2-ancestry mixing, positive discriminant

Percentage Accurate: 43.5% → 96.0%

Time: 14.1s

Alternatives: 5

Speedup: 2.6×

Specification

Math

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

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
   (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[((-g) - t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Initial Program: 43.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
   (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[((-g) - t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 96.0% accurate, 1.4× speedup

Math

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

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 42.3%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 29.0

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

5. Applied rewrites 29.0%

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

6. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 71.7

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

8. Applied rewrites 71.7%

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

   1. lift-/.f64 N/A

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

   2. cbrt-unprod N/A

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

   3. neg-mul-1 N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-frac-neg N/A

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

   6. cbrt-div N/A

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

   7. lower-/.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-cbrt.f64 96.9

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

10. Applied rewrites 96.9%

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

Alternative 2: 89.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq -5 \cdot 10^{-309}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (/ 1.0 (* a 2.0)) -5e-309)
   (* (pow (- a) -0.3333333333333333) (cbrt g))
   (* (cbrt (- g)) (pow a -0.3333333333333333))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (a * 2.0)) <= -5e-309) {
		tmp = pow(-a, -0.3333333333333333) * cbrt(g);
	} else {
		tmp = cbrt(-g) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (a * 2.0)) <= -5e-309) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt(g);
	} else {
		tmp = Math.cbrt(-g) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(1.0 / Float64(a * 2.0)) <= -5e-309)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(g));
	else
		tmp = Float64(cbrt(Float64(-g)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -5e-309], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[(-g), 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < -4.9999999999999995e-309

   1. Initial program 43.7%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 27.5

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

   5. Applied rewrites 27.5%

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

   6. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 72.5

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

   8. Applied rewrites 72.5%

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

      1. lift-/.f64 N/A

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

      2. cbrt-unprod N/A

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

      3. neg-mul-1 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-neg.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. cbrt-prod N/A

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

      10. pow1/3 N/A

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

      11. inv-pow N/A

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

      12. pow-pow N/A

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

      13. metadata-eval N/A

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

      14. sqr-pow N/A

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

      15. lift-cbrt.f64 N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

          \[\leadsto {\left(\mathsf{neg}\left(a\right)\right)}^{\color{blue}{\frac{-1}{6}}} \cdot \left({\left(\mathsf{neg}\left(a\right)\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} \cdot \sqrt[3]{g}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto {\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}} \cdot \color{blue}{\left({\left(\mathsf{neg}\left(a\right)\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} \cdot \sqrt[3]{g}\right)} \]

      21. lower-pow.f64 N/A

          \[\leadsto {\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}} \cdot \left(\color{blue}{{\left(\mathsf{neg}\left(a\right)\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)}} \cdot \sqrt[3]{g}\right) \]

      22. metadata-eval 90.6

          \[\leadsto {\left(-a\right)}^{-0.16666666666666666} \cdot \left({\left(-a\right)}^{\color{blue}{-0.16666666666666666}} \cdot \sqrt[3]{g}\right) \]

   10. Applied rewrites 90.6%

       \[\leadsto \color{blue}{{\left(-a\right)}^{-0.16666666666666666} \cdot \left({\left(-a\right)}^{-0.16666666666666666} \cdot \sqrt[3]{g}\right)} \]
   11. Step-by-step derivation

       1. lift-neg.f64 N/A

          \[\leadsto {\color{blue}{\left(\mathsf{neg}\left(a\right)\right)}}^{\frac{-1}{6}} \cdot \left({\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}} \cdot \sqrt[3]{g}\right) \]

       2. lift-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}}} \cdot \left({\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}} \cdot \sqrt[3]{g}\right) \]

       3. lift-neg.f64 N/A

          \[\leadsto {\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}} \cdot \left({\color{blue}{\left(\mathsf{neg}\left(a\right)\right)}}^{\frac{-1}{6}} \cdot \sqrt[3]{g}\right) \]

       4. lift-pow.f64 N/A

          \[\leadsto {\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}} \cdot \left(\color{blue}{{\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}}} \cdot \sqrt[3]{g}\right) \]

       5. lift-cbrt.f64 N/A

          \[\leadsto {\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}} \cdot \left({\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}} \cdot \color{blue}{\sqrt[3]{g}}\right) \]

       6. associate-*r* N/A

          \[\leadsto \color{blue}{\left({\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}} \cdot {\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}}\right) \cdot \sqrt[3]{g}} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}} \cdot {\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}}\right) \cdot \sqrt[3]{g}} \]

       8. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}}} \cdot {\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}}\right) \cdot \sqrt[3]{g} \]

       9. lift-pow.f64 N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}} \cdot \color{blue}{{\left(\mathsf{neg}\left(a\right)\right)}^{\frac{-1}{6}}}\right) \cdot \sqrt[3]{g} \]

       10. pow-prod-up N/A

           \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(a\right)\right)}^{\left(\frac{-1}{6} + \frac{-1}{6}\right)}} \cdot \sqrt[3]{g} \]

