2-ancestry mixing, positive discriminant

Percentage Accurate: 44.4% → 96.5%

Time: 22.4s

Alternatives: 11

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)\\ t_1 := \sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\\ \mathbf{if}\;h \cdot h \leq 10^{-200}:\\ \;\;\;\;t\_0 + \frac{1}{\sqrt[3]{g \cdot \frac{a}{{h}^{2}}}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 + t\_1 \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (* (cbrt g) (cbrt (/ 1.0 a))) (* (cbrt -0.5) (cbrt 2.0))))
        (t_1 (* (cbrt -0.5) (cbrt 0.5))))
   (if (<= (* h h) 1e-200)
     (+ t_0 (* (/ 1.0 (cbrt (* g (/ a (pow h 2.0))))) t_1))
     (+ t_0 (* t_1 (/ (pow (cbrt h) 2.0) (cbrt (* g a))))))))

C

double code(double g, double h, double a) {
	double t_0 = (cbrt(g) * cbrt((1.0 / a))) * (cbrt(-0.5) * cbrt(2.0));
	double t_1 = cbrt(-0.5) * cbrt(0.5);
	double tmp;
	if ((h * h) <= 1e-200) {
		tmp = t_0 + ((1.0 / cbrt((g * (a / pow(h, 2.0))))) * t_1);
	} else {
		tmp = t_0 + (t_1 * (pow(cbrt(h), 2.0) / cbrt((g * a))));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = (Math.cbrt(g) * Math.cbrt((1.0 / a))) * (Math.cbrt(-0.5) * Math.cbrt(2.0));
	double t_1 = Math.cbrt(-0.5) * Math.cbrt(0.5);
	double tmp;
	if ((h * h) <= 1e-200) {
		tmp = t_0 + ((1.0 / Math.cbrt((g * (a / Math.pow(h, 2.0))))) * t_1);
	} else {
		tmp = t_0 + (t_1 * (Math.pow(Math.cbrt(h), 2.0) / Math.cbrt((g * a))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(cbrt(g) * cbrt(Float64(1.0 / a))) * Float64(cbrt(-0.5) * cbrt(2.0)))
	t_1 = Float64(cbrt(-0.5) * cbrt(0.5))
	tmp = 0.0
	if (Float64(h * h) <= 1e-200)
		tmp = Float64(t_0 + Float64(Float64(1.0 / cbrt(Float64(g * Float64(a / (h ^ 2.0))))) * t_1));
	else
		tmp = Float64(t_0 + Float64(t_1 * Float64((cbrt(h) ^ 2.0) / cbrt(Float64(g * a)))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h * h), $MachinePrecision], 1e-200], N[(t$95$0 + N[(N[(1.0 / N[Power[N[(g * N[(a / N[Power[h, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(t$95$1 * N[(N[Power[N[Power[h, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(g * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 9.9999999999999998e-201

   1. Initial program 54.1%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in 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. pow1/3 33.2%

         \[\leadsto \color{blue}{{\left(\frac{g}{a}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      2. div-inv 33.2%

         \[\leadsto {\color{blue}{\left(g \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      3. unpow-prod-down 17.5%

         \[\leadsto \color{blue}{\left({g}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      4. pow1/3 37.5%

         \[\leadsto \left(\color{blue}{\sqrt[3]{g}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

   7. Applied egg-rr 37.5%

      \[\leadsto \color{blue}{\left(\sqrt[3]{g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]
   8. Step-by-step derivation

      1. unpow1/3 89.0%

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

   9. Simplified 89.0%

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

       1. clear-num 89.0%

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

       2. *-commutative 89.0%

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

       3. cbrt-div 89.0%

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

       4. metadata-eval 89.0%

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

   11. Applied egg-rr 89.0%

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

       1. associate-/l* 96.9%

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

   13. Simplified 96.9%

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

   if 9.9999999999999998e-201 < (*.f64 h h)

   1. Initial program 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in h around 0 52.7%

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

      1. pow1/3 27.0%

         \[\leadsto \color{blue}{{\left(\frac{g}{a}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      2. div-inv 27.0%

         \[\leadsto {\color{blue}{\left(g \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      3. unpow-prod-down 24.8%

         \[\leadsto \color{blue}{\left({g}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      4. pow1/3 41.2%

         \[\leadsto \left(\color{blue}{\sqrt[3]{g}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

   7. Applied egg-rr 41.2%

      \[\leadsto \color{blue}{\left(\sqrt[3]{g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]
   8. Step-by-step derivation

      1. unpow1/3 77.8%

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

   9. Simplified 77.8%

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

       1. cbrt-div 81.0%

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

       2. *-commutative 81.0%

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

       3. unpow2 81.0%

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

       4. cbrt-prod 95.1%

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

       5. pow2 95.1%

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

   11. Applied egg-rr 95.1%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 10^{-200}:\\ \;\;\;\;\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \frac{1}{\sqrt[3]{g \cdot \frac{a}{{h}^{2}}}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{\frac{g}{a}}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;g \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;g \leq -1.1 \cdot 10^{-141}:\\ \;\;\;\;\sqrt[3]{0.5} \cdot \left(\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}\right) + \sqrt[3]{\left(g + t\_1\right) \cdot \frac{-0.5}{a}}\\ \mathbf{elif}\;g \leq 4.3 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{g \cdot -0.5}{a}}, \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{2}}{g} \cdot \frac{-0.25}{a}}\right)\\ \mathbf{elif}\;g \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(t\_1 - g\right)} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \sqrt{{g}^{2} - {h}^{2}}\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0
         (+
          (* (* (cbrt -0.5) (cbrt 0.5)) (/ (pow (cbrt h) 2.0) (cbrt (* g a))))
          (* (* (cbrt -0.5) (cbrt 2.0)) (cbrt (/ g a)))))
        (t_1 (sqrt (- (* g g) (* h h)))))
   (if (<= g -1.34e+154)
     t_0
     (if (<= g -1.1e-141)
       (+
        (*
         (cbrt 0.5)
         (* (cbrt (/ 1.0 a)) (cbrt (- (sqrt (* (- g h) (+ h g))) g))))
        (cbrt (* (+ g t_1) (/ -0.5 a))))
       (if (<= g 4.3e-172)
         (fma
          (cbrt (/ (* g -0.5) a))
          (cbrt 2.0)
          (cbrt (* (/ (pow h 2.0) g) (/ -0.25 a))))
         (if (<= g 1.35e+154)
           (+
            (cbrt (* (/ 0.5 a) (- t_1 g)))
            (/
             (cbrt (* -0.5 (+ g (sqrt (- (pow g 2.0) (pow h 2.0))))))
             (cbrt a)))
           t_0))))))

