2-ancestry mixing, positive discriminant

Percentage Accurate: 44.0% → 77.1%

Time: 14.3s

Alternatives: 12

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.0% 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: 77.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{h}{g} \cdot h\\ t_1 := \sqrt{\sqrt{\left(g - h\right) \cdot \left(h + g\right)}}\\ \mathbf{if}\;g \leq -1.02 \cdot 10^{+136}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-124}:\\ \;\;\;\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)} + \sqrt[3]{0.5 \cdot \left(t\_1 \cdot t\_1 - g\right)} \cdot \sqrt[3]{{a}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.25 \cdot t\_0 - g}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (/ h g) h)) (t_1 (sqrt (sqrt (* (- g h) (+ h g))))))
   (if (<= g -1.02e+136)
     (+ (cbrt (* (* (/ h a) (/ h g)) -0.25)) (cbrt (/ (- g) a)))
     (if (<= g -6e-124)
       (+
        (cbrt (* (/ -1.0 (* a 2.0)) (+ (sqrt (- (* g g) (* h h))) g)))
        (* (cbrt (* 0.5 (- (* t_1 t_1) g))) (cbrt (pow a -1.0))))
       (+
        (cbrt (/ (- (* 0.25 t_0) g) a))
        (cbrt (* (/ 0.5 a) (* -0.5 t_0))))))))

C

double code(double g, double h, double a) {
	double t_0 = (h / g) * h;
	double t_1 = sqrt(sqrt(((g - h) * (h + g))));
	double tmp;
	if (g <= -1.02e+136) {
		tmp = cbrt((((h / a) * (h / g)) * -0.25)) + cbrt((-g / a));
	} else if (g <= -6e-124) {
		tmp = cbrt(((-1.0 / (a * 2.0)) * (sqrt(((g * g) - (h * h))) + g))) + (cbrt((0.5 * ((t_1 * t_1) - g))) * cbrt(pow(a, -1.0)));
	} else {
		tmp = cbrt((((0.25 * t_0) - g) / a)) + cbrt(((0.5 / a) * (-0.5 * t_0)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = (h / g) * h;
	double t_1 = Math.sqrt(Math.sqrt(((g - h) * (h + g))));
	double tmp;
	if (g <= -1.02e+136) {
		tmp = Math.cbrt((((h / a) * (h / g)) * -0.25)) + Math.cbrt((-g / a));
	} else if (g <= -6e-124) {
		tmp = Math.cbrt(((-1.0 / (a * 2.0)) * (Math.sqrt(((g * g) - (h * h))) + g))) + (Math.cbrt((0.5 * ((t_1 * t_1) - g))) * Math.cbrt(Math.pow(a, -1.0)));
	} else {
		tmp = Math.cbrt((((0.25 * t_0) - g) / a)) + Math.cbrt(((0.5 / a) * (-0.5 * t_0)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(h / g) * h)
	t_1 = sqrt(sqrt(Float64(Float64(g - h) * Float64(h + g))))
	tmp = 0.0
	if (g <= -1.02e+136)
		tmp = Float64(cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25)) + cbrt(Float64(Float64(-g) / a)));
	elseif (g <= -6e-124)
		tmp = Float64(cbrt(Float64(Float64(-1.0 / Float64(a * 2.0)) * Float64(sqrt(Float64(Float64(g * g) - Float64(h * h))) + g))) + Float64(cbrt(Float64(0.5 * Float64(Float64(t_1 * t_1) - g))) * cbrt((a ^ -1.0))));
	else
		tmp = Float64(cbrt(Float64(Float64(Float64(0.25 * t_0) - g) / a)) + cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * t_0))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[g, -1.02e+136], N[(N[Power[N[(N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, -6e-124], N[(N[Power[N[(N[(-1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(0.5 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[a, -1.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(0.25 * t$95$0), $MachinePrecision] - g), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if g < -1.01999999999999996e136

   1. Initial program 7.1%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 4.5

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 64.1

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

   8. Applied rewrites 64.1%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower-cbrt.f64 N/A

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

       3. unpow2 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-cbrt.f64 N/A

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

       10. lower-cbrt.f64 65.5

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

   11. Applied rewrites 65.5%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 65.5

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

   13. Applied rewrites 65.5%

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

   if -1.01999999999999996e136 < g < -6e-124

   1. Initial program 74.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. cbrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. div-inv N/A

