2-ancestry mixing, positive discriminant

Percentage Accurate: 44.8% → 95.5%

Time: 19.6s

Alternatives: 4

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: 44.8% 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: 95.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{g} \cdot \frac{-1}{\sqrt[3]{a}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (* 0.0 (/ -0.5 a))) (* (cbrt g) (/ -1.0 (cbrt a)))))

C

double code(double g, double h, double a) {
	return cbrt((0.0 * (-0.5 / a))) + (cbrt(g) * (-1.0 / cbrt(a)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((0.0 * (-0.5 / a))) + (Math.cbrt(g) * (-1.0 / Math.cbrt(a)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(0.0 * Float64(-0.5 / a))) + Float64(cbrt(g) * Float64(-1.0 / cbrt(a))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(0.0 * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[g, 1/3], $MachinePrecision] * N[(-1.0 / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.7%

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

3. Simplified 42.7%

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

4. Taylor expanded in g around inf 27.0%

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

   1. distribute-rgt1-in 27.0%

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

   2. metadata-eval 27.0%

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

   3. mul0-lft 27.0%

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

   4. metadata-eval 27.0%

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

6. Simplified 27.0%

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

7. Taylor expanded in g around inf 76.0%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
8. Step-by-step derivation

   1. associate-*r/ 76.0%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

   2. neg-mul-1 76.0%

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

9. Simplified 76.0%

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

    1. frac-2neg 76.0%

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

    2. cbrt-div 95.9%

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

    3. remove-double-neg 95.9%

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

11. Applied egg-rr 95.9%

    \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{-a}}} \]
12. Step-by-step derivation

    1. frac-2neg 95.9%

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

    2. div-inv 95.9%

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

    3. neg-mul-1 95.9%

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

    4. add-cbrt-cube 95.9%

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

    5. metadata-eval 95.9%

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

    6. metadata-eval 95.9%

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

    7. add-sqr-sqrt 47.8%

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

    8. sqrt-unprod 26.6%

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

    9. sqr-neg 26.6%

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

    10. sqrt-unprod 0.7%

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

    11. add-sqr-sqrt 1.4%

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

    12. cbrt-prod 1.4%

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

    13. neg-mul-1 1.4%

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

    14. pow1/3 0.6%

        \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(-\sqrt[3]{g}\right) \cdot \frac{1}{\color{blue}{{\left(-a\right)}^{0.3333333333333333}}} \]

    15. pow-flip 0.6%

        \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(-\sqrt[3]{g}\right) \cdot \color{blue}{{\left(-a\right)}^{\left(-0.3333333333333333\right)}} \]

    16. add-sqr-sqrt 0.6%

        \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(-\sqrt[3]{g}\right) \cdot {\color{blue}{\left(\sqrt{-a} \cdot \sqrt{-a}\right)}}^{\left(-0.3333333333333333\right)} \]

    17. sqrt-unprod 26.3%

        \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(-\sqrt[3]{g}\right) \cdot {\color{blue}{\left(\sqrt{\left(-a\right) \cdot \left(-a\right)}\right)}}^{\left(-0.3333333333333333\right)} \]

    18. sqr-neg 26.3%

        \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(-\sqrt[3]{g}\right) \cdot {\left(\sqrt{\color{blue}{a \cdot a}}\right)}^{\left(-0.3333333333333333\right)} \]

    19. sqrt-unprod 45.1%

        \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(-\sqrt[3]{g}\right) \cdot {\color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)}}^{\left(-0.3333333333333333\right)} \]

    20. add-sqr-sqrt 45.1%

        \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(-\sqrt[3]{g}\right) \cdot {\color{blue}{a}}^{\left(-0.3333333333333333\right)} \]

    21. metadata-eval 45.1%

        \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(-\sqrt[3]{g}\right) \cdot {a}^{\color{blue}{-0.3333333333333333}} \]

    22. metadata-eval 45.1%

        \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(-\sqrt[3]{g}\right) \cdot {a}^{\color{blue}{\left(-1 \cdot 0.3333333333333333\right)}} \]

    23. pow-pow 45.1%

        \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(-\sqrt[3]{g}\right) \cdot \color{blue}{{\left({a}^{-1}\right)}^{0.3333333333333333}} \]

13. Applied egg-rr 95.9%

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

14. Final simplification 95.9%

    \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{g} \cdot \frac{-1}{\sqrt[3]{a}} \]

Alternative 2: 95.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(0.0 * 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 42.7%

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

3. Simplified 42.7%

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

4. Taylor expanded in g around inf 27.0%

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

   1. distribute-rgt1-in 27.0%

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

   2. metadata-eval 27.0%

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

   3. mul0-lft 27.0%

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

   4. metadata-eval 27.0%

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

6. Simplified 27.0%

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

7. Taylor expanded in g around inf 76.0%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
8. Step-by-step derivation

   1. associate-*r/ 76.0%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

   2. neg-mul-1 76.0%

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

9. Simplified 76.0%

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

    1. frac-2neg 76.0%

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

    2. cbrt-div 95.9%

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

    3. remove-double-neg 95.9%

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

11. Applied egg-rr 95.9%

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

12. Final simplification 95.9%

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

Alternative 3: 73.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.7%

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

3. Simplified 42.7%

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

4. Taylor expanded in g around inf 27.0%

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

   1. distribute-rgt1-in 27.0%

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

   2. metadata-eval 27.0%

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

   3. mul0-lft 27.0%

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

   4. metadata-eval 27.0%

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

6. Simplified 27.0%

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

7. Taylor expanded in g around inf 76.0%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
8. Step-by-step derivation

   1. associate-*r/ 76.0%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

   2. neg-mul-1 76.0%

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

9. Simplified 76.0%

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

10. Final simplification 76.0%

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

Alternative 4: 1.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.7%

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

3. Simplified 42.7%

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

4. Taylor expanded in g around inf 27.0%

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

   1. distribute-rgt1-in 27.0%

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

   2. metadata-eval 27.0%

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

   3. mul0-lft 27.0%

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

   4. metadata-eval 27.0%

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

6. Simplified 27.0%

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

7. Taylor expanded in g around inf 76.0%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
8. Step-by-step derivation

   1. associate-*r/ 76.0%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

   2. neg-mul-1 76.0%

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

9. Simplified 76.0%

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

    1. expm1-log1p-u 50.5%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{-g}{a}}\right)\right)} \]

    2. expm1-udef 31.5%

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

    3. add-sqr-sqrt 14.8%

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

    4. sqrt-unprod 8.1%

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

    5. sqr-neg 8.1%

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

    6. sqrt-unprod 0.7%

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

    7. add-sqr-sqrt 1.3%

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

11. Applied egg-rr 1.3%

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

    1. expm1-def 1.0%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{g}{a}}\right)\right)} \]

    2. expm1-log1p 1.3%

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

13. Simplified 1.3%

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

14. Final simplification 1.3%

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

Reproduce

herbie shell --seed 2023194 
(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))))))))