C

double code(double g, double h, double a) {
	double t_0 = ((cbrt(-0.5) * cbrt(0.5)) * (pow(cbrt(h), 2.0) / cbrt((g * a)))) + ((cbrt(-0.5) * cbrt(2.0)) * cbrt((g / a)));
	double t_1 = sqrt(((g * g) - (h * h)));
	double tmp;
	if (g <= -1.34e+154) {
		tmp = t_0;
	} else if (g <= -1.1e-141) {
		tmp = (cbrt(0.5) * (cbrt((1.0 / a)) * cbrt((sqrt(((g - h) * (h + g))) - g)))) + cbrt(((g + t_1) * (-0.5 / a)));
	} else if (g <= 4.3e-172) {
		tmp = fma(cbrt(((g * -0.5) / a)), cbrt(2.0), cbrt(((pow(h, 2.0) / g) * (-0.25 / a))));
	} else if (g <= 1.35e+154) {
		tmp = cbrt(((0.5 / a) * (t_1 - g))) + (cbrt((-0.5 * (g + sqrt((pow(g, 2.0) - pow(h, 2.0)))))) / cbrt(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(Float64(cbrt(-0.5) * cbrt(0.5)) * Float64((cbrt(h) ^ 2.0) / cbrt(Float64(g * a)))) + Float64(Float64(cbrt(-0.5) * cbrt(2.0)) * cbrt(Float64(g / a))))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	tmp = 0.0
	if (g <= -1.34e+154)
		tmp = t_0;
	elseif (g <= -1.1e-141)
		tmp = Float64(Float64(cbrt(0.5) * Float64(cbrt(Float64(1.0 / a)) * cbrt(Float64(sqrt(Float64(Float64(g - h) * Float64(h + g))) - g)))) + cbrt(Float64(Float64(g + t_1) * Float64(-0.5 / a))));
	elseif (g <= 4.3e-172)
		tmp = fma(cbrt(Float64(Float64(g * -0.5) / a)), cbrt(2.0), cbrt(Float64(Float64((h ^ 2.0) / g) * Float64(-0.25 / a))));
	elseif (g <= 1.35e+154)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(t_1 - g))) + Float64(cbrt(Float64(-0.5 * Float64(g + sqrt(Float64((g ^ 2.0) - (h ^ 2.0)))))) / cbrt(a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Power[h, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(g * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[g, -1.34e+154], t$95$0, If[LessEqual[g, -1.1e-141], N[(N[(N[Power[0.5, 1/3], $MachinePrecision] * N[(N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(g + t$95$1), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 4.3e-172], N[(N[Power[N[(N[(g * -0.5), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision] + N[Power[N[(N[(N[Power[h, 2.0], $MachinePrecision] / g), $MachinePrecision] * N[(-0.25 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 1.35e+154], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$1 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 * N[(g + N[Sqrt[N[(N[Power[g, 2.0], $MachinePrecision] - N[Power[h, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if g < -1.34000000000000001e154 or 1.35000000000000003e154 < g

   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 46.8%

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

      1. cbrt-div 83.6%

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

      2. *-commutative 83.6%

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

      3. unpow2 83.6%

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

      4. cbrt-prod 96.7%

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

      5. pow2 96.7%

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

   7. Applied egg-rr 54.3%

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

   if -1.34000000000000001e154 < g < -1.10000000000000005e-141

   1. Initial program 85.3%

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

   3. Simplified 85.3%

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

      1. pow1/2 85.3%

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

      2. difference-of-squares 85.3%

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

      3. unpow-prod-down 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. unpow1/2 0.0%

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

      2. unpow1/2 0.0%

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

      3. +-commutative 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in a around 0 85.3%

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

       1. +-commutative 85.3%

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

   11. Simplified 85.3%

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

       1. pow1/3 36.8%

          \[\leadsto \color{blue}{{\left(\frac{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}{a}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

       2. div-inv 36.8%

          \[\leadsto {\color{blue}{\left(\left(\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g\right) \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \cdot \sqrt[3]{0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

       3. unpow-prod-down 39.4%

          \[\leadsto \color{blue}{\left({\left(\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \sqrt[3]{0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

       4. pow1/3 39.3%

          \[\leadsto \left(\color{blue}{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

   13. Applied egg-rr 39.3%

       \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \sqrt[3]{0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   14. Step-by-step derivation

       1. unpow1/3 98.0%

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

       2. *-commutative 98.0%

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

       3. +-commutative 98.0%

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

   15. Simplified 98.0%

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

   if -1.10000000000000005e-141 < g < 4.2999999999999997e-172

   1. Initial program 18.6%

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

   3. Simplified 18.6%

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

   5. Taylor expanded in h around 0 31.9%

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

      1. add-cube-cbrt 31.4%

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

      2. pow3 31.5%

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

   7. Applied egg-rr 31.5%

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

      1. rem-cube-cbrt 31.9%

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

      2. associate-*r* 31.9%

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

      3. fma-define 31.9%

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

      4. cbrt-unprod 31.9%

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

      5. rem-3cbrt-lft 31.9%

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

      6. cbrt-unprod 31.9%

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

      7. cbrt-prod 31.9%

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

   9. Applied egg-rr 31.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{g}{a} \cdot -0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{2}}{g \cdot a} \cdot -0.25}\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 31.9%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\color{blue}{\frac{g \cdot -0.5}{a}}}, \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{2}}{g \cdot a} \cdot -0.25}\right) \]

       2. associate-*l/ 31.9%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{g \cdot -0.5}{a}}, \sqrt[3]{2}, \sqrt[3]{\color{blue}{\frac{{h}^{2} \cdot -0.25}{g \cdot a}}}\right) \]

       3. times-frac 68.1%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{g \cdot -0.5}{a}}, \sqrt[3]{2}, \sqrt[3]{\color{blue}{\frac{{h}^{2}}{g} \cdot \frac{-0.25}{a}}}\right) \]