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

      15. metadata-eval N/A

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

   4. Applied rewrites 97.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift--.f64 N/A

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

      7. difference-of-squares N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-prod N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 98.0%

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

   if -6e-124 < g

   1. Initial program 46.6%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 43.4

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

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.4

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

   8. Applied rewrites 77.4%

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

   9. Taylor expanded in h around 0

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

       1. +-commutative N/A

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

       2. associate-*r/ N/A

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

       3. times-frac N/A

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

       4. lower-fma.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)}} \]

       5. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{4}}{a}}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       6. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \color{blue}{\frac{{h}^{2}}{g}}, -1 \cdot \frac{g}{a}\right)} \]

       7. unpow2 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       9. associate-*r/ N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{-1 \cdot g}{a}}\right)} \]

       10. mul-1-neg N/A

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}\right)} \]

       11. lower-/.f64 N/A

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}\right)} \]

       12. lower-neg.f64 77.5

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{-g}}{a}\right)} \]

   11. Applied rewrites 77.5%

       \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{-g}{a}\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 81.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.02 \cdot 10^{+136}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-124}:\\ \;\;\;\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)} + \sqrt[3]{0.5 \cdot \left(\sqrt{\sqrt{\left(g - h\right) \cdot \left(h + g\right)}} \cdot \sqrt{\sqrt{\left(g - h\right) \cdot \left(h + g\right)}} - g\right)} \cdot \sqrt[3]{{a}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.25 \cdot \left(\frac{h}{g} \cdot h\right) - g}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(\frac{h}{g} \cdot h\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{h}{g} \cdot h\\ \mathbf{if}\;g \leq -4.6 \cdot 10^{+145}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.25 \cdot t\_0 - g}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (/ h g) h)))
   (if (<= g -4.6e+145)
     (+ (cbrt (* (* (/ h a) (/ h g)) -0.25)) (cbrt (/ (- g) a)))
     (if (<= g -6e-124)
       (+
        (/ (cbrt (- (sqrt (* (- g h) (+ h g))) g)) (cbrt (* a 2.0)))
        (cbrt (* (/ -1.0 (* a 2.0)) (+ (sqrt (- (* g g) (* h h))) g))))
       (+
        (cbrt (/ (- (* 0.25 t_0) g) a))
        (cbrt (* (/ 0.5 a) (* -0.5 t_0))))))))

C

double code(double g, double h, double a) {
	double t_0 = (h / g) * h;
	double tmp;
	if (g <= -4.6e+145) {
		tmp = cbrt((((h / a) * (h / g)) * -0.25)) + cbrt((-g / a));
	} else if (g <= -6e-124) {
		tmp = (cbrt((sqrt(((g - h) * (h + g))) - g)) / cbrt((a * 2.0))) + cbrt(((-1.0 / (a * 2.0)) * (sqrt(((g * g) - (h * h))) + g)));
	} else {
		tmp = cbrt((((0.25 * t_0) - g) / a)) + cbrt(((0.5 / a) * (-0.5 * t_0)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = (h / g) * h;
	double tmp;
	if (g <= -4.6e+145) {
		tmp = Math.cbrt((((h / a) * (h / g)) * -0.25)) + Math.cbrt((-g / a));
	} else if (g <= -6e-124) {
		tmp = (Math.cbrt((Math.sqrt(((g - h) * (h + g))) - g)) / Math.cbrt((a * 2.0))) + Math.cbrt(((-1.0 / (a * 2.0)) * (Math.sqrt(((g * g) - (h * h))) + g)));
	} else {
		tmp = Math.cbrt((((0.25 * t_0) - g) / a)) + Math.cbrt(((0.5 / a) * (-0.5 * t_0)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(h / g) * h)
	tmp = 0.0
	if (g <= -4.6e+145)
		tmp = Float64(cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25)) + cbrt(Float64(Float64(-g) / a)));
	elseif (g <= -6e-124)
		tmp = Float64(Float64(cbrt(Float64(sqrt(Float64(Float64(g - h) * Float64(h + g))) - g)) / cbrt(Float64(a * 2.0))) + cbrt(Float64(Float64(-1.0 / Float64(a * 2.0)) * Float64(sqrt(Float64(Float64(g * g) - Float64(h * h))) + g))));
	else
		tmp = Float64(cbrt(Float64(Float64(Float64(0.25 * t_0) - g) / a)) + cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * t_0))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision]}, If[LessEqual[g, -4.6e+145], N[(N[Power[N[(N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, -6e-124], N[(N[(N[Power[N[(N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(-1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(0.25 * t$95$0), $MachinePrecision] - g), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if g < -4.6e145

