2-ancestry mixing, positive discriminant

Percentage Accurate: 43.9% → 97.4%

Time: 17.3s

Alternatives: 14

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 43.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 97.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g} \cdot \sqrt[3]{a}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (* (* (cbrt g) (cbrt (/ 1.0 a))) (* (cbrt -0.5) (cbrt 2.0)))
  (* (/ (pow (cbrt h) 2.0) (* (cbrt g) (cbrt a))) (* (cbrt -0.5) (cbrt 0.5)))))

C

double code(double g, double h, double a) {
	return ((cbrt(g) * cbrt((1.0 / a))) * (cbrt(-0.5) * cbrt(2.0))) + ((pow(cbrt(h), 2.0) / (cbrt(g) * cbrt(a))) * (cbrt(-0.5) * cbrt(0.5)));
}

Java

public static double code(double g, double h, double a) {
	return ((Math.cbrt(g) * Math.cbrt((1.0 / a))) * (Math.cbrt(-0.5) * Math.cbrt(2.0))) + ((Math.pow(Math.cbrt(h), 2.0) / (Math.cbrt(g) * Math.cbrt(a))) * (Math.cbrt(-0.5) * Math.cbrt(0.5)));
}

Julia

function code(g, h, a)
	return Float64(Float64(Float64(cbrt(g) * cbrt(Float64(1.0 / a))) * Float64(cbrt(-0.5) * cbrt(2.0))) + Float64(Float64((cbrt(h) ^ 2.0) / Float64(cbrt(g) * cbrt(a))) * Float64(cbrt(-0.5) * cbrt(0.5))))
end

Wolfram

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

Derivation

1. Initial program 42.7%

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

3. Simplified 42.7%

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

5. Taylor expanded in h around 0 66.4%

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

   1. pow1/3 33.8%

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

   2. div-inv 33.8%

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

   3. unpow-prod-down 22.4%

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

   4. pow1/3 43.6%

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

7. Applied egg-rr 43.6%

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

   1. unpow1/3 86.8%

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

9. Simplified 86.8%

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

    1. pow2 86.8%

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

    2. cbrt-div 87.7%

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

    3. cbrt-prod 93.7%

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

    4. pow2 93.7%

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

11. Applied egg-rr 93.7%

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

    1. *-commutative 93.7%

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

    2. cbrt-prod 97.7%

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

13. Applied egg-rr 97.7%

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

Alternative 2: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(t\_1 - g\right)} + \sqrt[3]{\left(g + t\_1\right) \cdot \frac{-1}{a \cdot 2}} \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}} + t\_0 \cdot \frac{1}{\sqrt[3]{a \cdot \frac{g}{{h}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot {2}^{0.3333333333333333}\right) + t\_0 \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (cbrt -0.5) (cbrt 0.5))) (t_1 (sqrt (- (* g g) (* h h)))))
   (if (<=
        (+
         (cbrt (* (/ 1.0 (* a 2.0)) (- t_1 g)))
         (cbrt (* (+ g t_1) (/ -1.0 (* a 2.0)))))
        2e+95)
     (+
      (* (* (cbrt -0.5) (cbrt 2.0)) (/ (cbrt g) (cbrt a)))
      (* t_0 (/ 1.0 (cbrt (* a (/ g (pow h 2.0)))))))
     (+
      (*
       (* (cbrt g) (cbrt (/ 1.0 a)))
       (* (cbrt -0.5) (pow 2.0 0.3333333333333333)))
      (* t_0 (/ (pow (cbrt h) 2.0) (cbrt (* g a))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(-0.5) * cbrt(0.5);
	double t_1 = sqrt(((g * g) - (h * h)));
	double tmp;
	if ((cbrt(((1.0 / (a * 2.0)) * (t_1 - g))) + cbrt(((g + t_1) * (-1.0 / (a * 2.0))))) <= 2e+95) {
		tmp = ((cbrt(-0.5) * cbrt(2.0)) * (cbrt(g) / cbrt(a))) + (t_0 * (1.0 / cbrt((a * (g / pow(h, 2.0))))));
	} else {
		tmp = ((cbrt(g) * cbrt((1.0 / a))) * (cbrt(-0.5) * pow(2.0, 0.3333333333333333))) + (t_0 * (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(-0.5) * Math.cbrt(0.5);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	double tmp;
	if ((Math.cbrt(((1.0 / (a * 2.0)) * (t_1 - g))) + Math.cbrt(((g + t_1) * (-1.0 / (a * 2.0))))) <= 2e+95) {
		tmp = ((Math.cbrt(-0.5) * Math.cbrt(2.0)) * (Math.cbrt(g) / Math.cbrt(a))) + (t_0 * (1.0 / Math.cbrt((a * (g / Math.pow(h, 2.0))))));
	} else {
		tmp = ((Math.cbrt(g) * Math.cbrt((1.0 / a))) * (Math.cbrt(-0.5) * Math.pow(2.0, 0.3333333333333333))) + (t_0 * (Math.pow(Math.cbrt(h), 2.0) / Math.cbrt((g * a))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(cbrt(-0.5) * cbrt(0.5))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	tmp = 0.0
	if (Float64(cbrt(Float64(Float64(1.0 / Float64(a * 2.0)) * Float64(t_1 - g))) + cbrt(Float64(Float64(g + t_1) * Float64(-1.0 / Float64(a * 2.0))))) <= 2e+95)
		tmp = Float64(Float64(Float64(cbrt(-0.5) * cbrt(2.0)) * Float64(cbrt(g) / cbrt(a))) + Float64(t_0 * Float64(1.0 / cbrt(Float64(a * Float64(g / (h ^ 2.0)))))));
	else
		tmp = Float64(Float64(Float64(cbrt(g) * cbrt(Float64(1.0 / a))) * Float64(cbrt(-0.5) * (2.0 ^ 0.3333333333333333))) + Float64(t_0 * Float64((cbrt(h) ^ 2.0) / 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[0.5, 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(g + t$95$1), $MachinePrecision] * N[(-1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 2e+95], N[(N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(1.0 / N[Power[N[(a * N[(g / N[Power[h, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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, 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 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 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 2.00000000000000004e95

