2-ancestry mixing, positive discriminant

Percentage Accurate: 44.0% → 95.8%

Time: 12.5s

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: 44.0% 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, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.7%

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

3. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-rgt-neg-in N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. cbrt-lowering-cbrt.f64 78.2%

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

5. Simplified 78.2%

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

   1. cbrt-div N/A

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

   2. cbrt-unprod N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

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

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

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

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

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

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

   10. cbrt-lowering-cbrt.f64 95.4%

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

7. Applied egg-rr 95.4%

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

   1. *-commutative N/A

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

   2. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt[3]{g}\right)\right), \mathsf{cbrt.f64}\left(\color{blue}{a}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt[3]{g}\right)\right), \mathsf{cbrt.f64}\left(\color{blue}{a}\right)\right) \]

   4. cbrt-lowering-cbrt.f64 95.4%

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

9. Applied egg-rr 95.4%

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

10. Final simplification 95.4%

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

Alternative 2: 74.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.7%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 78.0%

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

5. Simplified 78.0%

   \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-0.5}\right) \cdot \sqrt[3]{2}} \]
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 79.9%

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

8. Final simplification 79.9%

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

Alternative 3: 73.3% accurate, 4.1× speedup

Math

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

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(0.0 - Float64(g / a)))
end

Wolfram

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

Derivation

1. Initial program 44.7%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 78.0%

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

5. Simplified 78.0%

   \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-0.5}\right) \cdot \sqrt[3]{2}} \]
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. frac-2neg N/A

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

   2. neg-mul-1 N/A

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

   3. associate-/l* N/A

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

   4. distribute-neg-frac2 N/A

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

   5. sub0-neg N/A

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

   6. metadata-eval N/A

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

   7. flip3-- N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{{0}^{3} - {\left(\frac{g}{a}\right)}^{3}}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{0 - {\left(\frac{g}{a}\right)}^{3}}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   9. sub0-neg N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{\mathsf{neg}\left({\left(\frac{g}{a}\right)}^{3}\right)}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   10. cube-neg N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{{\left(\mathsf{neg}\left(\frac{g}{a}\right)\right)}^{3}}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   11. distribute-neg-frac N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{{\left(\frac{\mathsf{neg}\left(g\right)}{a}\right)}^{3}}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   12. cube-div N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{\frac{{\left(\mathsf{neg}\left(g\right)\right)}^{3}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{\frac{{\left(\mathsf{neg}\left(g\right)\right)}^{\left(2 \cdot \frac{3}{2}\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{\frac{{\left(\mathsf{neg}\left(g\right)\right)}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   15. pow-pow N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{\frac{{\left({\left(\mathsf{neg}\left(g\right)\right)}^{2}\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   16. pow2 N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{\frac{{\left(\left(\mathsf{neg}\left(g\right)\right) \cdot \left(\mathsf{neg}\left(g\right)\right)\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   17. sqr-neg N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{\frac{{\left(g \cdot g\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   18. pow2 N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{\frac{{\left({g}^{2}\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   19. pow-pow N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{\frac{{g}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{\frac{{g}^{\left(2 \cdot \frac{3}{2}\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   21. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{\frac{{g}^{3}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   22. cube-div N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{{\left(\frac{g}{a}\right)}^{3}}{0 \cdot 0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

   23. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left({-1}^{1} \cdot \frac{{\left(\frac{g}{a}\right)}^{3}}{0 + \left(\frac{g}{a} \cdot \frac{g}{a} + 0 \cdot \frac{g}{a}\right)}\right)\right) \]

9. Applied egg-rr 78.9%

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

10. Final simplification 78.9%

    \[\leadsto \sqrt[3]{0 - \frac{g}{a}} \]
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 44.7%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 78.0%

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

5. Simplified 78.0%

   \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-0.5}\right) \cdot \sqrt[3]{2}} \]
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 2024192 
(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))))))))