2-ancestry mixing, positive discriminant

Percentage Accurate: 44.4% → 96.0%

Time: 22.5s

Alternatives: 6

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: 44.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 96.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

3. Simplified 39.7%

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

5. Taylor expanded in g around -inf 24.8%

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

6. Taylor expanded in g around -inf 68.0%

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

   1. associate-*l/ 68.0%

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

   2. cbrt-div 95.7%

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

   3. associate-*r* 95.9%

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

   4. metadata-eval 95.9%

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

   5. neg-mul-1 95.9%

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

8. Applied egg-rr 95.9%

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

   1. div-inv 95.9%

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

10. Applied egg-rr 95.9%

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

Alternative 2: 96.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Derivation

1. Initial program 39.7%

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

3. Simplified 39.7%

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

5. Taylor expanded in g around -inf 24.8%

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

6. Taylor expanded in g around -inf 68.0%

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

   1. associate-*l/ 68.0%

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

   2. cbrt-div 95.7%

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

   3. associate-*r* 95.9%

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

   4. metadata-eval 95.9%

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

   5. neg-mul-1 95.9%

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

8. Applied egg-rr 95.9%

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

Alternative 3: 73.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

3. Simplified 39.7%

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

5. Taylor expanded in g around -inf 24.8%

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

6. Taylor expanded in g around -inf 68.0%

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

   1. associate-*l/ 68.0%

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

   2. add-sqr-sqrt 38.5%

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

   3. associate-*r* 38.7%

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

   4. metadata-eval 38.7%

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

   5. neg-mul-1 38.7%

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

   6. associate-/r* 38.8%

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

8. Applied egg-rr 38.8%

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

   1. associate-/l/ 38.7%

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

   2. add-sqr-sqrt 68.2%

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

   3. div-inv 68.2%

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

10. Applied egg-rr 68.2%

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

Alternative 4: 74.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

3. Simplified 39.7%

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

5. Taylor expanded in g around -inf 24.8%

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

6. Taylor expanded in g around -inf 68.0%

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

   1. associate-*l/ 68.0%

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

   2. add-sqr-sqrt 38.5%

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

   3. associate-*r* 38.7%

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

   4. metadata-eval 38.7%

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

   5. neg-mul-1 38.7%

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

   6. associate-/r* 38.8%

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

8. Applied egg-rr 38.8%

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

   1. associate-/l/ 38.7%

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

   2. add-sqr-sqrt 68.2%

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

   3. clear-num 67.7%

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

   4. cbrt-div 68.9%

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

   5. metadata-eval 68.9%

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

10. Applied egg-rr 68.9%

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

Alternative 5: 73.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

3. Simplified 39.7%

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

5. Taylor expanded in g around -inf 24.8%

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

6. Taylor expanded in g around -inf 68.0%

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

   1. associate-*l/ 14.4%

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

   2. associate-*r* 14.4%

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

   3. metadata-eval 14.4%

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

   4. neg-mul-1 14.4%

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

8. Applied egg-rr 68.2%

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

Alternative 6: 15.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

3. Simplified 39.7%

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

5. Taylor expanded in g around -inf 24.8%

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

6. Taylor expanded in g around inf 14.4%

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

   1. associate-*l/ 14.4%

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

   2. associate-*r* 14.4%

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

   3. metadata-eval 14.4%

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

   4. neg-mul-1 14.4%

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

8. Applied egg-rr 14.4%

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

9. Taylor expanded in g around -inf 14.5%

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

Reproduce

herbie shell --seed 2024050 -o generate:simplify
(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))))))))