2-ancestry mixing, positive discriminant

Percentage Accurate: 44.1% → 96.4%

Time: 17.9s

Alternatives: 5

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 96.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \cdot h \leq 10^{+295}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g}}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (* h h) 1e+295)
   (+
    (cbrt (* (/ 0.5 a) (* -0.5 (/ (pow h 2.0) g))))
    (/ (cbrt (* -0.5 (+ g g))) (cbrt a)))
   (+
    (cbrt (* (/ 0.5 a) (- g g)))
    (* (/ 1.0 (/ (cbrt a) (cbrt g))) (* (cbrt -0.5) (cbrt 2.0))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((h * h) <= 1e+295) {
		tmp = cbrt(((0.5 / a) * (-0.5 * (pow(h, 2.0) / g)))) + (cbrt((-0.5 * (g + g))) / cbrt(a));
	} else {
		tmp = cbrt(((0.5 / a) * (g - g))) + ((1.0 / (cbrt(a) / cbrt(g))) * (cbrt(-0.5) * cbrt(2.0)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((h * h) <= 1e+295) {
		tmp = Math.cbrt(((0.5 / a) * (-0.5 * (Math.pow(h, 2.0) / g)))) + (Math.cbrt((-0.5 * (g + g))) / Math.cbrt(a));
	} else {
		tmp = Math.cbrt(((0.5 / a) * (g - g))) + ((1.0 / (Math.cbrt(a) / Math.cbrt(g))) * (Math.cbrt(-0.5) * Math.cbrt(2.0)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(h * h) <= 1e+295)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * Float64((h ^ 2.0) / g)))) + Float64(cbrt(Float64(-0.5 * Float64(g + g))) / cbrt(a)));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + Float64(Float64(1.0 / Float64(cbrt(a) / cbrt(g))) * Float64(cbrt(-0.5) * cbrt(2.0))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(h * h), $MachinePrecision], 1e+295], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * N[(N[Power[h, 2.0], $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[(1.0 / N[(N[Power[a, 1/3], $MachinePrecision] / N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 9.9999999999999998e294

   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. Step-by-step derivation

      1. associate-*r/ 48.3%

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

      2. cbrt-div 53.1%

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

      3. pow2 53.1%

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

      4. pow2 53.1%

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

   5. Applied egg-rr 53.1%

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

   6. Taylor expanded in g around inf 33.3%

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

   7. Taylor expanded in g around inf 97.4%

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

   if 9.9999999999999998e294 < (*.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. Taylor expanded in g around inf 0.0%

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

   5. Taylor expanded in h around 0 29.1%

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

      1. unpow1/3 45.8%

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

      2. *-lft-identity 45.8%

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

   7. Simplified 45.8%

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

      1. cbrt-div 82.8%

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

      2. clear-num 82.8%

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

   9. Applied egg-rr 82.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 10^{+295}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g}}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)\\ \end{array} \]

Alternative 2: 93.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 1.1 \cdot 10^{+150}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= h 1.1e+150)
   (+
    (cbrt (* (/ 0.5 a) (* -0.5 (/ (pow h 2.0) g))))
    (/ (cbrt (* -0.5 (+ g g))) (cbrt a)))
   (+
    (cbrt (* (/ 0.5 a) (- g g)))
    (* (* (cbrt -0.5) (cbrt 2.0)) (/ (cbrt g) (cbrt a))))))

C

double code(double g, double h, double a) {
	double tmp;
	if (h <= 1.1e+150) {
		tmp = cbrt(((0.5 / a) * (-0.5 * (pow(h, 2.0) / g)))) + (cbrt((-0.5 * (g + g))) / cbrt(a));
	} else {
		tmp = cbrt(((0.5 / a) * (g - g))) + ((cbrt(-0.5) * cbrt(2.0)) * (cbrt(g) / cbrt(a)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (h <= 1.1e+150) {
		tmp = Math.cbrt(((0.5 / a) * (-0.5 * (Math.pow(h, 2.0) / g)))) + (Math.cbrt((-0.5 * (g + g))) / Math.cbrt(a));
	} else {
		tmp = Math.cbrt(((0.5 / a) * (g - g))) + ((Math.cbrt(-0.5) * Math.cbrt(2.0)) * (Math.cbrt(g) / Math.cbrt(a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (h <= 1.1e+150)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * Float64((h ^ 2.0) / g)))) + Float64(cbrt(Float64(-0.5 * Float64(g + g))) / cbrt(a)));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + Float64(Float64(cbrt(-0.5) * cbrt(2.0)) * Float64(cbrt(g) / cbrt(a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[h, 1.1e+150], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * N[(N[Power[h, 2.0], $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - 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]]

Derivation

1. Split input into 2 regimes

2. if h < 1.1e150

   1. Initial program 46.9%

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

   3. Simplified 46.9%

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

      1. associate-*r/ 46.9%

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

      2. cbrt-div 51.6%

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

      3. pow2 51.6%

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

      4. pow2 51.6%

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

   5. Applied egg-rr 51.6%

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

   6. Taylor expanded in g around inf 32.4%

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

   7. Taylor expanded in g around inf 94.7%

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

   if 1.1e150 < 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. Taylor expanded in g around inf 0.0%

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

   5. Taylor expanded in h around 0 39.5%

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

      1. unpow1/3 50.9%

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

      2. *-lft-identity 50.9%

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

   7. Simplified 50.9%

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

      1. cbrt-div 84.2%

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

      2. div-inv 84.3%

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

   9. Applied egg-rr 84.3%

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

       1. associate-*r/ 84.2%

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

       2. *-rgt-identity 84.2%

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

   11. Simplified 84.2%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 1.1 \cdot 10^{+150}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{g}}{\sqrt[3]{a}}\\ \end{array} \]

