2-ancestry mixing, positive discriminant

Percentage Accurate: 43.2% → 96.0%

Time: 15.2s

Alternatives: 5

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: 43.2% 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: 96.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{g}}{0 - \sqrt[3]{a}} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (/ (cbrt g) (- 0.0 (cbrt a))))

C

double code(double g, double h, double a) {
	return cbrt(g) / (0.0 - cbrt(a));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(g) / (0.0 - Math.cbrt(a));
}

Julia

function code(g, h, a)
	return Float64(cbrt(g) / Float64(0.0 - cbrt(a)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[g, 1/3], $MachinePrecision] / N[(0.0 - N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.3%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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. neg-sub0 N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{0 - \frac{g}{a}}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{0 - \frac{g}{a}}} \]

   4. /-lowering-/.f64 27.4

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{0 - \color{blue}{\frac{g}{a}}} \]

5. Simplified 27.4%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{0 - \frac{g}{a}}} \]

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. *-lowering-*.f64 N/A

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

   2. cbrt-lowering-cbrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. cbrt-lowering-cbrt.f64 75.8

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

8. Simplified 75.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
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. associate-*l/ N/A

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

   3. /-lowering-/.f64 N/A

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

   4. pow1/3 N/A

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

   5. pow1/3 N/A

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

   6. pow-prod-down N/A

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

   7. pow-lowering-pow.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. cbrt-lowering-cbrt.f64 47.1

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

10. Applied egg-rr 47.1%

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

11. Taylor expanded in g around -inf

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

    1. mul-1-neg N/A

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

    2. neg-sub0 N/A

       \[\leadsto \frac{\color{blue}{0 - \sqrt[3]{g}}}{\sqrt[3]{a}} \]

    3. --lowering--.f64 N/A

       \[\leadsto \frac{\color{blue}{0 - \sqrt[3]{g}}}{\sqrt[3]{a}} \]

    4. cbrt-lowering-cbrt.f64 96.2

       \[\leadsto \frac{0 - \color{blue}{\sqrt[3]{g}}}{\sqrt[3]{a}} \]

13. Simplified 96.2%

    \[\leadsto \frac{\color{blue}{0 - \sqrt[3]{g}}}{\sqrt[3]{a}} \]

14. Final simplification 96.2%

    \[\leadsto \frac{\sqrt[3]{g}}{0 - \sqrt[3]{a}} \]
15. Add Preprocessing

Alternative 2: 73.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt[3]{0 - \frac{a}{g}}} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (/ 1.0 (cbrt (- 0.0 (/ a g)))))

C

double code(double g, double h, double a) {
	return 1.0 / cbrt((0.0 - (a / g)));
}

Java

public static double code(double g, double h, double a) {
	return 1.0 / Math.cbrt((0.0 - (a / g)));
}

Julia

function code(g, h, a)
	return Float64(1.0 / cbrt(Float64(0.0 - Float64(a / g))))
end

Wolfram

code[g_, h_, a_] := N[(1.0 / N[Power[N[(0.0 - N[(a / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.3%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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. neg-sub0 N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{0 - \frac{g}{a}}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{0 - \frac{g}{a}}} \]

   4. /-lowering-/.f64 27.4

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{0 - \color{blue}{\frac{g}{a}}} \]

5. Simplified 27.4%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{0 - \frac{g}{a}}} \]

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. *-lowering-*.f64 N/A

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

   2. cbrt-lowering-cbrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. cbrt-lowering-cbrt.f64 75.8

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

8. Simplified 75.8%

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

   1. *-commutative N/A

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

   2. cbrt-unprod N/A

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

   3. neg-mul-1 N/A

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

   4. clear-num N/A

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

   5. distribute-neg-frac2 N/A

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

   6. cbrt-div N/A

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

   7. metadata-eval N/A

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

   8. /-lowering-/.f64 N/A

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

   9. cbrt-lowering-cbrt.f64 N/A

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

   10. neg-lowering-neg.f64 N/A

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

   11. /-lowering-/.f64 76.1

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

10. Applied egg-rr 76.1%

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

11. Final simplification 75.7%

    \[\leadsto \frac{1}{\sqrt[3]{0 - \frac{a}{g}}} \]
12. Add Preprocessing

Alternative 3: 73.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ 0 - \sqrt[3]{\frac{g}{a}} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (- 0.0 (cbrt (/ g a))))

C

double code(double g, double h, double a) {
	return 0.0 - cbrt((g / a));
}

Java

public static double code(double g, double h, double a) {
	return 0.0 - Math.cbrt((g / a));
}

Julia

function code(g, h, a)
	return Float64(0.0 - cbrt(Float64(g / a)))
end

Wolfram

code[g_, h_, a_] := N[(0.0 - N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.3%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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. neg-sub0 N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{0 - \frac{g}{a}}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{0 - \frac{g}{a}}} \]

   4. /-lowering-/.f64 27.4

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{0 - \color{blue}{\frac{g}{a}}} \]

