2-ancestry mixing, positive discriminant

Percentage Accurate: 44.2% → 95.7%

Time: 14.0s

Alternatives: 6

Speedup: 4.2×

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: 44.2% 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: 95.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.0%

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

3. Simplified 47.0%

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

5. Taylor expanded in g around inf 71.4%

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

   1. pow1/3 32.8%

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

   2. div-inv 32.8%

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

   3. unpow-prod-down 18.8%

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

   4. pow1/3 43.1%

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

7. Applied egg-rr 43.1%

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

   1. unpow1/3 94.0%

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

9. Simplified 94.0%

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

    1. *-un-lft-identity 94.0%

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

    2. metadata-eval 94.0%

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

    3. metadata-eval 94.0%

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

    4. metadata-eval 94.0%

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

    5. metadata-eval 94.0%

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

    6. cbrt-unprod 94.0%

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

    7. cbrt-unprod 94.5%

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

    8. rem-3cbrt-lft 94.5%

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

    9. metadata-eval 94.5%

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

11. Applied egg-rr 94.5%

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

12. Final simplification 94.5%

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

Alternative 2: 95.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.0%

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

3. Simplified 47.0%

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

5. Taylor expanded in g around inf 71.4%

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

   1. pow1/3 32.8%

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

   2. div-inv 32.8%

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

   3. unpow-prod-down 18.8%

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

   4. pow1/3 43.1%

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

7. Applied egg-rr 43.1%

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

   1. unpow1/3 94.0%

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

9. Simplified 94.0%

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

11. Applied egg-rr 94.3%

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

Alternative 3: 95.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.0%

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

3. Simplified 47.0%

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

5. Taylor expanded in g around inf 71.4%

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

   1. pow1/3 32.8%

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

   2. div-inv 32.8%

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

   3. unpow-prod-down 18.8%

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

   4. pow1/3 43.1%

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

7. Applied egg-rr 43.1%

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

   1. unpow1/3 94.0%

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

9. Simplified 94.0%

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

11. Applied egg-rr 94.3%

    \[\leadsto \color{blue}{\frac{-1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g}}}} \]
12. Step-by-step derivation

    1. associate-/r/ 94.3%

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

    2. associate-*l/ 94.3%

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

    3. neg-mul-1 94.3%

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

13. Simplified 94.3%

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

Alternative 4: 7.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (if (<= a -2e-310) (cbrt g) (- (cbrt g))))

C

double code(double g, double h, double a) {
	double tmp;
	if (a <= -2e-310) {
		tmp = cbrt(g);
	} else {
		tmp = -cbrt(g);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (a <= -2e-310) {
		tmp = Math.cbrt(g);
	} else {
		tmp = -Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (a <= -2e-310)
		tmp = cbrt(g);
	else
		tmp = Float64(-cbrt(g));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[a, -2e-310], N[Power[g, 1/3], $MachinePrecision], (-N[Power[g, 1/3], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if a < -1.999999999999994e-310

   1. Initial program 47.0%

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

   3. Simplified 47.0%

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

   5. Taylor expanded in g around inf 71.9%

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

      1. pow1 71.9%

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

      2. cbrt-unprod 72.4%

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

      3. cbrt-unprod 72.4%

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

      4. metadata-eval 72.4%

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

   7. Applied egg-rr 72.4%

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

      1. unpow1 72.4%

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

      2. *-commutative 72.4%

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

      3. neg-mul-1 72.4%

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

      4. distribute-neg-frac2 72.4%

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

   9. Simplified 72.4%

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

   11. Applied egg-rr 0.0%

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

       1. *-commutative 0.0%

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

       2. associate-*l/ 0.0%

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

       3. associate-/l* 0.0%

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

       4. *-inverses 7.9%

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

       5. *-rgt-identity 7.9%

          \[\leadsto \sqrt[3]{\color{blue}{g}} \]

   13. Simplified 7.9%

       \[\leadsto \sqrt[3]{\color{blue}{g}} \]

   if -1.999999999999994e-310 < a

   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. Add Preprocessing

   5. Taylor expanded in g around inf 70.9%

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

      1. pow1 70.9%

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

      2. cbrt-unprod 71.5%

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

      3. cbrt-unprod 71.5%

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

      4. metadata-eval 71.5%

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

   7. Applied egg-rr 71.5%

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

      1. unpow1 71.5%

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

      2. *-commutative 71.5%

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

      3. neg-mul-1 71.5%

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

      4. distribute-neg-frac2 71.5%

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

   9. Simplified 71.5%

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

   11. Applied egg-rr 1.2%

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

       1. *-commutative 1.2%

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

       2. associate-*l/ 1.2%

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

       3. associate-/l* 1.3%

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

       4. *-inverses 1.3%

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

       5. *-rgt-identity 1.3%

          \[\leadsto \sqrt[3]{\color{blue}{g}} \]

   13. Simplified 1.3%

       \[\leadsto \sqrt[3]{\color{blue}{g}} \]

   15. Applied egg-rr 8.0%

       \[\leadsto \color{blue}{\sqrt[3]{g} \cdot -1} \]
   16. Step-by-step derivation

       1. *-commutative 8.0%

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

       2. neg-mul-1 8.0%

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

   17. Simplified 8.0%

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

Alternative 5: 73.4% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.0%

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

3. Simplified 47.0%

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

5. Taylor expanded in g around inf 71.4%

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

   1. pow1 71.4%

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

   2. cbrt-unprod 71.9%

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

   3. cbrt-unprod 71.9%

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

   4. metadata-eval 71.9%

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

7. Applied egg-rr 71.9%

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

   1. unpow1 71.9%

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

   2. *-commutative 71.9%

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

   3. neg-mul-1 71.9%

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

   4. distribute-neg-frac2 71.9%

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

9. Simplified 71.9%

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

Alternative 6: 4.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{g} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (cbrt g))

C

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

Java

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

Julia

function code(g, h, a)
	return cbrt(g)
end

Wolfram

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

Derivation

1. Initial program 47.0%

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

3. Simplified 47.0%

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

5. Taylor expanded in g around inf 71.4%

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

   1. pow1 71.4%

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

   2. cbrt-unprod 71.9%

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

   3. cbrt-unprod 71.9%

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

   4. metadata-eval 71.9%

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

7. Applied egg-rr 71.9%

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

   1. unpow1 71.9%

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

   2. *-commutative 71.9%

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

   3. neg-mul-1 71.9%

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

   4. distribute-neg-frac2 71.9%

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

9. Simplified 71.9%

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

11. Applied egg-rr 0.6%

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

    1. *-commutative 0.6%

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

    2. associate-*l/ 0.6%

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

    3. associate-/l* 0.6%

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

    4. *-inverses 4.7%

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

    5. *-rgt-identity 4.7%

       \[\leadsto \sqrt[3]{\color{blue}{g}} \]

13. Simplified 4.7%

    \[\leadsto \sqrt[3]{\color{blue}{g}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024184 
(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))))))))