2-ancestry mixing, positive discriminant

Percentage Accurate: 44.0% → 95.5%

Time: 15.7s

Alternatives: 5

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: 44.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 95.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.7%

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

3. Simplified 41.7%

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

5. Taylor expanded in g around -inf 28.6%

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

   1. associate-*r/ 28.6%

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

   2. neg-mul-1 28.6%

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

7. Simplified 28.6%

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

   1. pow1/3 13.1%

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

   2. div-inv 13.1%

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

   3. unpow-prod-down 13.5%

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

   4. pow1/3 16.6%

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

9. Applied egg-rr 16.6%

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

    1. *-commutative 16.6%

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

    2. unpow1/3 34.6%

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

11. Simplified 34.6%

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

12. Taylor expanded in g around -inf 95.9%

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

    1. neg-mul-1 95.9%

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

14. Simplified 95.9%

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

    1. *-un-lft-identity 95.9%

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

    2. cbrt-div 95.9%

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

    3. metadata-eval 95.9%

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

16. Applied egg-rr 95.9%

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

18. Simplified 95.9%

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

19. Final simplification 95.9%

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

Alternative 2: 95.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.7%

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

3. Simplified 41.7%

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

5. Taylor expanded in g around -inf 28.6%

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

   1. associate-*r/ 28.6%

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

   2. neg-mul-1 28.6%

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

7. Simplified 28.6%

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

   1. pow1/3 13.1%

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

   2. div-inv 13.1%

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

   3. unpow-prod-down 13.5%

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

   4. pow1/3 16.6%

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

9. Applied egg-rr 16.6%

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

    1. *-commutative 16.6%

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

    2. unpow1/3 34.6%

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

11. Simplified 34.6%

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

12. Taylor expanded in g around -inf 95.9%

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

    1. neg-mul-1 95.9%

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

14. Simplified 95.9%

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

15. Taylor expanded in g around 0 95.9%

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

16. Final simplification 95.9%

    \[\leadsto \sqrt[3]{-g} \cdot \sqrt[3]{\frac{1}{a}} \]
17. Add Preprocessing

Alternative 3: 15.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{a}} \cdot -2 \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.7%

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

3. Simplified 41.7%

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

5. Taylor expanded in g around -inf 28.6%

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

   1. associate-*r/ 28.6%

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

   2. neg-mul-1 28.6%

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

7. Simplified 28.6%

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

8. Taylor expanded in g around inf 15.5%

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

9. Taylor expanded in g around -inf 15.5%

   \[\leadsto \color{blue}{-2 \cdot \sqrt[3]{\frac{g}{a}}} \]

11. Simplified 15.5%

    \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot -2} \]
12. Add Preprocessing

Alternative 4: 6.2% accurate, 61.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-310}:\\ \;\;\;\;g\\ \mathbf{else}:\\ \;\;\;\;-g\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (if (<= a -2e-310) g (- g)))

C

double code(double g, double h, double a) {
	double tmp;
	if (a <= -2e-310) {
		tmp = g;
	} else {
		tmp = -g;
	}
	return tmp;
}

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-2d-310)) then
        tmp = g
    else
        tmp = -g
    end if
    code = tmp
end function

Java

public static double code(double g, double h, double a) {
	double tmp;
	if (a <= -2e-310) {
		tmp = g;
	} else {
		tmp = -g;
	}
	return tmp;
}

Python

def code(g, h, a):
	tmp = 0
	if a <= -2e-310:
		tmp = g
	else:
		tmp = -g
	return tmp

Julia

function code(g, h, a)
	tmp = 0.0
	if (a <= -2e-310)
		tmp = g;
	else
		tmp = Float64(-g);
	end
	return tmp
end

MATLAB

function tmp_2 = code(g, h, a)
	tmp = 0.0;
	if (a <= -2e-310)
		tmp = g;
	else
		tmp = -g;
	end
	tmp_2 = tmp;
end

Wolfram

code[g_, h_, a_] := If[LessEqual[a, -2e-310], g, (-g)]

Derivation

1. Split input into 2 regimes

2. if a < -1.999999999999994e-310

   1. Initial program 44.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in g around -inf 30.3%

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

      1. associate-*r/ 30.3%

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

      2. neg-mul-1 30.3%

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

   7. Simplified 30.3%

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

   8. Taylor expanded in g around inf 8.9%

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

      1. flip-+ 7.2%

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

   10. Applied egg-rr 0.0%

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

   12. Simplified 0.0%

       \[\leadsto g \cdot \color{blue}{\frac{0}{0}} \]

   14. Applied egg-rr 3.7%

       \[\leadsto \color{blue}{\log \left(e^{g}\right)} \]
   15. Step-by-step derivation

       1. rem-log-exp 6.1%

          \[\leadsto \color{blue}{g} \]

   16. Simplified 6.1%

       \[\leadsto \color{blue}{g} \]

   if -1.999999999999994e-310 < a

   1. Initial program 38.6%

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

   3. Simplified 38.6%

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

   5. Taylor expanded in g around -inf 27.0%

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

      1. associate-*r/ 27.0%

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

      2. neg-mul-1 27.0%

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

   7. Simplified 27.0%

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

   8. Taylor expanded in g around inf 9.5%

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

      1. flip-+ 8.0%

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

   10. Applied egg-rr 0.0%

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

   12. Simplified 0.0%

       \[\leadsto g \cdot \color{blue}{\frac{0}{0}} \]

   14. Applied egg-rr 6.4%

       \[\leadsto \color{blue}{-g} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 3.7% accurate, 433.0× speedup

Math

\[\begin{array}{l} \\ g \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 g)

C

double code(double g, double h, double a) {
	return g;
}

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    code = g
end function

Java

public static double code(double g, double h, double a) {
	return g;
}

Python

def code(g, h, a):
	return g

Julia

function code(g, h, a)
	return g
end

MATLAB

function tmp = code(g, h, a)
	tmp = g;
end

Wolfram

code[g_, h_, a_] := g

Derivation

1. Initial program 41.7%

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

3. Simplified 41.7%

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

5. Taylor expanded in g around -inf 28.6%

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

   1. associate-*r/ 28.6%

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

   2. neg-mul-1 28.6%

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

7. Simplified 28.6%

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

8. Taylor expanded in g around inf 9.2%

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

   1. flip-+ 7.6%

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

10. Applied egg-rr 0.0%

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

12. Simplified 0.0%

    \[\leadsto g \cdot \color{blue}{\frac{0}{0}} \]

14. Applied egg-rr 2.5%

    \[\leadsto \color{blue}{\log \left(e^{g}\right)} \]
15. Step-by-step derivation

    1. rem-log-exp 3.7%

       \[\leadsto \color{blue}{g} \]

16. Simplified 3.7%

    \[\leadsto \color{blue}{g} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(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))))))))