   11. Simplified 68.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{g \cdot -0.5}{a}}, \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{2}}{g} \cdot \frac{-0.25}{a}}\right)} \]

   if 4.2999999999999997e-172 < g < 1.35000000000000003e154

   1. Initial program 80.1%

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

   3. Simplified 80.1%

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

      1. associate-*r/ 80.1%

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

      2. cbrt-div 95.5%

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

      3. pow2 95.5%

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

      4. pow2 95.5%

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

   6. Applied egg-rr 95.5%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{\frac{g}{a}}\\ \mathbf{elif}\;g \leq -1.1 \cdot 10^{-141}:\\ \;\;\;\;\sqrt[3]{0.5} \cdot \left(\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}\right) + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{elif}\;g \leq 4.3 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{g \cdot -0.5}{a}}, \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{2}}{g} \cdot \frac{-0.25}{a}}\right)\\ \mathbf{elif}\;g \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \sqrt{{g}^{2} - {h}^{2}}\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{\frac{g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{-0.5} \cdot \sqrt[3]{2}\\ t_1 := \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot t\_0\\ t_2 := \sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\\ \mathbf{if}\;h \cdot h \leq 10^{-200}:\\ \;\;\;\;t\_1 + \frac{1}{\sqrt[3]{g \cdot \frac{a}{{h}^{2}}}} \cdot t\_2\\ \mathbf{elif}\;h \cdot h \leq 2 \cdot 10^{+282}:\\ \;\;\;\;t\_1 + t\_2 \cdot \sqrt[3]{\frac{{h}^{2}}{g \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}} + t\_0 \cdot \sqrt[3]{\frac{g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (cbrt -0.5) (cbrt 2.0)))
        (t_1 (* (* (cbrt g) (cbrt (/ 1.0 a))) t_0))
        (t_2 (* (cbrt -0.5) (cbrt 0.5))))
   (if (<= (* h h) 1e-200)
     (+ t_1 (* (/ 1.0 (cbrt (* g (/ a (pow h 2.0))))) t_2))
     (if (<= (* h h) 2e+282)
       (+ t_1 (* t_2 (cbrt (/ (pow h 2.0) (* g a)))))
       (+
        (* t_2 (/ (pow (cbrt h) 2.0) (cbrt (* g a))))
        (* t_0 (cbrt (/ g a))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(-0.5) * cbrt(2.0);
	double t_1 = (cbrt(g) * cbrt((1.0 / a))) * t_0;
	double t_2 = cbrt(-0.5) * cbrt(0.5);
	double tmp;
	if ((h * h) <= 1e-200) {
		tmp = t_1 + ((1.0 / cbrt((g * (a / pow(h, 2.0))))) * t_2);
	} else if ((h * h) <= 2e+282) {
		tmp = t_1 + (t_2 * cbrt((pow(h, 2.0) / (g * a))));
	} else {
		tmp = (t_2 * (pow(cbrt(h), 2.0) / cbrt((g * a)))) + (t_0 * cbrt((g / a)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(-0.5) * Math.cbrt(2.0);
	double t_1 = (Math.cbrt(g) * Math.cbrt((1.0 / a))) * t_0;
	double t_2 = Math.cbrt(-0.5) * Math.cbrt(0.5);
	double tmp;
	if ((h * h) <= 1e-200) {
		tmp = t_1 + ((1.0 / Math.cbrt((g * (a / Math.pow(h, 2.0))))) * t_2);
	} else if ((h * h) <= 2e+282) {
		tmp = t_1 + (t_2 * Math.cbrt((Math.pow(h, 2.0) / (g * a))));
	} else {
		tmp = (t_2 * (Math.pow(Math.cbrt(h), 2.0) / Math.cbrt((g * a)))) + (t_0 * Math.cbrt((g / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(cbrt(-0.5) * cbrt(2.0))
	t_1 = Float64(Float64(cbrt(g) * cbrt(Float64(1.0 / a))) * t_0)
	t_2 = Float64(cbrt(-0.5) * cbrt(0.5))
	tmp = 0.0
	if (Float64(h * h) <= 1e-200)
		tmp = Float64(t_1 + Float64(Float64(1.0 / cbrt(Float64(g * Float64(a / (h ^ 2.0))))) * t_2));
	elseif (Float64(h * h) <= 2e+282)
		tmp = Float64(t_1 + Float64(t_2 * cbrt(Float64((h ^ 2.0) / Float64(g * a)))));
	else
		tmp = Float64(Float64(t_2 * Float64((cbrt(h) ^ 2.0) / cbrt(Float64(g * a)))) + Float64(t_0 * cbrt(Float64(g / a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h * h), $MachinePrecision], 1e-200], N[(t$95$1 + N[(N[(1.0 / N[Power[N[(g * N[(a / N[Power[h, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h * h), $MachinePrecision], 2e+282], N[(t$95$1 + N[(t$95$2 * N[Power[N[(N[Power[h, 2.0], $MachinePrecision] / N[(g * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(N[Power[N[Power[h, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(g * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 h h) < 9.9999999999999998e-201

   1. Initial program 54.1%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in 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. pow1/3 33.2%

         \[\leadsto \color{blue}{{\left(\frac{g}{a}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      2. div-inv 33.2%

         \[\leadsto {\color{blue}{\left(g \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      3. unpow-prod-down 17.5%

         \[\leadsto \color{blue}{\left({g}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      4. pow1/3 37.5%

         \[\leadsto \left(\color{blue}{\sqrt[3]{g}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

   7. Applied egg-rr 37.5%

      \[\leadsto \color{blue}{\left(\sqrt[3]{g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]
   8. Step-by-step derivation

      1. unpow1/3 89.0%

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

   9. Simplified 89.0%

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

       1. clear-num 89.0%

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

       2. *-commutative 89.0%

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

       3. cbrt-div 89.0%

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

       4. metadata-eval 89.0%

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

   11. Applied egg-rr 89.0%

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

       1. associate-/l* 96.9%

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

   13. Simplified 96.9%

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

   if 9.9999999999999998e-201 < (*.f64 h h) < 2.00000000000000007e282

   1. Initial program 34.1%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in h around 0 61.8%

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

      1. pow1/3 29.8%

         \[\leadsto \color{blue}{{\left(\frac{g}{a}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      2. div-inv 29.8%

         \[\leadsto {\color{blue}{\left(g \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      3. unpow-prod-down 28.4%

         \[\leadsto \color{blue}{\left({g}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      4. pow1/3 49.1%

         \[\leadsto \left(\color{blue}{\sqrt[3]{g}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

   7. Applied egg-rr 49.1%

      \[\leadsto \color{blue}{\left(\sqrt[3]{g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]
   8. Step-by-step derivation