   1. Initial program 4.4%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 4.1

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 63.0

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

   8. Applied rewrites 63.0%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower-cbrt.f64 N/A

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

       3. unpow2 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-cbrt.f64 N/A

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

       10. lower-cbrt.f64 64.5

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

   11. Applied rewrites 64.5%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 64.5

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

   13. Applied rewrites 64.5%

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

   if -4.6e145 < g < -6e-124

   1. Initial program 75.3%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. pow1/3 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 98.0%

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

   if -6e-124 < g

   1. Initial program 46.6%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 43.4

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

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.4

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

   8. Applied rewrites 77.4%

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

   9. Taylor expanded in h around 0

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

       1. +-commutative N/A

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

       2. associate-*r/ N/A

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

       3. times-frac N/A

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

       4. lower-fma.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)}} \]

       5. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{4}}{a}}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       6. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \color{blue}{\frac{{h}^{2}}{g}}, -1 \cdot \frac{g}{a}\right)} \]

       7. unpow2 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       9. associate-*r/ N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{-1 \cdot g}{a}}\right)} \]

       10. mul-1-neg N/A

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}\right)} \]

       11. lower-/.f64 N/A

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}\right)} \]

       12. lower-neg.f64 77.5

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{-g}}{a}\right)} \]

   11. Applied rewrites 77.5%

       \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{-g}{a}\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 81.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -4.6 \cdot 10^{+145}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.25 \cdot \left(\frac{h}{g} \cdot h\right) - g}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(\frac{h}{g} \cdot h\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{h}{g} \cdot h\\ \mathbf{if}\;g \leq -2.5 \cdot 10^{+150}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right) \cdot 0.5}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.25 \cdot t\_0 - g}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (/ h g) h)))
   (if (<= g -2.5e+150)
     (+ (cbrt (* (* (/ h a) (/ h g)) -0.25)) (cbrt (/ (- g) a)))
     (if (<= g -6e-124)
       (+
        (/ (cbrt (* (- (sqrt (* (- g h) (+ h g))) g) 0.5)) (cbrt a))
        (cbrt (* (/ -1.0 (* a 2.0)) (+ (sqrt (- (* g g) (* h h))) g))))
       (+
        (cbrt (/ (- (* 0.25 t_0) g) a))
        (cbrt (* (/ 0.5 a) (* -0.5 t_0))))))))

