2-ancestry mixing, positive discriminant

Percentage Accurate: 45.0% → 95.8%

Time: 7.3s

Alternatives: 4

Speedup: 3.9×

Specification

Math

\[\begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{t\_0 \cdot \left(\left(-g\right) - t\_1\right)} \end{array} \]

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: 45.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{t\_0 \cdot \left(\left(-g\right) - t\_1\right)} \end{array} \]

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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.0%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.2%

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

4. Applied rewrites 95.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. cbrt-unprod N/A

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

   9. metadata-eval N/A

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

   10. lift-cbrt.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. cbrt-neg N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lift-cbrt.f64 N/A

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

   17. lower-/.f64 95.8%

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

6. Applied rewrites 95.8%

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

Alternative 2: 95.8% accurate, 2.3× speedup

Math

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

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(g) / Float64(-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 45.0%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.2%

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

4. Applied rewrites 95.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. cbrt-unprod N/A

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

   9. metadata-eval N/A

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

   10. lift-cbrt.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. cbrt-neg N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lift-cbrt.f64 N/A

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

   17. lower-/.f64 95.8%

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

6. Applied rewrites 95.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. mult-flip-rev N/A

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

   7. lower-/.f64 N/A

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

   8. lower-neg.f64 95.8%

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

8. Applied rewrites 95.8%

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

Alternative 3: 74.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.0%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.2%

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

4. Applied rewrites 95.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. cbrt-unprod N/A

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

   9. metadata-eval N/A

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

   10. lift-cbrt.f64 N/A

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

   11. cbrt-undiv N/A

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

   12. lift-cbrt.f64 N/A

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

   13. cbrt-unprod N/A

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

   14. lower-cbrt.f64 N/A

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

   15. frac-2neg-rev N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval N/A

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

   19. frac-2neg-rev N/A

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

   20. lower-/.f64 74.6%

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

6. Applied rewrites 74.6%

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

Alternative 4: 74.6% accurate, 3.9× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 45.0%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.2%

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

4. Applied rewrites 95.2%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. cbrt-unprod N/A

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

   11. metadata-eval N/A

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

   12. cbrt-unprod N/A

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

   13. *-commutative N/A

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

   14. mul-1-neg N/A

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

   15. lift-neg.f64 N/A

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

   16. lift-cbrt.f64 N/A

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

   17. cbrt-neg-rev N/A

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

   18. cbrt-undiv N/A

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

   19. lift-neg.f64 N/A

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

   20. frac-2neg N/A

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

   21. lift-/.f64 N/A

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

6. Applied rewrites 74.6%

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

Reproduce

herbie shell --seed 2025185 
(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))))))))