2-ancestry mixing, positive discriminant

Percentage Accurate: 44.4% → 95.6%

Time: 34.2s

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: 44.4% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

3. Simplified 49.4%

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

5. Taylor expanded in g around -inf 28.2%

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

   1. *-commutative 28.2%

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

7. Simplified 28.2%

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

8. Taylor expanded in g around -inf 74.7%

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

   1. neg-mul-1 74.7%

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

10. Simplified 74.7%

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

    1. associate-*l/ 74.7%

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

    2. frac-2neg 74.7%

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

    3. *-commutative 74.7%

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

    4. associate-*r* 74.7%

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

    5. metadata-eval 74.7%

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

    6. neg-mul-1 74.7%

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

    7. add-sqr-sqrt 34.1%

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

    8. sqrt-unprod 22.5%

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

    9. sqr-neg 22.5%

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

    10. sqrt-prod 0.7%

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

    11. add-sqr-sqrt 1.4%

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

    12. cbrt-div 1.4%

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

    13. add-sqr-sqrt 0.7%

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

    14. sqrt-unprod 31.7%

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

    15. sqr-neg 31.7%

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

    16. pow2 31.7%

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

    17. sqrt-pow1 95.4%

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

    18. metadata-eval 95.4%

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

    19. pow1 95.4%

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

12. Applied egg-rr 95.4%

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

13. Final simplification 95.4%

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

Alternative 2: 89.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \mathbf{if}\;a \leq -4.8 \cdot 10^{-86}:\\ \;\;\;\;t\_0 + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-69}:\\ \;\;\;\;\sqrt[3]{-1} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \sqrt[3]{\frac{g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (- g g) (/ -0.5 a)))))
   (if (<= a -4.8e-86)
     (+ t_0 (cbrt (* (/ 0.5 a) (* g -2.0))))
     (if (<= a 7.2e-69)
       (+ (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 <= -4.8e-86) {
		tmp = t_0 + cbrt(((0.5 / a) * (g * -2.0)));
	} else if (a <= 7.2e-69) {
		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 <= -4.8e-86) {
		tmp = t_0 + Math.cbrt(((0.5 / a) * (g * -2.0)));
	} else if (a <= 7.2e-69) {
		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 <= -4.8e-86)
		tmp = Float64(t_0 + cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))));
	elseif (a <= 7.2e-69)
		tmp = Float64(cbrt(-1.0) + Float64(cbrt(Float64(-g)) / cbrt(a)));
	else
		tmp = Float64(t_0 - cbrt(Float64(g / 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, -4.8e-86], 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, 7.2e-69], 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 < -4.80000000000000026e-86

   1. Initial program 58.9%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in g around -inf 37.1%

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

      1. *-commutative 37.1%

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

   7. Simplified 37.1%

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

   8. Taylor expanded in g around -inf 94.3%

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

      1. neg-mul-1 94.3%

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

   10. Simplified 94.3%

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

   if -4.80000000000000026e-86 < a < 7.20000000000000035e-69

   1. Initial program 41.1%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in g around -inf 23.7%

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

      1. *-commutative 23.7%

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

   7. Simplified 23.7%

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

   8. Taylor expanded in g around inf 11.6%

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

   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 45.8%

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

       1. add-sqr-sqrt 22.8%

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

       2. sqrt-unprod 10.3%

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

       3. swap-sqr 7.3%

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

       4. count-2 7.3%

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

       5. count-2 7.3%

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

       6. swap-sqr 7.3%

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

       7. metadata-eval 7.3%

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

       8. metadata-eval 7.3%

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

       9. swap-sqr 7.3%

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

       10. *-commutative 7.3%

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

       11. *-commutative 7.3%

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

       12. frac-times 7.3%

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

       13. metadata-eval 7.3%

           \[\leadsto \sqrt[3]{-1} + \sqrt[3]{\sqrt{\left(\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)\right) \cdot \frac{\color{blue}{0.25}}{a \cdot a}}} \]

