2-ancestry mixing, positive discriminant

Percentage Accurate: 44.2% → 95.7%

Time: 28.9s

Alternatives: 10

Speedup: 2.1×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{t\_0 \cdot \left(\left(-g\right) - t\_1\right)} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.2% 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: 95.7% 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 50.4%

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

3. Simplified 50.4%

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

5. Taylor expanded in g around -inf 29.4%

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

   1. *-commutative 29.4%

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

7. Simplified 29.4%

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

8. Taylor expanded in g around -inf 75.8%

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

   1. neg-mul-1 75.8%

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

10. Simplified 75.8%

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

    1. cbrt-prod 95.5%

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

12. Applied egg-rr 95.5%

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

13. Final simplification 95.5%

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

Alternative 2: 88.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \mathbf{if}\;a \leq -4.6 \cdot 10^{-11}:\\ \;\;\;\;t\_0 - \sqrt[3]{\frac{g}{a}}\\ \mathbf{elif}\;a \leq 1.32 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \sqrt[3]{\frac{-1}{\frac{a}{g}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (- g g) (/ -0.5 a)))))
   (if (<= a -4.6e-11)
     (- t_0 (cbrt (/ g a)))
     (if (<= a 1.32e-64)
       (+ (/ (cbrt (- g)) (cbrt a)) (cbrt -1.0))
       (+ t_0 (cbrt (/ -1.0 (/ a g))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((g - g) * (-0.5 / a)));
	double tmp;
	if (a <= -4.6e-11) {
		tmp = t_0 - cbrt((g / a));
	} else if (a <= 1.32e-64) {
		tmp = (cbrt(-g) / cbrt(a)) + cbrt(-1.0);
	} else {
		tmp = t_0 + cbrt((-1.0 / (a / g)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((g - g) * (-0.5 / a)));
	double tmp;
	if (a <= -4.6e-11) {
		tmp = t_0 - Math.cbrt((g / a));
	} else if (a <= 1.32e-64) {
		tmp = (Math.cbrt(-g) / Math.cbrt(a)) + Math.cbrt(-1.0);
	} else {
		tmp = t_0 + Math.cbrt((-1.0 / (a / g)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(g - g) * Float64(-0.5 / a)))
	tmp = 0.0
	if (a <= -4.6e-11)
		tmp = Float64(t_0 - cbrt(Float64(g / a)));
	elseif (a <= 1.32e-64)
		tmp = Float64(Float64(cbrt(Float64(-g)) / cbrt(a)) + cbrt(-1.0));
	else
		tmp = Float64(t_0 + cbrt(Float64(-1.0 / Float64(a / g))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[a, -4.6e-11], N[(t$95$0 - N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.32e-64], N[(N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Power[N[(-1.0 / N[(a / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.60000000000000027e-11

