2-ancestry mixing, positive discriminant

Percentage Accurate: 44.0% → 95.6%

Time: 20.1s

Alternatives: 7

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: 44.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 95.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.6%

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

3. Simplified 47.6%

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

4. Taylor expanded in g around -inf 26.8%

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

   1. *-commutative 26.8%

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

6. Simplified 26.8%

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

7. Taylor expanded in g around -inf 74.6%

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

   1. neg-mul-1 74.6%

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

9. Simplified 74.6%

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

    1. associate-*l/ 74.6%

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

    2. cbrt-div 96.2%

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

    3. *-commutative 96.2%

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

    4. associate-*r* 96.2%

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

    5. metadata-eval 96.2%

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

    6. neg-mul-1 96.2%

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

11. Applied egg-rr 96.2%

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

12. Final simplification 96.2%

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

Alternative 2: 89.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \mathbf{if}\;a \leq -6.8 \cdot 10^{-81}:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-2}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \sqrt[3]{g \cdot \frac{1}{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (- g g) (/ -0.5 a)))))
   (if (<= a -6.8e-81)
     (+ t_0 (cbrt (/ (- g) a)))
     (if (<= a 7.5e-40)
       (+ (/ (cbrt (- g)) (cbrt a)) (cbrt -2.0))
       (+ t_0 (cbrt (* g (/ 1.0 (- a)))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((g - g) * (-0.5 / a)));
	double tmp;
	if (a <= -6.8e-81) {
		tmp = t_0 + cbrt((-g / a));
	} else if (a <= 7.5e-40) {
		tmp = (cbrt(-g) / cbrt(a)) + cbrt(-2.0);
	} else {
		tmp = t_0 + cbrt((g * (1.0 / -a)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((g - g) * (-0.5 / a)));
	double tmp;
	if (a <= -6.8e-81) {
		tmp = t_0 + Math.cbrt((-g / a));
	} else if (a <= 7.5e-40) {
		tmp = (Math.cbrt(-g) / Math.cbrt(a)) + Math.cbrt(-2.0);
	} else {
		tmp = t_0 + Math.cbrt((g * (1.0 / -a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(g - g) * Float64(-0.5 / a)))
	tmp = 0.0
	if (a <= -6.8e-81)
		tmp = Float64(t_0 + cbrt(Float64(Float64(-g) / a)));
	elseif (a <= 7.5e-40)
		tmp = Float64(Float64(cbrt(Float64(-g)) / cbrt(a)) + cbrt(-2.0));
	else
		tmp = Float64(t_0 + cbrt(Float64(g * Float64(1.0 / Float64(-a)))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[a, -6.8e-81], N[(t$95$0 + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 7.5e-40], N[(N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[-2.0, 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Power[N[(g * N[(1.0 / (-a)), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.7999999999999997e-81

