2-ancestry mixing, positive discriminant

Percentage Accurate: 43.7% → 95.8%

Time: 22.5s

Alternatives: 6

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: 43.7% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

3. Simplified 40.4%

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

   1. associate-*l/ 40.3%

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

   2. cbrt-div 44.6%

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

   3. pow2 44.6%

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

   4. pow2 44.6%

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

5. Applied egg-rr 44.6%

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

6. Taylor expanded in g around -inf 36.7%

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

   1. neg-mul-1 36.7%

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

8. Simplified 36.7%

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

9. Taylor expanded in g around -inf 95.5%

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

    1. mul-1-neg 95.5%

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

11. Simplified 95.5%

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

12. Taylor expanded in g around 0 95.9%

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

    1. mul-1-neg 95.9%

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

14. Simplified 95.9%

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

15. Final simplification 95.9%

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

Alternative 2: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-54} \lor \neg \left(a \leq 9.4 \cdot 10^{-38}\right):\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} + \frac{\sqrt[3]{0.5 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= a -2.7e-54) (not (<= a 9.4e-38)))
   (+ (cbrt (* (/ 0.5 a) (- g g))) (cbrt (/ (- g) a)))
   (+ (cbrt (- g)) (/ (cbrt (* 0.5 (- (- g) g))) (cbrt a)))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((a <= -2.7e-54) || !(a <= 9.4e-38)) {
		tmp = cbrt(((0.5 / a) * (g - g))) + cbrt((-g / a));
	} else {
		tmp = cbrt(-g) + (cbrt((0.5 * (-g - g))) / cbrt(a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((a <= -2.7e-54) || !(a <= 9.4e-38)) {
		tmp = Math.cbrt(((0.5 / a) * (g - g))) + Math.cbrt((-g / a));
	} else {
		tmp = Math.cbrt(-g) + (Math.cbrt((0.5 * (-g - g))) / Math.cbrt(a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((a <= -2.7e-54) || !(a <= 9.4e-38))
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + cbrt(Float64(Float64(-g) / a)));
	else
		tmp = Float64(cbrt(Float64(-g)) + Float64(cbrt(Float64(0.5 * Float64(Float64(-g) - g))) / cbrt(a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[a, -2.7e-54], N[Not[LessEqual[a, 9.4e-38]], $MachinePrecision]], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[(-g), 1/3], $MachinePrecision] + N[(N[Power[N[(0.5 * N[((-g) - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.70000000000000026e-54 or 9.39999999999999996e-38 < a

   1. Initial program 43.5%

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

   3. Simplified 43.5%

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

   4. Taylor expanded in g around inf 17.4%

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

   5. Taylor expanded in g around inf 85.5%

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

   6. Taylor expanded in g around 0 86.2%

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

   if -2.70000000000000026e-54 < a < 9.39999999999999996e-38

   1. Initial program 35.7%

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

   3. Simplified 35.7%

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

      1. associate-*l/ 35.7%

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

      2. cbrt-div 41.5%

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

      3. pow2 41.5%

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

      4. pow2 41.5%

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

   5. Applied egg-rr 41.5%

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

   6. Taylor expanded in g around -inf 30.7%

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

      1. neg-mul-1 30.7%

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

   8. Simplified 30.7%

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

   9. Taylor expanded in g around inf 11.1%

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

   10. Taylor expanded in g around 0 11.1%

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

   12. Simplified 96.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-54} \lor \neg \left(a \leq 9.4 \cdot 10^{-38}\right):\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} + \frac{\sqrt[3]{0.5 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{a}}\\ \end{array} \]

Alternative 3: 89.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-86} \lor \neg \left(a \leq 2.65 \cdot 10^{-42}\right):\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\sqrt[3]{0.5 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= a -6e-86) (not (<= a 2.65e-42)))
   (+ (cbrt (* (/ 0.5 a) (- g g))) (cbrt (/ (- g) a)))
   (+ -1.0 (/ (cbrt (* 0.5 (- (- g) g))) (cbrt a)))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((a <= -6e-86) || !(a <= 2.65e-42)) {
		tmp = cbrt(((0.5 / a) * (g - g))) + cbrt((-g / a));
	} else {
		tmp = -1.0 + (cbrt((0.5 * (-g - g))) / cbrt(a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((a <= -6e-86) || !(a <= 2.65e-42)) {
		tmp = Math.cbrt(((0.5 / a) * (g - g))) + Math.cbrt((-g / a));
	} else {
		tmp = -1.0 + (Math.cbrt((0.5 * (-g - g))) / Math.cbrt(a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((a <= -6e-86) || !(a <= 2.65e-42))
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + cbrt(Float64(Float64(-g) / a)));
	else
		tmp = Float64(-1.0 + Float64(cbrt(Float64(0.5 * Float64(Float64(-g) - g))) / cbrt(a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[a, -6e-86], N[Not[LessEqual[a, 2.65e-42]], $MachinePrecision]], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[Power[N[(0.5 * N[((-g) - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.0000000000000002e-86 or 2.65e-42 < a

   1. Initial program 43.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in g around inf 18.6%

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

   5. Taylor expanded in g around inf 85.6%

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

   6. Taylor expanded in g around 0 86.2%

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

   if -6.0000000000000002e-86 < a < 2.65e-42

   1. Initial program 35.4%

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

   3. Simplified 35.4%

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

      1. associate-*l/ 35.3%

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

      2. cbrt-div 41.7%

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

      3. pow2 41.7%

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

      4. pow2 41.7%

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

   5. Applied egg-rr 41.7%

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

   6. Taylor expanded in g around -inf 32.4%

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

      1. neg-mul-1 32.4%

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

   8. Simplified 32.4%

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

   9. Taylor expanded in g around inf 10.6%

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

       1. cbrt-prod 96.5%

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

   11. Applied egg-rr 0.0%

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

   13. Simplified 93.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-86} \lor \neg \left(a \leq 2.65 \cdot 10^{-42}\right):\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\sqrt[3]{0.5 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{a}}\\ \end{array} \]

Alternative 4: 4.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

3. Simplified 40.4%

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

4. Taylor expanded in g around inf 17.4%

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

5. Taylor expanded in g around inf 70.2%

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

6. Taylor expanded in g around 0 70.6%

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

8. Simplified 5.1%

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

9. Final simplification 5.1%

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

Alternative 5: 62.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

3. Simplified 40.4%

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

   1. associate-*l/ 40.3%

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

   2. cbrt-div 44.6%

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

   3. pow2 44.6%

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

   4. pow2 44.6%

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

5. Applied egg-rr 44.6%

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

6. Taylor expanded in g around -inf 36.7%

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

   1. neg-mul-1 36.7%

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

8. Simplified 36.7%

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

9. Taylor expanded in g around inf 19.8%

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

    1. cbrt-prod 95.6%

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

11. Applied egg-rr 0.0%

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

13. Simplified 62.6%

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

14. Final simplification 62.6%

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

Alternative 6: 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 40.4%

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

3. Simplified 40.4%

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

4. Taylor expanded in g around inf 17.4%

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

5. Taylor expanded in g around inf 70.2%

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

   1. cbrt-prod 95.6%

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

7. Applied egg-rr 0.0%

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

9. Simplified 4.3%

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

10. Final simplification 4.3%

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

Reproduce

herbie shell --seed 2023298 
(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))))))))