2-ancestry mixing, positive discriminant

Percentage Accurate: 43.5% → 97.2%

Time: 17.8s

Alternatives: 13

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 43.5% 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: 97.2% 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 42.5%

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

3. Simplified 42.5%

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

5. Taylor expanded in h around 0 67.4%

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

   1. pow1/3 32.5%

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

   2. div-inv 32.5%

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

   3. unpow-prod-down 18.3%

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

   4. pow1/3 40.1%

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

7. Applied egg-rr 40.1%

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

   1. unpow1/3 88.5%

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

9. Simplified 88.5%

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

    1. cbrt-div 89.4%

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

    2. pow2 89.4%

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

    3. cbrt-prod 93.5%

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

    4. pow2 93.5%

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

11. Applied egg-rr 93.5%

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

13. Applied egg-rr 98.0%

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

Alternative 2: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \cdot h \leq 10^{-188}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, \frac{\sqrt[3]{-1}}{\sqrt[3]{a}}, \sqrt[3]{-0.25} \cdot \sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot {2}^{0.3333333333333333}\right) + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (* h h) 1e-188)
   (fma
    (cbrt g)
    (/ (cbrt -1.0) (cbrt a))
    (* (cbrt -0.25) (cbrt (* (/ h g) (/ h a)))))
   (+
    (*
     (* (cbrt g) (cbrt (/ 1.0 a)))
     (* (cbrt -0.5) (pow 2.0 0.3333333333333333)))
    (* (* (cbrt -0.5) (cbrt 0.5)) (/ (pow (cbrt h) 2.0) (cbrt (* g a)))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((h * h) <= 1e-188) {
		tmp = fma(cbrt(g), (cbrt(-1.0) / cbrt(a)), (cbrt(-0.25) * cbrt(((h / g) * (h / a)))));
	} else {
		tmp = ((cbrt(g) * cbrt((1.0 / a))) * (cbrt(-0.5) * pow(2.0, 0.3333333333333333))) + ((cbrt(-0.5) * cbrt(0.5)) * (pow(cbrt(h), 2.0) / cbrt((g * a))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(h * h) <= 1e-188)
		tmp = fma(cbrt(g), Float64(cbrt(-1.0) / cbrt(a)), Float64(cbrt(-0.25) * cbrt(Float64(Float64(h / g) * Float64(h / a)))));
	else
		tmp = Float64(Float64(Float64(cbrt(g) * cbrt(Float64(1.0 / a))) * Float64(cbrt(-0.5) * (2.0 ^ 0.3333333333333333))) + Float64(Float64(cbrt(-0.5) * cbrt(0.5)) * Float64((cbrt(h) ^ 2.0) / cbrt(Float64(g * a)))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(h * h), $MachinePrecision], 1e-188], N[(N[Power[g, 1/3], $MachinePrecision] * N[(N[Power[-1.0, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[-0.25, 1/3], $MachinePrecision] * N[Power[N[(N[(h / g), $MachinePrecision] * N[(h / a), $MachinePrecision]), $MachinePrecision], 1/3], $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[(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]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 9.9999999999999995e-189

   1. Initial program 49.3%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in h around 0 71.0%

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

      1. pow1/3 33.4%

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

      2. div-inv 33.4%

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

      3. unpow-prod-down 17.8%

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

      4. pow1/3 42.8%

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

   7. Applied egg-rr 42.8%

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

      1. unpow1/3 90.2%

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

   9. Simplified 90.2%

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

       1. associate-*l* 90.3%

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

       2. fma-define 90.3%

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

       3. cbrt-div 90.1%

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

       4. metadata-eval 90.1%

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

       5. cbrt-unprod 90.7%

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

       6. metadata-eval 90.7%

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

       7. *-commutative 90.7%

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

       8. cbrt-unprod 90.7%

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

       9. metadata-eval 90.7%

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

   11. Applied egg-rr 90.7%

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

       1. associate-*l/ 90.7%

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

       2. *-lft-identity 90.7%

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

       3. *-commutative 90.7%

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

       4. associate-/r* 97.6%

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

   13. Simplified 97.6%

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

   15. Applied egg-rr 98.1%

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

   if 9.9999999999999995e-189 < (*.f64 h h)