Alternative 3: 92.8% accurate, 1.0× speedup

Math

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

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 (+ g g))) (cbrt a)))
   (+
    (cbrt (* (/ 0.5 a) (- g g)))
    (* (* (cbrt -0.5) (cbrt 2.0)) (/ 1.0 (cbrt (/ a g)))))))

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 * (g + g))) / cbrt(a));
	} else {
		tmp = cbrt(((0.5 / a) * (g - g))) + ((cbrt(-0.5) * cbrt(2.0)) * (1.0 / cbrt((a / g))));
	}
	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 * (g + g))) / Math.cbrt(a));
	} else {
		tmp = Math.cbrt(((0.5 / a) * (g - g))) + ((Math.cbrt(-0.5) * Math.cbrt(2.0)) * (1.0 / Math.cbrt((a / g))));
	}
	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(-0.5 * Float64((h ^ 2.0) / g)))) + Float64(cbrt(Float64(-0.5 * Float64(g + g))) / cbrt(a)));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + Float64(Float64(cbrt(-0.5) * cbrt(2.0)) * Float64(1.0 / cbrt(Float64(a / g)))));
	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[(-0.5 * N[(N[Power[h, 2.0], $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Power[N[(a / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 1.35000000000000003e154

   1. Initial program 46.9%

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

   3. Simplified 46.9%

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

      1. associate-*r/ 46.9%

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

      2. cbrt-div 51.6%

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

      3. pow2 51.6%

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

      4. pow2 51.6%

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

   5. Applied egg-rr 51.6%

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

   6. Taylor expanded in g around inf 32.4%

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

   7. Taylor expanded in g around inf 94.7%

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

   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. Taylor expanded in g around inf 0.0%

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

   5. Taylor expanded in h around 0 39.5%

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

      1. unpow1/3 50.9%

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

      2. *-lft-identity 50.9%

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

   7. Simplified 50.9%

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

      1. clear-num 50.9%

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

      2. cbrt-div 67.3%

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

      3. metadata-eval 67.3%

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

   9. Applied egg-rr 67.3%

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

4. Final simplification 93.7%

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

Alternative 4: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 7.4 \cdot 10^{+146}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\frac{0.5 \cdot {h}^{2}}{g} \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{1}{\sqrt[3]{\frac{a}{g}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= h 7.4e+146)
   (+
    (cbrt (* (/ 0.5 a) (* g -2.0)))
    (cbrt (* (/ (* 0.5 (pow h 2.0)) g) (/ -0.5 a))))
   (+
    (cbrt (* (/ 0.5 a) (- g g)))
    (* (* (cbrt -0.5) (cbrt 2.0)) (/ 1.0 (cbrt (/ a g)))))))

C

double code(double g, double h, double a) {
	double tmp;
	if (h <= 7.4e+146) {
		tmp = cbrt(((0.5 / a) * (g * -2.0))) + cbrt((((0.5 * pow(h, 2.0)) / g) * (-0.5 / a)));
	} else {
		tmp = cbrt(((0.5 / a) * (g - g))) + ((cbrt(-0.5) * cbrt(2.0)) * (1.0 / cbrt((a / g))));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (h <= 7.4e+146) {
		tmp = Math.cbrt(((0.5 / a) * (g * -2.0))) + Math.cbrt((((0.5 * Math.pow(h, 2.0)) / g) * (-0.5 / a)));
	} else {
		tmp = Math.cbrt(((0.5 / a) * (g - g))) + ((Math.cbrt(-0.5) * Math.cbrt(2.0)) * (1.0 / Math.cbrt((a / g))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (h <= 7.4e+146)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))) + cbrt(Float64(Float64(Float64(0.5 * (h ^ 2.0)) / g) * Float64(-0.5 / a))));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + Float64(Float64(cbrt(-0.5) * cbrt(2.0)) * Float64(1.0 / cbrt(Float64(a / g)))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[h, 7.4e+146], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(N[(0.5 * N[Power[h, 2.0], $MachinePrecision]), $MachinePrecision] / g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Power[N[(a / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 7.40000000000000009e146

   1. Initial program 47.3%

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

   3. Simplified 47.3%

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

   4. Taylor expanded in g around -inf 26.1%

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

      1. associate-*r/ 26.1%

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

   6. Simplified 26.1%

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

   7. Taylor expanded in g around -inf 74.0%

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

      1. *-commutative 74.0%

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

   9. Simplified 74.0%

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

   if 7.40000000000000009e146 < 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. Taylor expanded in g around inf 0.1%

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

   5. Taylor expanded in h around 0 33.3%

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

      1. unpow1/3 43.2%

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

      2. *-lft-identity 43.2%

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

   7. Simplified 43.2%

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

      1. clear-num 43.2%

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

      2. cbrt-div 56.9%

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

      3. metadata-eval 56.9%

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

   9. Applied egg-rr 56.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 7.4 \cdot 10^{+146}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\frac{0.5 \cdot {h}^{2}}{g} \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{1}{\sqrt[3]{\frac{a}{g}}}\\ \end{array} \]

Alternative 5: 73.4% accurate, 2.0× 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(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 45.1%

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

3. Simplified 45.1%

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

4. Taylor expanded in g around inf 25.5%

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

5. Taylor expanded in g around inf 72.2%

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

   1. associate-*r/ 72.2%

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

   2. neg-mul-1 72.2%

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

7. Simplified 72.2%

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

8. Final simplification 72.2%

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

Reproduce

herbie shell --seed 2023337 
(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))))))))