2-ancestry mixing, positive discriminant

Percentage Accurate: 43.3% → 95.7%

Time: 14.5s

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: 43.3% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ {4}^{0.16666666666666666} \cdot \left(\sqrt[3]{-0.5} \cdot \left(\sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{-g}\right)\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (*
  (pow 4.0 0.16666666666666666)
  (* (cbrt -0.5) (* (cbrt (/ -1.0 a)) (cbrt (- g))))))

C

double code(double g, double h, double a) {
	return pow(4.0, 0.16666666666666666) * (cbrt(-0.5) * (cbrt((-1.0 / a)) * cbrt(-g)));
}

Java

public static double code(double g, double h, double a) {
	return Math.pow(4.0, 0.16666666666666666) * (Math.cbrt(-0.5) * (Math.cbrt((-1.0 / a)) * Math.cbrt(-g)));
}

Julia

function code(g, h, a)
	return Float64((4.0 ^ 0.16666666666666666) * Float64(cbrt(-0.5) * Float64(cbrt(Float64(-1.0 / a)) * cbrt(Float64(-g)))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[4.0, 0.16666666666666666], $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[(N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[(-g), 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.1%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. cbrt-div N/A

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

   6. *-lft-identity N/A

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

   7. pow1/3 N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 50.1%

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

5. Taylor expanded in g around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 76.4

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

7. Applied rewrites 76.4%

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

9. Applied rewrites 95.0%

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

11. Applied rewrites 95.6%

    \[\leadsto \left(\left(\sqrt[3]{-g} \cdot \sqrt[3]{\frac{1}{-a}}\right) \cdot \sqrt[3]{-0.5}\right) \cdot {4}^{\color{blue}{0.16666666666666666}} \]

12. Final simplification 95.6%

    \[\leadsto {4}^{0.16666666666666666} \cdot \left(\sqrt[3]{-0.5} \cdot \left(\sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{-g}\right)\right) \]
13. Add Preprocessing

Alternative 2: 89.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{2 \cdot a} \leq -4 \cdot 10^{-306}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{-g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (/ 1.0 (* 2.0 a)) -4e-306)
   (* (pow (- a) -0.3333333333333333) (cbrt g))
   (* (pow a -0.3333333333333333) (cbrt (- g)))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (2.0 * a)) <= -4e-306) {
		tmp = pow(-a, -0.3333333333333333) * cbrt(g);
	} else {
		tmp = pow(a, -0.3333333333333333) * cbrt(-g);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (2.0 * a)) <= -4e-306) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt(g);
	} else {
		tmp = Math.pow(a, -0.3333333333333333) * Math.cbrt(-g);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(1.0 / Float64(2.0 * a)) <= -4e-306)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(g));
	else
		tmp = Float64((a ^ -0.3333333333333333) * cbrt(Float64(-g)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -4e-306], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -0.3333333333333333], $MachinePrecision] * N[Power[(-g), 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < -4.00000000000000011e-306

   1. Initial program 51.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. pow1/3 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in g around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 83.6

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

   7. Applied rewrites 83.6%

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

   9. Applied rewrites 96.9%

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

   11. Applied rewrites 90.6%

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

   if -4.00000000000000011e-306 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 44.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. pow1/3 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 47.7%

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

   5. Taylor expanded in g around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 69.5

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

   7. Applied rewrites 69.5%

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

   9. Applied rewrites 94.4%

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

   11. Applied rewrites 88.4%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{2 \cdot a} \leq -4 \cdot 10^{-306}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{-g}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{2 \cdot a} \leq -5 \cdot 10^{+156}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-1}{a} \cdot g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (/ 1.0 (* 2.0 a)) -5e+156)
   (* (pow (- a) -0.3333333333333333) (cbrt g))
   (cbrt (* (/ -1.0 a) g))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (2.0 * a)) <= -5e+156) {
		tmp = pow(-a, -0.3333333333333333) * cbrt(g);
	} else {
		tmp = cbrt(((-1.0 / a) * g));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (2.0 * a)) <= -5e+156) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt(g);
	} else {
		tmp = Math.cbrt(((-1.0 / a) * g));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(1.0 / Float64(2.0 * a)) <= -5e+156)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(g));
	else
		tmp = cbrt(Float64(Float64(-1.0 / a) * g));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -5e+156], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(-1.0 / a), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < -4.99999999999999992e156

   1. Initial program 44.1%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. pow1/3 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in g around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 48.3

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

   7. Applied rewrites 48.3%

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

   9. Applied rewrites 95.8%

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

   11. Applied rewrites 88.4%

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

   if -4.99999999999999992e156 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.6%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. pow1/3 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in g around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 79.9

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

   7. Applied rewrites 79.9%

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

   9. Applied rewrites 80.7%

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

   11. Applied rewrites 80.8%

       \[\leadsto \sqrt[3]{g \cdot \frac{-1}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{2 \cdot a} \leq -5 \cdot 10^{+156}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-1}{a} \cdot g}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% 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 48.1%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. cbrt-div N/A

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

   6. *-lft-identity N/A

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

   7. pow1/3 N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 50.1%

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

5. Taylor expanded in g around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 76.4

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

7. Applied rewrites 76.4%

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

9. Applied rewrites 95.6%

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

Alternative 5: 73.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.1%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. cbrt-div N/A

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

   6. *-lft-identity N/A

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

   7. pow1/3 N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 50.1%

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

5. Taylor expanded in g around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 76.4

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

7. Applied rewrites 76.4%

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

9. Applied rewrites 77.3%

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

11. Applied rewrites 77.3%

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

12. Final simplification 77.3%

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

Alternative 6: 73.2% 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 48.1%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. cbrt-div N/A

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

   6. *-lft-identity N/A

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

   7. pow1/3 N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 50.1%

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

5. Taylor expanded in g around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 76.4

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

7. Applied rewrites 76.4%

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

9. Applied rewrites 77.3%

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

Reproduce

herbie shell --seed 2024240 
(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))))))))