0 + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.3e-50], N[(N[Power[-1.0, 1/3], $MachinePrecision] + N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Power[N[(g / (-a)), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.1999999999999998e-81

   1. Initial program 44.4%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in g around -inf 27.6%

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

      1. *-commutative 27.6%

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

   7. Simplified 27.6%

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

   8. Taylor expanded in g around -inf 90.5%

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

      1. neg-mul-1 90.5%

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

   10. Simplified 90.5%

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

   if -5.1999999999999998e-81 < a < 2.3000000000000002e-50

   1. Initial program 31.4%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in g around -inf 19.6%

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

      1. *-commutative 19.6%

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

   7. Simplified 19.6%

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

   8. Taylor expanded in g around inf 10.5%

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

      1. add-sqr-sqrt 5.9%

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

      2. sqrt-unprod 4.2%

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

      3. swap-sqr 4.3%

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

      4. frac-times 4.3%

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

      5. metadata-eval 4.3%

         \[\leadsto \sqrt[3]{\sqrt{\frac{\color{blue}{0.25}}{a \cdot a} \cdot \left(\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)\right)}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      6. metadata-eval 4.3%

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

      7. frac-times 4.3%

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

      8. *-commutative 4.3%

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

      9. *-commutative 4.3%

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

      10. swap-sqr 4.3%

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

      11. metadata-eval 4.3%

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

      12. metadata-eval 4.3%

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

      13. swap-sqr 4.3%

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

      14. count-2 4.3%

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

      15. count-2 4.3%

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

      16. swap-sqr 4.2%

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

      17. *-commutative 4.2%

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

      18. *-commutative 4.2%

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

      19. sqrt-unprod 5.9%

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

      20. add-sqr-sqrt 10.5%

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

      21. expm1-log1p-u 6.2%

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

   10. Applied egg-rr 0.0%

       \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a}\right)} - 1}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   12. Simplified 39.2%

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

       1. add-sqr-sqrt 23.9%

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

       2. sqrt-unprod 12.0%

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

       3. *-commutative 12.0%

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

       4. *-commutative 12.0%

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

       5. swap-sqr 4.8%

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

       6. frac-times 4.8%

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

       7. metadata-eval 4.8%

          \[\leadsto \sqrt[3]{-1} + \sqrt[3]{\sqrt{\frac{\color{blue}{0.25}}{a \cdot a} \cdot \left(\left(g + g\right) \cdot \left(g + g\right)\right)}} \]

       8. metadata-eval 4.8%

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

       9. frac-times 4.8%

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

       10. count-2 4.8%

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

       11. count-2 4.8%

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

       12. swap-sqr 4.8%

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

       13. metadata-eval 4.8%

           \[\leadsto \sqrt[3]{-1} + \sqrt[3]{\sqrt{\left(\frac{0.5}{a} \cdot \frac{0.5}{a}\right) \cdot \left(\color{blue}{4} \cdot \left(g \cdot g\right)\right)}} \]

       14. metadata-eval 4.8%

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

       15. swap-sqr 4.8%

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

       16. *-commutative 4.8%

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

       17. *-commutative 4.8%

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

       18. swap-sqr 12.0%

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

   14. Applied egg-rr 91.7%

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

   if 2.3000000000000002e-50 < a

   1. Initial program 40.1%

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

   3. Simplified 40.1%

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

   5. Taylor expanded in g around -inf 26.8%

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

      1. *-commutative 26.8%

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

   7. Simplified 26.8%

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

   8. Taylor expanded in g around -inf 90.2%

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

      1. neg-mul-1 90.2%

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

   10. Simplified 90.2%

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

       1. associate-*l/ 90.3%

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

       2. *-commutative 90.3%

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

       3. associate-*r* 90.3%

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

       4. metadata-eval 90.3%

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

       5. neg-mul-1 90.3%

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

   12. Applied egg-rr 90.3%

       \[\leadsto \sqrt[3]{\color{blue}{\frac{-g}{a}}} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-81}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-50}:\\ \;\;\;\;\sqrt[3]{-1} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{g}{-a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \mathbf{if}\;g \leq -21500000000000 \lor \neg \left(g \leq 7.5 \cdot 10^{+21}\right):\\ \;\;\;\;t\_0 + \frac{-1}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (/ 0.5 a) (* g -2.0)))))
   (if (or (<= g -21500000000000.0) (not (<= g 7.5e+21)))
     (+ t_0 (/ -1.0 (cbrt a)))
     (- t_0 (cbrt g)))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((0.5 / a) * (g * -2.0)));
	double tmp;
	if ((g <= -21500000000000.0) || !(g <= 7.5e+21)) {
		tmp = t_0 + (-1.0 / cbrt(a));
	} else {
		tmp = t_0 - cbrt(g);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((0.5 / a) * (g * -2.0)));
	double tmp;
	if ((g <= -21500000000000.0) || !(g <= 7.5e+21)) {
		tmp = t_0 + (-1.0 / Math.cbrt(a));
	} else {
		tmp = t_0 - Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0)))
	tmp = 0.0
	if ((g <= -21500000000000.0) || !(g <= 7.5e+21))
		tmp = Float64(t_0 + Float64(-1.0 / cbrt(a)));
	else
		tmp = Float64(t_0 - cbrt(g));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[Or[LessEqual[g, -21500000000000.0], N[Not[LessEqual[g, 7.5e+21]], $MachinePrecision]], N[(t$95$0 + N[(-1.0 / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if g < -2.15e13 or 7.5e21 < g

