2-ancestry mixing, positive discriminant

Percentage Accurate: 44.9% → 95.6%

Time: 23.6s

Alternatives: 13

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 95.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[g_, h_, a_] := N[(N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g * -2.0), $MachinePrecision], 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 43.3%

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

3. Simplified 43.3%

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

4. Taylor expanded in g around -inf 26.1%

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

   1. *-commutative 26.1%

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

6. Simplified 26.1%

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

7. Taylor expanded in g around -inf 69.7%

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

   1. neg-mul-1 69.7%

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

9. Simplified 69.7%

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

    1. cbrt-prod 95.2%

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

11. Applied egg-rr 95.2%

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

12. Final simplification 95.2%

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

Alternative 2: 95.7% 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(Float64(-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 43.3%

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

3. Simplified 43.3%

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

4. Taylor expanded in g around -inf 26.1%

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

   1. *-commutative 26.1%

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

6. Simplified 26.1%

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

7. Taylor expanded in g around -inf 69.7%

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

   1. neg-mul-1 69.7%

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

9. Simplified 69.7%

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

    1. associate-*l/ 69.7%

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

    2. cbrt-div 95.2%

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

    3. *-commutative 95.2%

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

    4. associate-*r* 95.2%

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

    5. metadata-eval 95.2%

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

    6. neg-mul-1 95.2%

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

11. Applied egg-rr 95.2%

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

12. Final simplification 95.2%

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

Alternative 3: 89.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-67} \lor \neg \left(a \leq 1.5 \cdot 10^{-30}\right):\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= a -1.2e-67) (not (<= a 1.5e-30)))
   (+ (cbrt (* (- g g) (/ -0.5 a))) (cbrt (* (/ 0.5 a) (* g -2.0))))
   (+ (/ (cbrt (- g)) (cbrt a)) (cbrt -2.0))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((a <= -1.2e-67) || !(a <= 1.5e-30)) {
		tmp = cbrt(((g - g) * (-0.5 / a))) + cbrt(((0.5 / a) * (g * -2.0)));
	} else {
		tmp = (cbrt(-g) / cbrt(a)) + cbrt(-2.0);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((a <= -1.2e-67) || !(a <= 1.5e-30)) {
		tmp = Math.cbrt(((g - g) * (-0.5 / a))) + Math.cbrt(((0.5 / a) * (g * -2.0)));
	} else {
		tmp = (Math.cbrt(-g) / Math.cbrt(a)) + Math.cbrt(-2.0);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((a <= -1.2e-67) || !(a <= 1.5e-30))
		tmp = Float64(cbrt(Float64(Float64(g - g) * Float64(-0.5 / a))) + cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))));
	else
		tmp = Float64(Float64(cbrt(Float64(-g)) / cbrt(a)) + cbrt(-2.0));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[a, -1.2e-67], N[Not[LessEqual[a, 1.5e-30]], $MachinePrecision]], 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], N[(N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[-2.0, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.2e-67 or 1.49999999999999995e-30 < a

   1. Initial program 46.7%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in g around -inf 26.7%

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

      1. *-commutative 26.7%

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

   6. Simplified 26.7%

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

   7. Taylor expanded in g around -inf 85.6%

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

      1. neg-mul-1 85.6%

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

   9. Simplified 85.6%

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

   if -1.2e-67 < a < 1.49999999999999995e-30

   1. Initial program 39.2%

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

   3. Simplified 39.2%

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

   4. Taylor expanded in g around -inf 25.4%

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

      1. *-commutative 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in g around inf 11.6%