   1. Initial program 53.9%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in g around -inf 28.8%

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

      1. *-commutative 28.8%

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

   6. Simplified 28.8%

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

   7. Taylor expanded in g around -inf 92.6%

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

      1. neg-mul-1 92.6%

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

   9. Simplified 92.6%

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

       1. associate-*l/ 92.6%

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

       2. *-commutative 92.6%

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

       3. associate-*r* 92.6%

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

       4. metadata-eval 92.6%

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

       5. neg-mul-1 92.6%

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

   11. Applied egg-rr 92.6%

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

   if -6.7999999999999997e-81 < a < 7.50000000000000069e-40

   1. Initial program 40.7%

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

   3. Simplified 40.7%

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

   4. Taylor expanded in g around -inf 25.5%

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

      1. *-commutative 25.5%

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

   6. Simplified 25.5%

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

   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. associate-*l/ 11.6%

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

      2. associate-/l* 11.6%

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

      3. add-sqr-sqrt 5.8%

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

      4. sqrt-unprod 7.5%

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

      5. *-commutative 7.5%

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

      6. *-commutative 7.5%

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

      7. swap-sqr 7.5%

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

      8. metadata-eval 7.5%

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

      9. metadata-eval 7.5%

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

      10. swap-sqr 7.5%

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

      11. count-2 7.5%

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

      12. count-2 7.5%

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

      13. sqrt-unprod 1.1%

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

      14. add-sqr-sqrt 1.8%

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

      15. flip-+ 0.0%

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

      16. +-inverses 0.0%

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

      17. +-inverses 0.0%

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

   9. Applied egg-rr 0.0%

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

   11. Simplified 48.1%

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

       1. add-sqr-sqrt 24.6%

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

       2. sqrt-unprod 12.4%

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

       3. swap-sqr 7.5%

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

       4. count-2 7.5%

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

       5. count-2 7.5%

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

       6. swap-sqr 7.5%

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

       7. metadata-eval 7.5%

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

       8. metadata-eval 7.5%

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

       9. swap-sqr 7.5%

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

       10. *-commutative 7.5%

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

       11. *-commutative 7.5%

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

       12. frac-times 7.5%

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

       13. metadata-eval 7.5%

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

       14. metadata-eval 7.5%

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

       15. frac-times 7.5%

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

       16. swap-sqr 12.4%

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

       17. *-commutative 12.4%

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

       18. *-commutative 12.4%

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

       19. sqrt-unprod 24.6%

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

       20. add-sqr-sqrt 48.1%

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

   13. Applied egg-rr 94.9%

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

   if 7.50000000000000069e-40 < a

   1. Initial program 50.8%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in g around -inf 26.4%

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

      1. *-commutative 26.4%

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

   6. Simplified 26.4%

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

   7. Taylor expanded in g around -inf 90.8%

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

      1. neg-mul-1 90.8%

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

   9. Simplified 90.8%

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

       1. associate-*l/ 90.7%

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

       2. *-commutative 90.7%

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

       3. associate-*r* 90.7%

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

       4. metadata-eval 90.7%

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

       5. neg-mul-1 90.7%

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

       6. add-sqr-sqrt 90.7%

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

       7. associate-/r* 90.8%

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

   11. Applied egg-rr 90.8%

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

       1. associate-/l/ 90.7%

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

       2. add-sqr-sqrt 90.7%

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

       3. frac-2neg 90.7%

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

       4. div-inv 90.8%

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

       5. remove-double-neg 90.8%

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

   13. Applied egg-rr 90.8%

       \[\leadsto \sqrt[3]{\color{blue}{g \cdot \frac{1}{-a}}} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-81}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{g \cdot \frac{1}{-a}}\\ \end{array} \]

Alternative 3: 47.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq -7.7 \lor \neg \left(g \leq 0.00035\right):\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{\frac{-2}{g}}} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= g -7.7) (not (<= g 0.00035)))
   (+ (cbrt (/ (- g) a)) (cbrt -2.0))
   (+ (cbrt (/ -0.5 (/ -2.0 g))) (cbrt (* (/ -0.5 a) (+ g g))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((g <= -7.7) || !(g <= 0.00035)) {
		tmp = cbrt((-g / a)) + cbrt(-2.0);
	} else {
		tmp = cbrt((-0.5 / (-2.0 / g))) + cbrt(((-0.5 / a) * (g + g)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((g <= -7.7) || !(g <= 0.00035)) {
		tmp = Math.cbrt((-g / a)) + Math.cbrt(-2.0);
	} else {
		tmp = Math.cbrt((-0.5 / (-2.0 / g))) + Math.cbrt(((-0.5 / a) * (g + g)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((g <= -7.7) || !(g <= 0.00035))
		tmp = Float64(cbrt(Float64(Float64(-g) / a)) + cbrt(-2.0));
	else
		tmp = Float64(cbrt(Float64(-0.5 / Float64(-2.0 / g))) + cbrt(Float64(Float64(-0.5 / a) * Float64(g + g))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[g, -7.7], N[Not[LessEqual[g, 0.00035]], $MachinePrecision]], N[(N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[-2.0, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(-0.5 / N[(-2.0 / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < -7.70000000000000018 or 3.49999999999999996e-4 < g