   1. Initial program 53.4%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in g around -inf 31.5%

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

      1. *-commutative 31.5%

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

   7. Simplified 31.5%

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

   8. Taylor expanded in g around -inf 93.4%

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

      1. neg-mul-1 93.4%

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

   10. Simplified 93.4%

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

   11. Taylor expanded in g around -inf 93.4%

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

       1. mul-1-neg 93.4%

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

   13. Simplified 93.4%

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

   if -4.60000000000000027e-11 < a < 1.32e-64

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in g around -inf 27.4%

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

      1. *-commutative 27.4%

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

   7. Simplified 27.4%

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

   8. Taylor expanded in g around inf 12.7%

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

      1. 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 4.5%

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

      3. swap-sqr 3.8%

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

      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 3.8%

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

      8. *-commutative 3.8%

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

      9. *-commutative 3.8%

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

      10. swap-sqr 3.8%

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

      11. metadata-eval 3.8%

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

      12. metadata-eval 3.8%

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

      13. swap-sqr 3.8%

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

      14. count-2 3.8%

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

      15. count-2 3.8%

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

      16. swap-sqr 4.5%

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

      17. *-commutative 4.5%

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

      18. *-commutative 4.5%

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

      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 12.7%

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

      21. expm1-log1p-u 6.8%

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

   10. Applied egg-rr 0.0%

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

   12. Simplified 53.4%

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

       1. add-sqr-sqrt 27.0%

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

       2. sqrt-unprod 12.0%

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

       3. *-commutative 12.0%

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

       4. *-commutative 12.0%

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

       5. swap-sqr 7.7%

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

       6. frac-times 7.7%

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

       7. metadata-eval 7.7%

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

       8. metadata-eval 7.7%

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

       9. frac-times 7.7%

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

       10. count-2 7.7%

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

       11. count-2 7.7%

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

       12. swap-sqr 7.7%

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

       13. metadata-eval 7.7%

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

       14. metadata-eval 7.7%

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

       15. swap-sqr 7.7%

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

       16. *-commutative 7.7%

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

       17. *-commutative 7.7%

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

       18. swap-sqr 12.0%

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

   14. Applied egg-rr 91.4%

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

   if 1.32e-64 < a

   1. Initial program 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in g around -inf 30.5%

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

      1. *-commutative 30.5%

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

   7. Simplified 30.5%

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

   8. Taylor expanded in g around -inf 86.7%

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

      1. neg-mul-1 86.7%

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

   10. Simplified 86.7%

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

       1. associate-*l/ 86.7%

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

       2. clear-num 86.7%

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

       3. *-commutative 86.7%

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

       4. associate-*r* 86.7%

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

       5. metadata-eval 86.7%

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

       6. neg-mul-1 86.7%

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

   12. Applied egg-rr 86.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-11}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} - \sqrt[3]{\frac{g}{a}}\\ \mathbf{elif}\;a \leq 1.32 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-1}{\frac{a}{g}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(g - g) * Float64(-0.5 / a))) + Float64(cbrt(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 50.4%

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

3. Simplified 50.4%

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

5. Taylor expanded in g around -inf 29.4%

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

   1. *-commutative 29.4%

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

7. Simplified 29.4%

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

8. Taylor expanded in g around -inf 75.8%

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

   1. neg-mul-1 75.8%

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

10. Simplified 75.8%

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

    1. associate-*l/ 75.8%

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

    2. cbrt-div 95.5%

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

    3. *-commutative 95.5%

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

    4. associate-*r* 95.5%

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

    5. metadata-eval 95.5%

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

    6. neg-mul-1 95.5%

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

12. Applied egg-rr 95.5%

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

13. Final simplification 95.5%

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

Alternative 4: 62.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-40} \lor \neg \left(a \leq 9 \cdot 10^{-10}\right):\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \frac{-1}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= a -1.35e-40) (not (<= a 9e-10)))
   (+ (cbrt (* (/ 0.5 a) (* g -2.0))) (/ -1.0 (cbrt a)))
   (+ (cbrt g) (cbrt (* (/ -0.5 a) (+ g g))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((a <= -1.35e-40) || !(a <= 9e-10)) {
		tmp = cbrt(((0.5 / a) * (g * -2.0))) + (-1.0 / cbrt(a));
	} else {
		tmp = cbrt(g) + cbrt(((-0.5 / a) * (g + g)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((a <= -1.35e-40) || !(a <= 9e-10)) {
		tmp = Math.cbrt(((0.5 / a) * (g * -2.0))) + (-1.0 / Math.cbrt(a));
	} else {
		tmp = Math.cbrt(g) + Math.cbrt(((-0.5 / a) * (g + g)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((a <= -1.35e-40) || !(a <= 9e-10))
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))) + Float64(-1.0 / cbrt(a)));
	else
		tmp = Float64(cbrt(g) + cbrt(Float64(Float64(-0.5 / a) * Float64(g + g))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[a, -1.35e-40], N[Not[LessEqual[a, 9e-10]], $MachinePrecision]], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(-1.0 / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[g, 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.35e-40 or 8.9999999999999999e-10 < a

