2-ancestry mixing, positive discriminant

Percentage Accurate: 44.8% → 97.4%

Time: 28.3s

Alternatives: 8

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{{\left(\sqrt[3]{h}\right)}^{2}}{\sqrt[3]{g}} \cdot \sqrt[3]{-0.25}}{\sqrt[3]{a}} + \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (/ (* (/ (pow (cbrt h) 2.0) (cbrt g)) (cbrt -0.25)) (cbrt a))
  (* (* (cbrt g) (cbrt (/ 1.0 a))) (* (cbrt -0.5) (cbrt 2.0)))))

C

double code(double g, double h, double a) {
	return (((pow(cbrt(h), 2.0) / cbrt(g)) * cbrt(-0.25)) / cbrt(a)) + ((cbrt(g) * cbrt((1.0 / a))) * (cbrt(-0.5) * cbrt(2.0)));
}

Java

public static double code(double g, double h, double a) {
	return (((Math.pow(Math.cbrt(h), 2.0) / Math.cbrt(g)) * Math.cbrt(-0.25)) / Math.cbrt(a)) + ((Math.cbrt(g) * Math.cbrt((1.0 / a))) * (Math.cbrt(-0.5) * Math.cbrt(2.0)));
}

Julia

function code(g, h, a)
	return Float64(Float64(Float64(Float64((cbrt(h) ^ 2.0) / cbrt(g)) * cbrt(-0.25)) / cbrt(a)) + Float64(Float64(cbrt(g) * cbrt(Float64(1.0 / a))) * Float64(cbrt(-0.5) * cbrt(2.0))))
end

Wolfram

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

Derivation

1. Initial program 45.6%

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

3. Simplified 45.6%

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

5. Taylor expanded in g around inf 27.6%

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

6. Taylor expanded in g around inf 73.3%

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

   1. associate-*r/ 73.3%

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

8. Simplified 73.3%

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

   1. pow1/3 37.9%

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

   2. div-inv 37.8%

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

   3. unpow-prod-down 24.1%

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

   4. pow1/3 39.1%

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

10. Applied egg-rr 39.1%

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

    1. unpow1/3 88.3%

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

12. Simplified 88.3%

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

    1. frac-times 83.9%

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

    2. *-commutative 83.9%

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

    3. cbrt-div 84.3%

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

    4. associate-*r* 84.3%

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

    5. metadata-eval 84.3%

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

    6. *-commutative 84.3%

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

    7. cbrt-prod 84.3%

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

    8. cbrt-prod 88.7%

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

    9. times-frac 88.7%

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

    10. unpow2 88.7%

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

    11. cbrt-prod 97.0%

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

    12. pow2 97.0%

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

14. Applied egg-rr 97.0%

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

    1. associate-*r/ 97.0%

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

16. Simplified 97.0%

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

Alternative 2: 94.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\\ \mathbf{if}\;h \leq 2.8 \cdot 10^{+100}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \frac{-0.5 \cdot {h}^{2}}{g}} + t\_0 \cdot \left(\sqrt[3]{-0.5} \cdot {2}^{0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \frac{{\left(\sqrt[3]{h}\right)}^{2} \cdot \sqrt[3]{-0.25}}{\sqrt[3]{g \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (cbrt g) (cbrt (/ 1.0 a)))))
   (if (<= h 2.8e+100)
     (+
      (cbrt (* (/ 0.5 a) (/ (* -0.5 (pow h 2.0)) g)))
      (* t_0 (* (cbrt -0.5) (pow 2.0 0.3333333333333333))))
     (+
      (* t_0 (* (cbrt -0.5) (cbrt 2.0)))
      (/ (* (pow (cbrt h) 2.0) (cbrt -0.25)) (cbrt (* g a)))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(g) * cbrt((1.0 / a));
	double tmp;
	if (h <= 2.8e+100) {
		tmp = cbrt(((0.5 / a) * ((-0.5 * pow(h, 2.0)) / g))) + (t_0 * (cbrt(-0.5) * pow(2.0, 0.3333333333333333)));
	} else {
		tmp = (t_0 * (cbrt(-0.5) * cbrt(2.0))) + ((pow(cbrt(h), 2.0) * cbrt(-0.25)) / cbrt((g * a)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(g) * Math.cbrt((1.0 / a));
	double tmp;
	if (h <= 2.8e+100) {
		tmp = Math.cbrt(((0.5 / a) * ((-0.5 * Math.pow(h, 2.0)) / g))) + (t_0 * (Math.cbrt(-0.5) * Math.pow(2.0, 0.3333333333333333)));
	} else {
		tmp = (t_0 * (Math.cbrt(-0.5) * Math.cbrt(2.0))) + ((Math.pow(Math.cbrt(h), 2.0) * Math.cbrt(-0.25)) / Math.cbrt((g * a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(cbrt(g) * cbrt(Float64(1.0 / a)))
	tmp = 0.0
	if (h <= 2.8e+100)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(Float64(-0.5 * (h ^ 2.0)) / g))) + Float64(t_0 * Float64(cbrt(-0.5) * (2.0 ^ 0.3333333333333333))));
	else
		tmp = Float64(Float64(t_0 * Float64(cbrt(-0.5) * cbrt(2.0))) + Float64(Float64((cbrt(h) ^ 2.0) * cbrt(-0.25)) / cbrt(Float64(g * a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, 2.8e+100], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(N[(-0.5 * N[Power[h, 2.0], $MachinePrecision]), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(t$95$0 * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * 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[Power[-0.25, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[(g * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < 2.7999999999999998e100