   1. Initial program 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in h around 0 80.2%

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

      1. cbrt-div 89.0%

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

   7. Applied egg-rr 89.0%

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

      1. pow2 89.0%

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

      2. clear-num 89.0%

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

      3. cbrt-div 89.0%

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

      4. metadata-eval 89.0%

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

      5. pow2 89.0%

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

   9. Applied egg-rr 89.0%

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

       1. associate-/l* 95.9%

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

   11. Simplified 95.9%

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

   if 2.00000000000000004e95 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 0.2%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in h around 0 51.7%

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

      1. pow1/3 26.4%

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

      2. div-inv 26.4%

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

      3. unpow-prod-down 21.7%

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

      4. pow1/3 46.9%

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

   7. Applied egg-rr 46.9%

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

      1. unpow1/3 84.3%

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

   9. Simplified 84.3%

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

       1. pow2 84.3%

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

       2. cbrt-div 85.8%

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

       3. cbrt-prod 97.3%

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

       4. pow2 97.3%

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

   11. Applied egg-rr 97.3%

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

       1. pow1/3 98.0%

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

   13. Applied egg-rr 98.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{a \cdot 2}} \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \frac{1}{\sqrt[3]{a \cdot \frac{g}{{h}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot {2}^{0.3333333333333333}\right) + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := g + t\_0\\ t_2 := \sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(t\_0 - g\right)} + \sqrt[3]{t\_1 \cdot \frac{-1}{a \cdot 2}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-94}:\\ \;\;\;\;\sqrt[3]{\frac{{h}^{2}}{g} \cdot \frac{-0.25}{a}} - \sqrt[3]{\frac{g}{a}}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{t\_1 \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g - g\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h))))
        (t_1 (+ g t_0))
        (t_2
         (+
          (cbrt (* (/ 1.0 (* a 2.0)) (- t_0 g)))
          (cbrt (* t_1 (/ -1.0 (* a 2.0)))))))
   (if (<= t_2 -5e-94)
     (- (cbrt (* (/ (pow h 2.0) g) (/ -0.25 a))) (cbrt (/ g a)))
     (if (<= t_2 2e-104)
       (+ (/ (cbrt (- g)) (cbrt a)) (cbrt (* t_1 (/ -0.5 a))))
       (+ (cbrt (/ (- g) a)) (cbrt (* (/ -0.5 a) (- g g))))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double t_1 = g + t_0;
	double t_2 = cbrt(((1.0 / (a * 2.0)) * (t_0 - g))) + cbrt((t_1 * (-1.0 / (a * 2.0))));
	double tmp;
	if (t_2 <= -5e-94) {
		tmp = cbrt(((pow(h, 2.0) / g) * (-0.25 / a))) - cbrt((g / a));
	} else if (t_2 <= 2e-104) {
		tmp = (cbrt(-g) / cbrt(a)) + cbrt((t_1 * (-0.5 / a)));
	} else {
		tmp = cbrt((-g / a)) + cbrt(((-0.5 / a) * (g - g)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt(((g * g) - (h * h)));
	double t_1 = g + t_0;
	double t_2 = Math.cbrt(((1.0 / (a * 2.0)) * (t_0 - g))) + Math.cbrt((t_1 * (-1.0 / (a * 2.0))));
	double tmp;
	if (t_2 <= -5e-94) {
		tmp = Math.cbrt(((Math.pow(h, 2.0) / g) * (-0.25 / a))) - Math.cbrt((g / a));
	} else if (t_2 <= 2e-104) {
		tmp = (Math.cbrt(-g) / Math.cbrt(a)) + Math.cbrt((t_1 * (-0.5 / a)));
	} else {
		tmp = Math.cbrt((-g / a)) + Math.cbrt(((-0.5 / a) * (g - g)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = Float64(g + t_0)
	t_2 = Float64(cbrt(Float64(Float64(1.0 / Float64(a * 2.0)) * Float64(t_0 - g))) + cbrt(Float64(t_1 * Float64(-1.0 / Float64(a * 2.0)))))
	tmp = 0.0
	if (t_2 <= -5e-94)
		tmp = Float64(cbrt(Float64(Float64((h ^ 2.0) / g) * Float64(-0.25 / a))) - cbrt(Float64(g / a)));
	elseif (t_2 <= 2e-104)
		tmp = Float64(Float64(cbrt(Float64(-g)) / cbrt(a)) + cbrt(Float64(t_1 * Float64(-0.5 / a))));
	else
		tmp = Float64(cbrt(Float64(Float64(-g) / a)) + cbrt(Float64(Float64(-0.5 / a) * Float64(g - g))));
	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[(g + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$1 * N[(-1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-94], N[(N[Power[N[(N[(N[Power[h, 2.0], $MachinePrecision] / g), $MachinePrecision] * N[(-0.25 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-104], N[(N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(t$95$1 * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -4.9999999999999995e-94