       14. metadata-eval 7.3%

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

       15. frac-times 7.3%

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

       16. swap-sqr 10.3%

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

       17. *-commutative 10.3%

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

       18. *-commutative 10.3%

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

       19. sqrt-unprod 22.8%

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

       20. add-sqr-sqrt 45.8%

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

   14. Applied egg-rr 91.7%

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

   16. Simplified 91.7%

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

   if 7.20000000000000035e-69 < a

   1. Initial program 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in g around -inf 26.3%

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

      1. *-commutative 26.3%

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

   7. Simplified 26.3%

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

   8. Taylor expanded in g around -inf 87.5%

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

      1. neg-mul-1 87.5%

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

   10. Simplified 87.5%

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

   11. Taylor expanded in g around -inf 87.5%

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

       1. mul-1-neg 87.5%

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

   13. Simplified 87.5%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{-86}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-69}:\\ \;\;\;\;\sqrt[3]{-1} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} - \sqrt[3]{\frac{g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{-16} \lor \neg \left(a \leq 1.6 \cdot 10^{-42}\right):\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \frac{-1}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= a -2.1e-16) (not (<= a 1.6e-42)))
   (+ (cbrt (* (/ 0.5 a) (* g -2.0))) (/ -1.0 (cbrt a)))
   (+ (cbrt (- g)) (cbrt (* (/ -0.5 a) (+ g g))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((a <= -2.1e-16) || !(a <= 1.6e-42)) {
		tmp = cbrt(((0.5 / a) * (g * -2.0))) + (-1.0 / cbrt(a));
	} else {
		tmp = cbrt(-g) + cbrt(((-0.5 / a) * (g + g)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((a <= -2.1e-16) || !(a <= 1.6e-42)) {
		tmp = Math.cbrt(((0.5 / a) * (g * -2.0))) + (-1.0 / Math.cbrt(a));
	} else {
		tmp = Math.cbrt(-g) + Math.cbrt(((-0.5 / a) * (g + g)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((a <= -2.1e-16) || !(a <= 1.6e-42))
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))) + Float64(-1.0 / cbrt(a)));
	else
		tmp = Float64(cbrt(Float64(-g)) + cbrt(Float64(Float64(-0.5 / a) * Float64(g + g))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[a, -2.1e-16], N[Not[LessEqual[a, 1.6e-42]], $MachinePrecision]], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(-1.0 / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[(-g), 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.1000000000000001e-16 or 1.60000000000000012e-42 < a

   1. Initial program 52.3%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in g around -inf 30.9%

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

      1. *-commutative 30.9%

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

   7. Simplified 30.9%

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

   8. Taylor expanded in g around inf 17.9%

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

   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 70.1%

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

   if -2.1000000000000001e-16 < a < 1.60000000000000012e-42

   1. Initial program 45.7%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in g around -inf 24.7%

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

      1. *-commutative 24.7%

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

   7. Simplified 24.7%

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

   8. Taylor expanded in g around inf 12.5%

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

   9. Taylor expanded in a around 0 12.5%

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

   11. Simplified 54.5%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{-16} \lor \neg \left(a \leq 1.6 \cdot 10^{-42}\right):\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \frac{-1}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \mathbf{if}\;g \leq -0.0021:\\ \;\;\;\;\sqrt[3]{-1} + t\_0\\ \mathbf{elif}\;g \leq 6 \cdot 10^{-12}:\\ \;\;\;\;\sqrt[3]{-g} + t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-1} - \sqrt[3]{\frac{g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (/ -0.5 a) (+ g g)))))
   (if (<= g -0.0021)
     (+ (cbrt -1.0) t_0)
     (if (<= g 6e-12) (+ (cbrt (- g)) t_0) (- (cbrt -1.0) (cbrt (/ g a)))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((-0.5 / a) * (g + g)));
	double tmp;
	if (g <= -0.0021) {
		tmp = cbrt(-1.0) + t_0;
	} else if (g <= 6e-12) {
		tmp = cbrt(-g) + t_0;
	} else {
		tmp = cbrt(-1.0) - cbrt((g / a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((-0.5 / a) * (g + g)));
	double tmp;
	if (g <= -0.0021) {
		tmp = Math.cbrt(-1.0) + t_0;
	} else if (g <= 6e-12) {
		tmp = Math.cbrt(-g) + t_0;
	} else {
		tmp = Math.cbrt(-1.0) - Math.cbrt((g / a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(-0.5 / a) * Float64(g + g)))
	tmp = 0.0
	if (g <= -0.0021)
		tmp = Float64(cbrt(-1.0) + t_0);
	elseif (g <= 6e-12)
		tmp = Float64(cbrt(Float64(-g)) + t_0);
	else
		tmp = Float64(cbrt(-1.0) - cbrt(Float64(g / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[g, -0.0021], N[(N[Power[-1.0, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[g, 6e-12], N[(N[Power[(-g), 1/3], $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[Power[-1.0, 1/3], $MachinePrecision] - N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if g < -0.00209999999999999987

