2-ancestry mixing, positive discriminant

Percentage Accurate: 44.9% → 96.4%

Time: 16.6s

Alternatives: 5

Speedup: 4.2×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \cdot h \leq 4 \cdot 10^{+247}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \frac{-0.5 \cdot {h}^{2}}{g}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{2 \cdot g}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}, \sqrt[3]{-0.5} \cdot \sqrt{\sqrt[3]{4}}, \sqrt[3]{\frac{{h}^{-2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (* h h) 4e+247)
   (+
    (cbrt (* (/ 0.5 a) (/ (* -0.5 (pow h 2.0)) g)))
    (* (cbrt (/ -0.5 a)) (cbrt (* 2.0 g))))
   (fma
    (/ (cbrt g) (cbrt a))
    (* (cbrt -0.5) (sqrt (cbrt 4.0)))
    (* (cbrt (/ (pow h -2.0) (* a g))) (* (cbrt -0.5) (cbrt 0.5))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((h * h) <= 4e+247) {
		tmp = cbrt(((0.5 / a) * ((-0.5 * pow(h, 2.0)) / g))) + (cbrt((-0.5 / a)) * cbrt((2.0 * g)));
	} else {
		tmp = fma((cbrt(g) / cbrt(a)), (cbrt(-0.5) * sqrt(cbrt(4.0))), (cbrt((pow(h, -2.0) / (a * g))) * (cbrt(-0.5) * cbrt(0.5))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(h * h) <= 4e+247)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(Float64(-0.5 * (h ^ 2.0)) / g))) + Float64(cbrt(Float64(-0.5 / a)) * cbrt(Float64(2.0 * g))));
	else
		tmp = fma(Float64(cbrt(g) / cbrt(a)), Float64(cbrt(-0.5) * sqrt(cbrt(4.0))), Float64(cbrt(Float64((h ^ -2.0) / Float64(a * g))) * Float64(cbrt(-0.5) * cbrt(0.5))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(h * h), $MachinePrecision], 4e+247], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(N[(-0.5 * N[Power[h, 2.0], $MachinePrecision]), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Sqrt[N[Power[4.0, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[Power[h, -2.0], $MachinePrecision] / N[(a * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 3.99999999999999981e247

   1. Initial program 48.5%

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

   3. Simplified 48.5%

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

   5. Taylor expanded in g around inf 29.5%

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

   6. Taylor expanded in g around inf 77.3%

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

      1. associate-*r/ 77.3%

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

   8. Simplified 77.3%

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

      1. *-commutative 77.3%

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

      2. cbrt-prod 97.2%

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

      3. count-2 97.2%

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

   10. Applied egg-rr 97.2%

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

   if 3.99999999999999981e247 < (*.f64 h h)

   1. Initial program 0.2%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in h around 0 12.6%

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

      1. fma-define 12.6%

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

      2. *-commutative 12.6%

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

      3. *-commutative 12.6%

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

   7. Simplified 12.6%

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

   9. Applied egg-rr 19.5%

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

   11. Applied egg-rr 19.8%

       \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}, \sqrt[3]{-0.5} \cdot \color{blue}{\sqrt{\sqrt[3]{4}}}, \sqrt[3]{\frac{{h}^{2}}{g \cdot a}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \]

   13. Applied egg-rr 88.5%

       \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}, \sqrt[3]{-0.5} \cdot \sqrt{\sqrt[3]{4}}, \color{blue}{{\left(\sqrt[3]{\sqrt[3]{\frac{\frac{{h}^{-2}}{g}}{a}}}\right)}^{3}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \]
   14. Step-by-step derivation

       1. rem-cube-cbrt 88.5%

          \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}, \sqrt[3]{-0.5} \cdot \sqrt{\sqrt[3]{4}}, \color{blue}{\sqrt[3]{\frac{\frac{{h}^{-2}}{g}}{a}}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \]

       2. associate-/l/ 88.5%

          \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}, \sqrt[3]{-0.5} \cdot \sqrt{\sqrt[3]{4}}, \sqrt[3]{\color{blue}{\frac{{h}^{-2}}{a \cdot g}}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \]