   1. Initial program 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in h around 0 84.5%

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

      1. add-cube-cbrt 83.9%

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

      2. pow3 83.9%

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

   7. Applied egg-rr 83.8%

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

   9. Applied egg-rr 83.8%

      \[\leadsto \color{blue}{\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, -1, \sqrt[3]{\frac{{h}^{2}}{a \cdot g} \cdot -0.25}\right)} \cdot {\left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, -1, \sqrt[3]{\frac{{h}^{2}}{a \cdot g} \cdot -0.25}\right)}\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 83.8%

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

       2. rem-3cbrt-rft 85.1%

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

       3. fma-undefine 85.1%

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

       4. *-commutative 85.1%

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

       5. +-commutative 85.1%

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

       6. mul-1-neg 85.1%

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

       7. unsub-neg 85.1%

          \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g} \cdot -0.25} - \sqrt[3]{\frac{g}{a}}} \]

       8. associate-*l/ 85.1%

          \[\leadsto \sqrt[3]{\color{blue}{\frac{{h}^{2} \cdot -0.25}{a \cdot g}}} - \sqrt[3]{\frac{g}{a}} \]

       9. *-commutative 85.1%

          \[\leadsto \sqrt[3]{\frac{{h}^{2} \cdot -0.25}{\color{blue}{g \cdot a}}} - \sqrt[3]{\frac{g}{a}} \]

       10. times-frac 93.9%

           \[\leadsto \sqrt[3]{\color{blue}{\frac{{h}^{2}}{g} \cdot \frac{-0.25}{a}}} - \sqrt[3]{\frac{g}{a}} \]

   11. Simplified 93.9%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{g} \cdot \frac{-0.25}{a}} - \sqrt[3]{\frac{g}{a}}} \]

   if -4.9999999999999995e-94 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 1.99999999999999985e-104

   1. Initial program 11.0%

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

   3. Simplified 11.0%

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

   5. Taylor expanded in g around -inf 11.0%

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

      1. associate-*r/ 11.0%

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

      2. neg-mul-1 11.0%

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

   7. Simplified 11.0%

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

      1. cbrt-div 85.2%

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

   9. Applied egg-rr 85.2%

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

   if 1.99999999999999985e-104 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 28.7%

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

   3. Simplified 28.7%

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

   5. Taylor expanded in g around -inf 15.8%

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

      1. associate-*r/ 15.8%

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

      2. neg-mul-1 15.8%

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

   7. Simplified 15.8%

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

   8. Taylor expanded in g around -inf 71.0%

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

      1. neg-mul-1 71.0%

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

   10. Simplified 71.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{a \cdot 2}} \leq -5 \cdot 10^{-94}:\\ \;\;\;\;\sqrt[3]{\frac{{h}^{2}}{g} \cdot \frac{-0.25}{a}} - \sqrt[3]{\frac{g}{a}}\\ \mathbf{elif}\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{a \cdot 2}} \leq 2 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g - g\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(t\_1 - g\right)} + \sqrt[3]{\left(g + t\_1\right) \cdot \frac{-1}{a \cdot 2}} \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}} + t\_0 \cdot \frac{1}{\sqrt[3]{a \cdot \frac{g}{{h}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (cbrt -0.5) (cbrt 0.5))) (t_1 (sqrt (- (* g g) (* h h)))))
   (if (<=
        (+
         (cbrt (* (/ 1.0 (* a 2.0)) (- t_1 g)))
         (cbrt (* (+ g t_1) (/ -1.0 (* a 2.0)))))
        2e+95)
     (+
      (* (* (cbrt -0.5) (cbrt 2.0)) (/ (cbrt g) (cbrt a)))
      (* t_0 (/ 1.0 (cbrt (* a (/ g (pow h 2.0)))))))
     (+
      (* t_0 (/ (pow (cbrt h) 2.0) (cbrt (* g a))))
      (/ (cbrt (- g)) (cbrt a))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(-0.5) * cbrt(0.5);
	double t_1 = sqrt(((g * g) - (h * h)));
	double tmp;
	if ((cbrt(((1.0 / (a * 2.0)) * (t_1 - g))) + cbrt(((g + t_1) * (-1.0 / (a * 2.0))))) <= 2e+95) {
		tmp = ((cbrt(-0.5) * cbrt(2.0)) * (cbrt(g) / cbrt(a))) + (t_0 * (1.0 / cbrt((a * (g / pow(h, 2.0))))));
	} else {
		tmp = (t_0 * (pow(cbrt(h), 2.0) / cbrt((g * a)))) + (cbrt(-g) / cbrt(a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(-0.5) * Math.cbrt(0.5);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	double tmp;
	if ((Math.cbrt(((1.0 / (a * 2.0)) * (t_1 - g))) + Math.cbrt(((g + t_1) * (-1.0 / (a * 2.0))))) <= 2e+95) {
		tmp = ((Math.cbrt(-0.5) * Math.cbrt(2.0)) * (Math.cbrt(g) / Math.cbrt(a))) + (t_0 * (1.0 / Math.cbrt((a * (g / Math.pow(h, 2.0))))));
	} else {
		tmp = (t_0 * (Math.pow(Math.cbrt(h), 2.0) / Math.cbrt((g * a)))) + (Math.cbrt(-g) / Math.cbrt(a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(cbrt(-0.5) * cbrt(0.5))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	tmp = 0.0
	if (Float64(cbrt(Float64(Float64(1.0 / Float64(a * 2.0)) * Float64(t_1 - g))) + cbrt(Float64(Float64(g + t_1) * Float64(-1.0 / Float64(a * 2.0))))) <= 2e+95)
		tmp = Float64(Float64(Float64(cbrt(-0.5) * cbrt(2.0)) * Float64(cbrt(g) / cbrt(a))) + Float64(t_0 * Float64(1.0 / cbrt(Float64(a * Float64(g / (h ^ 2.0)))))));
	else
		tmp = Float64(Float64(t_0 * Float64((cbrt(h) ^ 2.0) / cbrt(Float64(g * a)))) + Float64(cbrt(Float64(-g)) / cbrt(a)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 2.00000000000000004e95