   1. Initial program 39.0%

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

   3. Simplified 39.0%

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

   5. Taylor expanded in g around -inf 39.2%

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

      1. *-commutative 39.2%

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

   7. Simplified 39.2%

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

   8. Taylor expanded in g around inf 15.0%

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

   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 41.9%

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

   if -0.00209999999999999987 < g < 6.0000000000000003e-12

   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 43.3%

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

      1. *-commutative 43.3%

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

   7. Simplified 43.3%

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

   8. Taylor expanded in g around inf 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 a around 0 16.9%

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

   11. Simplified 50.1%

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

   if 6.0000000000000003e-12 < g

   1. Initial program 43.1%

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

   3. Simplified 43.1%

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

   5. Taylor expanded in g around -inf 10.2%

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

      1. *-commutative 10.2%

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

   7. Simplified 10.2%

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

   8. Taylor expanded in g around inf 15.0%

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

   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 45.6%

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

   13. Taylor expanded in g around 0 45.6%

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

   15. Simplified 45.6%

       \[\leadsto \sqrt[3]{-1} + \color{blue}{\left(-\sqrt[3]{\frac{g}{a}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -0.0021:\\ \;\;\;\;\sqrt[3]{-1} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \mathbf{elif}\;g \leq 6 \cdot 10^{-12}:\\ \;\;\;\;\sqrt[3]{-g} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-1} - \sqrt[3]{\frac{g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.7% 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 49.4%

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

3. Simplified 49.4%

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

5. Taylor expanded in g around -inf 28.2%

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

   1. *-commutative 28.2%

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

7. Simplified 28.2%

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

8. Taylor expanded in g around -inf 74.7%

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

   1. neg-mul-1 74.7%

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

10. Simplified 74.7%

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

11. Final simplification 74.7%

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

Alternative 6: 72.8% accurate, 2.1× 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 / 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 49.4%

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

3. Simplified 49.4%

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

5. Taylor expanded in g around -inf 28.2%

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

   1. *-commutative 28.2%

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

7. Simplified 28.2%

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

8. Taylor expanded in g around -inf 74.7%

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

   1. neg-mul-1 74.7%

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

10. Simplified 74.7%

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

11. Taylor expanded in g around -inf 74.7%

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

    1. mul-1-neg 74.7%

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

13. Simplified 74.7%

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

14. Final simplification 74.7%

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

Alternative 7: 43.0% 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 / 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 49.4%

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

3. Simplified 49.4%

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

5. Taylor expanded in g around -inf 28.2%

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

   1. *-commutative 28.2%

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

7. Simplified 28.2%

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

8. Taylor expanded in g around inf 15.5%

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

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 42.4%

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

13. Taylor expanded in g around 0 42.4%

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

15. Simplified 42.4%

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

16. Final simplification 42.4%

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

Alternative 8: 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 49.4%

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

3. Simplified 49.4%

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

5. Taylor expanded in g around -inf 28.2%

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

   1. *-commutative 28.2%

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

7. Simplified 28.2%

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

8. Taylor expanded in g around inf 15.5%

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

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 42.4%

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

14. Applied egg-rr 0.0%

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

16. Simplified 5.2%

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

Alternative 9: 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 49.4%

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

3. Simplified 49.4%

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

5. Taylor expanded in g around -inf 28.2%

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

   1. *-commutative 28.2%

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

7. Simplified 28.2%

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

8. Taylor expanded in g around inf 15.5%

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

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 42.4%

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

14. Applied egg-rr 4.6%

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

Reproduce

herbie shell --seed 2024088 
(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))))))))