2-ancestry mixing, positive discriminant

Percentage Accurate: 44.3% → 95.9%

Time: 18.3s

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: 44.3% 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: 95.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

3. Simplified 41.4%

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

4. Taylor expanded in g around inf 19.3%

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

5. Taylor expanded in g around inf 71.9%

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

   1. associate-*r/ 71.9%

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

   2. neg-mul-1 71.9%

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

7. Simplified 71.9%

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

   1. pow1/3 30.1%

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

   2. div-inv 30.1%

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

   3. unpow-prod-down 21.3%

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

   4. pow1/3 45.9%

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

9. Applied egg-rr 45.9%

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

    1. unpow1/3 96.0%

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

11. Simplified 96.0%

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

12. Final simplification 96.0%

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

Alternative 2: 95.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

3. Simplified 41.4%

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

4. Taylor expanded in g around inf 19.3%

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

5. Taylor expanded in g around inf 71.9%

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

   1. associate-*r/ 71.9%

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

   2. neg-mul-1 71.9%

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

7. Simplified 71.9%

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

   1. cbrt-div 95.8%

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

9. Applied egg-rr 95.8%

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

10. Final simplification 95.8%

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

Alternative 3: 73.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{-0.5}{a} \cdot 0} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (* (/ -0.5 a) 0.0)) (cbrt (* (/ -0.5 a) (+ g g)))))

C

double code(double g, double h, double a) {
	return cbrt(((-0.5 / a) * 0.0)) + cbrt(((-0.5 / a) * (g + g)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((-0.5 / a) * 0.0)) + Math.cbrt(((-0.5 / a) * (g + g)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(-0.5 / a) * 0.0)) + cbrt(Float64(Float64(-0.5 / a) * Float64(g + g))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(-0.5 / a), $MachinePrecision] * 0.0), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.4%

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

3. Simplified 41.4%

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

   1. pow1/2 41.4%

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

   2. pow-to-exp 28.4%

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

5. Applied egg-rr 28.4%

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

6. Taylor expanded in g around inf 19.8%

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

   1. distribute-rgt1-in 19.8%

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

   2. metadata-eval 19.8%

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

   3. mul0-lft 19.8%

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

   4. metadata-eval 19.8%

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

8. Simplified 19.8%

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

9. Taylor expanded in g around inf 71.9%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{\left(2 \cdot g\right)} \cdot \frac{-0.5}{a}} \]
10. Step-by-step derivation

    1. count-2 71.9%

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

11. Simplified 71.9%

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

12. Final simplification 71.9%

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

Alternative 4: 73.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

3. Simplified 41.4%

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

4. Taylor expanded in g around inf 19.3%

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

5. Taylor expanded in g around inf 71.9%

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

   1. associate-*r/ 71.9%

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

   2. neg-mul-1 71.9%

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

7. Simplified 71.9%

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

8. Final simplification 71.9%

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

Alternative 5: 3.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-0.5}{a} \cdot 0}\\ t_0 + t_0 \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (/ -0.5 a) 0.0)))) (+ t_0 t_0)))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((-0.5 / a) * 0.0));
	return t_0 + t_0;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((-0.5 / a) * 0.0));
	return t_0 + t_0;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(-0.5 / a) * 0.0))
	return Float64(t_0 + t_0)
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(-0.5 / a), $MachinePrecision] * 0.0), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$0 + t$95$0), $MachinePrecision]]

Derivation

1. Initial program 41.4%

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

3. Simplified 41.4%

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

   1. pow1/2 41.4%

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

   2. pow-to-exp 28.4%

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

5. Applied egg-rr 28.4%

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

6. Taylor expanded in g around inf 19.8%

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

   1. distribute-rgt1-in 19.8%

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

   2. metadata-eval 19.8%

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

   3. mul0-lft 19.8%

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

   4. metadata-eval 19.8%

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

8. Simplified 19.8%

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

9. Taylor expanded in g around -inf 2.9%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{\left(-0.5 \cdot \left(h + -1 \cdot h\right)\right)} \cdot \frac{-0.5}{a}} \]
10. Step-by-step derivation

    1. *-commutative 2.9%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{\left(\left(h + -1 \cdot h\right) \cdot -0.5\right)} \cdot \frac{-0.5}{a}} \]

    2. distribute-rgt1-in 2.9%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(\color{blue}{\left(\left(-1 + 1\right) \cdot h\right)} \cdot -0.5\right) \cdot \frac{-0.5}{a}} \]

    3. metadata-eval 2.9%

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

    4. mul0-lft 2.9%

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

    5. metadata-eval 2.9%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{0} \cdot \frac{-0.5}{a}} \]

11. Simplified 2.9%

    \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{0} \cdot \frac{-0.5}{a}} \]

12. Final simplification 2.9%

    \[\leadsto \sqrt[3]{\frac{-0.5}{a} \cdot 0} + \sqrt[3]{\frac{-0.5}{a} \cdot 0} \]

Reproduce

herbie shell --seed 2023222 
(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))))))))