   15. Simplified 88.5%

       \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}, \sqrt[3]{-0.5} \cdot \sqrt{\sqrt[3]{4}}, \color{blue}{\sqrt[3]{\frac{{h}^{-2}}{a \cdot g}}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 4 \cdot 10^{+247}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \frac{-0.5 \cdot {h}^{2}}{g}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{2 \cdot g}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}, \sqrt[3]{-0.5} \cdot \sqrt{\sqrt[3]{4}}, \sqrt[3]{\frac{{h}^{-2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \cdot h \leq 4 \cdot 10^{+247}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \frac{-0.5 \cdot {h}^{2}}{g}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{2 \cdot g}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}, \sqrt[3]{-0.5} \cdot \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{-2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (* h h) 4e+247)
   (+
    (cbrt (* (/ 0.5 a) (/ (* -0.5 (pow h 2.0)) g)))
    (* (cbrt (/ -0.5 a)) (cbrt (* 2.0 g))))
   (fma
    (/ (cbrt g) (cbrt a))
    (* (cbrt -0.5) (cbrt 2.0))
    (* (cbrt (/ (pow h -2.0) (* a g))) (* (cbrt -0.5) (cbrt 0.5))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((h * h) <= 4e+247) {
		tmp = cbrt(((0.5 / a) * ((-0.5 * pow(h, 2.0)) / g))) + (cbrt((-0.5 / a)) * cbrt((2.0 * g)));
	} else {
		tmp = fma((cbrt(g) / cbrt(a)), (cbrt(-0.5) * cbrt(2.0)), (cbrt((pow(h, -2.0) / (a * g))) * (cbrt(-0.5) * cbrt(0.5))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(h * h) <= 4e+247)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(Float64(-0.5 * (h ^ 2.0)) / g))) + Float64(cbrt(Float64(-0.5 / a)) * cbrt(Float64(2.0 * g))));
	else
		tmp = fma(Float64(cbrt(g) / cbrt(a)), Float64(cbrt(-0.5) * cbrt(2.0)), Float64(cbrt(Float64((h ^ -2.0) / Float64(a * g))) * Float64(cbrt(-0.5) * cbrt(0.5))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(h * h), $MachinePrecision], 4e+247], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(N[(-0.5 * N[Power[h, 2.0], $MachinePrecision]), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[Power[h, -2.0], $MachinePrecision] / N[(a * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 3.99999999999999981e247

   1. Initial program 48.5%

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

   3. Simplified 48.5%

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

   5. Taylor expanded in g around inf 29.5%

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

   6. Taylor expanded in g around inf 77.3%

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

      1. associate-*r/ 77.3%

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

   8. Simplified 77.3%

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

      1. *-commutative 77.3%

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

      2. cbrt-prod 97.2%

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

      3. count-2 97.2%

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

   10. Applied egg-rr 97.2%

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

   if 3.99999999999999981e247 < (*.f64 h h)

   1. Initial program 0.2%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in h around 0 12.6%

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

      1. fma-define 12.6%

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

      2. *-commutative 12.6%

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

      3. *-commutative 12.6%

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

   7. Simplified 12.6%

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

   9. Applied egg-rr 19.5%

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

   11. Applied egg-rr 87.7%

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

       1. rem-cube-cbrt 88.5%

          \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}, \sqrt[3]{-0.5} \cdot \sqrt{\sqrt[3]{4}}, \color{blue}{\sqrt[3]{\frac{\frac{{h}^{-2}}{g}}{a}}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \]

       2. associate-/l/ 88.5%

          \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}, \sqrt[3]{-0.5} \cdot \sqrt{\sqrt[3]{4}}, \sqrt[3]{\color{blue}{\frac{{h}^{-2}}{a \cdot g}}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \]

   13. Simplified 87.7%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 4 \cdot 10^{+247}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \frac{-0.5 \cdot {h}^{2}}{g}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{2 \cdot g}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}, \sqrt[3]{-0.5} \cdot \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{-2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \frac{-0.5 \cdot {h}^{2}}{g}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{2 \cdot g}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt[3]{\frac{a}{g}}}, \sqrt[3]{-0.5} \cdot \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{-2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= h 1.35e+154)
   (+
    (cbrt (* (/ 0.5 a) (/ (* -0.5 (pow h 2.0)) g)))
    (* (cbrt (/ -0.5 a)) (cbrt (* 2.0 g))))
   (fma
    (/ 1.0 (cbrt (/ a g)))
    (* (cbrt -0.5) (cbrt 2.0))
    (* (cbrt (/ (pow h -2.0) (* a g))) (* (cbrt -0.5) (cbrt 0.5))))))

C

double code(double g, double h, double a) {
	double tmp;
	if (h <= 1.35e+154) {
		tmp = cbrt(((0.5 / a) * ((-0.5 * pow(h, 2.0)) / g))) + (cbrt((-0.5 / a)) * cbrt((2.0 * g)));
	} else {
		tmp = fma((1.0 / cbrt((a / g))), (cbrt(-0.5) * cbrt(2.0)), (cbrt((pow(h, -2.0) / (a * g))) * (cbrt(-0.5) * cbrt(0.5))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (h <= 1.35e+154)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(Float64(-0.5 * (h ^ 2.0)) / g))) + Float64(cbrt(Float64(-0.5 / a)) * cbrt(Float64(2.0 * g))));
	else
		tmp = fma(Float64(1.0 / cbrt(Float64(a / g))), Float64(cbrt(-0.5) * cbrt(2.0)), Float64(cbrt(Float64((h ^ -2.0) / Float64(a * g))) * Float64(cbrt(-0.5) * cbrt(0.5))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[h, 1.35e+154], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(N[(-0.5 * N[Power[h, 2.0], $MachinePrecision]), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Power[N[(a / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[Power[h, -2.0], $MachinePrecision] / N[(a * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 1.35000000000000003e154