      1. unpow1/3 93.4%

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

   9. Simplified 93.4%

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

   if 2.00000000000000007e282 < (*.f64 h 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 17.7%

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

      1. cbrt-div 22.5%

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

      2. *-commutative 22.5%

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

      3. unpow2 22.5%

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

      4. cbrt-prod 90.7%

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

      5. pow2 90.7%

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

   7. Applied egg-rr 56.5%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 10^{-200}:\\ \;\;\;\;\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \frac{1}{\sqrt[3]{g \cdot \frac{a}{{h}^{2}}}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)\\ \mathbf{elif}\;h \cdot h \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{\frac{{h}^{2}}{g \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{\frac{g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)\\ t_1 := \sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\\ \mathbf{if}\;h \cdot h \leq 4 \cdot 10^{+73}:\\ \;\;\;\;t\_0 + \frac{1}{\sqrt[3]{g \cdot \frac{a}{{h}^{2}}}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 + t\_1 \cdot \frac{1}{\sqrt[3]{\frac{g \cdot a}{{h}^{-2}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (* (cbrt g) (cbrt (/ 1.0 a))) (* (cbrt -0.5) (cbrt 2.0))))
        (t_1 (* (cbrt -0.5) (cbrt 0.5))))
   (if (<= (* h h) 4e+73)
     (+ t_0 (* (/ 1.0 (cbrt (* g (/ a (pow h 2.0))))) t_1))
     (+ t_0 (* t_1 (/ 1.0 (cbrt (/ (* g a) (pow h -2.0)))))))))

C

double code(double g, double h, double a) {
	double t_0 = (cbrt(g) * cbrt((1.0 / a))) * (cbrt(-0.5) * cbrt(2.0));
	double t_1 = cbrt(-0.5) * cbrt(0.5);
	double tmp;
	if ((h * h) <= 4e+73) {
		tmp = t_0 + ((1.0 / cbrt((g * (a / pow(h, 2.0))))) * t_1);
	} else {
		tmp = t_0 + (t_1 * (1.0 / cbrt(((g * a) / pow(h, -2.0)))));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = (Math.cbrt(g) * Math.cbrt((1.0 / a))) * (Math.cbrt(-0.5) * Math.cbrt(2.0));
	double t_1 = Math.cbrt(-0.5) * Math.cbrt(0.5);
	double tmp;
	if ((h * h) <= 4e+73) {
		tmp = t_0 + ((1.0 / Math.cbrt((g * (a / Math.pow(h, 2.0))))) * t_1);
	} else {
		tmp = t_0 + (t_1 * (1.0 / Math.cbrt(((g * a) / Math.pow(h, -2.0)))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(cbrt(g) * cbrt(Float64(1.0 / a))) * Float64(cbrt(-0.5) * cbrt(2.0)))
	t_1 = Float64(cbrt(-0.5) * cbrt(0.5))
	tmp = 0.0
	if (Float64(h * h) <= 4e+73)
		tmp = Float64(t_0 + Float64(Float64(1.0 / cbrt(Float64(g * Float64(a / (h ^ 2.0))))) * t_1));
	else
		tmp = Float64(t_0 + Float64(t_1 * Float64(1.0 / cbrt(Float64(Float64(g * a) / (h ^ -2.0))))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h * h), $MachinePrecision], 4e+73], N[(t$95$0 + N[(N[(1.0 / N[Power[N[(g * N[(a / N[Power[h, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(t$95$1 * N[(1.0 / N[Power[N[(N[(g * a), $MachinePrecision] / N[Power[h, -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 3.99999999999999993e73

   1. Initial program 50.7%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in h around 0 67.2%

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

      1. pow1/3 33.0%

         \[\leadsto \color{blue}{{\left(\frac{g}{a}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      2. div-inv 33.0%

         \[\leadsto {\color{blue}{\left(g \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      3. unpow-prod-down 20.2%

         \[\leadsto \color{blue}{\left({g}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      4. pow1/3 41.0%

         \[\leadsto \left(\color{blue}{\sqrt[3]{g}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

   7. Applied egg-rr 41.0%

      \[\leadsto \color{blue}{\left(\sqrt[3]{g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]
   8. Step-by-step derivation

      1. unpow1/3 90.5%

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

   9. Simplified 90.5%

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

       1. clear-num 90.5%

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

       2. *-commutative 90.5%

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

       3. cbrt-div 90.5%

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

       4. metadata-eval 90.5%

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

   11. Applied egg-rr 90.5%

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

       1. associate-/l* 96.6%

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

   13. Simplified 96.6%

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

   if 3.99999999999999993e73 < (*.f64 h h)

   1. Initial program 12.9%

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

   3. Simplified 12.9%

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

   5. Taylor expanded in h around 0 35.7%

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

      1. pow1/3 20.8%

         \[\leadsto \color{blue}{{\left(\frac{g}{a}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      2. div-inv 20.8%

         \[\leadsto {\color{blue}{\left(g \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      3. unpow-prod-down 19.4%

         \[\leadsto \color{blue}{\left({g}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      4. pow1/3 27.8%

         \[\leadsto \left(\color{blue}{\sqrt[3]{g}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

