2-ancestry mixing, positive discriminant

Percentage Accurate: 43.8% → 95.7%

Time: 15.0s

Alternatives: 4

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: 43.8% 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: 95.7% 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(g) / cbrt(Float64(-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 47.2%

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

3. Simplified 47.2%

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

4. Taylor expanded in g around inf 22.2%

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

5. Taylor expanded in g around -inf 0.0%

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

   1. *-commutative 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 77.7%

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

   4. metadata-eval 77.7%

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

7. Simplified 77.7%

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

   1. *-un-lft-identity 77.7%

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

   2. associate-/l* 77.7%

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

9. Applied egg-rr 77.7%

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

    1. *-lft-identity 77.7%

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

    2. *-commutative 77.7%

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

    3. associate-/r/ 77.7%

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

    4. associate-*l* 77.7%

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

    5. metadata-eval 77.7%

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

    6. *-commutative 77.7%

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

    7. associate-*r/ 77.7%

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

    8. mul-1-neg 77.7%

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

11. Simplified 77.7%

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

    1. frac-2neg 77.7%

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

    2. cbrt-div 95.9%

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

    3. remove-double-neg 95.9%

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

13. Applied egg-rr 95.9%

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

14. Final simplification 95.9%

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

Alternative 2: 73.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Derivation

1. Initial program 47.2%

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

3. Simplified 47.2%

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

4. Taylor expanded in g around inf 22.2%

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

5. Taylor expanded in g around inf 77.7%

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

6. Final simplification 77.7%

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

Alternative 3: 73.0% 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 47.2%

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

3. Simplified 47.2%

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

4. Taylor expanded in g around inf 22.2%

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

5. Taylor expanded in g around -inf 0.0%

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

   1. *-commutative 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 77.7%

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

   4. metadata-eval 77.7%

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

7. Simplified 77.7%

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

   1. *-un-lft-identity 77.7%

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

   2. associate-/l* 77.7%

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

9. Applied egg-rr 77.7%

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

    1. *-lft-identity 77.7%

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

    2. *-commutative 77.7%

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

    3. associate-/r/ 77.7%

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

    4. associate-*l* 77.7%

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

    5. metadata-eval 77.7%

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

    6. *-commutative 77.7%

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

    7. associate-*r/ 77.7%

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

    8. mul-1-neg 77.7%

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

11. Simplified 77.7%

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

12. Final simplification 77.7%

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

Alternative 4: 1.3% accurate, 2.1× 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(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 47.2%

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

3. Simplified 47.2%

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

4. Taylor expanded in g around inf 22.2%

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

5. Taylor expanded in g around -inf 0.0%

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

   1. *-commutative 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 77.7%

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

   4. metadata-eval 77.7%

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

7. Simplified 77.7%

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

   1. *-un-lft-identity 77.7%

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

   2. associate-/l* 77.7%

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

9. Applied egg-rr 77.7%

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

    1. *-lft-identity 77.7%

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

    2. *-commutative 77.7%

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

    3. associate-/r/ 77.7%

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

    4. associate-*l* 77.7%

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

    5. metadata-eval 77.7%

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

    6. *-commutative 77.7%

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

    7. associate-*r/ 77.7%

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

    8. mul-1-neg 77.7%

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

11. Simplified 77.7%

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

    1. cbrt-div 95.9%

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

    2. add-sqr-sqrt 50.2%

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

    3. sqrt-unprod 29.5%

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

    4. sqr-neg 29.5%

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

    5. sqrt-unprod 0.7%

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

    6. add-sqr-sqrt 1.4%

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

    7. cbrt-div 1.3%

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

    8. expm1-log1p-u 1.0%

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

    9. expm1-udef 1.3%

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

13. Applied egg-rr 1.3%

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

    1. expm1-def 1.0%

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

    2. expm1-log1p 1.3%

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

15. Simplified 1.3%

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

16. Final simplification 1.3%

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

Reproduce

herbie shell --seed 2023174 
(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))))))))