2-ancestry mixing, positive discriminant

Percentage Accurate: 44.4% → 95.7%

Time: 36.1s

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: 44.4% 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: 95.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Simplified 37.8%

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

5. Taylor expanded in g around inf 20.0%

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

6. Taylor expanded in g around inf 67.5%

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

7. Taylor expanded in g around 0 67.6%

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

9. Simplified 67.6%

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

    1. cbrt-div 94.9%

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

    2. div-inv 94.8%

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

11. Applied egg-rr 94.8%

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

    1. associate-*r/ 94.9%

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

    2. *-rgt-identity 94.9%

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

13. Simplified 94.9%

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

14. Final simplification 94.9%

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

Alternative 2: 6.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \mathbf{if}\;g \leq 7.3 \cdot 10^{-240}:\\ \;\;\;\;t\_0 + -1\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \sqrt[3]{\frac{-1}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (/ 0.5 a) (- g g)))))
   (if (<= g 7.3e-240) (+ t_0 -1.0) (+ t_0 (cbrt (/ -1.0 a))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((0.5 / a) * (g - g)));
	double tmp;
	if (g <= 7.3e-240) {
		tmp = t_0 + -1.0;
	} else {
		tmp = t_0 + cbrt((-1.0 / a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((0.5 / a) * (g - g)));
	double tmp;
	if (g <= 7.3e-240) {
		tmp = t_0 + -1.0;
	} else {
		tmp = t_0 + Math.cbrt((-1.0 / a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(0.5 / a) * Float64(g - g)))
	tmp = 0.0
	if (g <= 7.3e-240)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64(t_0 + cbrt(Float64(-1.0 / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[g, 7.3e-240], N[(t$95$0 + -1.0), $MachinePrecision], N[(t$95$0 + N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if g < 7.30000000000000004e-240

   1. Initial program 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in g around inf 2.6%

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

   6. Taylor expanded in g around inf 66.1%

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

      1. cbrt-prod 93.0%

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

      2. flip-+ 0.0%

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

      3. cbrt-div 0.0%

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

      4. unpow2 0.0%

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

      5. unpow2 0.0%

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

      6. +-inverses 0.0%

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

      7. metadata-eval 0.0%

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

      8. metadata-eval 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. +-inverses 0.0%

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

      12. add-cbrt-cube 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. metadata-eval 0.0%

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

      16. metadata-eval 0.0%

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

      17. +-inverses 0.0%

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

      18. +-inverses 0.0%

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

      19. +-inverses 0.0%

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

      20. add-cbrt-cube 0.0%

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

      21. +-inverses 0.0%

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

   8. Applied egg-rr 0.0%

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

   10. Simplified 4.7%

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

   if 7.30000000000000004e-240 < g

   1. Initial program 36.8%

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

   3. Simplified 36.8%

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

   5. Taylor expanded in g around inf 38.3%

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

   6. Taylor expanded in g around inf 69.0%

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

      1. associate-*r/ 69.1%

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

      2. add-cube-cbrt 68.7%

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

      3. associate-/r* 68.7%

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

      4. flip-+ 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. unpow2 0.0%

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

      8. unpow2 0.0%

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

      9. associate-*l/ 0.0%

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

      10. unpow2 0.0%

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

      11. unpow2 0.0%

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

      12. +-inverses 0.0%

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

      13. metadata-eval 0.0%

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

      14. unpow2 0.0%

          \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{\frac{\frac{0}{\color{blue}{{g}^{2}} - g \cdot g}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}}} \]

      15. unpow2 0.0%

          \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{\frac{\frac{0}{{g}^{2} - \color{blue}{{g}^{2}}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}}} \]

      16. +-inverses 0.0%

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

      17. pow2 0.0%

          \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{\frac{\frac{0}{0}}{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}}{\sqrt[3]{a}}} \]

   8. Applied egg-rr 0.0%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\color{blue}{\frac{\frac{\frac{0}{0}}{{\left(\sqrt[3]{a}\right)}^{2}}}{\sqrt[3]{a}}}} \]

   10. Simplified 7.5%

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

   11. Taylor expanded in a around 0 7.5%

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

4. Final simplification 6.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 7.3 \cdot 10^{-240}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{-1}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Simplified 37.8%

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

5. Taylor expanded in g around inf 20.0%

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

6. Taylor expanded in g around inf 67.5%

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

7. Taylor expanded in g around 0 67.6%

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

9. Simplified 67.6%

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

    1. clear-num 67.3%

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

    2. cbrt-div 69.2%

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

    3. metadata-eval 69.2%

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

11. Applied egg-rr 69.2%

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

12. Final simplification 69.2%

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

Alternative 4: 74.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Simplified 37.8%

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

5. Taylor expanded in g around inf 20.0%

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

6. Taylor expanded in g around inf 67.5%

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

7. Taylor expanded in g around 0 67.6%

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

9. Simplified 67.6%

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

10. Final simplification 67.6%

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

Alternative 5: 4.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Simplified 37.8%

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

5. Taylor expanded in g around inf 20.0%

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

6. Taylor expanded in g around inf 67.5%

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

   1. cbrt-prod 95.0%

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

   2. flip-+ 0.0%

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

   3. cbrt-div 0.0%

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

   4. unpow2 0.0%

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

   5. unpow2 0.0%

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

   6. +-inverses 0.0%

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

   7. metadata-eval 0.0%

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

   8. metadata-eval 0.0%

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

   9. +-inverses 0.0%

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

   10. +-inverses 0.0%

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

   11. +-inverses 0.0%

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

   12. add-cbrt-cube 0.0%

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

   13. +-inverses 0.0%

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

   14. +-inverses 0.0%

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

   15. metadata-eval 0.0%

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

   16. metadata-eval 0.0%

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

   17. +-inverses 0.0%

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

   18. +-inverses 0.0%

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

   19. +-inverses 0.0%

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

   20. add-cbrt-cube 0.0%

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

   21. +-inverses 0.0%

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

8. Applied egg-rr 0.0%

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

10. Simplified 4.4%

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

11. Final simplification 4.4%

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

Reproduce

herbie shell --seed 2024075 
(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))))))))