   1. Initial program 48.2%

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

   3. Simplified 48.2%

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

   5. Taylor expanded in g around -inf 29.9%

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

      1. *-commutative 29.9%

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

   7. Simplified 29.9%

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

   8. Taylor expanded in g around inf 17.7%

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

      1. *-un-lft-identity 46.1%

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

      2. *-commutative 46.1%

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

   10. Applied egg-rr 0.0%

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

   12. Simplified 69.6%

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

   if -1.35e-40 < a < 8.9999999999999999e-10

   1. Initial program 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in g around -inf 28.9%

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

      1. *-commutative 28.9%

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

   7. Simplified 28.9%

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

   8. Taylor expanded in g around inf 13.4%

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

   9. Taylor expanded in a around 0 13.4%

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

   11. Simplified 57.9%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-40} \lor \neg \left(a \leq 9 \cdot 10^{-10}\right):\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \frac{-1}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq -4.1 \cdot 10^{-38} \lor \neg \left(g \leq 6.5 \cdot 10^{+28}\right):\\ \;\;\;\;\sqrt[3]{-1} + \sqrt[3]{\frac{g}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= g -4.1e-38) (not (<= g 6.5e+28)))
   (+ (cbrt -1.0) (cbrt (/ g (- a))))
   (+ (cbrt g) (cbrt (* (/ -0.5 a) (+ g g))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((g <= -4.1e-38) || !(g <= 6.5e+28)) {
		tmp = cbrt(-1.0) + cbrt((g / -a));
	} else {
		tmp = cbrt(g) + cbrt(((-0.5 / a) * (g + g)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((g <= -4.1e-38) || !(g <= 6.5e+28)) {
		tmp = Math.cbrt(-1.0) + Math.cbrt((g / -a));
	} else {
		tmp = Math.cbrt(g) + Math.cbrt(((-0.5 / a) * (g + g)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((g <= -4.1e-38) || !(g <= 6.5e+28))
		tmp = Float64(cbrt(-1.0) + cbrt(Float64(g / Float64(-a))));
	else
		tmp = Float64(cbrt(g) + cbrt(Float64(Float64(-0.5 / a) * Float64(g + g))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[g, -4.1e-38], N[Not[LessEqual[g, 6.5e+28]], $MachinePrecision]], N[(N[Power[-1.0, 1/3], $MachinePrecision] + N[Power[N[(g / (-a)), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[g, 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < -4.0999999999999998e-38 or 6.5000000000000001e28 < g

   1. Initial program 41.4%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in g around -inf 28.7%

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

      1. *-commutative 28.7%

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

   7. Simplified 28.7%

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

   8. Taylor expanded in g around inf 15.1%

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

      1. add-sqr-sqrt 7.5%

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

      2. sqrt-unprod 11.5%

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

      3. swap-sqr 12.3%

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

      4. frac-times 12.3%

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

      5. metadata-eval 12.3%

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

      6. metadata-eval 12.3%

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

      7. frac-times 12.3%

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

      8. *-commutative 12.3%

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

      9. *-commutative 12.3%

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

      10. swap-sqr 12.3%

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

      11. metadata-eval 12.3%

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

      12. metadata-eval 12.3%

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

      13. swap-sqr 12.3%

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

      14. count-2 12.3%

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

      15. count-2 12.3%

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

      16. swap-sqr 11.5%

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

      17. *-commutative 11.5%

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

      18. *-commutative 11.5%

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

      19. sqrt-unprod 7.5%

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

      20. add-sqr-sqrt 15.1%

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

      21. expm1-log1p-u 9.3%

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

   10. Applied egg-rr 0.0%

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

   12. Simplified 53.7%

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

   13. Taylor expanded in g around 0 53.7%

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

       1. neg-mul-1 53.7%

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

       2. distribute-neg-frac2 53.7%

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

   15. Simplified 53.7%

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

   if -4.0999999999999998e-38 < g < 6.5000000000000001e28

   1. Initial program 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in g around -inf 31.2%

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

      1. *-commutative 31.2%

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

   7. Simplified 31.2%

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

   8. Taylor expanded in g around inf 17.0%

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

   9. Taylor expanded in a around 0 17.0%

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

   11. Simplified 48.6%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -4.1 \cdot 10^{-38} \lor \neg \left(g \leq 6.5 \cdot 10^{+28}\right):\\ \;\;\;\;\sqrt[3]{-1} + \sqrt[3]{\frac{g}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq -2.8 \cdot 10^{-13}:\\ \;\;\;\;\sqrt[3]{-1} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \mathbf{elif}\;g \leq 115000000000:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} - \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-1} + \sqrt[3]{\frac{g}{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= g -2.8e-13)
   (+ (cbrt -1.0) (cbrt (* (/ -0.5 a) (+ g g))))
   (if (<= g 115000000000.0)
     (- (cbrt (* (/ 0.5 a) (* g -2.0))) (cbrt g))
     (+ (cbrt -1.0) (cbrt (/ g (- a)))))))