       11. metadata-eval N/A

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

       12. lower-pow.f64 90.6

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

   12. Applied rewrites 90.6%

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

   if -4.9999999999999995e-309 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 40.6%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 30.8

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 70.6

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

   8. Applied rewrites 70.6%

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

      1. lift-/.f64 N/A

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

      2. cbrt-unprod N/A

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

      3. neg-mul-1 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cbrt-prod N/A

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

      8. pow1/3 N/A

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

      9. inv-pow N/A

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

      10. pow-pow N/A

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

      11. metadata-eval N/A

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

      12. sqr-pow N/A

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

      13. pow-prod-down N/A

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

      14. sqr-neg N/A

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

      15. lift-neg.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. pow-prod-down N/A

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

      18. sqr-pow N/A

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

      19. lift-pow.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-cbrt.f64 N/A

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

      22. lower-neg.f64 0.0

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

      23. lift-pow.f64 N/A

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

      24. sqr-pow N/A

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

      25. pow-prod-down N/A

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

   10. Applied rewrites 90.5%

       \[\leadsto \color{blue}{\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq -5 \cdot 10^{-309}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;-\sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (/ 1.0 (* a 2.0)) 2e+63)
   (- (cbrt (/ g a)))
   (* (cbrt (- g)) (pow a -0.3333333333333333))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (a * 2.0)) <= 2e+63) {
		tmp = -cbrt((g / a));
	} else {
		tmp = cbrt(-g) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (a * 2.0)) <= 2e+63) {
		tmp = -Math.cbrt((g / a));
	} else {
		tmp = Math.cbrt(-g) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(1.0 / Float64(a * 2.0)) <= 2e+63)
		tmp = Float64(-cbrt(Float64(g / a)));
	else
		tmp = Float64(cbrt(Float64(-g)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 2e+63], (-N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), N[(N[Power[(-g), 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 2.00000000000000012e63

   1. Initial program 47.9%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 31.6

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

   5. Applied rewrites 31.6%

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

   6. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 80.1

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

   8. Applied rewrites 80.1%

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

   9. Taylor expanded in a around -inf

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

       1. rem-cube-cbrt N/A

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

       2. *-commutative N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-cbrt.f64 N/A

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

       6. lower-/.f64 80.1

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

   11. Applied rewrites 80.1%

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

   if 2.00000000000000012e63 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 19.0%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 18.4

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

   5. Applied rewrites 18.4%

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

   6. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 36.8

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

   8. Applied rewrites 36.8%

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

      1. lift-/.f64 N/A

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

      2. cbrt-unprod N/A

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

      3. neg-mul-1 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cbrt-prod N/A

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

      8. pow1/3 N/A

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

      9. inv-pow N/A

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

      10. pow-pow N/A

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

      11. metadata-eval N/A

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

      12. sqr-pow N/A

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

      13. pow-prod-down N/A

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

      14. sqr-neg N/A

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

      15. lift-neg.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. pow-prod-down N/A

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

      18. sqr-pow N/A

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

      19. lift-pow.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-cbrt.f64 N/A

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

      22. lower-neg.f64 0.0

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

      23. lift-pow.f64 N/A

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

      24. sqr-pow N/A

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

      25. pow-prod-down N/A

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

   10. Applied rewrites 88.2%

       \[\leadsto \color{blue}{\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;-\sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 2.6× 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 42.3%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 29.0

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

5. Applied rewrites 29.0%

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

6. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 71.7

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

8. Applied rewrites 71.7%

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

9. Taylor expanded in a around -inf

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

    1. rem-cube-cbrt N/A

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

    2. *-commutative N/A

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

    3. mul-1-neg N/A

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

    4. lower-neg.f64 N/A

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

    5. lower-cbrt.f64 N/A

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

    6. lower-/.f64 71.7

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

11. Applied rewrites 71.7%

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

Alternative 5: 1.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 29.0

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

5. Applied rewrites 29.0%

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

6. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 71.7

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

8. Applied rewrites 71.7%

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

   1. lift-/.f64 N/A

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

   2. cbrt-unprod N/A

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

   3. pow1/3 N/A

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

   4. neg-mul-1 N/A

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

   5. lift-/.f64 N/A

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

   6. distribute-frac-neg2 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. sqr-pow N/A

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

   10. pow-prod-down N/A

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

   11. lift-/.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. distribute-frac-neg2 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-neg.f64 N/A

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

   17. distribute-frac-neg2 N/A

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

   18. lift-/.f64 N/A

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

   19. sqr-neg N/A

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

   20. pow-prod-down N/A

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

   21. sqr-pow N/A

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

   22. pow1/3 N/A

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

   23. lift-cbrt.f64 1.3

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

10. Applied rewrites 1.3%

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

Reproduce

herbie shell --seed 2024214 
(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))))))))