   7. Applied egg-rr 27.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]
   8. Step-by-step derivation

      1. unpow1/3 57.5%

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

   9. Simplified 57.5%

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

       1. clear-num 57.5%

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

       2. *-commutative 57.5%

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

       3. cbrt-div 59.6%

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

       4. metadata-eval 59.6%

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

   11. Applied egg-rr 59.6%

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

   13. Applied egg-rr 91.7%

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

       1. rem-cube-cbrt 91.7%

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

       2. associate-*r/ 91.7%

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

       3. *-commutative 91.7%

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

   15. Simplified 91.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \frac{1}{\sqrt[3]{g \cdot \frac{a}{{h}^{2}}}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \frac{1}{\sqrt[3]{\frac{g \cdot a}{{h}^{-2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{-0.5} \cdot \sqrt[3]{2}\\ t_1 := \sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\\ \mathbf{if}\;h \cdot h \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot t\_0 + t\_1 \cdot \sqrt[3]{\frac{{h}^{2}}{g \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}} + t\_0 \cdot \sqrt[3]{\frac{g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (cbrt -0.5) (cbrt 2.0))) (t_1 (* (cbrt -0.5) (cbrt 0.5))))
   (if (<= (* h h) 2e+282)
     (+
      (* (* (cbrt g) (cbrt (/ 1.0 a))) t_0)
      (* t_1 (cbrt (/ (pow h 2.0) (* g a)))))
     (+
      (* t_1 (/ (pow (cbrt h) 2.0) (cbrt (* g a))))
      (* t_0 (cbrt (/ g a)))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(-0.5) * cbrt(2.0);
	double t_1 = cbrt(-0.5) * cbrt(0.5);
	double tmp;
	if ((h * h) <= 2e+282) {
		tmp = ((cbrt(g) * cbrt((1.0 / a))) * t_0) + (t_1 * cbrt((pow(h, 2.0) / (g * a))));
	} else {
		tmp = (t_1 * (pow(cbrt(h), 2.0) / cbrt((g * a)))) + (t_0 * cbrt((g / a)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(-0.5) * Math.cbrt(2.0);
	double t_1 = Math.cbrt(-0.5) * Math.cbrt(0.5);
	double tmp;
	if ((h * h) <= 2e+282) {
		tmp = ((Math.cbrt(g) * Math.cbrt((1.0 / a))) * t_0) + (t_1 * Math.cbrt((Math.pow(h, 2.0) / (g * a))));
	} else {
		tmp = (t_1 * (Math.pow(Math.cbrt(h), 2.0) / Math.cbrt((g * a)))) + (t_0 * Math.cbrt((g / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(cbrt(-0.5) * cbrt(2.0))
	t_1 = Float64(cbrt(-0.5) * cbrt(0.5))
	tmp = 0.0
	if (Float64(h * h) <= 2e+282)
		tmp = Float64(Float64(Float64(cbrt(g) * cbrt(Float64(1.0 / a))) * t_0) + Float64(t_1 * cbrt(Float64((h ^ 2.0) / Float64(g * a)))));
	else
		tmp = Float64(Float64(t_1 * Float64((cbrt(h) ^ 2.0) / cbrt(Float64(g * a)))) + Float64(t_0 * cbrt(Float64(g / a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h * h), $MachinePrecision], 2e+282], N[(N[(N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(t$95$1 * N[Power[N[(N[Power[h, 2.0], $MachinePrecision] / N[(g * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[Power[N[Power[h, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(g * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 2.00000000000000007e282

   1. Initial program 48.0%

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

   3. Simplified 48.0%

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

   5. Taylor expanded in h around 0 65.4%

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

      1. pow1/3 32.2%

         \[\leadsto \color{blue}{{\left(\frac{g}{a}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      2. div-inv 32.2%

         \[\leadsto {\color{blue}{\left(g \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      3. unpow-prod-down 20.8%

         \[\leadsto \color{blue}{\left({g}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

      4. pow1/3 41.0%

         \[\leadsto \left(\color{blue}{\sqrt[3]{g}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]

   7. Applied egg-rr 41.0%

      \[\leadsto \color{blue}{\left(\sqrt[3]{g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \]
   8. Step-by-step derivation

      1. unpow1/3 90.4%

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

   9. Simplified 90.4%

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

   if 2.00000000000000007e282 < (*.f64 h 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 17.7%

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

      1. cbrt-div 22.5%

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

      2. *-commutative 22.5%

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

      3. unpow2 22.5%

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

      4. cbrt-prod 90.7%

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

      5. pow2 90.7%

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

   7. Applied egg-rr 56.5%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{\frac{{h}^{2}}{g \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{\frac{g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{-0.5} \cdot \sqrt[3]{2}\\ t_1 := \sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\\ \mathbf{if}\;h \cdot h \leq 4 \cdot 10^{+295}:\\ \;\;\;\;t\_1 \cdot \sqrt[3]{\frac{{h}^{2}}{g \cdot a}} + t\_0 \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}} + t\_0 \cdot \sqrt[3]{\frac{g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (cbrt -0.5) (cbrt 2.0))) (t_1 (* (cbrt -0.5) (cbrt 0.5))))
   (if (<= (* h h) 4e+295)
     (+ (* t_1 (cbrt (/ (pow h 2.0) (* g a)))) (* t_0 (/ (cbrt g) (cbrt a))))
     (+
      (* t_1 (/ (pow (cbrt h) 2.0) (cbrt (* g a))))
      (* t_0 (cbrt (/ g a)))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(-0.5) * cbrt(2.0);
	double t_1 = cbrt(-0.5) * cbrt(0.5);
	double tmp;
	if ((h * h) <= 4e+295) {
		tmp = (t_1 * cbrt((pow(h, 2.0) / (g * a)))) + (t_0 * (cbrt(g) / cbrt(a)));
	} else {
		tmp = (t_1 * (pow(cbrt(h), 2.0) / cbrt((g * a)))) + (t_0 * cbrt((g / a)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(-0.5) * Math.cbrt(2.0);
	double t_1 = Math.cbrt(-0.5) * Math.cbrt(0.5);
	double tmp;
	if ((h * h) <= 4e+295) {
		tmp = (t_1 * Math.cbrt((Math.pow(h, 2.0) / (g * a)))) + (t_0 * (Math.cbrt(g) / Math.cbrt(a)));
	} else {
		tmp = (t_1 * (Math.pow(Math.cbrt(h), 2.0) / Math.cbrt((g * a)))) + (t_0 * Math.cbrt((g / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(cbrt(-0.5) * cbrt(2.0))
	t_1 = Float64(cbrt(-0.5) * cbrt(0.5))
	tmp = 0.0
	if (Float64(h * h) <= 4e+295)
		tmp = Float64(Float64(t_1 * cbrt(Float64((h ^ 2.0) / Float64(g * a)))) + Float64(t_0 * Float64(cbrt(g) / cbrt(a))));
	else
		tmp = Float64(Float64(t_1 * Float64((cbrt(h) ^ 2.0) / cbrt(Float64(g * a)))) + Float64(t_0 * cbrt(Float64(g / a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h * h), $MachinePrecision], 4e+295], N[(N[(t$95$1 * N[Power[N[(N[Power[h, 2.0], $MachinePrecision] / N[(g * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[Power[N[Power[h, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(g * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 3.9999999999999999e295

   1. Initial program 47.2%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in h around 0 65.6%

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

      1. cbrt-div 89.9%

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

   7. Applied egg-rr 89.9%

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

   if 3.9999999999999999e295 < (*.f64 h 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 2.5%