C

double code(double g, double h, double a) {
	double tmp;
	if (g <= -2.8e-13) {
		tmp = cbrt(-1.0) + cbrt(((-0.5 / a) * (g + g)));
	} else if (g <= 115000000000.0) {
		tmp = cbrt(((0.5 / a) * (g * -2.0))) - cbrt(g);
	} else {
		tmp = cbrt(-1.0) + cbrt((g / -a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (g <= -2.8e-13) {
		tmp = Math.cbrt(-1.0) + Math.cbrt(((-0.5 / a) * (g + g)));
	} else if (g <= 115000000000.0) {
		tmp = Math.cbrt(((0.5 / a) * (g * -2.0))) - Math.cbrt(g);
	} else {
		tmp = Math.cbrt(-1.0) + Math.cbrt((g / -a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (g <= -2.8e-13)
		tmp = Float64(cbrt(-1.0) + cbrt(Float64(Float64(-0.5 / a) * Float64(g + g))));
	elseif (g <= 115000000000.0)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))) - cbrt(g));
	else
		tmp = Float64(cbrt(-1.0) + cbrt(Float64(g / Float64(-a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[g, -2.8e-13], N[(N[Power[-1.0, 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 115000000000.0], 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], N[(N[Power[-1.0, 1/3], $MachinePrecision] + N[Power[N[(g / (-a)), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if g < -2.8000000000000002e-13

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in g around -inf 45.5%

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

      1. *-commutative 45.5%

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

   7. Simplified 45.5%

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

   8. Taylor expanded in g around inf 16.3%

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

      1. add-sqr-sqrt 7.7%

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

      2. sqrt-unprod 9.6%

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

      3. swap-sqr 10.4%

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

      4. frac-times 10.5%

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

      5. metadata-eval 10.5%

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

      6. metadata-eval 10.5%

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

      7. frac-times 10.4%

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

      8. *-commutative 10.4%

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

      9. *-commutative 10.4%

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

      10. swap-sqr 10.4%

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

      11. metadata-eval 10.4%

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

      12. metadata-eval 10.4%

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

      13. swap-sqr 10.4%

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

      14. count-2 10.4%

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

      15. count-2 10.4%

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

      16. swap-sqr 9.6%

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

      17. *-commutative 9.6%

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

      18. *-commutative 9.6%

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

      19. sqrt-unprod 7.7%

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

      20. add-sqr-sqrt 16.3%

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

      21. expm1-log1p-u 10.0%

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

   10. Applied egg-rr 0.0%

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

   12. Simplified 58.0%

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

   if -2.8000000000000002e-13 < g < 1.15e11

   1. Initial program 73.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in g around -inf 35.8%

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

      1. *-commutative 35.8%

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

   7. Simplified 35.8%

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

   8. Taylor expanded in g around inf 17.0%

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

   9. Taylor expanded in g around 0 17.0%

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

   11. Simplified 50.1%

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

   if 1.15e11 < g

   1. Initial program 37.0%

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

   3. Simplified 37.0%

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

   5. Taylor expanded in g around -inf 9.2%

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

      1. *-commutative 9.2%

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

   7. Simplified 9.2%

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

   8. Taylor expanded in g around inf 13.9%

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

      1. add-sqr-sqrt 7.6%

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

      2. sqrt-unprod 12.2%

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

      3. swap-sqr 13.0%

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

      4. frac-times 13.0%

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

      5. metadata-eval 13.0%

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

      6. metadata-eval 13.0%

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

      7. frac-times 13.0%

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

      8. *-commutative 13.0%

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

      9. *-commutative 13.0%

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

      10. swap-sqr 13.0%

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

      11. metadata-eval 13.0%

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

      12. metadata-eval 13.0%

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

      13. swap-sqr 13.0%

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

      14. count-2 13.0%

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

      15. count-2 13.0%

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

      16. swap-sqr 12.2%

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

      17. *-commutative 12.2%

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

      18. *-commutative 12.2%

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

      19. sqrt-unprod 7.6%

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

      20. add-sqr-sqrt 13.9%

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

      21. expm1-log1p-u 8.8%

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

   10. Applied egg-rr 0.0%

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

   12. Simplified 48.6%

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

   13. Taylor expanded in g around 0 48.6%

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

       1. neg-mul-1 48.6%

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

       2. distribute-neg-frac2 48.6%

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

   15. Simplified 48.6%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -2.8 \cdot 10^{-13}:\\ \;\;\;\;\sqrt[3]{-1} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \mathbf{elif}\;g \leq 115000000000:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} - \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-1} + \sqrt[3]{\frac{g}{-a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Simplified 50.4%

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

5. Taylor expanded in g around -inf 29.4%

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

   1. *-commutative 29.4%

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

7. Simplified 29.4%

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

8. Taylor expanded in g around -inf 75.8%

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

   1. neg-mul-1 75.8%

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

10. Simplified 75.8%

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

11. Taylor expanded in g around -inf 75.8%

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

    1. mul-1-neg 75.8%

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

13. Simplified 75.8%

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

14. Final simplification 75.8%

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

Alternative 8: 43.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Simplified 50.4%