   1. Initial program 49.0%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in g around inf 29.7%

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

   6. Taylor expanded in g around inf 77.7%

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

      1. associate-*r/ 77.7%

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

   8. Simplified 77.7%

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

      1. pow1/3 40.0%

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

      2. div-inv 40.0%

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

      3. unpow-prod-down 25.6%

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

      4. pow1/3 41.0%

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

   10. Applied egg-rr 41.0%

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

       1. unpow1/3 93.1%

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

   12. Simplified 93.1%

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

       1. pow1/3 93.6%

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

   14. Applied egg-rr 93.6%

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

   if 2.7999999999999998e100 < h

   1. Initial program 3.9%

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

   3. Simplified 3.9%

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

   5. Taylor expanded in g around inf 1.6%

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

   6. Taylor expanded in g around inf 18.6%

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

      1. associate-*r/ 18.6%

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

   8. Simplified 18.6%

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

      1. pow1/3 11.1%

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

      2. div-inv 11.1%

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

      3. unpow-prod-down 5.2%

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

      4. pow1/3 15.8%

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

   10. Applied egg-rr 15.8%

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

       1. unpow1/3 28.7%

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

   12. Simplified 28.7%

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

       1. frac-times 27.3%

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

       2. *-commutative 27.3%

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

       3. cbrt-div 27.3%

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

       4. associate-*r* 27.3%

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

       5. metadata-eval 27.3%

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

       6. *-commutative 27.3%

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

       7. cbrt-prod 27.3%

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

       8. associate-/l* 27.3%

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

       9. unpow2 27.3%

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

       10. cbrt-prod 89.3%

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

       11. pow2 89.3%

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

       12. *-commutative 89.3%

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

   14. Applied egg-rr 89.3%

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

       1. associate-*r/ 89.3%

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

       2. *-commutative 89.3%

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

   16. Simplified 89.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 2.8 \cdot 10^{+100}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \frac{-0.5 \cdot {h}^{2}}{g}} + \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot {2}^{0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) + \frac{{\left(\sqrt[3]{h}\right)}^{2} \cdot \sqrt[3]{-0.25}}{\sqrt[3]{g \cdot a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \frac{-0.5 \cdot {h}^{2}}{g}} + \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot {2}^{0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} - \sqrt[3]{\frac{g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= h 1.35e+154)
   (+
    (cbrt (* (/ 0.5 a) (/ (* -0.5 (pow h 2.0)) g)))
    (*
     (* (cbrt g) (cbrt (/ 1.0 a)))
     (* (cbrt -0.5) (pow 2.0 0.3333333333333333))))
   (- (cbrt (* (/ 0.5 a) (- g g))) (cbrt (/ g a)))))

C

double code(double g, double h, double a) {
	double tmp;
	if (h <= 1.35e+154) {
		tmp = cbrt(((0.5 / a) * ((-0.5 * pow(h, 2.0)) / g))) + ((cbrt(g) * cbrt((1.0 / a))) * (cbrt(-0.5) * pow(2.0, 0.3333333333333333)));
	} else {
		tmp = cbrt(((0.5 / a) * (g - g))) - cbrt((g / a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (h <= 1.35e+154) {
		tmp = Math.cbrt(((0.5 / a) * ((-0.5 * Math.pow(h, 2.0)) / g))) + ((Math.cbrt(g) * Math.cbrt((1.0 / a))) * (Math.cbrt(-0.5) * Math.pow(2.0, 0.3333333333333333)));
	} else {
		tmp = Math.cbrt(((0.5 / a) * (g - g))) - Math.cbrt((g / a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (h <= 1.35e+154)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(Float64(-0.5 * (h ^ 2.0)) / g))) + Float64(Float64(cbrt(g) * cbrt(Float64(1.0 / a))) * Float64(cbrt(-0.5) * (2.0 ^ 0.3333333333333333))));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) - cbrt(Float64(g / a)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if h < 1.35000000000000003e154