   1. Initial program 32.0%

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

   3. Simplified 32.0%

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

   5. Taylor expanded in h around 0 61.9%

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

      1. pow1/3 31.0%

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

      2. div-inv 31.0%

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

      3. unpow-prod-down 19.1%

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

      4. pow1/3 35.9%

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

   7. Applied egg-rr 35.9%

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

      1. unpow1/3 85.9%

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

   9. Simplified 85.9%

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

       1. cbrt-div 87.7%

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

       2. pow2 87.7%

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

       3. cbrt-prod 96.7%

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

       4. pow2 96.7%

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

   11. Applied egg-rr 96.7%

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

       1. pow1/3 97.4%

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

   13. Applied egg-rr 97.4%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 10^{-188}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, \frac{\sqrt[3]{-1}}{\sqrt[3]{a}}, \sqrt[3]{-0.25} \cdot \sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot {2}^{0.3333333333333333}\right) + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h)))) (t_1 (+ g t_0)))
   (if (<=
        (+
         (cbrt (* (/ 1.0 (* a 2.0)) (- t_0 g)))
         (cbrt (* t_1 (/ -1.0 (* a 2.0)))))
        (- INFINITY))
     (+ (* (cbrt (/ 1.0 a)) (cbrt (- g))) (cbrt (* t_1 (/ -0.5 a))))
     (+ (cbrt (* (/ 0.5 a) (* g -2.0))) (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 tmp;
	if ((cbrt(((1.0 / (a * 2.0)) * (t_0 - g))) + cbrt((t_1 * (-1.0 / (a * 2.0))))) <= -((double) INFINITY)) {
		tmp = (cbrt((1.0 / a)) * cbrt(-g)) + cbrt((t_1 * (-0.5 / a)));
	} else {
		tmp = cbrt(((0.5 / a) * (g * -2.0))) + 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 tmp;
	if ((Math.cbrt(((1.0 / (a * 2.0)) * (t_0 - g))) + Math.cbrt((t_1 * (-1.0 / (a * 2.0))))) <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.cbrt((1.0 / a)) * Math.cbrt(-g)) + Math.cbrt((t_1 * (-0.5 / a)));
	} else {
		tmp = Math.cbrt(((0.5 / a) * (g * -2.0))) + 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)
	tmp = 0.0
	if (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))))) <= Float64(-Inf))
		tmp = Float64(Float64(cbrt(Float64(1.0 / a)) * cbrt(Float64(-g))) + cbrt(Float64(t_1 * Float64(-0.5 / a))));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))) + 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]}, If[LessEqual[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], (-Infinity)], N[(N[(N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[(-g), 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(t$95$1 * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.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))))))) < -inf.0

   1. Initial program 4.2%

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

   3. Simplified 4.2%

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

   5. Taylor expanded in g around -inf 4.2%

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

      1. associate-*r/ 4.2%

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

      2. neg-mul-1 4.2%

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

   7. Simplified 4.2%

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

      1. pow1/3 0.2%

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

      2. div-inv 0.2%

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

      3. unpow-prod-down 0.0%

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

      4. pow1/3 1.9%

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

   9. Applied egg-rr 1.9%

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

       1. *-commutative 1.9%

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

       2. unpow1/3 55.8%

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

   11. Simplified 55.8%

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

   if -inf.0 < (+.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 44.2%

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

   3. Simplified 44.2%

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

      1. add-cbrt-cube 30.6%

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

      2. pow1/3 26.5%

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

      3. pow3 26.5%

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

      4. sqrt-pow2 26.5%

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

      5. pow2 26.5%

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

      6. pow2 26.5%

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

      7. metadata-eval 26.5%

         \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left({\left({\left({g}^{2} - {h}^{2}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

   6. Applied egg-rr 26.5%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\color{blue}{{\left({\left({g}^{2} - {h}^{2}\right)}^{1.5}\right)}^{0.3333333333333333}} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   7. Step-by-step derivation

      1. unpow1/3 31.3%

         \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\color{blue}{\sqrt[3]{{\left({g}^{2} - {h}^{2}\right)}^{1.5}}} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

   8. Simplified 31.3%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\color{blue}{\sqrt[3]{{\left({g}^{2} - {h}^{2}\right)}^{1.5}}} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

   9. Taylor expanded in g around -inf 22.7%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt[3]{{\left({g}^{2} - {h}^{2}\right)}^{1.5}} - g\right)} + \sqrt[3]{\left(g + \color{blue}{-1 \cdot g}\right) \cdot \frac{-0.5}{a}} \]
   10. Step-by-step derivation

