2-ancestry mixing, positive discriminant

Percentage Accurate: 43.4% → 95.7%

Time: 13.0s

Alternatives: 4

Speedup: 4.1×

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: 43.4% 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, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.9%

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

3. Taylor expanded in g around inf

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

   1. associate-*r* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\frac{-1}{2}}\right), \color{blue}{\left(\sqrt[3]{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}}\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right), \left(\sqrt[3]{\color{blue}{2}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{g}{a}\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right), \left(\sqrt[3]{2}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right), \left(\sqrt[3]{2}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right), \left(\sqrt[3]{2}\right)\right) \]

   7. cbrt-lowering-cbrt.f64 71.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right), \mathsf{cbrt.f64}\left(2\right)\right) \]

5. Simplified 71.4%

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

   1. associate-*l* N/A

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

   2. pow1/3 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \left({\frac{1}{4}}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot \sqrt[3]{2}\right) \]

   6. metadata-eval N/A

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

   7. pow-prod-down N/A

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

   8. sqr-pow N/A

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

   9. pow1/3 N/A

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

   10. cbrt-unprod N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\frac{1}{2} \cdot 2} \]

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. frac-2neg N/A

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

   15. div-inv N/A

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

7. Applied egg-rr 96.0%

   \[\leadsto \color{blue}{\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{0 - a}}} \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\left(\frac{\mathsf{neg}\left(-1\right)}{0 - a}\right)\right)\right) \]

   2. sub0-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\left(\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(a\right)}\right)\right)\right) \]

   3. frac-2neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\left(\frac{-1}{a}\right)\right)\right) \]

   4. /-lowering-/.f64 96.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, a\right)\right)\right) \]

9. Applied egg-rr 96.0%

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

Alternative 2: 73.9% accurate, 4.0× 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 39.9%

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

3. Taylor expanded in g around inf

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

   1. associate-*r* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\frac{-1}{2}}\right), \color{blue}{\left(\sqrt[3]{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}}\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right), \left(\sqrt[3]{\color{blue}{2}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{g}{a}\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right), \left(\sqrt[3]{2}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right), \left(\sqrt[3]{2}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right), \left(\sqrt[3]{2}\right)\right) \]

   7. cbrt-lowering-cbrt.f64 71.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right), \mathsf{cbrt.f64}\left(2\right)\right) \]

5. Simplified 71.4%

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

   1. associate-*l* N/A

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

   2. pow1/3 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \left({\frac{1}{4}}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot \sqrt[3]{2}\right) \]

   6. metadata-eval N/A

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

   7. pow-prod-down N/A

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

   8. sqr-pow N/A

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

   9. pow1/3 N/A

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

   10. cbrt-unprod N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\frac{1}{2} \cdot 2} \]

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. clear-num N/A

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

   15. frac-2neg N/A

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

   16. metadata-eval N/A

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

   17. cbrt-div N/A

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

7. Applied egg-rr 72.8%

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

8. Final simplification 72.5%

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

Alternative 3: 73.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.9%

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

3. Taylor expanded in g around inf

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

   1. associate-*r* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\frac{-1}{2}}\right), \color{blue}{\left(\sqrt[3]{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}}\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right), \left(\sqrt[3]{\color{blue}{2}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{g}{a}\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right), \left(\sqrt[3]{2}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right), \left(\sqrt[3]{2}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right), \left(\sqrt[3]{2}\right)\right) \]

   7. cbrt-lowering-cbrt.f64 71.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right), \mathsf{cbrt.f64}\left(2\right)\right) \]

5. Simplified 71.4%

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

   1. associate-*l* N/A

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

   2. pow1/3 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \left({\frac{1}{4}}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot \sqrt[3]{2}\right) \]

   6. metadata-eval N/A

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

   7. pow-prod-down N/A

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

   8. sqr-pow N/A

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

   9. pow1/3 N/A

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

   10. cbrt-unprod N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\frac{1}{2} \cdot 2} \]

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

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

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g}{a}\right)\right) \]

   15. /-lowering-/.f64 1.4%

       \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right) \]

7. Applied egg-rr 1.4%

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

   1. div-inv N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \frac{1}{a}\right)\right) \]

   2. inv-pow N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {a}^{-1}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {a}^{\left(2 \cdot \frac{-1}{2}\right)}\right)\right) \]

   4. pow-pow N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left({a}^{2}\right)}^{\frac{-1}{2}}\right)\right) \]

   5. pow2 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(a \cdot a\right)}^{\frac{-1}{2}}\right)\right) \]

   6. sqr-neg N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(\left(\mathsf{neg}\left(a\right)\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right)}^{\frac{-1}{2}}\right)\right) \]

   7. sub0-neg N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(\left(0 - a\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right)}^{\frac{-1}{2}}\right)\right) \]

   8. sub0-neg N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(\left(0 - a\right) \cdot \left(0 - a\right)\right)}^{\frac{-1}{2}}\right)\right) \]

   9. pow-prod-down N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \left({\left(0 - a\right)}^{\frac{-1}{2}} \cdot {\left(0 - a\right)}^{\frac{-1}{2}}\right)\right)\right) \]

   10. pow-prod-up N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(0 - a\right)}^{\left(\frac{-1}{2} + \frac{-1}{2}\right)}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(0 - a\right)}^{-1}\right)\right) \]

   12. inv-pow N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \frac{1}{0 - a}\right)\right) \]

   13. sub0-neg N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \frac{1}{\mathsf{neg}\left(a\right)}\right)\right) \]

   14. distribute-frac-neg2 N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)\right) \]

   15. distribute-rgt-neg-out N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(g \cdot \frac{1}{a}\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(\frac{1}{a} \cdot g\right)\right)\right) \]

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

       \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\left(\frac{1}{a} \cdot g\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\left(g \cdot \frac{1}{a}\right)\right)\right) \]

   19. div-inv N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\left(\frac{g}{a}\right)\right)\right) \]

   20. /-lowering-/.f64 72.1%

       \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(g, a\right)\right)\right) \]

9. Applied egg-rr 72.1%

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

10. Final simplification 72.1%

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

Alternative 4: 1.4% accurate, 4.2× 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 39.9%

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

3. Taylor expanded in g around inf

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

   1. associate-*r* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\frac{-1}{2}}\right), \color{blue}{\left(\sqrt[3]{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}}\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right), \left(\sqrt[3]{\color{blue}{2}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{g}{a}\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right), \left(\sqrt[3]{2}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right), \left(\sqrt[3]{2}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right), \left(\sqrt[3]{2}\right)\right) \]

   7. cbrt-lowering-cbrt.f64 71.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right), \mathsf{cbrt.f64}\left(2\right)\right) \]

5. Simplified 71.4%

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

   1. associate-*l* N/A

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

   2. pow1/3 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \left({\frac{1}{4}}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot \sqrt[3]{2}\right) \]

   6. metadata-eval N/A

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

   7. pow-prod-down N/A

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

   8. sqr-pow N/A

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

   9. pow1/3 N/A

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

   10. cbrt-unprod N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\frac{1}{2} \cdot 2} \]

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

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

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g}{a}\right)\right) \]

   15. /-lowering-/.f64 1.4%

       \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right) \]

7. Applied egg-rr 1.4%

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

Reproduce

herbie shell --seed 2024191 
(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))))))))