   1. Initial program 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in g around -inf 21.3%

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

      1. *-commutative 21.3%

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

   6. Simplified 21.3%

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

   7. Taylor expanded in g around inf 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. associate-*l/ 14.7%

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

      2. associate-/l* 14.7%

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

      3. add-sqr-sqrt 7.1%

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

      4. sqrt-unprod 5.1%

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

      5. *-commutative 5.1%

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

      6. *-commutative 5.1%

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

      7. swap-sqr 5.1%

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

      8. metadata-eval 5.1%

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

      9. metadata-eval 5.1%

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

      10. swap-sqr 5.1%

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

      11. count-2 5.1%

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

      12. count-2 5.1%

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

      13. sqrt-unprod 1.4%

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

      14. add-sqr-sqrt 2.4%

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

      15. flip-+ 0.0%

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

      16. +-inverses 0.0%

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

      17. +-inverses 0.0%

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

   9. Applied egg-rr 0.0%

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

   11. Simplified 49.4%

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

   12. Taylor expanded in g around 0 49.4%

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

       1. neg-mul-1 49.4%

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

       2. distribute-neg-frac 49.4%

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

   14. Simplified 49.4%

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

   if -7.70000000000000018 < g < 3.49999999999999996e-4

   1. Initial program 78.4%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in g around -inf 41.9%

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

      1. *-commutative 41.9%

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

   6. Simplified 41.9%

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

   7. Taylor expanded in g around inf 17.5%

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

   9. Applied egg-rr 15.8%

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

   11. Simplified 48.5%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -7.7 \lor \neg \left(g \leq 0.00035\right):\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{\frac{-2}{g}}} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \end{array} \]

Alternative 4: 47.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq -0.3 \lor \neg \left(g \leq 0.00035\right):\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= g -0.3) (not (<= g 0.00035)))
   (+ (cbrt (/ (- g) a)) (cbrt -2.0))
   (+ (cbrt (* (/ 0.5 a) (* g -2.0))) (cbrt g))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((g <= -0.3) || !(g <= 0.00035)) {
		tmp = cbrt((-g / a)) + cbrt(-2.0);
	} else {
		tmp = cbrt(((0.5 / a) * (g * -2.0))) + cbrt(g);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((g <= -0.3) || !(g <= 0.00035)) {
		tmp = Math.cbrt((-g / a)) + Math.cbrt(-2.0);
	} else {
		tmp = Math.cbrt(((0.5 / a) * (g * -2.0))) + Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((g <= -0.3) || !(g <= 0.00035))
		tmp = Float64(cbrt(Float64(Float64(-g) / a)) + cbrt(-2.0));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))) + cbrt(g));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[g, -0.3], N[Not[LessEqual[g, 0.00035]], $MachinePrecision]], N[(N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[-2.0, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < -0.299999999999999989 or 3.49999999999999996e-4 < g

