2-ancestry mixing, positive discriminant

Percentage Accurate: 44.1% → 95.7%

Time: 15.0s

Alternatives: 6

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: 44.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 95.7% accurate, 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 45.1%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 28.9

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

5. Simplified 28.9%

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

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. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 75.8

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

8. Simplified 75.8%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. cbrt-unprod N/A

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

   8. neg-mul-1 N/A

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

   9. lift-/.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. cbrt-div N/A

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

   12. pow1/3 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-cbrt.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. pow1/3 N/A

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

   17. lower-cbrt.f64 95.6

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

10. Applied egg-rr 95.6%

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

Alternative 2: 89.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (* a 2.0) -1e-307)
   (* (pow (- a) -0.3333333333333333) (cbrt g))
   (* (cbrt (- g)) (pow a -0.3333333333333333))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((a * 2.0) <= -1e-307) {
		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 ((a * 2.0) <= -1e-307) {
		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(a * 2.0) <= -1e-307)
		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[(a * 2.0), $MachinePrecision], -1e-307], 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 2 binary64) a) < -9.99999999999999909e-308

   1. Initial program 43.4%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. 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 29.2

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

   5. Simplified 29.2%

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

   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. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 72.0

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

   8. Simplified 72.0%

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

      1. lift-/.f64 N/A

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

      2. cbrt-unprod N/A

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

      3. pow1/3 N/A

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

      4. *-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lift-neg.f64 N/A

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

      9. clear-num N/A

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

      10. associate-/r/ N/A

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

      11. unpow-prod-down N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. pow-pow N/A

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

      15. metadata-eval N/A

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

      16. lower-pow.f64 N/A

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

      17. pow1/3 N/A

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

      18. lower-cbrt.f64 89.1

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

   10. Applied egg-rr 89.1%

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

   if -9.99999999999999909e-308 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 47.1%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 28.5

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

   5. Simplified 28.5%

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

   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. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 80.4

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

   8. Simplified 80.4%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. cbrt-unprod N/A

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

      8. neg-mul-1 N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

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

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

      12. cbrt-prod N/A

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

      13. pow1/3 N/A

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

      14. inv-pow N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-pow.f64 89.8

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

   10. Applied egg-rr 89.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 3 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{-g}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (* a 2.0) 3e-299)
   (/ 1.0 (cbrt (/ a (- g))))
   (* (cbrt (- g)) (pow a -0.3333333333333333))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((a * 2.0) <= 3e-299) {
		tmp = 1.0 / cbrt((a / -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 ((a * 2.0) <= 3e-299) {
		tmp = 1.0 / Math.cbrt((a / -g));
	} else {
		tmp = Math.cbrt(-g) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= 3e-299)
		tmp = Float64(1.0 / cbrt(Float64(a / Float64(-g))));
	else
		tmp = Float64(cbrt(Float64(-g)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], 3e-299], N[(1.0 / N[Power[N[(a / (-g)), $MachinePrecision], 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 2 binary64) a) < 2.99999999999999984e-299

   1. Initial program 44.2%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 29.1

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

   5. Simplified 29.1%

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

   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. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 72.4

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

   8. Simplified 72.4%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. cbrt-unprod N/A

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

      8. neg-mul-1 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. cbrt-div N/A

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

      12. pow1/3 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-cbrt.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. pow1/3 N/A

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

      17. lower-cbrt.f64 95.4

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

   10. Applied egg-rr 95.4%

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

       1. lift-neg.f64 N/A

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

       2. lift-cbrt.f64 N/A

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

       3. lift-cbrt.f64 N/A

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

       4. clear-num N/A

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

       5. lower-/.f64 N/A

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

       6. lift-cbrt.f64 N/A

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

       7. lift-cbrt.f64 N/A

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

       8. cbrt-undiv N/A

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

       9. lower-cbrt.f64 N/A

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

       10. lower-/.f64 72.6

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

   12. Applied egg-rr 72.6%

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

   if 2.99999999999999984e-299 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 46.2%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 28.7

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

   5. Simplified 28.7%

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

   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. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 80.1

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

   8. Simplified 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. cbrt-unprod N/A

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

      8. neg-mul-1 N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

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

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

      12. cbrt-prod N/A

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

      13. pow1/3 N/A

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

      14. inv-pow N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-pow.f64 89.8

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

   10. Applied egg-rr 89.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 3 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{-g}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.1%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 28.9

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

5. Simplified 28.9%

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

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. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 75.8

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

8. Simplified 75.8%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. cbrt-unprod N/A

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

   8. neg-mul-1 N/A

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

   9. lift-/.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. cbrt-div N/A

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

   12. pow1/3 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-cbrt.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. pow1/3 N/A

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

   17. lower-cbrt.f64 95.6

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

10. Applied egg-rr 95.6%

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

    1. lift-neg.f64 N/A

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

    2. lift-cbrt.f64 N/A

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

    3. lift-cbrt.f64 N/A

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

    4. clear-num N/A

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

    5. lower-/.f64 N/A

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

    6. lift-cbrt.f64 N/A

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

    7. lift-cbrt.f64 N/A

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

    8. cbrt-undiv N/A

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

    9. lower-cbrt.f64 N/A

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

    10. lower-/.f64 75.9

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

12. Applied egg-rr 75.9%

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

Alternative 5: 73.3% 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 cbrt(Float64(-Float64(g / a)))
end

Wolfram

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

Derivation

1. Initial program 45.1%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 28.9

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

5. Simplified 28.9%

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

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. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 75.8

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

8. Simplified 75.8%

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

   1. lift-/.f64 N/A

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

   2. cbrt-unprod N/A

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

   3. *-commutative N/A

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

   4. neg-mul-1 N/A

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

   5. lift-/.f64 N/A

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

   6. distribute-frac-neg2 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-cbrt.f64 75.8

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

10. Applied egg-rr 75.8%

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

11. Final simplification 75.8%

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

Alternative 6: 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 45.1%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 28.9

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

5. Simplified 28.9%

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

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. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 75.8

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

8. Simplified 75.8%

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

   1. lift-/.f64 N/A

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

   2. cbrt-unprod N/A

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

   3. pow1/3 N/A

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

   4. *-commutative N/A

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

   5. neg-mul-1 N/A

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

   6. lift-/.f64 N/A

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

   7. distribute-frac-neg2 N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. lift-/.f64 N/A

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

   13. lift-neg.f64 N/A

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

   14. distribute-frac-neg2 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. lift-neg.f64 N/A

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

   18. distribute-frac-neg2 N/A

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

   19. lift-/.f64 N/A

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

   20. sqr-neg N/A

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

   21. pow-prod-down N/A

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

   22. sqr-pow N/A

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

   23. pow1/3 N/A

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

   24. lift-cbrt.f64 1.5

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

10. Applied egg-rr 1.5%

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

Reproduce

herbie shell --seed 2024208 
(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))))))))