2-ancestry mixing, positive discriminant

Percentage Accurate: 43.8% → 95.7%

Time: 8.4s

Alternatives: 5

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 43.8% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot {\left(\sqrt[3]{\frac{a}{\frac{h}{g} \cdot h}}\right)}^{-1} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-2 \cdot g} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (* (* (cbrt -0.5) (cbrt 0.5)) (pow (cbrt (/ a (* (/ h g) h))) -1.0))
  (* (cbrt (/ 0.5 a)) (cbrt (* -2.0 g)))))

C

double code(double g, double h, double a) {
	return ((cbrt(-0.5) * cbrt(0.5)) * pow(cbrt((a / ((h / g) * h))), -1.0)) + (cbrt((0.5 / a)) * cbrt((-2.0 * g)));
}

Java

public static double code(double g, double h, double a) {
	return ((Math.cbrt(-0.5) * Math.cbrt(0.5)) * Math.pow(Math.cbrt((a / ((h / g) * h))), -1.0)) + (Math.cbrt((0.5 / a)) * Math.cbrt((-2.0 * g)));
}

Julia

function code(g, h, a)
	return Float64(Float64(Float64(cbrt(-0.5) * cbrt(0.5)) * (cbrt(Float64(a / Float64(Float64(h / g) * h))) ^ -1.0)) + Float64(cbrt(Float64(0.5 / a)) * cbrt(Float64(-2.0 * g))))
end

Wolfram

code[g_, h_, a_] := N[(N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[(a / N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(-2.0 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.8%

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

3. Taylor expanded in g around inf

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

   1. lower-*.f64 27.1

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

5. Applied rewrites 27.1%

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

6. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. unpow2 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 73.3

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

8. Applied rewrites 73.3%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lift-*.f64 N/A

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

   4. unpow-prod-down N/A

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

   5. lower-*.f64 N/A

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

   6. pow1/3 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. pow1/3 N/A

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

   14. lower-cbrt.f64 94.5

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

10. Applied rewrites 94.5%

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

12. Applied rewrites 96.3%

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

13. Final simplification 96.3%

    \[\leadsto \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot {\left(\sqrt[3]{\frac{a}{\frac{h}{g} \cdot h}}\right)}^{-1} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-2 \cdot g} \]
14. Add Preprocessing

Alternative 2: 39.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \frac{0.5}{a} \cdot \left(-2 \cdot g\right)\\ t_2 := \sqrt[3]{t\_1}\\ \mathbf{if}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + t\_0\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + t\_0\right)} \leq -1 \cdot 10^{-100}:\\ \;\;\;\;t\_2 + t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + {t\_1}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h))))
        (t_1 (* (/ 0.5 a) (* -2.0 g)))
        (t_2 (cbrt t_1)))
   (if (<=
        (+
         (cbrt (* (pow (* 2.0 a) -1.0) (+ (- g) t_0)))
         (cbrt (* (/ -1.0 (* 2.0 a)) (+ g t_0))))
        -1e-100)
     (+ t_2 t_2)
     (+ (cbrt (* (* (/ h a) (/ h g)) -0.25)) (pow t_1 0.3333333333333333)))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double t_1 = (0.5 / a) * (-2.0 * g);
	double t_2 = cbrt(t_1);
	double tmp;
	if ((cbrt((pow((2.0 * a), -1.0) * (-g + t_0))) + cbrt(((-1.0 / (2.0 * a)) * (g + t_0)))) <= -1e-100) {
		tmp = t_2 + t_2;
	} else {
		tmp = cbrt((((h / a) * (h / g)) * -0.25)) + pow(t_1, 0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt(((g * g) - (h * h)));
	double t_1 = (0.5 / a) * (-2.0 * g);
	double t_2 = Math.cbrt(t_1);
	double tmp;
	if ((Math.cbrt((Math.pow((2.0 * a), -1.0) * (-g + t_0))) + Math.cbrt(((-1.0 / (2.0 * a)) * (g + t_0)))) <= -1e-100) {
		tmp = t_2 + t_2;
	} else {
		tmp = Math.cbrt((((h / a) * (h / g)) * -0.25)) + Math.pow(t_1, 0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = Float64(Float64(0.5 / a) * Float64(-2.0 * g))
	t_2 = cbrt(t_1)
	tmp = 0.0
	if (Float64(cbrt(Float64((Float64(2.0 * a) ^ -1.0) * Float64(Float64(-g) + t_0))) + cbrt(Float64(Float64(-1.0 / Float64(2.0 * a)) * Float64(g + t_0)))) <= -1e-100)
		tmp = Float64(t_2 + t_2);
	else
		tmp = Float64(cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25)) + (t_1 ^ 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 / a), $MachinePrecision] * N[(-2.0 * g), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 1/3], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[Power[N[(2.0 * a), $MachinePrecision], -1.0], $MachinePrecision] * N[((-g) + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(g + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], -1e-100], N[(t$95$2 + t$95$2), $MachinePrecision], N[(N[Power[N[(N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[t$95$1, 0.3333333333333333], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -1e-100

   1. Initial program 83.8%

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

   3. Taylor expanded in g around inf

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

      1. lower-*.f64 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in g around -inf