       1. neg-mul-1 22.7%

          \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt[3]{{\left({g}^{2} - {h}^{2}\right)}^{1.5}} - g\right)} + \sqrt[3]{\left(g + \color{blue}{\left(-g\right)}\right) \cdot \frac{-0.5}{a}} \]

   11. Simplified 22.7%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt[3]{{\left({g}^{2} - {h}^{2}\right)}^{1.5}} - g\right)} + \sqrt[3]{\left(g + \color{blue}{\left(-g\right)}\right) \cdot \frac{-0.5}{a}} \]

   12. Taylor expanded in g around -inf 76.6%

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

       1. *-commutative 76.6%

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

   14. Simplified 76.6%

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

4. Final simplification 75.7%

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

Alternative 4: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{1}{a}}\\ \mathbf{if}\;h \cdot h \leq 2 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, t\_0 \cdot \sqrt[3]{-1}, \sqrt[3]{-0.25} \cdot \sqrt[3]{\frac{\frac{{h}^{2}}{g}}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{g} \cdot t\_0\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{-0.25} \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (/ 1.0 a))))
   (if (<= (* h h) 2e+227)
     (fma
      (cbrt g)
      (* t_0 (cbrt -1.0))
      (* (cbrt -0.25) (cbrt (/ (/ (pow h 2.0) g) a))))
     (+
      (* (* (cbrt g) t_0) (* (cbrt -0.5) (cbrt 2.0)))
      (* (cbrt -0.25) (/ (pow (cbrt h) 2.0) (cbrt (* g a))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((1.0 / a));
	double tmp;
	if ((h * h) <= 2e+227) {
		tmp = fma(cbrt(g), (t_0 * cbrt(-1.0)), (cbrt(-0.25) * cbrt(((pow(h, 2.0) / g) / a))));
	} else {
		tmp = ((cbrt(g) * t_0) * (cbrt(-0.5) * cbrt(2.0))) + (cbrt(-0.25) * (pow(cbrt(h), 2.0) / cbrt((g * a))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(1.0 / a))
	tmp = 0.0
	if (Float64(h * h) <= 2e+227)
		tmp = fma(cbrt(g), Float64(t_0 * cbrt(-1.0)), Float64(cbrt(-0.25) * cbrt(Float64(Float64((h ^ 2.0) / g) / a))));
	else
		tmp = Float64(Float64(Float64(cbrt(g) * t_0) * Float64(cbrt(-0.5) * cbrt(2.0))) + Float64(cbrt(-0.25) * Float64((cbrt(h) ^ 2.0) / cbrt(Float64(g * a)))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(h * h), $MachinePrecision], 2e+227], N[(N[Power[g, 1/3], $MachinePrecision] * N[(t$95$0 * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[-0.25, 1/3], $MachinePrecision] * N[Power[N[(N[(N[Power[h, 2.0], $MachinePrecision] / g), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[g, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[-0.25, 1/3], $MachinePrecision] * N[(N[Power[N[Power[h, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(g * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 2.0000000000000002e227

   1. Initial program 46.1%

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

   3. Simplified 46.1%

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

   5. Taylor expanded in h around 0 70.9%

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

      1. pow1/3 34.3%

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

      2. div-inv 34.3%

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

      3. unpow-prod-down 19.1%

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

      4. pow1/3 42.3%

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

   7. Applied egg-rr 42.3%

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

      1. unpow1/3 92.6%

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

   9. Simplified 92.6%

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

       1. associate-*l* 92.6%

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

       2. fma-define 92.6%

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

       3. cbrt-div 92.4%

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

       4. metadata-eval 92.4%

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

       5. cbrt-unprod 93.0%

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

       6. metadata-eval 93.0%

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

       7. *-commutative 93.0%

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

       8. cbrt-unprod 93.0%

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

       9. metadata-eval 93.0%

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

   11. Applied egg-rr 93.0%

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

       1. associate-*l/ 93.0%

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

       2. *-lft-identity 93.0%

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

       3. *-commutative 93.0%

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

       4. associate-/r* 97.8%

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

   13. Simplified 97.8%

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

   14. Taylor expanded in a around 0 97.9%

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

   if 2.0000000000000002e227 < (*.f64 h h)

