2-ancestry mixing, positive discriminant

Percentage Accurate: 44.2% → 96.1%

Time: 17.7s

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: 44.2% 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: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \left({2}^{0.3333333333333333} \cdot \sqrt[3]{g}\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (cbrt (* (- g g) (/ -0.5 a)))
  (* (cbrt (/ -0.5 a)) (* (pow 2.0 0.3333333333333333) (cbrt g)))))

C

double code(double g, double h, double a) {
	return cbrt(((g - g) * (-0.5 / a))) + (cbrt((-0.5 / a)) * (pow(2.0, 0.3333333333333333) * cbrt(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)) * (Math.pow(2.0, 0.3333333333333333) * Math.cbrt(g)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(g - g) * Float64(-0.5 / a))) + Float64(cbrt(Float64(-0.5 / a)) * Float64((2.0 ^ 0.3333333333333333) * cbrt(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[(N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[2.0, 0.3333333333333333], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.1%

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

3. Simplified 44.1%

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

4. Taylor expanded in g around inf 23.8%

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

5. Taylor expanded in g around inf 76.1%

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

   1. pow1/3 32.4%

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

   2. *-commutative 32.4%

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

   3. unpow-prod-down 19.4%

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

   4. pow1/3 42.3%

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

   5. pow1/3 95.8%

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

7. Applied egg-rr 95.8%

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

   1. pow1/3 42.3%

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

   2. count-2 42.3%

      \[\leadsto \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot {\color{blue}{\left(2 \cdot g\right)}}^{0.3333333333333333} \]

   3. unpow-prod-down 42.3%

      \[\leadsto \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \color{blue}{\left({2}^{0.3333333333333333} \cdot {g}^{0.3333333333333333}\right)} \]

   4. pow1/3 95.8%

      \[\leadsto \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \left({2}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{g}}\right) \]

9. Applied egg-rr 95.8%

   \[\leadsto \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \color{blue}{\left({2}^{0.3333333333333333} \cdot \sqrt[3]{g}\right)} \]

10. Final simplification 95.8%

    \[\leadsto \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \left({2}^{0.3333333333333333} \cdot \sqrt[3]{g}\right) \]

Alternative 2: 96.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 44.1%

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

3. Simplified 44.1%

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

4. Taylor expanded in g around inf 23.8%

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

5. Taylor expanded in g around inf 76.1%

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

   1. pow1/3 32.4%

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

   2. *-commutative 32.4%

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

   3. unpow-prod-down 19.4%

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

   4. pow1/3 42.3%

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

   5. pow1/3 95.8%

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

7. Applied egg-rr 95.8%

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

8. Final simplification 95.8%

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

Alternative 3: 74.3% 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 44.1%

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

3. Simplified 44.1%

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

4. Taylor expanded in g around inf 23.8%

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

5. Taylor expanded in g around inf 76.1%

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

6. Taylor expanded in g around 0 76.1%

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

   1. associate-*r/ 76.1%

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

   2. neg-mul-1 76.1%

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

8. Simplified 76.1%

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

9. Final simplification 76.1%

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

Alternative 4: 4.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.1%

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

   1. associate-/r* 44.1%

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

   2. metadata-eval 44.1%

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

   3. +-commutative 44.1%

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

   4. unsub-neg 44.1%

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

   5. fma-neg 44.1%

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

   6. sub-neg 44.1%

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

   7. distribute-neg-out 44.1%

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

   8. neg-mul-1 44.1%

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

   9. associate-*r* 44.1%

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

3. Simplified 44.1%

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

   1. cbrt-div 46.3%

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

   2. fma-udef 46.3%

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

   3. add-sqr-sqrt 23.9%

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

   4. hypot-def 25.1%

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

   5. add-sqr-sqrt 25.1%

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

   6. sqrt-unprod 47.5%

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

   7. sqr-neg 47.5%

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

   8. sqrt-unprod 47.9%

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

   9. add-sqr-sqrt 47.9%

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

   10. sqrt-prod 21.4%

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

   11. add-sqr-sqrt 47.9%

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

   12. div-inv 47.9%

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

   13. metadata-eval 47.9%

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

5. Applied egg-rr 47.9%

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

6. Taylor expanded in g around inf 46.1%

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

   1. unpow2 46.1%

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

8. Simplified 46.1%

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

9. Taylor expanded in g around 0 2.4%

   \[\leadsto \color{blue}{{\left(\frac{1 \cdot h}{a}\right)}^{0.3333333333333333} \cdot \frac{1}{\sqrt[3]{-2}}} \]
10. Step-by-step derivation

    1. associate-*r/ 2.4%

       \[\leadsto \color{blue}{\frac{{\left(\frac{1 \cdot h}{a}\right)}^{0.3333333333333333} \cdot 1}{\sqrt[3]{-2}}} \]

    2. *-lft-identity 2.4%

       \[\leadsto \frac{{\left(\frac{\color{blue}{h}}{a}\right)}^{0.3333333333333333} \cdot 1}{\sqrt[3]{-2}} \]

    3. *-rgt-identity 2.4%

       \[\leadsto \frac{\color{blue}{{\left(\frac{h}{a}\right)}^{0.3333333333333333}}}{\sqrt[3]{-2}} \]

    4. unpow1/3 4.2%

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

11. Simplified 4.2%

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{h}{a}}}{\sqrt[3]{-2}}} \]
12. Step-by-step derivation

    1. cbrt-undiv 4.2%

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

13. Applied egg-rr 4.2%

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

14. Final simplification 4.2%

    \[\leadsto \sqrt[3]{\frac{\frac{h}{a}}{-2}} \]

Reproduce

herbie shell --seed 2023230 
(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))))))))