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

      1. add-sqr-sqrt 5.6%

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

      2. sqrt-unprod 3.2%

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

      3. swap-sqr 3.7%

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

      4. frac-times 3.8%

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

      5. metadata-eval 3.8%

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

      6. metadata-eval 3.8%

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

      7. frac-times 3.7%

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

      8. *-commutative 3.7%

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

      9. *-commutative 3.7%

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

      10. swap-sqr 3.7%

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

      11. metadata-eval 3.7%

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

      12. metadata-eval 3.7%

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

      13. swap-sqr 3.7%

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

      14. count-2 3.7%

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

      15. count-2 3.7%

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

      16. swap-sqr 3.2%

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

      17. *-commutative 3.2%

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

      18. *-commutative 3.2%

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

      19. sqrt-unprod 5.6%

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

      20. add-sqr-sqrt 11.6%

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

      21. expm1-log1p-u 5.9%

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

   9. Applied egg-rr 0.0%

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

   11. Simplified 45.3%

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

       1. add-sqr-sqrt 22.5%

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

       2. sqrt-unprod 5.4%

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

       3. *-commutative 5.4%

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

       4. *-commutative 5.4%

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

       5. swap-sqr 4.6%

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

       6. frac-times 4.6%

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

       7. metadata-eval 4.6%

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

       8. metadata-eval 4.6%

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

       9. frac-times 4.6%

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

       10. count-2 4.6%

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

       11. count-2 4.6%

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

       12. swap-sqr 4.6%

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

       13. metadata-eval 4.6%

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

       14. metadata-eval 4.6%

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

       15. swap-sqr 4.6%

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

       16. *-commutative 4.6%

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

       17. *-commutative 4.6%

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

       18. swap-sqr 5.4%

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

       19. sqrt-unprod 22.5%

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

       20. add-sqr-sqrt 45.3%

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

   13. Applied egg-rr 91.3%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-67} \lor \neg \left(a \leq 1.5 \cdot 10^{-30}\right):\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-2}\\ \end{array} \]

Alternative 4: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \mathbf{if}\;a \leq -1 \cdot 10^{-32} \lor \neg \left(a \leq 4.2 \cdot 10^{-38}\right):\\ \;\;\;\;\sqrt[3]{\frac{-0.25}{a \cdot g}} + t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 + \sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (/ -0.5 a) (+ g g)))))
   (if (or (<= a -1e-32) (not (<= a 4.2e-38)))
     (+ (cbrt (/ -0.25 (* a g))) t_0)
     (+ t_0 (cbrt g)))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((-0.5 / a) * (g + g)));
	double tmp;
	if ((a <= -1e-32) || !(a <= 4.2e-38)) {
		tmp = cbrt((-0.25 / (a * g))) + t_0;
	} 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 + g)));
	double tmp;
	if ((a <= -1e-32) || !(a <= 4.2e-38)) {
		tmp = Math.cbrt((-0.25 / (a * g))) + t_0;
	} 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 + g)))
	tmp = 0.0
	if ((a <= -1e-32) || !(a <= 4.2e-38))
		tmp = Float64(cbrt(Float64(-0.25 / Float64(a * g))) + t_0);
	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 + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[Or[LessEqual[a, -1e-32], N[Not[LessEqual[a, 4.2e-38]], $MachinePrecision]], N[(N[Power[N[(-0.25 / N[(a * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + t$95$0), $MachinePrecision], N[(t$95$0 + N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.00000000000000006e-32 or 4.20000000000000026e-38 < a

   1. Initial program 44.9%

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

   3. Simplified 44.9%

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

   4. Taylor expanded in g around -inf 25.9%

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

      1. *-commutative 25.9%

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

   6. Simplified 25.9%

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

   7. Taylor expanded in g around inf 17.1%

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

   9. Applied egg-rr 69.0%

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

   if -1.00000000000000006e-32 < a < 4.20000000000000026e-38

   1. Initial program 41.6%

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

   3. Simplified 41.6%

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

   4. Taylor expanded in g around -inf 26.3%

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

      1. *-commutative 26.3%

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

   6. Simplified 26.3%

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

   7. Taylor expanded in g around inf 12.1%

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

   8. Taylor expanded in a around 0 12.1%

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

   10. Simplified 52.8%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-32} \lor \neg \left(a \leq 4.2 \cdot 10^{-38}\right):\\ \;\;\;\;\sqrt[3]{\frac{-0.25}{a \cdot g}} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)} + \sqrt[3]{g}\\ \end{array} \]