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in h around 0 26.4%

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

      1. pow1/3 10.5%

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

      2. div-inv 10.5%

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

      3. unpow-prod-down 9.6%

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

      4. pow1/3 14.1%

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

   7. Applied egg-rr 14.1%

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

      1. unpow1/3 40.3%

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

   9. Simplified 40.3%

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

       1. cbrt-div 49.7%

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

       2. pow2 49.7%

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

       3. cbrt-prod 94.9%

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

       4. pow2 94.9%

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

   11. Applied egg-rr 94.9%

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

       1. pow1 94.9%

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

       2. cbrt-unprod 94.9%

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

       3. metadata-eval 94.9%

          \[\leadsto \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{a \cdot g}} \cdot {\left(\sqrt[3]{\color{blue}{-0.25}}\right)}^{1} \]

   13. Applied egg-rr 94.9%

       \[\leadsto \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{a \cdot g}} \cdot \color{blue}{{\left(\sqrt[3]{-0.25}\right)}^{1}} \]
   14. Step-by-step derivation

       1. unpow1 94.9%

          \[\leadsto \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{a \cdot g}} \cdot \color{blue}{\sqrt[3]{-0.25}} \]

   15. Simplified 94.9%

       \[\leadsto \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{a \cdot g}} \cdot \color{blue}{\sqrt[3]{-0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 2 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{-1}, \sqrt[3]{-0.25} \cdot \sqrt[3]{\frac{\frac{{h}^{2}}{g}}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \sqrt[3]{-0.25} \cdot \frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g \cdot a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \cdot h \leq 4 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{-1}, \sqrt[3]{-0.25} \cdot \sqrt[3]{\frac{\frac{{h}^{2}}{g}}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, \frac{\sqrt[3]{-1}}{\sqrt[3]{a}}, \sqrt[3]{-0.25} \cdot \sqrt[3]{\frac{\frac{\frac{g}{h}}{h}}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (* h h) 4e+176)
   (fma
    (cbrt g)
    (* (cbrt (/ 1.0 a)) (cbrt -1.0))
    (* (cbrt -0.25) (cbrt (/ (/ (pow h 2.0) g) a))))
   (fma
    (cbrt g)
    (/ (cbrt -1.0) (cbrt a))
    (* (cbrt -0.25) (cbrt (/ (/ (/ g h) h) a))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((h * h) <= 4e+176) {
		tmp = fma(cbrt(g), (cbrt((1.0 / a)) * cbrt(-1.0)), (cbrt(-0.25) * cbrt(((pow(h, 2.0) / g) / a))));
	} else {
		tmp = fma(cbrt(g), (cbrt(-1.0) / cbrt(a)), (cbrt(-0.25) * cbrt((((g / h) / h) / a))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(h * h) <= 4e+176)
		tmp = fma(cbrt(g), Float64(cbrt(Float64(1.0 / a)) * cbrt(-1.0)), Float64(cbrt(-0.25) * cbrt(Float64(Float64((h ^ 2.0) / g) / a))));
	else
		tmp = fma(cbrt(g), Float64(cbrt(-1.0) / cbrt(a)), Float64(cbrt(-0.25) * cbrt(Float64(Float64(Float64(g / h) / h) / a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(h * h), $MachinePrecision], 4e+176], N[(N[Power[g, 1/3], $MachinePrecision] * N[(N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[-0.25, 1/3], $MachinePrecision] * N[Power[N[(N[(N[Power[h, 2.0], $MachinePrecision] / g), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[g, 1/3], $MachinePrecision] * N[(N[Power[-1.0, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[-0.25, 1/3], $MachinePrecision] * N[Power[N[(N[(N[(g / h), $MachinePrecision] / h), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 4e176