5. Simplified 27.4%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{0 - \frac{g}{a}}} \]

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. *-lowering-*.f64 N/A

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

   2. cbrt-lowering-cbrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. cbrt-lowering-cbrt.f64 75.8

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

8. Simplified 75.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}} \]
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. associate-*l/ N/A

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

   3. /-lowering-/.f64 N/A

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

   4. pow1/3 N/A

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

   5. pow1/3 N/A

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

   6. pow-prod-down N/A

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

   7. pow-lowering-pow.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. cbrt-lowering-cbrt.f64 47.1

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

10. Applied egg-rr 47.1%

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

11. Taylor expanded in g around -inf

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

    1. mul-1-neg N/A

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

    2. neg-sub0 N/A

       \[\leadsto \color{blue}{0 - \sqrt[3]{\frac{g}{a}}} \]

    3. --lowering--.f64 N/A

       \[\leadsto \color{blue}{0 - \sqrt[3]{\frac{g}{a}}} \]

    4. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto 0 - \color{blue}{\sqrt[3]{\frac{g}{a}}} \]

    5. /-lowering-/.f64 75.8

       \[\leadsto 0 - \sqrt[3]{\color{blue}{\frac{g}{a}}} \]

13. Simplified 75.8%

    \[\leadsto \color{blue}{0 - \sqrt[3]{\frac{g}{a}}} \]
14. Add Preprocessing

Alternative 4: 1.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.3%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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. neg-sub0 N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{0 - \frac{g}{a}}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{0 - \frac{g}{a}}} \]

   4. /-lowering-/.f64 27.4

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{0 - \color{blue}{\frac{g}{a}}} \]

5. Simplified 27.4%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{0 - \frac{g}{a}}} \]

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. *-lowering-*.f64 N/A

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

   2. cbrt-lowering-cbrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. cbrt-lowering-cbrt.f64 75.8

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

8. Simplified 75.8%

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

   1. *-commutative N/A

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

   2. cbrt-unprod N/A

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

   3. neg-mul-1 N/A

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

   4. sub0-neg N/A

      \[\leadsto \sqrt[3]{\color{blue}{0 - \frac{g}{a}}} \]

   5. pow1/3 N/A

      \[\leadsto \color{blue}{{\left(0 - \frac{g}{a}\right)}^{\frac{1}{3}}} \]

   6. sqr-pow N/A

      \[\leadsto \color{blue}{{\left(0 - \frac{g}{a}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot {\left(0 - \frac{g}{a}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}} \]

   7. pow-prod-down N/A

      \[\leadsto \color{blue}{{\left(\left(0 - \frac{g}{a}\right) \cdot \left(0 - \frac{g}{a}\right)\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}} \]

   8. sub0-neg N/A

      \[\leadsto {\left(\color{blue}{\left(\mathsf{neg}\left(\frac{g}{a}\right)\right)} \cdot \left(0 - \frac{g}{a}\right)\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} \]

   9. sub0-neg N/A

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

   10. sqr-neg N/A

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

   11. pow-prod-down N/A

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

   12. sqr-pow N/A

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

   13. pow1/3 N/A

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

   14. cbrt-lowering-cbrt.f64 N/A

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

   15. /-lowering-/.f64 1.4

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

10. Applied egg-rr 1.4%

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

Alternative 5: 0.0% accurate, 25.2× speedup

Math

\[\begin{array}{l} \\ \frac{0}{0} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (/ 0.0 0.0))

C

double code(double g, double h, double a) {
	return 0.0 / 0.0;
}

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    code = 0.0d0 / 0.0d0
end function

Java

public static double code(double g, double h, double a) {
	return 0.0 / 0.0;
}

Python

def code(g, h, a):
	return 0.0 / 0.0

Julia

function code(g, h, a)
	return Float64(0.0 / 0.0)
end

MATLAB

function tmp = code(g, h, a)
	tmp = 0.0 / 0.0;
end

Wolfram

code[g_, h_, a_] := N[(0.0 / 0.0), $MachinePrecision]

Derivation

1. Initial program 44.3%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 24.8

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

5. Simplified 24.8%

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

6. Taylor expanded in g around -inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 4.2

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

8. Simplified 4.2%

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r/ N/A

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

   5. flip-+ N/A

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

   6. +-inverses N/A

      \[\leadsto \frac{\color{blue}{0}}{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{\frac{-1}{2} \cdot \left(h \cdot h\right)}{g}} - \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{\frac{-1}{2} \cdot \left(h \cdot h\right)}{g}}} \]

   7. +-inverses N/A

      \[\leadsto \frac{0}{\color{blue}{0}} \]

   8. /-lowering-/.f64 0.0

      \[\leadsto \color{blue}{\frac{0}{0}} \]

10. Applied egg-rr 0.0%

    \[\leadsto \color{blue}{\frac{0}{0}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(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))))))))