2-ancestry mixing, positive discriminant

Percentage Accurate: 43.9% → 95.8%

Time: 31.8s

Alternatives: 9

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 43.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) - Float64(cbrt(g) * cbrt(Float64(1.0 / 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[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.2%

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

3. Simplified 40.2%

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

5. Taylor expanded in g around inf 22.6%

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

6. Taylor expanded in g around inf 71.5%

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

7. Taylor expanded in g around 0 71.6%

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

9. Simplified 71.6%

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

    1. pow1/3 34.0%

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

    2. div-inv 34.0%

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

    3. unpow-prod-down 20.6%

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

    4. pow1/3 44.3%

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

11. Applied egg-rr 44.3%

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

    1. unpow1/3 96.5%

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

13. Simplified 96.5%

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

14. Final simplification 96.5%

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

Alternative 2: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) - Float64(cbrt(g) / 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[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.2%

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

3. Simplified 40.2%

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

5. Taylor expanded in g around inf 22.6%

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

6. Taylor expanded in g around inf 71.5%

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

7. Taylor expanded in g around 0 71.6%

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

9. Simplified 71.6%

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

    1. cbrt-div 96.5%

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

    2. div-inv 96.4%

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

11. Applied egg-rr 96.4%

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

    1. associate-*r/ 96.5%

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

    2. *-rgt-identity 96.5%

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

13. Simplified 96.5%

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

14. Final simplification 96.5%

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

Alternative 3: 6.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \mathbf{if}\;a \leq 9 \cdot 10^{-309}:\\ \;\;\;\;t\_0 + \sqrt[3]{\frac{-1}{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \sqrt[3]{-g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (/ 0.5 a) (- g g)))))
   (if (<= a 9e-309) (+ t_0 (cbrt (/ -1.0 a))) (+ t_0 (cbrt (- g))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((0.5 / a) * (g - g)));
	double tmp;
	if (a <= 9e-309) {
		tmp = t_0 + cbrt((-1.0 / a));
	} else {
		tmp = t_0 + cbrt(-g);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((0.5 / a) * (g - g)));
	double tmp;
	if (a <= 9e-309) {
		tmp = t_0 + Math.cbrt((-1.0 / a));
	} else {
		tmp = t_0 + Math.cbrt(-g);
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(0.5 / a) * Float64(g - g)))
	tmp = 0.0
	if (a <= 9e-309)
		tmp = Float64(t_0 + cbrt(Float64(-1.0 / a)));
	else
		tmp = Float64(t_0 + cbrt(Float64(-g)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[a, 9e-309], N[(t$95$0 + N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Power[(-g), 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 9.0000000000000021e-309

   1. Initial program 44.0%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in g around inf 22.1%

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

   6. Taylor expanded in g around inf 72.8%

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

      1. add-sqr-sqrt 34.1%

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

      2. pow2 34.1%

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

   8. Applied egg-rr 0.0%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\color{blue}{{\left(\frac{\frac{0}{0}}{\sqrt{a}}\right)}^{2}}} \]

   10. Simplified 0.0%

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

       1. unpow2 0.0%

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

       2. div-inv 0.0%

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

       3. metadata-eval 0.0%

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

       4. sqrt-div 0.0%

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

       5. associate-*l* 0.0%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-prod 0.0%