   1. Initial program 47.1%

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

   3. Simplified 47.1%

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

   5. Taylor expanded in h around 0 71.1%

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

      1. pow1/3 35.1%

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

      2. div-inv 35.1%

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

      3. unpow-prod-down 19.5%

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

      4. pow1/3 41.6%

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

   7. Applied egg-rr 41.6%

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

      1. unpow1/3 92.5%

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

   9. Simplified 92.5%

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

       1. associate-*l* 92.5%

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

       2. fma-define 92.5%

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

       3. cbrt-div 92.3%

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

       4. metadata-eval 92.3%

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

       5. cbrt-unprod 92.9%

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

       6. metadata-eval 92.9%

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

       7. *-commutative 92.9%

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

       8. cbrt-unprod 92.9%

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

       9. metadata-eval 92.9%

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

   11. Applied egg-rr 92.9%

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

       1. associate-*l/ 92.9%

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

       2. *-lft-identity 92.9%

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

       3. *-commutative 92.9%

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

       4. associate-/r* 97.8%

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

   13. Simplified 97.8%

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

   14. Taylor expanded in a around 0 97.9%

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

   if 4e176 < (*.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 33.2%

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

      1. pow1/3 8.4%

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

      2. div-inv 8.4%

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

      3. unpow-prod-down 7.6%

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

      4. pow1/3 25.8%

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

   7. Applied egg-rr 25.8%

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

      1. unpow1/3 51.7%

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

   9. Simplified 51.7%

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

       1. associate-*l* 51.7%

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

       2. fma-define 51.7%

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

       3. cbrt-div 51.9%

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

       4. metadata-eval 51.9%

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

       5. cbrt-unprod 52.3%

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

       6. metadata-eval 52.3%

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

       7. *-commutative 52.3%

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

       8. cbrt-unprod 52.3%

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

       9. metadata-eval 52.3%

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

   11. Applied egg-rr 52.3%

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

       1. associate-*l/ 52.3%

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

       2. *-lft-identity 52.3%

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

       3. *-commutative 52.3%

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

       4. associate-/r* 53.1%

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

   13. Simplified 53.1%

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

   15. Applied egg-rr 47.3%

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

       1. associate-*l/ 47.3%

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

       2. associate-*r/ 47.3%

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

       3. rem-square-sqrt 94.5%

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

   17. Simplified 94.5%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{g}, \frac{\sqrt[3]{-1}}{\sqrt[3]{a}}, \sqrt[3]{-0.25} \cdot \sqrt[3]{\frac{\color{blue}{\frac{\frac{g}{h}}{h}}}{a}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt[3]{-1}}{\sqrt[3]{a}}\\ \mathbf{if}\;h \cdot h \leq 4 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, t\_0, \sqrt[3]{-0.25} \cdot \sqrt[3]{\frac{\frac{{h}^{2}}{g}}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, t\_0, \sqrt[3]{-0.25} \cdot \sqrt[3]{\frac{\frac{\frac{g}{h}}{h}}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ (cbrt -1.0) (cbrt a))))
   (if (<= (* h h) 4e+176)
     (fma (cbrt g) t_0 (* (cbrt -0.25) (cbrt (/ (/ (pow h 2.0) g) a))))
     (fma (cbrt g) t_0 (* (cbrt -0.25) (cbrt (/ (/ (/ g h) h) a)))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(-1.0) / cbrt(a);
	double tmp;
	if ((h * h) <= 4e+176) {
		tmp = fma(cbrt(g), t_0, (cbrt(-0.25) * cbrt(((pow(h, 2.0) / g) / a))));
	} else {
		tmp = fma(cbrt(g), t_0, (cbrt(-0.25) * cbrt((((g / h) / h) / a))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(cbrt(-1.0) / cbrt(a))
	tmp = 0.0
	if (Float64(h * h) <= 4e+176)
		tmp = fma(cbrt(g), t_0, Float64(cbrt(-0.25) * cbrt(Float64(Float64((h ^ 2.0) / g) / a))));
	else
		tmp = fma(cbrt(g), t_0, Float64(cbrt(-0.25) * cbrt(Float64(Float64(Float64(g / h) / h) / a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[Power[-1.0, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h * h), $MachinePrecision], 4e+176], N[(N[Power[g, 1/3], $MachinePrecision] * t$95$0 + N[(N[Power[-0.25, 1/3], $MachinePrecision] * N[Power[N[(N[(N[Power[h, 2.0], $MachinePrecision] / g), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[g, 1/3], $MachinePrecision] * t$95$0 + N[(N[Power[-0.25, 1/3], $MachinePrecision] * N[Power[N[(N[(N[(g / h), $MachinePrecision] / h), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 4e176