   1. Initial program 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in g around -inf 21.3%

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

      1. *-commutative 21.3%

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

   6. Simplified 21.3%

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

   7. Taylor expanded in g around inf 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. associate-*l/ 14.7%

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

      2. associate-/l* 14.7%

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

      3. add-sqr-sqrt 7.1%

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

      4. sqrt-unprod 5.1%

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

      5. *-commutative 5.1%

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

      6. *-commutative 5.1%

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

      7. swap-sqr 5.1%

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

      8. metadata-eval 5.1%

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

      9. metadata-eval 5.1%

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

      10. swap-sqr 5.1%

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

      11. count-2 5.1%

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

      12. count-2 5.1%

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

      13. sqrt-unprod 1.4%

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

      14. add-sqr-sqrt 2.4%

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

      15. flip-+ 0.0%

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

      16. +-inverses 0.0%

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

      17. +-inverses 0.0%

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

   9. Applied egg-rr 0.0%

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

   11. Simplified 49.4%

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

   12. Taylor expanded in g around 0 49.4%

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

       1. neg-mul-1 49.4%

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

       2. distribute-neg-frac 49.4%

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

   14. Simplified 49.4%

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

   if -0.299999999999999989 < g < 3.49999999999999996e-4

   1. Initial program 78.4%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in g around -inf 41.9%

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

      1. *-commutative 41.9%

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

   6. Simplified 41.9%

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

   7. Taylor expanded in g around inf 17.5%

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

      1. expm1-log1p-u 14.2%

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

      2. expm1-udef 51.9%

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

   9. Applied egg-rr 49.0%

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

   11. Simplified 48.4%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -0.3 \lor \neg \left(g \leq 0.00035\right):\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{g}\\ \end{array} \]

Alternative 5: 73.0% 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 47.6%

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

3. Simplified 47.6%

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

4. Taylor expanded in g around -inf 26.8%

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

   1. *-commutative 26.8%

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

6. Simplified 26.8%

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

7. Taylor expanded in g around -inf 74.6%

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

   1. neg-mul-1 74.6%

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

9. Simplified 74.6%

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

    1. associate-*l/ 74.6%

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

    2. *-commutative 74.6%

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

    3. associate-*r* 74.6%

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

    4. metadata-eval 74.6%

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

    5. neg-mul-1 74.6%

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

11. Applied egg-rr 74.6%

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

12. Final simplification 74.6%

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

Alternative 6: 43.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.6%

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

3. Simplified 47.6%

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

4. Taylor expanded in g around -inf 26.8%

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

   1. *-commutative 26.8%

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

6. Simplified 26.8%

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

7. Taylor expanded in g around inf 15.4%

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

   1. associate-*l/ 15.4%

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

   2. associate-/l* 15.6%

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

   3. add-sqr-sqrt 7.6%

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

   4. sqrt-unprod 8.1%

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

   5. *-commutative 8.1%

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

   6. *-commutative 8.1%

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

   7. swap-sqr 8.5%

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

   8. metadata-eval 8.5%

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

   9. metadata-eval 8.5%

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

   10. swap-sqr 8.1%

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

   11. count-2 8.1%

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

   12. count-2 8.1%

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

   13. sqrt-unprod 1.6%

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

   14. add-sqr-sqrt 3.0%

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

   15. flip-+ 0.0%

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

   16. +-inverses 0.0%

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

   17. +-inverses 0.0%

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

9. Applied egg-rr 0.0%

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

11. Simplified 45.2%

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

12. Taylor expanded in g around 0 45.2%

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

    1. neg-mul-1 45.2%

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

    2. distribute-neg-frac 45.2%

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

14. Simplified 45.2%

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

15. Final simplification 45.2%

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

Alternative 7: 4.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.6%

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

3. Simplified 47.6%

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

4. Taylor expanded in g around -inf 26.8%

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

   1. *-commutative 26.8%

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

6. Simplified 26.8%

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

7. Taylor expanded in g around inf 15.4%

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

   1. associate-*l/ 15.4%

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

   2. associate-/l* 15.6%

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

   3. add-sqr-sqrt 7.6%

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

   4. sqrt-unprod 8.1%

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

   5. *-commutative 8.1%

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

   6. *-commutative 8.1%

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

   7. swap-sqr 8.5%

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

   8. metadata-eval 8.5%

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

   9. metadata-eval 8.5%

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

   10. swap-sqr 8.1%

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

   11. count-2 8.1%

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

   12. count-2 8.1%

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

   13. sqrt-unprod 1.6%

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

   14. add-sqr-sqrt 3.0%

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

   15. flip-+ 0.0%

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

   16. +-inverses 0.0%

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

   17. +-inverses 0.0%

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

9. Applied egg-rr 0.0%

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

11. Simplified 45.2%

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

12. Taylor expanded in g around 0 4.6%

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

13. Final simplification 4.6%

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

Reproduce

herbie shell --seed 2023318 
(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))))))))