   1. Initial program 48.3%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in g around inf 29.2%

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

   6. Taylor expanded in g around inf 77.3%

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

      1. associate-*r/ 77.3%

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

   8. Simplified 77.3%

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

      1. pow1/3 39.9%

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

      2. div-inv 39.9%

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

      3. unpow-prod-down 25.5%

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

      4. pow1/3 41.3%

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

   10. Applied egg-rr 41.3%

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

       1. unpow1/3 93.2%

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

   12. Simplified 93.2%

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

       1. pow1/3 93.7%

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

   14. Applied egg-rr 93.7%

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

   if 1.35000000000000003e154 < h

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in g around inf 0.0%

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

   6. Taylor expanded in g around inf 63.1%

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

   7. Taylor expanded in g around 0 63.1%

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

   9. Simplified 63.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \frac{-0.5 \cdot {h}^{2}}{g}} + \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot {2}^{0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} - \sqrt[3]{\frac{g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if h < 1.35000000000000003e154

   1. Initial program 48.3%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in g around inf 29.2%

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

   6. Taylor expanded in g around inf 77.3%

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

      1. associate-*r/ 77.3%

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

   8. Simplified 77.3%

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

      1. pow1/3 39.9%

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

      2. div-inv 39.9%

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

      3. unpow-prod-down 25.5%

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

      4. pow1/3 41.3%

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

   10. Applied egg-rr 41.3%

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

       1. unpow1/3 93.2%

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

   12. Simplified 93.2%

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

   if 1.35000000000000003e154 < h

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in g around inf 0.0%

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

   6. Taylor expanded in g around inf 63.1%

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

   7. Taylor expanded in g around 0 63.1%

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

   9. Simplified 63.1%

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

4. Final simplification 91.6%

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

Alternative 5: 92.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if h < 1.35000000000000003e154

   1. Initial program 48.3%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in g around inf 29.2%

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

   6. Taylor expanded in g around inf 77.3%

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

      1. associate-*r/ 77.3%

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

   8. Simplified 77.3%

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

      1. cbrt-div 92.9%

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

      2. div-inv 92.9%

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

   10. Applied egg-rr 92.9%

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

       1. associate-*r/ 92.9%

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

       2. *-rgt-identity 92.9%

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

   12. Simplified 92.9%

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

   if 1.35000000000000003e154 < h

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in g around inf 0.0%

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

   6. Taylor expanded in g around inf 63.1%

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

   7. Taylor expanded in g around 0 63.1%

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

   9. Simplified 63.1%

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

4. Final simplification 91.3%

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

Alternative 6: 73.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.6%

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

3. Simplified 45.6%

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

5. Taylor expanded in g around inf 24.3%

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

6. Taylor expanded in g around inf 78.5%

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

7. Taylor expanded in g around 0 78.5%

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

9. Simplified 78.5%

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

10. Final simplification 78.5%

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

Alternative 7: 4.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.6%

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

3. Simplified 45.6%

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

5. Taylor expanded in g around inf 24.3%

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

6. Taylor expanded in g around inf 78.5%

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

   1. associate-*r/ 78.5%

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

   2. cbrt-div 95.0%

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

   3. flip-+ 0.0%

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

   4. unpow2 0.0%

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

   5. unpow2 0.0%

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

   6. +-inverses 0.0%

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

   7. +-inverses 0.0%

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

8. Applied egg-rr 0.0%

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

10. Simplified 4.8%

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

Alternative 8: 4.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.6%

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

3. Simplified 45.6%

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

5. Taylor expanded in g around inf 24.3%

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

6. Taylor expanded in g around inf 78.5%

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

   1. expm1-log1p-u 53.5%

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

   2. expm1-undefine 28.5%

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

   3. flip-+ 0.0%

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

   4. frac-times 0.0%

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

   5. unpow2 0.0%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\left(\color{blue}{{g}^{2}} - g \cdot g\right) \cdot -0.5}{\left(g - g\right) \cdot a}}\right)} - 1\right) \]

   6. unpow2 0.0%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\left({g}^{2} - \color{blue}{{g}^{2}}\right) \cdot -0.5}{\left(g - g\right) \cdot a}}\right)} - 1\right) \]

   7. +-inverses 0.0%

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

   8. metadata-eval 0.0%

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

   9. +-inverses 0.0%

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

8. Applied egg-rr 0.0%

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

10. Simplified 4.7%

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

Reproduce

herbie shell --seed 2024108 
(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))))))))