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

      1. cbrt-div 2.5%

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

      2. *-commutative 2.5%

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

      3. unpow2 2.5%

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

      4. cbrt-prod 88.9%

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

      5. pow2 88.9%

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

   7. Applied egg-rr 51.8%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{\frac{{h}^{2}}{g \cdot a}} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{\frac{g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{\frac{g}{a}}\\ \mathbf{if}\;g \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;g \leq -2.15 \cdot 10^{-138}:\\ \;\;\;\;\sqrt[3]{0.5} \cdot \left(\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}\right) + \sqrt[3]{\left(g + t\_0\right) \cdot \frac{-0.5}{a}}\\ \mathbf{elif}\;g \leq 4.3 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{g \cdot -0.5}{a}}, \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{2}}{g} \cdot \frac{-0.25}{a}}\right)\\ \mathbf{elif}\;g \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(t\_0 - g\right)} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \sqrt{{g}^{2} - {h}^{2}}\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h))))
        (t_1 (* (* (cbrt -0.5) (cbrt 2.0)) (cbrt (/ g a)))))
   (if (<= g -1.34e+154)
     t_1
     (if (<= g -2.15e-138)
       (+
        (*
         (cbrt 0.5)
         (* (cbrt (/ 1.0 a)) (cbrt (- (sqrt (* (- g h) (+ h g))) g))))
        (cbrt (* (+ g t_0) (/ -0.5 a))))
       (if (<= g 4.3e-172)
         (fma
          (cbrt (/ (* g -0.5) a))
          (cbrt 2.0)
          (cbrt (* (/ (pow h 2.0) g) (/ -0.25 a))))
         (if (<= g 1.35e+154)
           (+
            (cbrt (* (/ 0.5 a) (- t_0 g)))
            (/
             (cbrt (* -0.5 (+ g (sqrt (- (pow g 2.0) (pow h 2.0))))))
             (cbrt a)))
           t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double t_1 = (cbrt(-0.5) * cbrt(2.0)) * cbrt((g / a));
	double tmp;
	if (g <= -1.34e+154) {
		tmp = t_1;
	} else if (g <= -2.15e-138) {
		tmp = (cbrt(0.5) * (cbrt((1.0 / a)) * cbrt((sqrt(((g - h) * (h + g))) - g)))) + cbrt(((g + t_0) * (-0.5 / a)));
	} else if (g <= 4.3e-172) {
		tmp = fma(cbrt(((g * -0.5) / a)), cbrt(2.0), cbrt(((pow(h, 2.0) / g) * (-0.25 / a))));
	} else if (g <= 1.35e+154) {
		tmp = cbrt(((0.5 / a) * (t_0 - g))) + (cbrt((-0.5 * (g + sqrt((pow(g, 2.0) - pow(h, 2.0)))))) / cbrt(a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = Float64(Float64(cbrt(-0.5) * cbrt(2.0)) * cbrt(Float64(g / a)))
	tmp = 0.0
	if (g <= -1.34e+154)
		tmp = t_1;
	elseif (g <= -2.15e-138)
		tmp = Float64(Float64(cbrt(0.5) * Float64(cbrt(Float64(1.0 / a)) * cbrt(Float64(sqrt(Float64(Float64(g - h) * Float64(h + g))) - g)))) + cbrt(Float64(Float64(g + t_0) * Float64(-0.5 / a))));
	elseif (g <= 4.3e-172)
		tmp = fma(cbrt(Float64(Float64(g * -0.5) / a)), cbrt(2.0), cbrt(Float64(Float64((h ^ 2.0) / g) * Float64(-0.25 / a))));
	elseif (g <= 1.35e+154)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(t_0 - g))) + Float64(cbrt(Float64(-0.5 * Float64(g + sqrt(Float64((g ^ 2.0) - (h ^ 2.0)))))) / cbrt(a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[g, -1.34e+154], t$95$1, If[LessEqual[g, -2.15e-138], N[(N[(N[Power[0.5, 1/3], $MachinePrecision] * N[(N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(g + t$95$0), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 4.3e-172], N[(N[Power[N[(N[(g * -0.5), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision] + N[Power[N[(N[(N[Power[h, 2.0], $MachinePrecision] / g), $MachinePrecision] * N[(-0.25 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 1.35e+154], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$0 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 * N[(g + N[Sqrt[N[(N[Power[g, 2.0], $MachinePrecision] - N[Power[h, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if g < -1.34000000000000001e154 or 1.35000000000000003e154 < g

   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 46.8%

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

      1. fma-define 46.8%

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

      2. *-commutative 46.8%

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

      3. *-commutative 46.8%

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

   7. Simplified 46.8%

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

   8. Taylor expanded in g around inf 54.1%

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

      1. *-commutative 54.1%

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

   10. Simplified 54.1%

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

   if -1.34000000000000001e154 < g < -2.15e-138

   1. Initial program 85.3%

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

   3. Simplified 85.3%

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

      1. pow1/2 85.3%

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

      2. difference-of-squares 85.3%

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

      3. unpow-prod-down 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. unpow1/2 0.0%

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

      2. unpow1/2 0.0%

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

      3. +-commutative 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in a around 0 85.3%

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

       1. +-commutative 85.3%

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

   11. Simplified 85.3%

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

       1. pow1/3 36.8%

          \[\leadsto \color{blue}{{\left(\frac{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}{a}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

       2. div-inv 36.8%

          \[\leadsto {\color{blue}{\left(\left(\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g\right) \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \cdot \sqrt[3]{0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

       3. unpow-prod-down 39.4%

          \[\leadsto \color{blue}{\left({\left(\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \sqrt[3]{0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

       4. pow1/3 39.3%

          \[\leadsto \left(\color{blue}{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

   13. Applied egg-rr 39.3%

       \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \sqrt[3]{0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   14. Step-by-step derivation

       1. unpow1/3 98.0%

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

       2. *-commutative 98.0%

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

       3. +-commutative 98.0%

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

   15. Simplified 98.0%

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

   if -2.15e-138 < g < 4.2999999999999997e-172

   1. Initial program 18.6%

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

   3. Simplified 18.6%

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

   5. Taylor expanded in h around 0 31.9%

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

      1. add-cube-cbrt 31.4%

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

      2. pow3 31.5%

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

   7. Applied egg-rr 31.5%

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

      1. rem-cube-cbrt 31.9%

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

      2. associate-*r* 31.9%

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

      3. fma-define 31.9%

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

      4. cbrt-unprod 31.9%

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

      5. rem-3cbrt-lft 31.9%

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

      6. cbrt-unprod 31.9%

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

      7. cbrt-prod 31.9%

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

   9. Applied egg-rr 31.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{g}{a} \cdot -0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{2}}{g \cdot a} \cdot -0.25}\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 31.9%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\color{blue}{\frac{g \cdot -0.5}{a}}}, \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{2}}{g \cdot a} \cdot -0.25}\right) \]