Alternative 5: 61.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \mathbf{if}\;g \leq -1.12 \cdot 10^{+23} \lor \neg \left(g \leq 9.5 \cdot 10^{-5}\right):\\ \;\;\;\;t_0 + \sqrt[3]{\frac{-2}{a}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (/ -0.5 a) (+ g g)))))
   (if (or (<= g -1.12e+23) (not (<= g 9.5e-5)))
     (+ t_0 (cbrt (/ -2.0 a)))
     (+ t_0 (cbrt g)))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((-0.5 / a) * (g + g)));
	double tmp;
	if ((g <= -1.12e+23) || !(g <= 9.5e-5)) {
		tmp = t_0 + cbrt((-2.0 / 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 + g)));
	double tmp;
	if ((g <= -1.12e+23) || !(g <= 9.5e-5)) {
		tmp = t_0 + Math.cbrt((-2.0 / 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 + g)))
	tmp = 0.0
	if ((g <= -1.12e+23) || !(g <= 9.5e-5))
		tmp = Float64(t_0 + cbrt(Float64(-2.0 / 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 + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[Or[LessEqual[g, -1.12e+23], N[Not[LessEqual[g, 9.5e-5]], $MachinePrecision]], N[(t$95$0 + N[Power[N[(-2.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if g < -1.12e23 or 9.5000000000000005e-5 < g

   1. Initial program 35.4%

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

   3. Simplified 35.4%

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

   4. Taylor expanded in g around -inf 20.8%

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

      1. *-commutative 20.8%

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

   6. Simplified 20.8%

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

   7. Taylor expanded in g around inf 14.4%

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

      1. add-sqr-sqrt 7.0%

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

      2. sqrt-unprod 10.1%

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

      3. swap-sqr 14.6%

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

      4. frac-times 15.1%

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

      5. metadata-eval 15.1%

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

      6. metadata-eval 15.1%

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

      7. frac-times 14.6%

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

      8. *-commutative 14.6%

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

      9. *-commutative 14.6%

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

      10. swap-sqr 14.6%

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

      11. metadata-eval 14.6%

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

      12. metadata-eval 14.6%

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

      13. swap-sqr 14.6%

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

      14. count-2 14.6%

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

      15. count-2 14.6%

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

      16. swap-sqr 10.1%

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

      17. *-commutative 10.1%

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

      18. *-commutative 10.1%

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

      19. sqrt-unprod 7.0%

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

      20. add-sqr-sqrt 14.4%

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

      21. associate-*r/ 14.4%

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

   9. Applied egg-rr 0.0%

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

   11. Simplified 35.3%

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

   12. Taylor expanded in a around 0 66.1%

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

   if -1.12e23 < g < 9.5000000000000005e-5

   1. Initial program 65.7%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in g around -inf 41.2%

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

      1. *-commutative 41.2%

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

   6. Simplified 41.2%

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

   7. Taylor expanded in g around inf 15.7%

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

   8. Taylor expanded in a around 0 15.7%

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

   10. Simplified 45.8%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.12 \cdot 10^{+23} \lor \neg \left(g \leq 9.5 \cdot 10^{-5}\right):\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)} + \sqrt[3]{\frac{-2}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)} + \sqrt[3]{g}\\ \end{array} \]

Alternative 6: 73.0% 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 43.3%

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

3. Simplified 43.3%

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

4. Taylor expanded in g around -inf 26.1%

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

   1. *-commutative 26.1%

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

6. Simplified 26.1%

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

7. Taylor expanded in g around -inf 69.7%

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

   1. neg-mul-1 69.7%

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

9. Simplified 69.7%

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

10. Final simplification 69.7%

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

Alternative 7: 46.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \mathbf{if}\;g \leq -55:\\ \;\;\;\;t_0 + \sqrt[3]{-0.5}\\ \mathbf{elif}\;g \leq 1.45 \cdot 10^{+73}:\\ \;\;\;\;t_0 + \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-2} + \sqrt[3]{\frac{-g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (/ -0.5 a) (+ g g)))))
   (if (<= g -55.0)
     (+ t_0 (cbrt -0.5))
     (if (<= g 1.45e+73)
       (+ t_0 (cbrt g))
       (+ (cbrt -2.0) (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 <= -55.0) {
		tmp = t_0 + cbrt(-0.5);
	} else if (g <= 1.45e+73) {
		tmp = t_0 + cbrt(g);
	} else {
		tmp = cbrt(-2.0) + 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 <= -55.0) {
		tmp = t_0 + Math.cbrt(-0.5);
	} else if (g <= 1.45e+73) {
		tmp = t_0 + Math.cbrt(g);
	} else {
		tmp = Math.cbrt(-2.0) + 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 <= -55.0)
		tmp = Float64(t_0 + cbrt(-0.5));
	elseif (g <= 1.45e+73)
		tmp = Float64(t_0 + cbrt(g));
	else
		tmp = Float64(cbrt(-2.0) + cbrt(Float64(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, -55.0], N[(t$95$0 + N[Power[-0.5, 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 1.45e+73], N[(t$95$0 + N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[-2.0, 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if g < -55