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

      1. lower-*.f64 17.4

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

   8. Applied rewrites 17.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 17.4

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

   10. Applied rewrites 17.4%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. metadata-eval N/A

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

       5. lift-/.f64 17.4

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

   12. Applied rewrites 17.4%

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

   if -1e-100 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 25.1%

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

   3. Taylor expanded in g around inf

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

      1. lower-*.f64 16.0

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

   5. Applied rewrites 16.0%

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

   6. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 67.0

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

   8. Applied rewrites 67.0%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lower-pow.f64 46.3

         \[\leadsto \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}} + \color{blue}{{\left(\frac{1}{2 \cdot a} \cdot \left(-2 \cdot g\right)\right)}^{0.3333333333333333}} \]

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 46.3

         \[\leadsto \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}} + {\left(\color{blue}{\frac{0.5}{a}} \cdot \left(-2 \cdot g\right)\right)}^{0.3333333333333333} \]

   10. Applied rewrites 46.3%

       \[\leadsto \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}} + \color{blue}{{\left(\frac{0.5}{a} \cdot \left(-2 \cdot g\right)\right)}^{0.3333333333333333}} \]
   11. Step-by-step derivation

   12. Applied rewrites 46.3%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25}} + {\left(\frac{0.5}{a} \cdot \left(-2 \cdot g\right)\right)}^{0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq -1 \cdot 10^{-100}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-2 \cdot g\right)} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-2 \cdot g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + {\left(\frac{0.5}{a} \cdot \left(-2 \cdot g\right)\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-2 \cdot g} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (cbrt (* (* (/ h a) (/ h g)) -0.25))
  (* (cbrt (/ 0.5 a)) (cbrt (* -2.0 g)))))

C

double code(double g, double h, double a) {
	return cbrt((((h / a) * (h / g)) * -0.25)) + (cbrt((0.5 / a)) * cbrt((-2.0 * g)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((((h / a) * (h / g)) * -0.25)) + (Math.cbrt((0.5 / a)) * Math.cbrt((-2.0 * g)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25)) + Float64(cbrt(Float64(0.5 / a)) * cbrt(Float64(-2.0 * g))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(-2.0 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.8%

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

3. Taylor expanded in g around inf

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

   1. lower-*.f64 27.1

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

5. Applied rewrites 27.1%

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

6. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. unpow2 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 73.3

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

8. Applied rewrites 73.3%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lift-*.f64 N/A

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

   4. unpow-prod-down N/A

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

   5. lower-*.f64 N/A

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

   6. pow1/3 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. pow1/3 N/A

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

   14. lower-cbrt.f64 94.5

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

10. Applied rewrites 94.5%

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

12. Applied rewrites 94.5%

    \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25}} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-2 \cdot g} \]
13. Add Preprocessing

Alternative 4: 74.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(-2 \cdot g\right)} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (cbrt (* (* (/ h a) (/ h g)) -0.25))
  (cbrt (* (pow (* 2.0 a) -1.0) (* -2.0 g)))))

C

double code(double g, double h, double a) {
	return cbrt((((h / a) * (h / g)) * -0.25)) + cbrt((pow((2.0 * a), -1.0) * (-2.0 * g)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((((h / a) * (h / g)) * -0.25)) + Math.cbrt((Math.pow((2.0 * a), -1.0) * (-2.0 * g)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25)) + cbrt(Float64((Float64(2.0 * a) ^ -1.0) * Float64(-2.0 * g))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[Power[N[(2.0 * a), $MachinePrecision], -1.0], $MachinePrecision] * N[(-2.0 * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.8%

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

3. Taylor expanded in g around inf

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

   1. lower-*.f64 27.1

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

5. Applied rewrites 27.1%

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

6. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. unpow2 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 73.3

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

8. Applied rewrites 73.3%

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

10. Applied rewrites 73.3%

    \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-2 \cdot g\right)} \]

11. Final simplification 73.3%

    \[\leadsto \sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(-2 \cdot g\right)} \]
12. Add Preprocessing

Alternative 5: 15.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a} \cdot \left(-2 \cdot g\right)}\\ t\_0 + t\_0 \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (/ 0.5 a) (* -2.0 g))))) (+ t_0 t_0)))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((0.5 / a) * (-2.0 * g)));
	return t_0 + t_0;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((0.5 / a) * (-2.0 * g)));
	return t_0 + t_0;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(0.5 / a) * Float64(-2.0 * g)))
	return Float64(t_0 + t_0)
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-2.0 * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$0 + t$95$0), $MachinePrecision]]

Derivation

1. Initial program 41.8%

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

3. Taylor expanded in g around inf

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

   1. lower-*.f64 27.1

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

5. Applied rewrites 27.1%

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

6. Taylor expanded in g around -inf

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

   1. lower-*.f64 14.9

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

8. Applied rewrites 14.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 14.9

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

10. Applied rewrites 14.9%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/r* N/A

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

    4. metadata-eval N/A

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

    5. lift-/.f64 14.9

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

12. Applied rewrites 14.9%

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

Reproduce

herbie shell --seed 2024298 
(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))))))))