   1. Initial program 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in h around 0 80.2%

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

      1. cbrt-div 89.0%

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

   7. Applied egg-rr 89.0%

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

      1. pow2 89.0%

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

      2. clear-num 89.0%

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

      3. cbrt-div 89.0%

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

      4. metadata-eval 89.0%

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

      5. pow2 89.0%

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

   9. Applied egg-rr 89.0%

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

       1. associate-/l* 95.9%

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

   11. Simplified 95.9%

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

   if 2.00000000000000004e95 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 0.2%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in h around 0 51.7%

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

      1. pow1/3 26.4%

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

      2. div-inv 26.4%

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

      3. unpow-prod-down 21.7%

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

      4. pow1/3 46.9%

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

   7. Applied egg-rr 46.9%

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

      1. unpow1/3 84.3%

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

   9. Simplified 84.3%

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

       1. pow2 84.3%

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

       2. cbrt-div 85.8%

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

       3. cbrt-prod 97.3%

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

       4. pow2 97.3%

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

   11. Applied egg-rr 97.3%

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

       1. cbrt-div 97.2%

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

       2. metadata-eval 97.2%

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

       3. div-inv 97.1%

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

       4. associate-*l/ 97.1%

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

       5. cbrt-unprod 97.9%

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

       6. metadata-eval 97.9%

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

       7. cbrt-unprod 97.9%

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

   13. Applied egg-rr 97.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{a \cdot 2}} \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \frac{1}{\sqrt[3]{a \cdot \frac{g}{{h}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.7%

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

3. Simplified 42.7%

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

5. Taylor expanded in h around 0 66.4%

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

   1. pow1/3 33.8%

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

   2. div-inv 33.8%

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

   3. unpow-prod-down 22.4%

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

   4. pow1/3 43.6%

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

7. Applied egg-rr 43.6%

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

   1. unpow1/3 86.8%

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

9. Simplified 86.8%

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

    1. pow2 86.8%

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

    2. cbrt-div 87.7%

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

    3. cbrt-prod 93.7%

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

    4. pow2 93.7%

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

11. Applied egg-rr 93.7%

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

    1. cbrt-div 93.6%

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

    2. metadata-eval 93.6%

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

    3. div-inv 93.6%

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

    4. associate-*l/ 93.5%

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

    5. cbrt-unprod 94.2%

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

    6. metadata-eval 94.2%

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

    7. cbrt-unprod 94.2%

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

13. Applied egg-rr 94.2%

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

14. Final simplification 94.2%

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

Alternative 6: 90.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\\ \mathbf{if}\;h \cdot h \leq 5 \cdot 10^{+283}:\\ \;\;\;\;t\_0 \cdot \sqrt[3]{\frac{{h}^{2}}{g \cdot a}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;h \cdot \left(t\_0 \cdot \sqrt[3]{\frac{1}{a \cdot \left(g \cdot h\right)}} + \frac{1}{\sqrt[3]{\frac{a}{g}}} \cdot \frac{\sqrt[3]{-0.5} \cdot \sqrt[3]{2}}{h}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (cbrt -0.5) (cbrt 0.5))))
   (if (<= (* h h) 5e+283)
     (+ (* t_0 (cbrt (/ (pow h 2.0) (* g a)))) (/ (cbrt (- g)) (cbrt a)))
     (*
      h
      (+
       (* t_0 (cbrt (/ 1.0 (* a (* g h)))))
       (* (/ 1.0 (cbrt (/ a g))) (/ (* (cbrt -0.5) (cbrt 2.0)) h)))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(-0.5) * cbrt(0.5);
	double tmp;
	if ((h * h) <= 5e+283) {
		tmp = (t_0 * cbrt((pow(h, 2.0) / (g * a)))) + (cbrt(-g) / cbrt(a));
	} else {
		tmp = h * ((t_0 * cbrt((1.0 / (a * (g * h))))) + ((1.0 / cbrt((a / g))) * ((cbrt(-0.5) * cbrt(2.0)) / h)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(-0.5) * Math.cbrt(0.5);
	double tmp;
	if ((h * h) <= 5e+283) {
		tmp = (t_0 * Math.cbrt((Math.pow(h, 2.0) / (g * a)))) + (Math.cbrt(-g) / Math.cbrt(a));
	} else {
		tmp = h * ((t_0 * Math.cbrt((1.0 / (a * (g * h))))) + ((1.0 / Math.cbrt((a / g))) * ((Math.cbrt(-0.5) * Math.cbrt(2.0)) / h)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(cbrt(-0.5) * cbrt(0.5))
	tmp = 0.0
	if (Float64(h * h) <= 5e+283)
		tmp = Float64(Float64(t_0 * cbrt(Float64((h ^ 2.0) / Float64(g * a)))) + Float64(cbrt(Float64(-g)) / cbrt(a)));
	else
		tmp = Float64(h * Float64(Float64(t_0 * cbrt(Float64(1.0 / Float64(a * Float64(g * h))))) + Float64(Float64(1.0 / cbrt(Float64(a / g))) * Float64(Float64(cbrt(-0.5) * cbrt(2.0)) / h))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h * h), $MachinePrecision], 5e+283], N[(N[(t$95$0 * N[Power[N[(N[Power[h, 2.0], $MachinePrecision] / N[(g * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(h * N[(N[(t$95$0 * N[Power[N[(1.0 / N[(a * N[(g * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[Power[N[(a / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 5.0000000000000004e283