   1. Initial program 26.9%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in g around -inf 15.6%

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

      1. *-commutative 15.6%

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

   7. Simplified 15.6%

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

   8. Taylor expanded in g around inf 13.8%

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

      1. *-un-lft-identity 13.8%

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

      2. *-commutative 13.8%

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

   10. Applied egg-rr 0.0%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \color{blue}{\frac{\frac{0}{0}}{\sqrt[3]{a}} \cdot 1} \]

   12. Simplified 63.5%

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

   if -2.15e13 < g < 7.5e21

   1. Initial program 71.2%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in g around -inf 49.8%

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

      1. *-commutative 49.8%

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

   7. Simplified 49.8%

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

   8. Taylor expanded in g around inf 16.9%

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

   9. Taylor expanded in g around 0 16.9%

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

   11. Simplified 51.0%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \color{blue}{\left(-\sqrt[3]{g}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -21500000000000 \lor \neg \left(g \leq 7.5 \cdot 10^{+21}\right):\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \frac{-1}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} - \sqrt[3]{g}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (* (- g g) (/ -0.5 a))) (cbrt (* (/ 0.5 a) (* g -2.0)))))

C

double code(double g, double h, double a) {
	return cbrt(((g - g) * (-0.5 / a))) + cbrt(((0.5 / a) * (g * -2.0)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((g - g) * (-0.5 / a))) + Math.cbrt(((0.5 / a) * (g * -2.0)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(g - g) * Float64(-0.5 / a))) + cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

3. Simplified 37.8%

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

5. Taylor expanded in g around -inf 24.0%

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

   1. *-commutative 24.0%

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

7. Simplified 24.0%

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

8. Taylor expanded in g around -inf 69.8%

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

   1. neg-mul-1 69.8%

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

10. Simplified 69.8%

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

11. Final simplification 69.8%

    \[\leadsto \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} \]
12. Add Preprocessing

Alternative 5: 73.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Simplified 37.8%

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

5. Taylor expanded in g around -inf 24.0%

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

   1. *-commutative 24.0%

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

7. Simplified 24.0%

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

8. Taylor expanded in g around -inf 69.8%

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

   1. neg-mul-1 69.8%

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

10. Simplified 69.8%

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

    1. associate-*l/ 69.8%

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

    2. *-commutative 69.8%

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

    3. associate-*r* 69.8%

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

    4. metadata-eval 69.8%

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

    5. neg-mul-1 69.8%

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

12. Applied egg-rr 69.8%

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

13. Final simplification 69.8%

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

Alternative 6: 44.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Simplified 37.8%

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

5. Taylor expanded in g around -inf 24.0%

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

   1. *-commutative 24.0%

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

7. Simplified 24.0%

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

8. Taylor expanded in g around inf 14.6%

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

   1. add-sqr-sqrt 8.0%

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

   2. sqrt-unprod 13.0%

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

   3. swap-sqr 13.3%

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

   4. frac-times 13.4%

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

   5. metadata-eval 13.4%

      \[\leadsto \sqrt[3]{\sqrt{\frac{\color{blue}{0.25}}{a \cdot a} \cdot \left(\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)\right)}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   6. metadata-eval 13.4%

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

   7. frac-times 13.3%

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

   8. *-commutative 13.3%

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

   9. *-commutative 13.3%

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

   10. swap-sqr 13.3%

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

   11. metadata-eval 13.3%

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

   12. metadata-eval 13.3%

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

   13. swap-sqr 13.3%

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

   14. count-2 13.3%

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

   15. count-2 13.3%

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

   16. swap-sqr 13.0%

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

   17. *-commutative 13.0%

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

   18. *-commutative 13.0%

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

   19. sqrt-unprod 8.0%

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

   20. add-sqr-sqrt 14.6%

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

   21. expm1-log1p-u 10.4%

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

10. Applied egg-rr 0.0%

    \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a}\right)} - 1}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

12. Simplified 42.6%

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

13. Taylor expanded in g around 0 42.6%

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

    1. mul-1-neg 42.6%

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

    2. distribute-neg-frac2 42.6%

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

15. Simplified 42.6%

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

Alternative 7: 4.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Simplified 37.8%

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

5. Taylor expanded in g around -inf 24.0%

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

   1. *-commutative 24.0%

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

7. Simplified 24.0%

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

8. Taylor expanded in g around inf 14.6%

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

   1. add-sqr-sqrt 8.0%

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

   2. sqrt-unprod 13.0%

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

   3. swap-sqr 13.3%

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

   4. frac-times 13.4%

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

   5. metadata-eval 13.4%

      \[\leadsto \sqrt[3]{\sqrt{\frac{\color{blue}{0.25}}{a \cdot a} \cdot \left(\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)\right)}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   6. metadata-eval 13.4%

