2-ancestry mixing, positive discriminant

Percentage Accurate: 44.1% → 95.8%

Time: 15.7s

Alternatives: 6

Speedup: 2.6×

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: 44.1% 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: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.8%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 27.3

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

5. Simplified 27.3%

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

6. Taylor expanded in g around inf

   \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 72.0

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

8. Simplified 72.0%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
9. Step-by-step derivation

   1. cbrt-div N/A

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

   2. lift-cbrt.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-cbrt.f64 N/A

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

   5. cbrt-prod N/A

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

   6. *-commutative N/A

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

   7. neg-mul-1 N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\mathsf{neg}\left(g\right)}}}{\sqrt[3]{a}} \]

   8. lift-neg.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\mathsf{neg}\left(g\right)}}}{\sqrt[3]{a}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a}}} \]

   10. lower-cbrt.f64 N/A

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\mathsf{neg}\left(g\right)}}}{\sqrt[3]{a}} \]

   11. lower-cbrt.f64 96.5

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

10. Applied egg-rr 96.5%

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

Alternative 2: 81.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 10^{-233}:\\ \;\;\;\;\frac{1}{\sqrt[3]{-\frac{a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (/ 1.0 (* a 2.0)) 1e-233)
   (/ 1.0 (cbrt (- (/ a g))))
   (* (cbrt (- g)) (pow a -0.3333333333333333))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (a * 2.0)) <= 1e-233) {
		tmp = 1.0 / cbrt(-(a / g));
	} else {
		tmp = cbrt(-g) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (a * 2.0)) <= 1e-233) {
		tmp = 1.0 / Math.cbrt(-(a / g));
	} else {
		tmp = Math.cbrt(-g) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(1.0 / Float64(a * 2.0)) <= 1e-233)
		tmp = Float64(1.0 / cbrt(Float64(-Float64(a / g))));
	else
		tmp = Float64(cbrt(Float64(-g)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 1e-233], N[(1.0 / N[Power[(-N[(a / g), $MachinePrecision]), 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[(-g), 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 9.99999999999999958e-234

   1. Initial program 43.1%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 26.9

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

   5. Simplified 26.9%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 74.3

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

   8. Simplified 74.3%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
   9. Step-by-step derivation

      1. cbrt-div N/A

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

      2. lift-cbrt.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-cbrt.f64 N/A

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

      5. cbrt-prod N/A

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

      6. *-commutative N/A

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

      7. neg-mul-1 N/A

         \[\leadsto \frac{\sqrt[3]{\color{blue}{\mathsf{neg}\left(g\right)}}}{\sqrt[3]{a}} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt[3]{\color{blue}{\mathsf{neg}\left(g\right)}}}{\sqrt[3]{a}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a}}} \]

      10. lower-cbrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt[3]{\mathsf{neg}\left(g\right)}}}{\sqrt[3]{a}} \]

      11. lower-cbrt.f64 95.8

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

   10. Applied egg-rr 95.8%

       \[\leadsto \color{blue}{\frac{\sqrt[3]{-g}}{\sqrt[3]{a}}} \]
   11. Step-by-step derivation

       1. lift-neg.f64 N/A

          \[\leadsto \frac{\sqrt[3]{\color{blue}{\mathsf{neg}\left(g\right)}}}{\sqrt[3]{a}} \]

       2. lift-cbrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt[3]{\mathsf{neg}\left(g\right)}}}{\sqrt[3]{a}} \]

       3. lift-cbrt.f64 N/A

          \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\color{blue}{\sqrt[3]{a}}} \]

       4. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{\mathsf{neg}\left(g\right)}}}} \]

       5. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{\mathsf{neg}\left(g\right)}}}} \]

       6. lift-cbrt.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt[3]{a}}}{\sqrt[3]{\mathsf{neg}\left(g\right)}}} \]

       7. lift-cbrt.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt[3]{a}}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(g\right)}}}} \]

       8. cbrt-undiv N/A

          \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\frac{a}{\mathsf{neg}\left(g\right)}}}} \]

       9. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{a}{\color{blue}{\mathsf{neg}\left(g\right)}}}} \]

       10. distribute-neg-frac2 N/A

           \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\mathsf{neg}\left(\frac{a}{g}\right)}}} \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{1}{\sqrt[3]{\mathsf{neg}\left(\color{blue}{\frac{a}{g}}\right)}} \]

       12. lower-cbrt.f64 N/A

           \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(\frac{a}{g}\right)}}} \]

       13. lift-/.f64 N/A

           \[\leadsto \frac{1}{\sqrt[3]{\mathsf{neg}\left(\color{blue}{\frac{a}{g}}\right)}} \]

       14. distribute-neg-frac2 N/A

           \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\frac{a}{\mathsf{neg}\left(g\right)}}}} \]

       15. lift-neg.f64 N/A

           \[\leadsto \frac{1}{\sqrt[3]{\frac{a}{\color{blue}{\mathsf{neg}\left(g\right)}}}} \]

       16. lower-/.f64 75.0

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

   12. Applied egg-rr 75.0%

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

   if 9.99999999999999958e-234 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 45.3%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 28.3

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

   5. Simplified 28.3%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 73.5

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

   8. Simplified 73.5%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. cbrt-unprod N/A

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

      8. neg-mul-1 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\mathsf{neg}\left(\frac{g}{a}\right)}} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt[3]{\mathsf{neg}\left(\color{blue}{\frac{g}{a}}\right)} \]

      10. distribute-frac-neg2 N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\mathsf{neg}\left(a\right)}}} \]

      11. lift-neg.f64 N/A

          \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{\mathsf{neg}\left(a\right)}}} \]

      12. frac-2neg N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(a\right)\right)\right)}}} \]

      13. lift-neg.f64 N/A

          \[\leadsto \sqrt[3]{\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(a\right)\right)\right)}} \]

      14. div-inv N/A

          \[\leadsto \sqrt[3]{\color{blue}{\left(\mathsf{neg}\left(g\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(a\right)\right)\right)}}} \]

      15. lift-neg.f64 N/A

          \[\leadsto \sqrt[3]{\left(\mathsf{neg}\left(g\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)}} \]

      16. remove-double-neg N/A

          \[\leadsto \sqrt[3]{\left(\mathsf{neg}\left(g\right)\right) \cdot \frac{1}{\color{blue}{a}}} \]

      17. cbrt-prod N/A

          \[\leadsto \color{blue}{\sqrt[3]{\mathsf{neg}\left(g\right)} \cdot \sqrt[3]{\frac{1}{a}}} \]

      18. pow1/3 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cbrt.f64 N/A

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

      21. inv-pow N/A

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

      22. pow-pow N/A

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

      23. lower-pow.f64 N/A

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

      24. metadata-eval 89.9

          \[\leadsto \sqrt[3]{-g} \cdot {a}^{\color{blue}{-0.3333333333333333}} \]

   10. Applied egg-rr 89.9%

       \[\leadsto \color{blue}{\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 10^{-233}:\\ \;\;\;\;\frac{1}{\sqrt[3]{-\frac{a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024218 
(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))))))))