   1. Initial program 45.9%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in h around 0 70.8%

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

      1. pow1/3 35.9%

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

      2. div-inv 35.9%

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

      3. unpow-prod-down 23.6%

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

      4. pow1/3 46.4%

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

   7. Applied egg-rr 46.4%

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

      1. unpow1/3 92.7%

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

   9. Simplified 92.7%

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

       1. cbrt-div 93.7%

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

       2. metadata-eval 93.7%

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

       3. div-inv 93.7%

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

       4. associate-*l/ 93.6%

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

       5. cbrt-unprod 94.3%

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

       6. metadata-eval 94.3%

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

       7. cbrt-unprod 94.3%

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

   11. Applied egg-rr 93.2%

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

   if 5.0000000000000004e283 < (*.f64 h h)

   1. Initial program 0.2%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in h around 0 7.9%

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

   6. Taylor expanded in h around inf 61.3%

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

      1. clear-num 61.3%

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

      2. cbrt-div 64.1%

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

      3. metadata-eval 64.1%

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

   8. Applied egg-rr 64.1%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{\frac{{h}^{2}}{g \cdot a}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;h \cdot \left(\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{\frac{1}{a \cdot \left(g \cdot h\right)}} + \frac{1}{\sqrt[3]{\frac{a}{g}}} \cdot \frac{\sqrt[3]{-0.5} \cdot \sqrt[3]{2}}{h}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \cdot h \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{\frac{{h}^{2}}{g \cdot a}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(h \cdot \left(\frac{\sqrt[3]{\frac{-g}{a}}}{h} + \sqrt[3]{\frac{\frac{1}{a} \cdot -0.25}{g \cdot h}}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (* h h) 5e+283)
   (+
    (* (* (cbrt -0.5) (cbrt 0.5)) (cbrt (/ (pow h 2.0) (* g a))))
    (/ (cbrt (- g)) (cbrt a)))
   (cbrt
    (pow
     (* h (+ (/ (cbrt (/ (- g) a)) h) (cbrt (/ (* (/ 1.0 a) -0.25) (* g h)))))
     3.0))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((h * h) <= 5e+283) {
		tmp = ((cbrt(-0.5) * cbrt(0.5)) * cbrt((pow(h, 2.0) / (g * a)))) + (cbrt(-g) / cbrt(a));
	} else {
		tmp = cbrt(pow((h * ((cbrt((-g / a)) / h) + cbrt((((1.0 / a) * -0.25) / (g * h))))), 3.0));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((h * h) <= 5e+283) {
		tmp = ((Math.cbrt(-0.5) * Math.cbrt(0.5)) * Math.cbrt((Math.pow(h, 2.0) / (g * a)))) + (Math.cbrt(-g) / Math.cbrt(a));
	} else {
		tmp = Math.cbrt(Math.pow((h * ((Math.cbrt((-g / a)) / h) + Math.cbrt((((1.0 / a) * -0.25) / (g * h))))), 3.0));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(h * h) <= 5e+283)
		tmp = Float64(Float64(Float64(cbrt(-0.5) * cbrt(0.5)) * cbrt(Float64((h ^ 2.0) / Float64(g * a)))) + Float64(cbrt(Float64(-g)) / cbrt(a)));
	else
		tmp = cbrt((Float64(h * Float64(Float64(cbrt(Float64(Float64(-g) / a)) / h) + cbrt(Float64(Float64(Float64(1.0 / a) * -0.25) / Float64(g * h))))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(h * h), $MachinePrecision], 5e+283], N[(N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[h, 2.0], $MachinePrecision] / N[(g * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(h * N[(N[(N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision] / h), $MachinePrecision] + N[Power[N[(N[(N[(1.0 / a), $MachinePrecision] * -0.25), $MachinePrecision] / N[(g * h), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 5.0000000000000004e283