   1. Initial program 47.1%

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

   3. Simplified 47.1%

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

   5. Taylor expanded in h around 0 71.1%

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

      1. pow1/3 35.1%

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

      2. div-inv 35.1%

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

      3. unpow-prod-down 19.5%

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

      4. pow1/3 41.6%

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

   7. Applied egg-rr 41.6%

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

      1. unpow1/3 92.5%

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

   9. Simplified 92.5%

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

       1. associate-*l* 92.5%

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

       2. fma-define 92.5%

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

       3. cbrt-div 92.3%

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

       4. metadata-eval 92.3%

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

       5. cbrt-unprod 92.9%

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

       6. metadata-eval 92.9%

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

       7. *-commutative 92.9%

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

       8. cbrt-unprod 92.9%

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

       9. metadata-eval 92.9%

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

   11. Applied egg-rr 92.9%

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

       1. associate-*l/ 92.9%

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

       2. *-lft-identity 92.9%

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

       3. *-commutative 92.9%

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

       4. associate-/r* 97.8%

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

   13. Simplified 97.8%

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

   if 4e176 < (*.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 33.2%

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

      1. pow1/3 8.4%

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

      2. div-inv 8.4%

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

      3. unpow-prod-down 7.6%

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

      4. pow1/3 25.8%

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

   7. Applied egg-rr 25.8%

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

      1. unpow1/3 51.7%

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

   9. Simplified 51.7%

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

       1. associate-*l* 51.7%

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

       2. fma-define 51.7%

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

       3. cbrt-div 51.9%

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

       4. metadata-eval 51.9%

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

       5. cbrt-unprod 52.3%

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

       6. metadata-eval 52.3%

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

       7. *-commutative 52.3%

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

       8. cbrt-unprod 52.3%

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

       9. metadata-eval 52.3%

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

   11. Applied egg-rr 52.3%

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

       1. associate-*l/ 52.3%

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

       2. *-lft-identity 52.3%

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

       3. *-commutative 52.3%

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

       4. associate-/r* 53.1%

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

   13. Simplified 53.1%

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

   15. Applied egg-rr 47.3%

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

       1. associate-*l/ 47.3%

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

       2. associate-*r/ 47.3%

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

       3. rem-square-sqrt 94.5%

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

   17. Simplified 94.5%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{g}, \frac{\sqrt[3]{-1}}{\sqrt[3]{a}}, \sqrt[3]{-0.25} \cdot \sqrt[3]{\frac{\color{blue}{\frac{\frac{g}{h}}{h}}}{a}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt[3]{-1}}{\sqrt[3]{a}}\\ \mathbf{if}\;h \cdot h \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, t\_0, \sqrt[3]{-0.25} \cdot \sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, t\_0, \sqrt[3]{-0.25} \cdot \sqrt[3]{\frac{\frac{\frac{g}{h}}{h}}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ (cbrt -1.0) (cbrt a))))
   (if (<= (* h h) 2e+41)
     (fma (cbrt g) t_0 (* (cbrt -0.25) (cbrt (* (/ h g) (/ h a)))))
     (fma (cbrt g) t_0 (* (cbrt -0.25) (cbrt (/ (/ (/ g h) h) a)))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(-1.0) / cbrt(a);
	double tmp;
	if ((h * h) <= 2e+41) {
		tmp = fma(cbrt(g), t_0, (cbrt(-0.25) * cbrt(((h / g) * (h / a)))));
	} else {
		tmp = fma(cbrt(g), t_0, (cbrt(-0.25) * cbrt((((g / h) / h) / a))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(cbrt(-1.0) / cbrt(a))
	tmp = 0.0
	if (Float64(h * h) <= 2e+41)
		tmp = fma(cbrt(g), t_0, Float64(cbrt(-0.25) * cbrt(Float64(Float64(h / g) * Float64(h / a)))));
	else
		tmp = fma(cbrt(g), t_0, Float64(cbrt(-0.25) * cbrt(Float64(Float64(Float64(g / h) / h) / a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[Power[-1.0, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h * h), $MachinePrecision], 2e+41], N[(N[Power[g, 1/3], $MachinePrecision] * t$95$0 + N[(N[Power[-0.25, 1/3], $MachinePrecision] * N[Power[N[(N[(h / g), $MachinePrecision] * N[(h / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[g, 1/3], $MachinePrecision] * t$95$0 + N[(N[Power[-0.25, 1/3], $MachinePrecision] * N[Power[N[(N[(N[(g / h), $MachinePrecision] / h), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 2.00000000000000001e41

