2-ancestry mixing, positive discriminant

Percentage Accurate: 44.4% → 95.8%

Time: 9.0s

Alternatives: 5

Speedup: 3.9×

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: 44.4% 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: 95.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(-\sqrt[3]{2 \cdot g}\right) \cdot \sqrt[3]{\frac{0.5}{a}} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (* (- (cbrt (* 2.0 g))) (cbrt (/ 0.5 a))))

C

double code(double g, double h, double a) {
	return -cbrt((2.0 * g)) * cbrt((0.5 / a));
}

Java

public static double code(double g, double h, double a) {
	return -Math.cbrt((2.0 * g)) * Math.cbrt((0.5 / a));
}

Julia

function code(g, h, a)
	return Float64(Float64(-cbrt(Float64(2.0 * g))) * cbrt(Float64(0.5 / a)))
end

Wolfram

code[g_, h_, a_] := N[((-N[Power[N[(2.0 * g), $MachinePrecision], 1/3], $MachinePrecision]) * N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.4%

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

   1. lift-+.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. cbrt-unprod N/A

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

   5. pow1/3 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. cbrt-unprod N/A

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

   9. pow1/3 N/A

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

3. Applied rewrites 50.1%

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

4. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 95.4

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

6. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 95.4

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

8. Applied rewrites 95.7%

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

Alternative 2: 95.8% accurate, 2.1× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 44.4%

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

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 95.1

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

4. Applied rewrites 95.1%

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

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

   3. lower-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. cbrt-unprod N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. cbrt-neg N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lift-cbrt.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. cbrt-neg N/A

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

   20. metadata-eval N/A

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

   21. lower-/.f64 N/A

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

6. Applied rewrites 95.8%

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

Alternative 3: 95.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.4%

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

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 95.1

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

4. Applied rewrites 95.1%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}}} \]
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. lift-*.f64 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. cbrt-unprod N/A

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

   8. metadata-eval N/A

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

   9. lift-cbrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. cbrt-neg N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lift-cbrt.f64 N/A

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

   15. associate-/l* N/A

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

6. Applied rewrites 95.8%

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

Alternative 4: 73.5% accurate, 3.9× 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 Float64(-cbrt(Float64(g / a)))
end

Wolfram

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

Derivation

1. Initial program 44.4%

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

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 95.1

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

4. Applied rewrites 95.1%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-log.f64 44.7

      \[\leadsto \frac{e^{\log g \cdot 0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}} \]

6. Applied rewrites 44.7%

   \[\leadsto \frac{e^{\log g \cdot 0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg N/A

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

8. Applied rewrites 73.5%

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

Alternative 5: 1.4% accurate, 4.1× 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 44.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. frac-times N/A

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

   11. lower-/.f64 N/A

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

3. Applied rewrites 12.6%

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

4. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-cbrt.f64 1.4

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

6. Applied rewrites 1.4%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. lift-/.f64 N/A

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

   6. lower-cbrt.f64 1.4

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

8. Applied rewrites 1.4%

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

Reproduce

herbie shell --seed 2025149 
(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))))))))