   1. Initial program 45.9%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in h around 0 70.8%

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

      1. pow1/3 35.9%

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

      2. div-inv 35.9%

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

      3. unpow-prod-down 23.6%

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

      4. pow1/3 46.4%

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

   7. Applied egg-rr 46.4%

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

      1. unpow1/3 92.7%

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

   9. Simplified 92.7%

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

       1. cbrt-div 93.7%

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

       2. metadata-eval 93.7%

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

       3. div-inv 93.7%

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

       4. associate-*l/ 93.6%

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

       5. cbrt-unprod 94.3%

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

       6. metadata-eval 94.3%

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

       7. cbrt-unprod 94.3%

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

   11. Applied egg-rr 93.2%

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

   if 5.0000000000000004e283 < (*.f64 h h)

   1. Initial program 0.2%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in h around 0 7.9%

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

   6. Taylor expanded in h around inf 61.3%

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

      1. +-commutative 61.3%

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

      2. cbrt-div 89.0%

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

      3. add-cube-cbrt 88.8%

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

      4. fma-define 88.8%

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

   8. Applied egg-rr 61.3%

      \[\leadsto h \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\frac{\sqrt[3]{-1 \cdot \frac{g}{a}}}{h}}\right)}^{2}, \sqrt[3]{\frac{\sqrt[3]{-1 \cdot \frac{g}{a}}}{h}}, \sqrt[3]{-0.25 \cdot \frac{\frac{1}{a}}{g \cdot h}}\right)} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 61.4%

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

      2. pow3 61.3%

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

   10. Applied egg-rr 62.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{\frac{{h}^{2}}{g \cdot a}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(h \cdot \left(\frac{\sqrt[3]{\frac{-g}{a}}}{h} + \sqrt[3]{\frac{\frac{1}{a} \cdot -0.25}{g \cdot h}}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 9.8 \cdot 10^{+142}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\frac{{h}^{2}}{g \cdot a} \cdot -0.25}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(h \cdot \left(\frac{\sqrt[3]{\frac{-g}{a}}}{h} + \sqrt[3]{\frac{\frac{1}{a} \cdot -0.25}{g \cdot h}}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= h 9.8e+142)
   (pow
    (cbrt
     (+ (/ (cbrt (- g)) (cbrt a)) (cbrt (* (/ (pow h 2.0) (* g a)) -0.25))))
    3.0)
   (cbrt
    (pow
     (* h (+ (/ (cbrt (/ (- g) a)) h) (cbrt (/ (* (/ 1.0 a) -0.25) (* g h)))))
     3.0))))

C

double code(double g, double h, double a) {
	double tmp;
	if (h <= 9.8e+142) {
		tmp = pow(cbrt(((cbrt(-g) / cbrt(a)) + cbrt(((pow(h, 2.0) / (g * a)) * -0.25)))), 3.0);
	} else {
		tmp = cbrt(pow((h * ((cbrt((-g / a)) / h) + cbrt((((1.0 / a) * -0.25) / (g * h))))), 3.0));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (h <= 9.8e+142) {
		tmp = Math.pow(Math.cbrt(((Math.cbrt(-g) / Math.cbrt(a)) + Math.cbrt(((Math.pow(h, 2.0) / (g * a)) * -0.25)))), 3.0);
	} else {
		tmp = Math.cbrt(Math.pow((h * ((Math.cbrt((-g / a)) / h) + Math.cbrt((((1.0 / a) * -0.25) / (g * h))))), 3.0));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (h <= 9.8e+142)
		tmp = cbrt(Float64(Float64(cbrt(Float64(-g)) / cbrt(a)) + cbrt(Float64(Float64((h ^ 2.0) / Float64(g * a)) * -0.25)))) ^ 3.0;
	else
		tmp = cbrt((Float64(h * Float64(Float64(cbrt(Float64(Float64(-g) / a)) / h) + cbrt(Float64(Float64(Float64(1.0 / a) * -0.25) / Float64(g * h))))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[h, 9.8e+142], N[Power[N[Power[N[(N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[Power[h, 2.0], $MachinePrecision] / N[(g * a), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], N[Power[N[Power[N[(h * N[(N[(N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision] / h), $MachinePrecision] + N[Power[N[(N[(N[(1.0 / a), $MachinePrecision] * -0.25), $MachinePrecision] / N[(g * h), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 9.80000000000000101e142

   1. Initial program 44.3%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in h around 0 68.3%

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

      1. add-cube-cbrt 67.9%

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

      2. pow3 67.9%

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

   7. Applied egg-rr 68.0%

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

      1. *-commutative 68.0%

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

      2. associate-*r/ 68.0%

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

      3. cbrt-div 89.0%

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

   9. Applied egg-rr 89.0%

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

       1. *-commutative 89.0%

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

       2. *-commutative 89.0%

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

       3. mul-1-neg 89.0%

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

   11. Simplified 89.0%

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

   if 9.80000000000000101e142 < h

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in h around 0 14.3%

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

   6. Taylor expanded in h around inf 45.3%

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

      1. +-commutative 45.3%

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

      2. cbrt-div 90.6%

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

      3. add-cube-cbrt 90.2%

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

      4. fma-define 90.1%

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

   8. Applied egg-rr 45.1%

      \[\leadsto h \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\frac{\sqrt[3]{-1 \cdot \frac{g}{a}}}{h}}\right)}^{2}, \sqrt[3]{\frac{\sqrt[3]{-1 \cdot \frac{g}{a}}}{h}}, \sqrt[3]{-0.25 \cdot \frac{\frac{1}{a}}{g \cdot h}}\right)} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 45.1%

