2-ancestry mixing, positive discriminant

Percentage Accurate: 43.6% → 95.7%

Time: 22.7s

Alternatives: 5

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 43.6% 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

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

FPCore

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

C

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + Float64(cbrt(g) * cbrt(Float64(-1.0 / 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[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.9%

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

3. Simplified 42.2%

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

5. Taylor expanded in g around inf 21.3%

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

6. Taylor expanded in g around inf 73.3%

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

   1. mul-1-neg 73.3%

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

   2. distribute-neg-frac2 73.3%

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

8. Simplified 73.3%

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

   1. pow1/3 34.1%

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

   2. div-inv 34.0%

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

   3. unpow-prod-down 18.9%

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

   4. pow1/3 45.6%

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

10. Applied egg-rr 45.6%

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

    1. unpow1/3 96.5%

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

    2. distribute-frac-neg2 96.5%

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

    3. distribute-neg-frac 96.5%

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

    4. metadata-eval 96.5%

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

12. Simplified 96.5%

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

Alternative 2: 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(Float64(-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 41.9%

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

3. Simplified 42.2%

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

5. Taylor expanded in g around inf 21.3%

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

6. Taylor expanded in g around inf 73.3%

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

   1. mul-1-neg 73.3%

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

   2. distribute-neg-frac2 73.3%

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

8. Simplified 73.3%

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

   1. frac-2neg 73.3%

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

   2. cbrt-div 96.4%

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

10. Applied egg-rr 96.4%

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

    1. remove-double-neg 96.4%

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

12. Simplified 96.4%

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

Alternative 3: 73.5% 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 / Float64(-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 41.9%

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

3. Simplified 42.2%

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

5. Taylor expanded in g around inf 21.3%

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

6. Taylor expanded in g around inf 73.3%

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

   1. mul-1-neg 73.3%

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

   2. distribute-neg-frac2 73.3%

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

8. Simplified 73.3%

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

   1. clear-num 73.1%

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

   2. cbrt-div 73.8%

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

   3. metadata-eval 73.8%

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

10. Applied egg-rr 73.8%

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

11. Final simplification 73.8%

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

Alternative 4: 72.8% accurate, 2.0× 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 / Float64(-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 41.9%

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

3. Simplified 42.2%

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

5. Taylor expanded in g around inf 21.3%

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

6. Taylor expanded in g around inf 73.3%

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

   1. mul-1-neg 73.3%

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

   2. distribute-neg-frac2 73.3%

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

8. Simplified 73.3%

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

Alternative 5: 1.4% 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 41.9%

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

3. Simplified 42.2%

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

5. Taylor expanded in g around inf 21.3%

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

6. Taylor expanded in g around inf 73.3%

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

   1. mul-1-neg 73.3%

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

   2. distribute-neg-frac2 73.3%

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

8. Simplified 73.3%

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

   1. pow1/3 34.1%

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

   2. div-inv 34.0%

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

   3. unpow-prod-down 18.9%

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

   4. pow1/3 45.6%

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

10. Applied egg-rr 45.6%

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

    1. unpow1/3 96.5%

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

    2. distribute-frac-neg2 96.5%

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

    3. distribute-neg-frac 96.5%

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

    4. metadata-eval 96.5%

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

12. Simplified 96.5%

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

    1. cbrt-unprod 73.3%

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

    2. frac-2neg 73.3%

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

    3. metadata-eval 73.3%

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

    4. div-inv 73.3%

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

    5. expm1-log1p-u 47.7%

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

    6. expm1-undefine 24.3%

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

    7. sub-neg 24.3%

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

    8. metadata-eval 24.3%

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

    9. log1p-undefine 24.3%

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

    10. rem-exp-log 49.9%

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

    11. +-commutative 49.9%

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

    12. add-sqr-sqrt 23.4%

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

    13. sqrt-unprod 14.1%

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

    14. distribute-frac-neg2 14.1%

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

    15. distribute-frac-neg2 14.1%

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

    16. sqr-neg 14.1%

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

    17. sqrt-unprod 1.0%

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

    18. add-sqr-sqrt 1.8%

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

14. Applied egg-rr 1.8%

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

    1. associate-+l+ 1.4%

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

    2. metadata-eval 1.4%

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

    3. +-rgt-identity 1.4%

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

16. Simplified 1.4%

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

Reproduce

herbie shell --seed 2024097 
(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))))))))