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

   7. frac-times 13.3%

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

   8. *-commutative 13.3%

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

   9. *-commutative 13.3%

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

   10. swap-sqr 13.3%

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

   11. metadata-eval 13.3%

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

   12. metadata-eval 13.3%

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

   13. swap-sqr 13.3%

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

   14. count-2 13.3%

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

   15. count-2 13.3%

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

   16. swap-sqr 13.0%

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

   17. *-commutative 13.0%

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

   18. *-commutative 13.0%

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

   19. sqrt-unprod 8.0%

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

   20. add-sqr-sqrt 14.6%

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

   21. expm1-log1p-u 10.4%

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

10. Applied egg-rr 0.0%

    \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a}\right)} - 1}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

12. Simplified 42.6%

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

    1. *-un-lft-identity 14.6%

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

    2. *-commutative 14.6%

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

14. Applied egg-rr 0.0%

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

16. Simplified 4.9%

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

Alternative 8: 4.4% accurate, 433.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 1.0)

C

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

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    code = 1.0d0
end function

Java

public static double code(double g, double h, double a) {
	return 1.0;
}

Python

def code(g, h, a):
	return 1.0

Julia

function code(g, h, a)
	return 1.0
end

MATLAB

function tmp = code(g, h, a)
	tmp = 1.0;
end

Wolfram

code[g_, h_, a_] := 1.0

Derivation

1. Initial program 37.8%

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

3. Simplified 37.8%

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

5. Taylor expanded in g around -inf 24.0%

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

   1. *-commutative 24.0%

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

7. Simplified 24.0%

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

8. Taylor expanded in g around inf 14.6%

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

   1. add-sqr-sqrt 8.0%

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

   2. sqrt-unprod 13.0%

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

   3. swap-sqr 13.3%

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

   4. frac-times 13.4%

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

   5. metadata-eval 13.4%

      \[\leadsto \sqrt[3]{\sqrt{\frac{\color{blue}{0.25}}{a \cdot a} \cdot \left(\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)\right)}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   6. metadata-eval 13.4%

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

   7. frac-times 13.3%

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

   8. *-commutative 13.3%

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

   9. *-commutative 13.3%

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

   10. swap-sqr 13.3%

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

   11. metadata-eval 13.3%

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

   12. metadata-eval 13.3%

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

   13. swap-sqr 13.3%

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

   14. count-2 13.3%

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

   15. count-2 13.3%

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

   16. swap-sqr 13.0%

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

   17. *-commutative 13.0%

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

   18. *-commutative 13.0%

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

   19. sqrt-unprod 8.0%

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

   20. add-sqr-sqrt 14.6%

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

   21. expm1-log1p-u 10.4%

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

10. Applied egg-rr 0.0%

    \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a}\right)} - 1}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

12. Simplified 42.6%

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

14. Applied egg-rr 4.7%

    \[\leadsto \color{blue}{1} \]
15. Add Preprocessing

Alternative 9: 4.5% accurate, 433.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    code = -1.0d0
end function

Java

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

Python

def code(g, h, a):
	return -1.0

Julia

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

MATLAB

function tmp = code(g, h, a)
	tmp = -1.0;
end

Wolfram

code[g_, h_, a_] := -1.0

Derivation

1. Initial program 37.8%

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

3. Simplified 37.8%

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

5. Taylor expanded in g around -inf 24.0%

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

   1. *-commutative 24.0%

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

7. Simplified 24.0%

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

8. Taylor expanded in g around inf 14.6%

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

   1. add-sqr-sqrt 8.0%

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

   2. sqrt-unprod 13.0%

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

   3. swap-sqr 13.3%

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

   4. frac-times 13.4%

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

   5. metadata-eval 13.4%

      \[\leadsto \sqrt[3]{\sqrt{\frac{\color{blue}{0.25}}{a \cdot a} \cdot \left(\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)\right)}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   6. metadata-eval 13.4%

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

   7. frac-times 13.3%

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

   8. *-commutative 13.3%

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

   9. *-commutative 13.3%

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

   10. swap-sqr 13.3%

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

   11. metadata-eval 13.3%

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

   12. metadata-eval 13.3%

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

   13. swap-sqr 13.3%

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

   14. count-2 13.3%

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

   15. count-2 13.3%

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

   16. swap-sqr 13.0%

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

   17. *-commutative 13.0%

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

   18. *-commutative 13.0%

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

   19. sqrt-unprod 8.0%

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

   20. add-sqr-sqrt 14.6%

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

   21. expm1-log1p-u 10.4%

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

10. Applied egg-rr 0.0%

    \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a}\right)} - 1}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

12. Simplified 42.6%

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

14. Applied egg-rr 4.3%

    \[\leadsto \color{blue}{-1} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024114 
(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))))))))