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

      2. pow3 45.1%

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

   10. Applied egg-rr 45.9%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 9.8 \cdot 10^{+142}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\frac{{h}^{2}}{g \cdot a} \cdot -0.25}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(h \cdot \left(\frac{\sqrt[3]{\frac{-g}{a}}}{h} + \sqrt[3]{\frac{\frac{1}{a} \cdot -0.25}{g \cdot h}}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 9.8 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, \sqrt[3]{\frac{-1}{a}}, \sqrt[3]{{h}^{2} \cdot \frac{-0.25}{g \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(h \cdot \left(\frac{\sqrt[3]{\frac{-g}{a}}}{h} + \sqrt[3]{\frac{\frac{1}{a} \cdot -0.25}{g \cdot h}}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= h 9.8e+142)
   (fma (cbrt g) (cbrt (/ -1.0 a)) (cbrt (* (pow h 2.0) (/ -0.25 (* g a)))))
   (cbrt
    (pow
     (* h (+ (/ (cbrt (/ (- g) a)) h) (cbrt (/ (* (/ 1.0 a) -0.25) (* g h)))))
     3.0))))

C

double code(double g, double h, double a) {
	double tmp;
	if (h <= 9.8e+142) {
		tmp = fma(cbrt(g), cbrt((-1.0 / a)), cbrt((pow(h, 2.0) * (-0.25 / (g * a)))));
	} else {
		tmp = cbrt(pow((h * ((cbrt((-g / a)) / h) + cbrt((((1.0 / a) * -0.25) / (g * h))))), 3.0));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (h <= 9.8e+142)
		tmp = fma(cbrt(g), cbrt(Float64(-1.0 / a)), cbrt(Float64((h ^ 2.0) * Float64(-0.25 / Float64(g * a)))));
	else
		tmp = cbrt((Float64(h * Float64(Float64(cbrt(Float64(Float64(-g) / a)) / h) + cbrt(Float64(Float64(Float64(1.0 / a) * -0.25) / Float64(g * h))))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[h, 9.8e+142], N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[Power[h, 2.0], $MachinePrecision] * N[(-0.25 / N[(g * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(h * N[(N[(N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision] / h), $MachinePrecision] + N[Power[N[(N[(N[(1.0 / a), $MachinePrecision] * -0.25), $MachinePrecision] / N[(g * h), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 9.80000000000000101e142

   1. Initial program 44.3%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in h around 0 68.3%

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

      1. pow1/3 34.6%

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

      2. div-inv 34.6%

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

      3. unpow-prod-down 22.8%

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

      4. pow1/3 44.8%

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

   7. Applied egg-rr 44.8%

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

      1. unpow1/3 89.4%

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

   9. Simplified 89.4%

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

       1. associate-*l* 89.4%

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

       2. fma-define 89.4%

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

       3. cbrt-unprod 89.9%

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

       4. metadata-eval 89.9%

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

       5. cbrt-unprod 89.9%

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

       6. *-commutative 89.9%

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

       7. cbrt-unprod 89.9%

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

       8. metadata-eval 89.9%

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

       9. cbrt-unprod 89.9%

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

       10. *-commutative 89.9%

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

   11. Applied egg-rr 89.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{g}, \sqrt[3]{\frac{1}{a} \cdot -1}, \sqrt[3]{-0.25 \cdot \frac{{h}^{2}}{g \cdot a}}\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 89.9%

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

       2. metadata-eval 89.9%

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

       3. *-commutative 89.9%

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

       4. associate-*l/ 89.9%

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

       5. associate-/l* 88.3%

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

   13. Simplified 88.3%

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

   if 9.80000000000000101e142 < h

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in h around 0 14.3%

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

   6. Taylor expanded in h around inf 45.3%

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

      1. +-commutative 45.3%

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

      2. cbrt-div 90.6%

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

      3. add-cube-cbrt 90.2%

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

      4. fma-define 90.1%

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

   8. Applied egg-rr 45.1%

      \[\leadsto h \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\frac{\sqrt[3]{-1 \cdot \frac{g}{a}}}{h}}\right)}^{2}, \sqrt[3]{\frac{\sqrt[3]{-1 \cdot \frac{g}{a}}}{h}}, \sqrt[3]{-0.25 \cdot \frac{\frac{1}{a}}{g \cdot h}}\right)} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 45.1%

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

      2. pow3 45.1%

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

   10. Applied egg-rr 45.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 9.8 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, \sqrt[3]{\frac{-1}{a}}, \sqrt[3]{{h}^{2} \cdot \frac{-0.25}{g \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(h \cdot \left(\frac{\sqrt[3]{\frac{-g}{a}}}{h} + \sqrt[3]{\frac{\frac{1}{a} \cdot -0.25}{g \cdot h}}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 7.8 \cdot 10^{-271}:\\ \;\;\;\;{\left(0 - {\left(\frac{g}{a}\right)}^{0.1111111111111111}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;h \cdot \frac{\sqrt[3]{\frac{g}{a}}}{-h}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= h 7.8e-271)
   (pow (- 0.0 (pow (/ g a) 0.1111111111111111)) 3.0)
   (* h (/ (cbrt (/ g a)) (- h)))))