   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 27.7%

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

   6. Taylor expanded in g around inf 73.7%

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

      1. associate-*r/ 73.7%

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

   8. Simplified 73.7%

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

      1. *-commutative 73.7%

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

      2. cbrt-prod 93.0%

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

      3. count-2 93.0%

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

   10. Applied egg-rr 93.0%

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

   if 1.35000000000000003e154 < h

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in h around 0 1.4%

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

      1. fma-define 1.4%

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

      2. *-commutative 1.4%

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

      3. *-commutative 1.4%

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

   7. Simplified 1.4%

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

   9. Applied egg-rr 1.4%

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

       1. clear-num 1.4%

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

       2. inv-pow 1.4%

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

       3. cbrt-undiv 1.4%

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

   11. Applied egg-rr 1.4%

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

   13. Simplified 1.4%

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

   15. Applied egg-rr 83.0%

       \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{\frac{a}{g}}}, \sqrt[3]{-0.5} \cdot \sqrt[3]{2}, \color{blue}{{\left(\sqrt[3]{\sqrt[3]{\frac{\frac{{h}^{-2}}{g}}{a}}}\right)}^{3}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \]
   16. Step-by-step derivation

       1. rem-cube-cbrt 83.4%

          \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}, \sqrt[3]{-0.5} \cdot \sqrt{\sqrt[3]{4}}, \color{blue}{\sqrt[3]{\frac{\frac{{h}^{-2}}{g}}{a}}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \]

       2. associate-/l/ 83.4%

          \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}, \sqrt[3]{-0.5} \cdot \sqrt{\sqrt[3]{4}}, \sqrt[3]{\color{blue}{\frac{{h}^{-2}}{a \cdot g}}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \]

   17. Simplified 83.0%

       \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{\frac{a}{g}}}, \sqrt[3]{-0.5} \cdot \sqrt[3]{2}, \color{blue}{\sqrt[3]{\frac{{h}^{-2}}{a \cdot g}}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \frac{-0.5 \cdot {h}^{2}}{g}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{2 \cdot g}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt[3]{\frac{a}{g}}}, \sqrt[3]{-0.5} \cdot \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{-2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \frac{-0.5 \cdot {h}^{2}}{g}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{2 \cdot g}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt[3]{\frac{g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= h 1.35e+154)
   (+
    (cbrt (* (/ 0.5 a) (/ (* -0.5 (pow h 2.0)) g)))
    (* (cbrt (/ -0.5 a)) (cbrt (* 2.0 g))))
   (- (cbrt (/ g a)))))

C

double code(double g, double h, double a) {
	double tmp;
	if (h <= 1.35e+154) {
		tmp = cbrt(((0.5 / a) * ((-0.5 * pow(h, 2.0)) / g))) + (cbrt((-0.5 / a)) * cbrt((2.0 * g)));
	} else {
		tmp = -cbrt((g / a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (h <= 1.35e+154) {
		tmp = Math.cbrt(((0.5 / a) * ((-0.5 * Math.pow(h, 2.0)) / g))) + (Math.cbrt((-0.5 / a)) * Math.cbrt((2.0 * g)));
	} else {
		tmp = -Math.cbrt((g / a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (h <= 1.35e+154)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(Float64(-0.5 * (h ^ 2.0)) / g))) + Float64(cbrt(Float64(-0.5 / a)) * cbrt(Float64(2.0 * g))));
	else
		tmp = Float64(-cbrt(Float64(g / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[h, 1.35e+154], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(N[(-0.5 * N[Power[h, 2.0], $MachinePrecision]), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if h < 1.35000000000000003e154

   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 27.7%

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

   6. Taylor expanded in g around inf 73.7%

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

      1. associate-*r/ 73.7%

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

   8. Simplified 73.7%

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

      1. *-commutative 73.7%

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

      2. cbrt-prod 93.0%

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

      3. count-2 93.0%

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

   10. Applied egg-rr 93.0%

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

   if 1.35000000000000003e154 < h

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in h around 0 1.4%

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

      1. fma-define 1.4%

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

      2. *-commutative 1.4%

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

      3. *-commutative 1.4%

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

   7. Simplified 1.4%

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

   9. Applied egg-rr 1.4%

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

       1. add-cbrt-cube 1.4%

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

       2. pow3 1.4%

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

   11. Applied egg-rr 1.4%

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

   12. Taylor expanded in g around -inf 81.0%

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

       1. mul-1-neg 81.0%

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

   14. Simplified 81.0%

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

Alternative 5: 74.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.4%

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

3. Simplified 43.4%

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

5. Taylor expanded in h around 0 67.1%

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

   1. fma-define 67.1%

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

   2. *-commutative 67.1%

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

   3. *-commutative 67.1%

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

7. Simplified 67.1%

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

9. Applied egg-rr 85.5%

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

    1. add-cbrt-cube 67.0%

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

    2. pow3 67.0%

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

11. Applied egg-rr 67.6%

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

12. Taylor expanded in g around -inf 75.9%

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

    1. mul-1-neg 75.9%

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

14. Simplified 75.9%

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

Reproduce

herbie shell --seed 2024144 
(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))))))))