   1. Initial program 47.8%

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

   3. Simplified 47.8%

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

   5. Taylor expanded in h around 0 71.8%

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

      1. pow1/3 34.2%

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

      2. div-inv 34.2%

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

      3. unpow-prod-down 19.2%

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

      4. pow1/3 42.4%

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

   7. Applied egg-rr 42.4%

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

      1. unpow1/3 92.1%

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

   9. Simplified 92.1%

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

       1. associate-*l* 92.1%

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

       2. fma-define 92.1%

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

       3. cbrt-div 91.9%

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

       4. metadata-eval 91.9%

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

       5. cbrt-unprod 92.5%

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

       6. metadata-eval 92.5%

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

       7. *-commutative 92.5%

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

       8. cbrt-unprod 92.5%

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

       9. metadata-eval 92.5%

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

   11. Applied egg-rr 92.5%

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

       1. associate-*l/ 92.5%

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

       2. *-lft-identity 92.5%

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

       3. *-commutative 92.5%

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

       4. associate-/r* 97.8%

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

   13. Simplified 97.8%

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

   15. Applied egg-rr 98.1%

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

   if 2.00000000000000001e41 < (*.f64 h h)

   1. Initial program 20.4%

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

   3. Simplified 20.4%

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

   5. Taylor expanded in h around 0 49.4%

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

      1. pow1/3 25.2%

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

      2. div-inv 25.2%

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

      3. unpow-prod-down 14.8%

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

      4. pow1/3 30.6%

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

   7. Applied egg-rr 30.6%

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

      1. unpow1/3 73.8%

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

   9. Simplified 73.8%

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

       1. associate-*l* 73.6%

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

       2. fma-define 73.6%

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

       3. cbrt-div 73.6%

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

       4. metadata-eval 73.6%

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

       5. cbrt-unprod 74.2%

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

       6. metadata-eval 74.2%

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

       7. *-commutative 74.2%

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

       8. cbrt-unprod 74.2%

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

       9. metadata-eval 74.2%

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

   11. Applied egg-rr 74.2%

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

       1. associate-*l/ 74.2%

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

       2. *-lft-identity 74.2%

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

       3. *-commutative 74.2%

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

       4. associate-/r* 75.6%

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

   13. Simplified 75.6%

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

   15. Applied egg-rr 42.3%

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

       1. associate-*l/ 42.3%

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

       2. associate-*r/ 42.3%

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

       3. rem-square-sqrt 94.1%

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

   17. Simplified 94.1%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{g}, \frac{\sqrt[3]{-1}}{\sqrt[3]{a}}, \sqrt[3]{-0.25} \cdot \sqrt[3]{\frac{\color{blue}{\frac{\frac{g}{h}}{h}}}{a}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 93.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt[3]{g}, \frac{\sqrt[3]{-1}}{\sqrt[3]{a}}, \sqrt[3]{-0.25} \cdot \sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}}\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (fma
  (cbrt g)
  (/ (cbrt -1.0) (cbrt a))
  (* (cbrt -0.25) (cbrt (* (/ h g) (/ h a))))))

C

double code(double g, double h, double a) {
	return fma(cbrt(g), (cbrt(-1.0) / cbrt(a)), (cbrt(-0.25) * cbrt(((h / g) * (h / a)))));
}

Julia

function code(g, h, a)
	return fma(cbrt(g), Float64(cbrt(-1.0) / cbrt(a)), Float64(cbrt(-0.25) * cbrt(Float64(Float64(h / g) * Float64(h / a)))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[g, 1/3], $MachinePrecision] * N[(N[Power[-1.0, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[-0.25, 1/3], $MachinePrecision] * N[Power[N[(N[(h / g), $MachinePrecision] * N[(h / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.5%

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

3. Simplified 42.5%

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

5. Taylor expanded in h around 0 67.4%

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

   1. pow1/3 32.5%

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

   2. div-inv 32.5%

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

   3. unpow-prod-down 18.3%

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

   4. pow1/3 40.1%

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

7. Applied egg-rr 40.1%

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

   1. unpow1/3 88.5%

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

9. Simplified 88.5%

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

    1. associate-*l* 88.5%

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

    2. fma-define 88.5%

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

    3. cbrt-div 88.3%

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

    4. metadata-eval 88.3%

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

    5. cbrt-unprod 89.0%

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

    6. metadata-eval 89.0%

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

    7. *-commutative 89.0%

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

    8. cbrt-unprod 89.0%

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

    9. metadata-eval 89.0%

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

11. Applied egg-rr 89.0%

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

    1. associate-*l/ 89.0%

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

    2. *-lft-identity 89.0%

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

    3. *-commutative 89.0%

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

    4. associate-/r* 93.4%

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

13. Simplified 93.4%

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

15. Applied egg-rr 94.8%

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

Alternative 9: 73.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $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 42.5%