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

       8. frac-times 0.0%

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

       9. metadata-eval 0.0%

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

       10. add-sqr-sqrt 0.0%

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

       11. add-sqr-sqrt 4.6%

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

   12. Applied egg-rr 4.6%

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

       1. associate-*r/ 4.6%

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

       2. metadata-eval 4.6%

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

   14. Simplified 4.6%

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

   if 9.0000000000000021e-309 < a

   1. Initial program 36.2%

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

   3. Simplified 36.2%

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

   5. Taylor expanded in g around inf 23.2%

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

   6. Taylor expanded in g around inf 70.2%

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

      1. *-commutative 70.2%

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

      2. distribute-lft-in 70.2%

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

   8. Applied egg-rr 70.2%

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

   10. Simplified 8.0%

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

4. Final simplification 6.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9 \cdot 10^{-309}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{-1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{-g}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 7.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \mathbf{if}\;g \leq -5 \cdot 10^{-300}:\\ \;\;\;\;t\_0 + \sqrt[3]{\frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \sqrt[3]{\frac{-1}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (/ 0.5 a) (- g g)))))
   (if (<= g -5e-300) (+ t_0 (cbrt (/ 1.0 a))) (+ t_0 (cbrt (/ -1.0 a))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((0.5 / a) * (g - g)));
	double tmp;
	if (g <= -5e-300) {
		tmp = t_0 + cbrt((1.0 / a));
	} else {
		tmp = t_0 + cbrt((-1.0 / a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((0.5 / a) * (g - g)));
	double tmp;
	if (g <= -5e-300) {
		tmp = t_0 + Math.cbrt((1.0 / a));
	} else {
		tmp = t_0 + Math.cbrt((-1.0 / a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(0.5 / a) * Float64(g - g)))
	tmp = 0.0
	if (g <= -5e-300)
		tmp = Float64(t_0 + cbrt(Float64(1.0 / a)));
	else
		tmp = Float64(t_0 + cbrt(Float64(-1.0 / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[g, -5e-300], N[(t$95$0 + N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if g < -4.99999999999999996e-300

   1. Initial program 38.3%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in g around inf 2.6%

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

   6. Taylor expanded in g around inf 73.7%

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

      1. add-sqr-sqrt 34.7%

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

      2. pow2 34.7%

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

   8. Applied egg-rr 0.0%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\color{blue}{{\left(\frac{\frac{0}{0}}{\sqrt{a}}\right)}^{2}}} \]

   10. Simplified 3.8%

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

   11. Taylor expanded in a around 0 7.9%

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

   if -4.99999999999999996e-300 < g

   1. Initial program 42.3%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in g around inf 43.6%

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

   6. Taylor expanded in g around inf 69.3%

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

      1. add-sqr-sqrt 36.0%

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

      2. pow2 36.0%

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

   8. Applied egg-rr 0.0%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\color{blue}{{\left(\frac{\frac{0}{0}}{\sqrt{a}}\right)}^{2}}} \]

   10. Simplified 0.7%

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

       1. unpow2 0.7%

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

       2. div-inv 0.7%

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

       3. metadata-eval 0.7%

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

       4. sqrt-div 0.7%

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

       5. associate-*l* 0.7%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-prod 3.7%

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

       8. frac-times 3.7%

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

       9. metadata-eval 3.7%

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

       10. add-sqr-sqrt 3.7%

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

       11. add-sqr-sqrt 7.8%

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

   12. Applied egg-rr 7.8%

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

       1. associate-*r/ 7.8%

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

       2. metadata-eval 7.8%

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

   14. Simplified 7.8%

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

4. Final simplification 7.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -5 \cdot 10^{-300}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{-1}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + Float64(-1.0 / cbrt(Float64(a / g))))
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[N[(a / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.2%

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

3. Simplified 40.2%

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

5. Taylor expanded in g around inf 22.6%

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

6. Taylor expanded in g around inf 71.5%

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

7. Taylor expanded in g around 0 71.6%

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

9. Simplified 71.6%

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

    1. clear-num 71.5%

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

    2. cbrt-div 72.8%

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

    3. metadata-eval 72.8%

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

11. Applied egg-rr 72.8%

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

12. Final simplification 72.8%

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

Alternative 6: 72.8% 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 40.2%

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

3. Simplified 40.2%

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

5. Taylor expanded in g around inf 22.6%

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

6. Taylor expanded in g around inf 71.5%

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

7. Taylor expanded in g around 0 71.6%

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

9. Simplified 71.6%

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

10. Final simplification 71.6%

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

Alternative 7: 4.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + cbrt(Float64(-g)))
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[(-g), 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.2%

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

3. Simplified 40.2%

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

5. Taylor expanded in g around inf 22.6%

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

6. Taylor expanded in g around inf 71.5%

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

   1. *-commutative 71.5%

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

   2. distribute-lft-in 71.5%

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

8. Applied egg-rr 71.5%

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

10. Simplified 4.6%

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

11. Final simplification 4.6%

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

Alternative 8: 4.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + cbrt(0.5))
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[0.5, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.2%

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

3. Simplified 40.2%

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

5. Taylor expanded in g around inf 22.6%

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

6. Taylor expanded in g around inf 71.5%

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

   1. frac-2neg 71.5%

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

   2. metadata-eval 71.5%

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

   3. associate-*r/ 71.6%

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

   4. flip-+ 0.0%

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

   5. +-inverses 0.0%

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

   6. +-inverses 0.0%

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

8. Applied egg-rr 0.0%

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

10. Simplified 4.4%

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

11. Final simplification 4.4%

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

Alternative 9: 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 40.2%

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

3. Simplified 40.2%

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

5. Taylor expanded in g around inf 22.6%

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

6. Taylor expanded in g around inf 71.5%

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

   1. cbrt-prod 96.5%

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

8. Applied egg-rr 0.0%

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

10. Simplified 4.2%

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

11. Final simplification 4.2%

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

Reproduce

herbie shell --seed 2024054 
(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))))))))