   1. Initial program 38.2%

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

   3. Simplified 38.2%

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

   4. Taylor expanded in g around -inf 38.4%

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

      1. *-commutative 38.4%

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

   6. Simplified 38.4%

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

   7. Taylor expanded in g around inf 15.0%

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

   9. Applied egg-rr 0.0%

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

   11. Simplified 52.7%

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

   if -55 < g < 1.4500000000000001e73

   1. Initial program 70.8%

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

   3. Simplified 70.8%

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

   4. Taylor expanded in g around -inf 35.6%

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

      1. *-commutative 35.6%

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

   6. Simplified 35.6%

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

   7. Taylor expanded in g around inf 16.3%

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

   8. Taylor expanded in a around 0 16.3%

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

   10. Simplified 43.1%

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

   if 1.4500000000000001e73 < g

   1. Initial program 25.5%

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

   3. Simplified 25.5%

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

   4. Taylor expanded in g around -inf 7.3%

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

      1. *-commutative 7.3%

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

   6. Simplified 7.3%

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

   7. Taylor expanded in g around inf 13.3%

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

      1. add-sqr-sqrt 6.0%

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

      2. sqrt-unprod 6.8%

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

      3. swap-sqr 8.7%

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

      4. frac-times 9.7%

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

      5. metadata-eval 9.7%

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

      6. metadata-eval 9.7%

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

      7. frac-times 8.7%

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

      8. *-commutative 8.7%

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

      9. *-commutative 8.7%

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

      10. swap-sqr 8.7%

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

      11. metadata-eval 8.7%

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

      12. metadata-eval 8.7%

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

      13. swap-sqr 8.7%

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

      14. count-2 8.7%

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

      15. count-2 8.7%

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

      16. swap-sqr 6.8%

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

      17. *-commutative 6.8%

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

      18. *-commutative 6.8%

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

      19. sqrt-unprod 6.0%

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

      20. add-sqr-sqrt 13.3%

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

      21. expm1-log1p-u 7.4%

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

   9. Applied egg-rr 0.0%

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

   11. Simplified 44.2%

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

   12. Taylor expanded in g around 0 44.3%

       \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -55:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)} + \sqrt[3]{-0.5}\\ \mathbf{elif}\;g \leq 1.45 \cdot 10^{+73}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)} + \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-2} + \sqrt[3]{\frac{-g}{a}}\\ \end{array} \]

Alternative 8: 73.1% 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(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 43.3%