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

3. Simplified 42.5%

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

   1. add-cbrt-cube 29.4%

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

   2. pow1/3 25.5%

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

   3. pow3 25.5%

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

   4. sqrt-pow2 25.5%

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

   5. pow2 25.5%

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

   6. pow2 25.5%

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

   7. metadata-eval 25.5%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left({\left({\left({g}^{2} - {h}^{2}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

6. Applied egg-rr 25.5%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\color{blue}{{\left({\left({g}^{2} - {h}^{2}\right)}^{1.5}\right)}^{0.3333333333333333}} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
7. Step-by-step derivation

   1. unpow1/3 30.0%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\color{blue}{\sqrt[3]{{\left({g}^{2} - {h}^{2}\right)}^{1.5}}} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

8. Simplified 30.0%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\color{blue}{\sqrt[3]{{\left({g}^{2} - {h}^{2}\right)}^{1.5}}} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

9. Taylor expanded in g around -inf 21.8%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt[3]{{\left({g}^{2} - {h}^{2}\right)}^{1.5}} - g\right)} + \sqrt[3]{\left(g + \color{blue}{-1 \cdot g}\right) \cdot \frac{-0.5}{a}} \]
10. Step-by-step derivation

    1. neg-mul-1 21.8%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt[3]{{\left({g}^{2} - {h}^{2}\right)}^{1.5}} - g\right)} + \sqrt[3]{\left(g + \color{blue}{\left(-g\right)}\right) \cdot \frac{-0.5}{a}} \]

11. Simplified 21.8%

    \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt[3]{{\left({g}^{2} - {h}^{2}\right)}^{1.5}} - g\right)} + \sqrt[3]{\left(g + \color{blue}{\left(-g\right)}\right) \cdot \frac{-0.5}{a}} \]

12. Taylor expanded in g around -inf 73.5%

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

    1. *-commutative 73.5%

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

14. Simplified 73.5%

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

15. Final simplification 73.5%

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

Alternative 10: 73.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.5%

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

3. Simplified 42.5%

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

5. Taylor expanded in h around 0 67.4%

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

   1. pow1/3 32.5%

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

   2. div-inv 32.5%

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

   3. unpow-prod-down 18.3%

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

   4. pow1/3 40.1%

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

7. Applied egg-rr 40.1%

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

   1. unpow1/3 88.5%

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

9. Simplified 88.5%

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

    1. associate-*l* 88.5%

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

    2. fma-define 88.5%

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

    3. cbrt-div 88.3%

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

    4. metadata-eval 88.3%

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

    5. cbrt-unprod 89.0%

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

    6. metadata-eval 89.0%

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

    7. *-commutative 89.0%

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

    8. cbrt-unprod 89.0%

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

    9. metadata-eval 89.0%

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

11. Applied egg-rr 89.0%

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

    1. associate-*l/ 89.0%

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

    2. *-lft-identity 89.0%

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

    3. *-commutative 89.0%

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

    4. associate-/r* 93.4%

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

13. Simplified 93.4%

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

14. Taylor expanded in g around inf 73.5%

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

15. Final simplification 73.5%

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

Alternative 11: 15.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.5%

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

3. Simplified 42.5%

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

5. Taylor expanded in g around -inf 30.0%

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

   1. associate-*r/ 30.0%

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

   2. neg-mul-1 30.0%

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

7. Simplified 30.0%

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

8. 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 15.0%

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

    2. cbrt-div 15.4%

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

    3. metadata-eval 15.4%

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

13. Applied egg-rr 15.4%

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

14. Final simplification 15.4%

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

Alternative 12: 15.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.5%

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

3. Simplified 42.5%

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

5. Taylor expanded in g around -inf 30.0%

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

   1. associate-*r/ 30.0%

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

   2. neg-mul-1 30.0%

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

7. Simplified 30.0%

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

8. 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. Final simplification 15.2%

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

Alternative 13: 5.8% accurate, 4.1× speedup

Math

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

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 42.5%

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

3. Simplified 42.5%

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

5. Taylor expanded in g around -inf 30.0%

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

   1. associate-*r/ 30.0%

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

   2. neg-mul-1 30.0%

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

7. Simplified 30.0%

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

8. 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%

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

13. Applied egg-rr 5.7%

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

    1. rem-cube-cbrt 5.7%

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

    2. *-commutative 5.7%

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

15. Simplified 5.7%

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

16. Final simplification 5.7%

    \[\leadsto \sqrt[3]{g \cdot a} \cdot -2 \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024121 
(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))))))))