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

5. Taylor expanded in g around -inf 29.4%

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

   1. *-commutative 29.4%

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

7. Simplified 29.4%

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

8. Taylor expanded in g around inf 15.6%

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

   1. add-sqr-sqrt 8.1%

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

   2. sqrt-unprod 17.4%

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

   3. swap-sqr 18.4%

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

   4. frac-times 18.4%

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

   5. metadata-eval 18.4%

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

   6. metadata-eval 18.4%

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

   7. frac-times 18.4%

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

   8. *-commutative 18.4%

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

   9. *-commutative 18.4%

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

   10. swap-sqr 18.4%

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

   11. metadata-eval 18.4%

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

   12. metadata-eval 18.4%

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

   13. swap-sqr 18.4%

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

   14. count-2 18.4%

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

   15. count-2 18.4%

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

   16. swap-sqr 17.4%

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

   17. *-commutative 17.4%

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

   18. *-commutative 17.4%

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

   19. sqrt-unprod 8.1%

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

   20. add-sqr-sqrt 15.6%

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

   21. expm1-log1p-u 10.4%

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

10. Applied egg-rr 0.0%

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

12. Simplified 48.7%

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

13. Taylor expanded in g around 0 48.7%

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

    1. neg-mul-1 48.7%

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

    2. distribute-neg-frac2 48.7%

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

15. Simplified 48.7%

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

16. Final simplification 48.7%

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

Alternative 9: 4.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Simplified 50.4%

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

5. Taylor expanded in g around -inf 29.4%

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

   1. *-commutative 29.4%

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

7. Simplified 29.4%

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

8. Taylor expanded in g around inf 15.6%

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

   1. add-sqr-sqrt 8.1%

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

   2. sqrt-unprod 17.4%

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

   3. swap-sqr 18.4%

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

   4. frac-times 18.4%

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

   5. metadata-eval 18.4%

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

   6. metadata-eval 18.4%

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

   7. frac-times 18.4%

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

   8. *-commutative 18.4%

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

   9. *-commutative 18.4%

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

   10. swap-sqr 18.4%

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

   11. metadata-eval 18.4%

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

   12. metadata-eval 18.4%

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

   13. swap-sqr 18.4%

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

   14. count-2 18.4%

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

   15. count-2 18.4%

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

   16. swap-sqr 17.4%

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

   17. *-commutative 17.4%

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

   18. *-commutative 17.4%

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

   19. sqrt-unprod 8.1%

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

   20. add-sqr-sqrt 15.6%

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

   21. expm1-log1p-u 10.4%

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

10. Applied egg-rr 0.0%

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

12. Simplified 48.7%

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

    1. *-un-lft-identity 48.7%

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

    2. *-commutative 48.7%

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

14. Applied egg-rr 0.0%

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

16. Simplified 5.0%

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

17. Final simplification 5.0%

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

Alternative 10: 4.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Simplified 50.4%

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

5. Taylor expanded in g around -inf 29.4%

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

   1. *-commutative 29.4%

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

7. Simplified 29.4%

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

8. Taylor expanded in g around inf 15.6%

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

   1. add-sqr-sqrt 8.1%

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

   2. sqrt-unprod 17.4%

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

   3. swap-sqr 18.4%

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

   4. frac-times 18.4%

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

   5. metadata-eval 18.4%

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

   6. metadata-eval 18.4%

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

   7. frac-times 18.4%

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

   8. *-commutative 18.4%

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

   9. *-commutative 18.4%

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

   10. swap-sqr 18.4%

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

   11. metadata-eval 18.4%

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

   12. metadata-eval 18.4%

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

   13. swap-sqr 18.4%

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

   14. count-2 18.4%

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

   15. count-2 18.4%

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

   16. swap-sqr 17.4%

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

   17. *-commutative 17.4%

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

   18. *-commutative 17.4%

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

   19. sqrt-unprod 8.1%

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

   20. add-sqr-sqrt 15.6%

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

   21. expm1-log1p-u 10.4%

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

10. Applied egg-rr 0.0%

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

12. Simplified 48.7%

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

13. Taylor expanded in g around 0 48.7%

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

15. Simplified 4.5%

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

16. Final simplification 4.5%

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

Reproduce

herbie shell --seed 2024051 
(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))))))))