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

3. Simplified 43.3%

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

4. Taylor expanded in g around -inf 26.1%

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

   1. *-commutative 26.1%

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

6. Simplified 26.1%

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

7. Taylor expanded in g around -inf 69.7%

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

   1. neg-mul-1 69.7%

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

9. Simplified 69.7%

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

    1. associate-*l/ 69.7%

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

    2. *-commutative 69.7%

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

    3. associate-*r* 69.7%

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

    4. metadata-eval 69.7%

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

    5. neg-mul-1 69.7%

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

11. Applied egg-rr 69.7%

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

12. Final simplification 69.7%

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

Alternative 9: 43.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.3%

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

3. Simplified 43.3%

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

4. Taylor expanded in g around -inf 26.1%

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

   1. *-commutative 26.1%

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

6. Simplified 26.1%

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

7. Taylor expanded in g around inf 14.8%

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

   1. add-sqr-sqrt 7.3%

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

   2. sqrt-unprod 14.0%

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

   3. swap-sqr 17.9%

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

   4. frac-times 18.2%

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

   5. metadata-eval 18.2%

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

   6. metadata-eval 18.2%

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

   7. frac-times 17.9%

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

   8. *-commutative 17.9%

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

   9. *-commutative 17.9%

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

   10. swap-sqr 17.9%

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

   11. metadata-eval 17.9%

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

   12. metadata-eval 17.9%

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

   13. swap-sqr 17.9%

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

   14. count-2 17.9%

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

   15. count-2 17.9%

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

   16. swap-sqr 14.0%

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

   17. *-commutative 14.0%

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

   18. *-commutative 14.0%

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

   19. sqrt-unprod 7.3%

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

   20. add-sqr-sqrt 14.8%

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

   21. expm1-log1p-u 9.9%

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

9. Applied egg-rr 0.0%

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

11. Simplified 43.2%

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

12. Final simplification 43.2%

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

Alternative 10: 43.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.3%

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

3. Simplified 43.3%

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

4. Taylor expanded in g around -inf 26.1%

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

   1. *-commutative 26.1%

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

6. Simplified 26.1%

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

7. Taylor expanded in g around inf 14.8%

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

9. Applied egg-rr 0.0%

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

11. Simplified 43.3%

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

12. Final simplification 43.3%

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

Alternative 11: 7.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-308}:\\ \;\;\;\;\sqrt[3]{-2} + \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-2} + -2 \cdot \sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= a -9e-308)
   (+ (cbrt -2.0) (cbrt g))
   (+ (cbrt -2.0) (* -2.0 (cbrt g)))))

C

double code(double g, double h, double a) {
	double tmp;
	if (a <= -9e-308) {
		tmp = cbrt(-2.0) + cbrt(g);
	} else {
		tmp = cbrt(-2.0) + (-2.0 * cbrt(g));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (a <= -9e-308) {
		tmp = Math.cbrt(-2.0) + Math.cbrt(g);
	} else {
		tmp = Math.cbrt(-2.0) + (-2.0 * Math.cbrt(g));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (a <= -9e-308)
		tmp = Float64(cbrt(-2.0) + cbrt(g));
	else
		tmp = Float64(cbrt(-2.0) + Float64(-2.0 * cbrt(g)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[a, -9e-308], N[(N[Power[-2.0, 1/3], $MachinePrecision] + N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[-2.0, 1/3], $MachinePrecision] + N[(-2.0 * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.00000000000000017e-308