       2. associate-*l/ 31.9%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{g \cdot -0.5}{a}}, \sqrt[3]{2}, \sqrt[3]{\color{blue}{\frac{{h}^{2} \cdot -0.25}{g \cdot a}}}\right) \]

       3. times-frac 68.1%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{g \cdot -0.5}{a}}, \sqrt[3]{2}, \sqrt[3]{\color{blue}{\frac{{h}^{2}}{g} \cdot \frac{-0.25}{a}}}\right) \]

   11. Simplified 68.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{g \cdot -0.5}{a}}, \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{2}}{g} \cdot \frac{-0.25}{a}}\right)} \]

   if 4.2999999999999997e-172 < g < 1.35000000000000003e154

   1. Initial program 80.1%

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

   3. Simplified 80.1%

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

      1. associate-*r/ 80.1%

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

      2. cbrt-div 95.5%

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

      3. pow2 95.5%

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

      4. pow2 95.5%

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

   6. Applied egg-rr 95.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{\frac{g}{a}}\\ \mathbf{elif}\;g \leq -2.15 \cdot 10^{-138}:\\ \;\;\;\;\sqrt[3]{0.5} \cdot \left(\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}\right) + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{elif}\;g \leq 4.3 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{g \cdot -0.5}{a}}, \sqrt[3]{2}, \sqrt[3]{\frac{{h}^{2}}{g} \cdot \frac{-0.25}{a}}\right)\\ \mathbf{elif}\;g \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \sqrt{{g}^{2} - {h}^{2}}\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{\frac{g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq -1.34 \cdot 10^{+154} \lor \neg \left(g \leq -1.1 \cdot 10^{-141}\right):\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{0.5} \cdot \left(\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}\right) + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= g -1.34e+154) (not (<= g -1.1e-141)))
   (* (* (cbrt -0.5) (cbrt 2.0)) (cbrt (/ g a)))
   (+
    (* (cbrt 0.5) (* (cbrt (/ 1.0 a)) (cbrt (- (sqrt (* (- g h) (+ h g))) g))))
    (cbrt (* (+ g (sqrt (- (* g g) (* h h)))) (/ -0.5 a))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((g <= -1.34e+154) || !(g <= -1.1e-141)) {
		tmp = (cbrt(-0.5) * cbrt(2.0)) * cbrt((g / a));
	} else {
		tmp = (cbrt(0.5) * (cbrt((1.0 / a)) * cbrt((sqrt(((g - h) * (h + g))) - g)))) + cbrt(((g + sqrt(((g * g) - (h * h)))) * (-0.5 / a)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((g <= -1.34e+154) || !(g <= -1.1e-141)) {
		tmp = (Math.cbrt(-0.5) * Math.cbrt(2.0)) * Math.cbrt((g / a));
	} else {
		tmp = (Math.cbrt(0.5) * (Math.cbrt((1.0 / a)) * Math.cbrt((Math.sqrt(((g - h) * (h + g))) - g)))) + Math.cbrt(((g + Math.sqrt(((g * g) - (h * h)))) * (-0.5 / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((g <= -1.34e+154) || !(g <= -1.1e-141))
		tmp = Float64(Float64(cbrt(-0.5) * cbrt(2.0)) * cbrt(Float64(g / a)));
	else
		tmp = Float64(Float64(cbrt(0.5) * Float64(cbrt(Float64(1.0 / a)) * cbrt(Float64(sqrt(Float64(Float64(g - h) * Float64(h + g))) - g)))) + cbrt(Float64(Float64(g + sqrt(Float64(Float64(g * g) - Float64(h * h)))) * Float64(-0.5 / a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[g, -1.34e+154], N[Not[LessEqual[g, -1.1e-141]], $MachinePrecision]], N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[0.5, 1/3], $MachinePrecision] * N[(N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(g + N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < -1.34000000000000001e154 or -1.10000000000000005e-141 < g

   1. Initial program 31.1%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in h around 0 54.9%

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

      1. fma-define 54.9%

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

      2. *-commutative 54.9%

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

      3. *-commutative 54.9%

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

   7. Simplified 54.9%

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

   8. Taylor expanded in g around inf 65.0%

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

      1. *-commutative 65.0%

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

   10. Simplified 65.0%

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

   if -1.34000000000000001e154 < g < -1.10000000000000005e-141

   1. Initial program 85.3%

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

   3. Simplified 85.3%

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

      1. pow1/2 85.3%

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

      2. difference-of-squares 85.3%

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

      3. unpow-prod-down 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. unpow1/2 0.0%

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

      2. unpow1/2 0.0%

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

      3. +-commutative 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in a around 0 85.3%

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

       1. +-commutative 85.3%

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

   11. Simplified 85.3%

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

       1. pow1/3 36.8%

          \[\leadsto \color{blue}{{\left(\frac{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}{a}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

       2. div-inv 36.8%

          \[\leadsto {\color{blue}{\left(\left(\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g\right) \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \cdot \sqrt[3]{0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

       3. unpow-prod-down 39.4%

          \[\leadsto \color{blue}{\left({\left(\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \sqrt[3]{0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

       4. pow1/3 39.3%

          \[\leadsto \left(\color{blue}{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

   13. Applied egg-rr 39.3%

       \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \sqrt[3]{0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   14. Step-by-step derivation

