2-ancestry mixing, positive discriminant

Percentage Accurate: 44.0% → 95.8%

Time: 23.0s

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: 44.0% 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

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

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(Float64(-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 42.2%

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

3. Simplified 42.2%

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

4. Taylor expanded in g around inf 21.9%

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

5. Taylor expanded in g around inf 74.3%

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

   1. add-sqr-sqrt 38.5%

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

   2. sqrt-unprod 23.0%

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

   3. swap-sqr 11.5%

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

   4. count-2 11.5%

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

   5. count-2 11.5%

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

   6. swap-sqr 11.5%

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

   7. metadata-eval 11.5%

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

   8. metadata-eval 11.5%

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

   9. swap-sqr 11.5%

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

   10. *-commutative 11.5%

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

   11. *-commutative 11.5%

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

   12. frac-times 11.5%

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

   13. metadata-eval 11.5%

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

   14. metadata-eval 11.5%

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

   15. frac-times 11.5%

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

   16. swap-sqr 23.0%

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

   17. *-commutative 23.0%

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

   18. *-commutative 23.0%

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

7. Applied egg-rr 96.5%

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

8. Final simplification 96.5%

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

Alternative 2: 73.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Derivation

1. Initial program 42.2%

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

3. Simplified 42.2%

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

4. Taylor expanded in g around inf 21.9%

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

5. Taylor expanded in g around inf 74.3%

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

6. Final simplification 74.3%

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

Alternative 3: 6.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \mathbf{if}\;a \leq -1.2 \cdot 10^{-295}:\\ \;\;\;\;t_0 + -2\\ \mathbf{else}:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{g}{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (/ 0.5 a) (- g g)))))
   (if (<= a -1.2e-295) (+ t_0 -2.0) (+ t_0 (cbrt (/ g -2.0))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((0.5 / a) * (g - g)));
	double tmp;
	if (a <= -1.2e-295) {
		tmp = t_0 + -2.0;
	} else {
		tmp = t_0 + cbrt((g / -2.0));
	}
	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 <= -1.2e-295) {
		tmp = t_0 + -2.0;
	} else {
		tmp = t_0 + Math.cbrt((g / -2.0));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(0.5 / a) * Float64(g - g)))
	tmp = 0.0
	if (a <= -1.2e-295)
		tmp = Float64(t_0 + -2.0);
	else
		tmp = Float64(t_0 + cbrt(Float64(g / -2.0)));
	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, -1.2e-295], N[(t$95$0 + -2.0), $MachinePrecision], N[(t$95$0 + N[Power[N[(g / -2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1999999999999999e-295

   1. Initial program 39.1%

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

   3. Simplified 39.1%

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

   4. Taylor expanded in g around inf 21.9%

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

   5. Taylor expanded in g around inf 75.6%

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

   7. Applied egg-rr 0.0%

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

   9. Simplified 5.0%

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

   if -1.1999999999999999e-295 < a

   1. Initial program 45.5%

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

   3. Simplified 45.5%

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

   4. Taylor expanded in g around inf 22.0%

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

   5. Taylor expanded in g around inf 73.0%

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

      1. add-sqr-sqrt 44.4%

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

      2. sqrt-unprod 23.2%

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

      3. swap-sqr 11.8%

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

      4. count-2 11.8%

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

      5. count-2 11.8%

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

      6. swap-sqr 11.8%

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

      7. metadata-eval 11.8%

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

      8. metadata-eval 11.8%

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

      9. swap-sqr 11.8%

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

      10. *-commutative 11.8%

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

      11. *-commutative 11.8%

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

      12. frac-times 11.8%

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

      13. metadata-eval 11.8%

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

      14. metadata-eval 11.8%

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

      15. frac-times 11.8%

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

      16. swap-sqr 23.2%

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

      17. *-commutative 23.2%

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

      18. *-commutative 23.2%

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

      19. sqrt-unprod 44.4%

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

      20. add-sqr-sqrt 73.0%

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

      21. associate-*l/ 72.9%

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

   7. Applied egg-rr 72.9%

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

   9. Simplified 7.8%

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

4. Final simplification 6.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-295}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + -2\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{g}{-2}}\\ \end{array} \]

Alternative 4: 73.6% 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 42.2%

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

3. Simplified 42.2%

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

4. Taylor expanded in g around inf 21.9%

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

5. Taylor expanded in g around inf 74.3%

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

   1. add-sqr-sqrt 38.5%

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

   2. sqrt-unprod 23.0%

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

   3. swap-sqr 11.5%

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

   4. count-2 11.5%

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

   5. count-2 11.5%

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

   6. swap-sqr 11.5%

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

   7. metadata-eval 11.5%

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

   8. metadata-eval 11.5%

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

   9. swap-sqr 11.5%

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

   10. *-commutative 11.5%

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

   11. *-commutative 11.5%

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

   12. frac-times 11.5%

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

   13. metadata-eval 11.5%

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

   14. metadata-eval 11.5%

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

   15. frac-times 11.5%

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

   16. swap-sqr 23.0%

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

   17. *-commutative 23.0%

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

   18. *-commutative 23.0%

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

   19. sqrt-unprod 38.5%

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

   20. add-sqr-sqrt 74.3%

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

   21. associate-*l/ 74.3%

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

7. Applied egg-rr 74.3%

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

8. Final simplification 74.3%

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

Alternative 5: 4.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.2%

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

3. Simplified 42.2%

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

4. Taylor expanded in g around inf 21.9%

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

5. Taylor expanded in g around inf 74.3%

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

7. Applied egg-rr 0.0%

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

9. Simplified 4.3%

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

10. Final simplification 4.3%

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

Reproduce

herbie shell --seed 2023334 
(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))))))))