   1. Initial program 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in g around -inf 26.2%

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

      1. *-commutative 26.2%

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

   6. Simplified 26.2%

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

   7. Taylor expanded in g around inf 14.9%

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

      1. add-sqr-sqrt 7.9%

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

      2. sqrt-unprod 17.0%

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

      3. swap-sqr 21.2%

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

      4. frac-times 21.2%

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

      5. metadata-eval 21.2%

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

      6. metadata-eval 21.2%

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

      7. frac-times 21.2%

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

      8. *-commutative 21.2%

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

      9. *-commutative 21.2%

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

      10. swap-sqr 21.2%

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

      11. metadata-eval 21.2%

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

      12. metadata-eval 21.2%

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

      13. swap-sqr 21.2%

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

      14. count-2 21.2%

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

      15. count-2 21.2%

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

      16. swap-sqr 17.0%

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

      17. *-commutative 17.0%

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

      18. *-commutative 17.0%

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

      19. sqrt-unprod 7.9%

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

      20. add-sqr-sqrt 14.9%

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

      21. expm1-log1p-u 11.0%

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

   9. Applied egg-rr 0.0%

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

   11. Simplified 38.7%

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

   12. Taylor expanded in g around 0 38.7%

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

   14. Simplified 6.9%

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

   if -9.00000000000000017e-308 < a

   1. Initial program 43.9%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in g around -inf 26.1%

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

      1. *-commutative 26.1%

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

   6. Simplified 26.1%

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

   7. Taylor expanded in g around inf 14.7%

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

      1. add-sqr-sqrt 6.7%

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

      2. sqrt-unprod 11.1%

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

      3. swap-sqr 14.9%

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

      4. frac-times 15.6%

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

      5. metadata-eval 15.6%

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

      6. metadata-eval 15.6%

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

      7. frac-times 14.9%

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

      8. *-commutative 14.9%

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

      9. *-commutative 14.9%

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

      10. swap-sqr 14.9%

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

      11. metadata-eval 14.9%

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

      12. metadata-eval 14.9%

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

      13. swap-sqr 14.9%

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

      14. count-2 14.9%

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

      15. count-2 14.9%

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

      16. swap-sqr 11.1%

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

      17. *-commutative 11.1%

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

      18. *-commutative 11.1%

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

      19. sqrt-unprod 6.7%

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

      20. add-sqr-sqrt 14.7%

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

      21. expm1-log1p-u 8.9%

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

   9. Applied egg-rr 0.0%

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

   11. Simplified 47.4%

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

       1. add-sqr-sqrt 22.3%

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

       2. sqrt-unprod 10.1%

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

       3. *-commutative 10.1%

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

       4. *-commutative 10.1%

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

       5. swap-sqr 6.9%

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

       6. frac-times 6.9%

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

       7. metadata-eval 6.9%

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

       8. metadata-eval 6.9%

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

       9. frac-times 6.9%

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

       10. count-2 6.9%

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

       11. count-2 6.9%

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

       12. swap-sqr 6.9%

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

       13. metadata-eval 6.9%

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

       14. metadata-eval 6.9%

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

       15. swap-sqr 6.9%

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

       16. *-commutative 6.9%

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

       17. *-commutative 6.9%

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

       18. swap-sqr 10.1%

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

       19. sqrt-unprod 22.3%

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

       20. add-sqr-sqrt 47.4%

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

   13. Applied egg-rr 46.8%

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

   15. Simplified 7.6%

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

4. Final simplification 7.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-308}:\\ \;\;\;\;\sqrt[3]{-2} + \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-2} + -2 \cdot \sqrt[3]{g}\\ \end{array} \]

Alternative 12: 43.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.3%

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

3. Simplified 43.3%

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

4. Taylor expanded in g around -inf 26.1%

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

   1. *-commutative 26.1%

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

6. Simplified 26.1%

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

7. Taylor expanded in g around inf 14.8%

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

   1. add-sqr-sqrt 7.3%

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

   2. sqrt-unprod 14.0%

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

   3. swap-sqr 17.9%

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

   4. frac-times 18.2%

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

   5. metadata-eval 18.2%

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

   6. metadata-eval 18.2%

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

   7. frac-times 17.9%

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

   8. *-commutative 17.9%

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

   9. *-commutative 17.9%

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

   10. swap-sqr 17.9%

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

   11. metadata-eval 17.9%

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

   12. metadata-eval 17.9%

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

   13. swap-sqr 17.9%

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

   14. count-2 17.9%

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

   15. count-2 17.9%

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

   16. swap-sqr 14.0%

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

   17. *-commutative 14.0%

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

   18. *-commutative 14.0%

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

   19. sqrt-unprod 7.3%

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

   20. add-sqr-sqrt 14.8%

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

   21. expm1-log1p-u 9.9%

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

9. Applied egg-rr 0.0%

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

11. Simplified 43.2%

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

12. Taylor expanded in g around 0 43.2%

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

13. Final simplification 43.2%

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

Alternative 13: 4.5% accurate, 433.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    code = -2.0d0
end function

Java

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

Python

def code(g, h, a):
	return -2.0

Julia

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

MATLAB

function tmp = code(g, h, a)
	tmp = -2.0;
end

Wolfram

code[g_, h_, a_] := -2.0

Derivation

1. Initial program 43.3%

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

3. Simplified 43.3%

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

4. Taylor expanded in g around -inf 26.1%

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

   1. *-commutative 26.1%

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

6. Simplified 26.1%

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

7. Taylor expanded in g around inf 14.8%

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

   1. add-sqr-sqrt 7.3%

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

   2. sqrt-unprod 14.0%

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

   3. swap-sqr 17.9%

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

   4. frac-times 18.2%

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

   5. metadata-eval 18.2%

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

   6. metadata-eval 18.2%

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

   7. frac-times 17.9%

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

   8. *-commutative 17.9%

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

   9. *-commutative 17.9%

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

   10. swap-sqr 17.9%

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

   11. metadata-eval 17.9%

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

   12. metadata-eval 17.9%

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

   13. swap-sqr 17.9%

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

   14. count-2 17.9%

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

   15. count-2 17.9%

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

   16. swap-sqr 14.0%

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

   17. *-commutative 14.0%

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

   18. *-commutative 14.0%

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

   19. sqrt-unprod 7.3%

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

   20. add-sqr-sqrt 14.8%

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

   21. expm1-log1p-u 9.9%

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

9. Applied egg-rr 0.0%

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

11. Simplified 43.2%

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

12. Taylor expanded in g around 0 4.5%

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

14. Simplified 4.5%

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

15. Final simplification 4.5%

    \[\leadsto -2 \]

Reproduce

herbie shell --seed 2023331 
(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))))))))