C

double code(double g, double h, double a) {
	double tmp;
	if (h <= 7.8e-271) {
		tmp = pow((0.0 - pow((g / a), 0.1111111111111111)), 3.0);
	} else {
		tmp = h * (cbrt((g / a)) / -h);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (h <= 7.8e-271) {
		tmp = Math.pow((0.0 - Math.pow((g / a), 0.1111111111111111)), 3.0);
	} else {
		tmp = h * (Math.cbrt((g / a)) / -h);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (h <= 7.8e-271)
		tmp = Float64(0.0 - (Float64(g / a) ^ 0.1111111111111111)) ^ 3.0;
	else
		tmp = Float64(h * Float64(cbrt(Float64(g / a)) / Float64(-h)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[h, 7.8e-271], N[Power[N[(0.0 - N[Power[N[(g / a), $MachinePrecision], 0.1111111111111111], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(h * N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] / (-h)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 7.79999999999999995e-271

   1. Initial program 42.2%

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

   3. Simplified 42.2%

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

   5. Taylor expanded in h around 0 66.3%

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

      1. add-cube-cbrt 65.9%

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

      2. pow3 65.9%

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

   7. Applied egg-rr 65.9%

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

   8. Taylor expanded in g around -inf 40.1%

      \[\leadsto {\color{blue}{\left(-1 \cdot {\left(\frac{1 \cdot g}{a}\right)}^{0.1111111111111111}\right)}}^{3} \]
   9. Step-by-step derivation

      1. mul-1-neg 40.1%

         \[\leadsto {\color{blue}{\left(-{\left(\frac{1 \cdot g}{a}\right)}^{0.1111111111111111}\right)}}^{3} \]

      2. neg-sub0 40.1%

         \[\leadsto {\color{blue}{\left(0 - {\left(\frac{1 \cdot g}{a}\right)}^{0.1111111111111111}\right)}}^{3} \]

      3. *-lft-identity 40.1%

         \[\leadsto {\left(0 - {\left(\frac{\color{blue}{g}}{a}\right)}^{0.1111111111111111}\right)}^{3} \]

   10. Simplified 40.1%

       \[\leadsto {\color{blue}{\left(0 - {\left(\frac{g}{a}\right)}^{0.1111111111111111}\right)}}^{3} \]

   if 7.79999999999999995e-271 < h

   1. Initial program 43.4%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in h around 0 66.6%

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

   6. Taylor expanded in h around inf 58.5%

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

      1. +-commutative 58.5%

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

      2. cbrt-div 76.7%

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

      3. add-cube-cbrt 76.2%

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

      4. fma-define 76.2%

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

   8. Applied egg-rr 58.4%

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

   9. Taylor expanded in g around -inf 66.9%

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

       1. mul-1-neg 66.9%

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

       2. associate-*r/ 67.1%

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

       3. *-rgt-identity 67.1%

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

       4. distribute-neg-frac2 67.1%

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

   11. Simplified 67.1%

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

Alternative 11: 74.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.7%

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

3. Simplified 42.7%

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

5. Taylor expanded in g around -inf 24.8%

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

   1. associate-*r/ 24.8%

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

   2. neg-mul-1 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in g around -inf 73.6%

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

   1. neg-mul-1 73.6%

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

10. Simplified 73.6%

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

11. Final simplification 73.6%

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

Alternative 12: 68.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.7%

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

3. Simplified 42.7%

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

5. Taylor expanded in h around 0 66.4%

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

6. Taylor expanded in h around inf 54.7%

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

   1. +-commutative 54.7%

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

   2. cbrt-div 67.8%

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

   3. add-cube-cbrt 67.5%

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

   4. fma-define 67.5%

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

8. Applied egg-rr 54.7%

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

9. Taylor expanded in g around -inf 65.7%

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

    1. mul-1-neg 65.7%

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

    2. associate-*r/ 65.8%

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

    3. *-rgt-identity 65.8%

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

    4. distribute-neg-frac2 65.8%

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

11. Simplified 65.8%

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

Alternative 13: 15.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt[3]{\frac{a}{g}}} \cdot -2 \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (* (/ 1.0 (cbrt (/ a g))) -2.0))

C

double code(double g, double h, double a) {
	return (1.0 / cbrt((a / g))) * -2.0;
}

Java

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

Julia

function code(g, h, a)
	return Float64(Float64(1.0 / cbrt(Float64(a / g))) * -2.0)
end

Wolfram

code[g_, h_, a_] := N[(N[(1.0 / N[Power[N[(a / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 42.7%

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

3. Simplified 42.7%

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

5. Taylor expanded in g around -inf 24.8%

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

   1. associate-*r/ 24.8%

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

   2. neg-mul-1 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in g around inf 15.2%

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

9. Taylor expanded in g around -inf 15.2%

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

    1. *-commutative 15.2%

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

11. Simplified 15.2%

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

    1. clear-num 54.5%

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

    2. cbrt-div 55.4%

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

    3. metadata-eval 55.4%

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

13. Applied egg-rr 15.5%

    \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{a}{g}}}} \cdot -2 \]
14. Add Preprocessing

Alternative 14: 15.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.7%

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

3. Simplified 42.7%

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

5. Taylor expanded in g around -inf 24.8%

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

   1. associate-*r/ 24.8%

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

   2. neg-mul-1 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in g around inf 15.2%

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

9. Taylor expanded in g around -inf 15.2%

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

    1. *-commutative 15.2%

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

11. Simplified 15.2%

    \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot -2} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024119 
(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))))))))