       1. unpow1/3 98.0%

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

       2. *-commutative 98.0%

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

       3. +-commutative 98.0%

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

   15. Simplified 98.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.34 \cdot 10^{+154} \lor \neg \left(g \leq -1.1 \cdot 10^{-141}\right):\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{0.5} \cdot \left(\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}\right) + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq -1.34 \cdot 10^{+154} \lor \neg \left(g \leq -1.1 \cdot 10^{-141}\right):\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= g -1.34e+154) (not (<= g -1.1e-141)))
   (* (* (cbrt -0.5) (cbrt 2.0)) (cbrt (/ g a)))
   (+
    (cbrt (* (+ g (sqrt (- (* g g) (* h h)))) (/ -0.5 a)))
    (* (cbrt (/ 0.5 a)) (cbrt (- (sqrt (* (- g h) (+ h g))) g))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((g <= -1.34e+154) || !(g <= -1.1e-141)) {
		tmp = (cbrt(-0.5) * cbrt(2.0)) * cbrt((g / a));
	} else {
		tmp = cbrt(((g + sqrt(((g * g) - (h * h)))) * (-0.5 / a))) + (cbrt((0.5 / a)) * cbrt((sqrt(((g - h) * (h + g))) - g)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((g <= -1.34e+154) || !(g <= -1.1e-141)) {
		tmp = (Math.cbrt(-0.5) * Math.cbrt(2.0)) * Math.cbrt((g / a));
	} else {
		tmp = Math.cbrt(((g + Math.sqrt(((g * g) - (h * h)))) * (-0.5 / a))) + (Math.cbrt((0.5 / a)) * Math.cbrt((Math.sqrt(((g - h) * (h + g))) - g)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((g <= -1.34e+154) || !(g <= -1.1e-141))
		tmp = Float64(Float64(cbrt(-0.5) * cbrt(2.0)) * cbrt(Float64(g / a)));
	else
		tmp = Float64(cbrt(Float64(Float64(g + sqrt(Float64(Float64(g * g) - Float64(h * h)))) * Float64(-0.5 / a))) + Float64(cbrt(Float64(0.5 / a)) * cbrt(Float64(sqrt(Float64(Float64(g - h) * Float64(h + g))) - g))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[g, -1.34e+154], N[Not[LessEqual[g, -1.1e-141]], $MachinePrecision]], N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(g + N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < -1.34000000000000001e154 or -1.10000000000000005e-141 < g

   1. Initial program 31.1%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in h around 0 54.9%

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

      1. fma-define 54.9%

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

      2. *-commutative 54.9%

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

      3. *-commutative 54.9%

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

   7. Simplified 54.9%

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

   8. Taylor expanded in g around inf 65.0%

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

      1. *-commutative 65.0%

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

   10. Simplified 65.0%

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

   if -1.34000000000000001e154 < g < -1.10000000000000005e-141

   1. Initial program 85.3%

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

   3. Simplified 85.3%

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

      1. pow1/2 85.3%

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

      2. difference-of-squares 85.3%

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

      3. unpow-prod-down 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. unpow1/2 0.0%

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

      2. unpow1/2 0.0%

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

      3. +-commutative 0.0%

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

   8. Simplified 0.0%

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

      1. cbrt-prod 0.0%

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

      2. sqrt-unprod 97.9%

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

      3. +-commutative 97.9%

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

   10. Applied egg-rr 97.9%

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

       1. +-commutative 97.9%

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

   12. Simplified 97.9%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.34 \cdot 10^{+154} \lor \neg \left(g \leq -1.1 \cdot 10^{-141}\right):\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq -1.34 \cdot 10^{+154} \lor \neg \left(g \leq -1.1 \cdot 10^{-141}\right):\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g - g\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= g -1.34e+154) (not (<= g -1.1e-141)))
   (* (* (cbrt -0.5) (cbrt 2.0)) (cbrt (/ g a)))
   (+
    (/ (cbrt (* 0.5 (- (sqrt (* (- g h) (+ h g))) g))) (cbrt a))
    (cbrt (* (/ -0.5 a) (- g g))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((g <= -1.34e+154) || !(g <= -1.1e-141)) {
		tmp = (cbrt(-0.5) * cbrt(2.0)) * cbrt((g / a));
	} else {
		tmp = (cbrt((0.5 * (sqrt(((g - h) * (h + g))) - g))) / cbrt(a)) + cbrt(((-0.5 / a) * (g - g)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((g <= -1.34e+154) || !(g <= -1.1e-141)) {
		tmp = (Math.cbrt(-0.5) * Math.cbrt(2.0)) * Math.cbrt((g / a));
	} else {
		tmp = (Math.cbrt((0.5 * (Math.sqrt(((g - h) * (h + g))) - g))) / Math.cbrt(a)) + Math.cbrt(((-0.5 / a) * (g - g)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((g <= -1.34e+154) || !(g <= -1.1e-141))
		tmp = Float64(Float64(cbrt(-0.5) * cbrt(2.0)) * cbrt(Float64(g / a)));
	else
		tmp = Float64(Float64(cbrt(Float64(0.5 * Float64(sqrt(Float64(Float64(g - h) * Float64(h + g))) - g))) / cbrt(a)) + cbrt(Float64(Float64(-0.5 / a) * Float64(g - g))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[g, -1.34e+154], N[Not[LessEqual[g, -1.1e-141]], $MachinePrecision]], N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(0.5 * N[(N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $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 < -1.34000000000000001e154 or -1.10000000000000005e-141 < g

   1. Initial program 31.1%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in h around 0 54.9%

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

      1. fma-define 54.9%

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

      2. *-commutative 54.9%

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

      3. *-commutative 54.9%

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

   7. Simplified 54.9%

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

   8. Taylor expanded in g around inf 65.0%

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

      1. *-commutative 65.0%

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

   10. Simplified 65.0%

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

   if -1.34000000000000001e154 < g < -1.10000000000000005e-141

   1. Initial program 85.3%

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

   3. Simplified 85.3%

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

      1. pow1/2 85.3%

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

      2. difference-of-squares 85.3%

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

      3. unpow-prod-down 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. unpow1/2 0.0%

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

      2. unpow1/2 0.0%

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

      3. +-commutative 0.0%

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

   8. Simplified 0.0%

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

      1. associate-*l/ 0.0%

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

      2. cbrt-div 0.0%

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

      3. sqrt-unprod 97.8%

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

      4. +-commutative 97.8%

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

   10. Applied egg-rr 97.8%

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

       1. +-commutative 97.8%

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

   12. Simplified 97.8%

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

   13. Taylor expanded in g around -inf 95.1%

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

       1. neg-mul-1 95.1%

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

   15. Simplified 95.1%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.34 \cdot 10^{+154} \lor \neg \left(g \leq -1.1 \cdot 10^{-141}\right):\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g - g\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.4%

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

3. Simplified 44.4%

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

5. Taylor expanded in h around 0 61.9%

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

   1. fma-define 61.9%

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

   2. *-commutative 61.9%

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

   3. *-commutative 61.9%

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

7. Simplified 61.9%

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

8. Taylor expanded in g around inf 69.1%

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

   1. *-commutative 69.1%

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

10. Simplified 69.1%

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

Reproduce

herbie shell --seed 2024111 
(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))))))))