C

double code(double g, double h, double a) {
	double t_0 = (h / g) * h;
	double tmp;
	if (g <= -2.5e+150) {
		tmp = cbrt((((h / a) * (h / g)) * -0.25)) + cbrt((-g / a));
	} else if (g <= -6e-124) {
		tmp = (cbrt(((sqrt(((g - h) * (h + g))) - g) * 0.5)) / cbrt(a)) + cbrt(((-1.0 / (a * 2.0)) * (sqrt(((g * g) - (h * h))) + g)));
	} else {
		tmp = cbrt((((0.25 * t_0) - g) / a)) + cbrt(((0.5 / a) * (-0.5 * t_0)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = (h / g) * h;
	double tmp;
	if (g <= -2.5e+150) {
		tmp = Math.cbrt((((h / a) * (h / g)) * -0.25)) + Math.cbrt((-g / a));
	} else if (g <= -6e-124) {
		tmp = (Math.cbrt(((Math.sqrt(((g - h) * (h + g))) - g) * 0.5)) / Math.cbrt(a)) + Math.cbrt(((-1.0 / (a * 2.0)) * (Math.sqrt(((g * g) - (h * h))) + g)));
	} else {
		tmp = Math.cbrt((((0.25 * t_0) - g) / a)) + Math.cbrt(((0.5 / a) * (-0.5 * t_0)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(h / g) * h)
	tmp = 0.0
	if (g <= -2.5e+150)
		tmp = Float64(cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25)) + cbrt(Float64(Float64(-g) / a)));
	elseif (g <= -6e-124)
		tmp = Float64(Float64(cbrt(Float64(Float64(sqrt(Float64(Float64(g - h) * Float64(h + g))) - g) * 0.5)) / cbrt(a)) + cbrt(Float64(Float64(-1.0 / Float64(a * 2.0)) * Float64(sqrt(Float64(Float64(g * g) - Float64(h * h))) + g))));
	else
		tmp = Float64(cbrt(Float64(Float64(Float64(0.25 * t_0) - g) / a)) + cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * t_0))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision]}, If[LessEqual[g, -2.5e+150], N[(N[Power[N[(N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, -6e-124], N[(N[(N[Power[N[(N[(N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision] * 0.5), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(-1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(0.25 * t$95$0), $MachinePrecision] - g), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if g < -2.50000000000000004e150

   1. Initial program 3.0%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 3.9

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

   5. Applied rewrites 3.9%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 62.5

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

   8. Applied rewrites 62.5%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower-cbrt.f64 N/A

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

       3. unpow2 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-cbrt.f64 N/A

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

       10. lower-cbrt.f64 64.0

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

   11. Applied rewrites 64.0%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 64.0

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

   13. Applied rewrites 64.0%

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

   if -2.50000000000000004e150 < g < -6e-124

   1. Initial program 75.7%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. cbrt-div N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 97.9%

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

   if -6e-124 < g

   1. Initial program 46.6%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 43.4

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

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.4

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

   8. Applied rewrites 77.4%

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

   9. Taylor expanded in h around 0

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

       1. +-commutative N/A

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

       2. associate-*r/ N/A

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

       3. times-frac N/A

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

       4. lower-fma.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)}} \]

       5. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{4}}{a}}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       6. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \color{blue}{\frac{{h}^{2}}{g}}, -1 \cdot \frac{g}{a}\right)} \]

       7. unpow2 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       9. associate-*r/ N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{-1 \cdot g}{a}}\right)} \]

       10. mul-1-neg N/A

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}\right)} \]

       11. lower-/.f64 N/A

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}\right)} \]

       12. lower-neg.f64 77.5

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{-g}}{a}\right)} \]

   11. Applied rewrites 77.5%

       \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{-g}{a}\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 81.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -2.5 \cdot 10^{+150}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right) \cdot 0.5}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.25 \cdot \left(\frac{h}{g} \cdot h\right) - g}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(\frac{h}{g} \cdot h\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{h}{g} \cdot h\\ t_1 := \sqrt{\left(g - h\right) \cdot \left(h + g\right)}\\ \mathbf{if}\;g \leq -2.1 \cdot 10^{+152}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{t\_1 - g}, \sqrt[3]{\frac{0.5}{-a} \cdot \left(t\_1 + g\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.25 \cdot t\_0 - g}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (/ h g) h)) (t_1 (sqrt (* (- g h) (+ h g)))))
   (if (<= g -2.1e+152)
     (+ (cbrt (* (* (/ h a) (/ h g)) -0.25)) (cbrt (/ (- g) a)))
     (if (<= g -6e-124)
       (fma
        (cbrt (/ 0.5 a))
        (cbrt (- t_1 g))
        (cbrt (* (/ 0.5 (- a)) (+ t_1 g))))
       (+
        (cbrt (/ (- (* 0.25 t_0) g) a))
        (cbrt (* (/ 0.5 a) (* -0.5 t_0))))))))

C

double code(double g, double h, double a) {
	double t_0 = (h / g) * h;
	double t_1 = sqrt(((g - h) * (h + g)));
	double tmp;
	if (g <= -2.1e+152) {
		tmp = cbrt((((h / a) * (h / g)) * -0.25)) + cbrt((-g / a));
	} else if (g <= -6e-124) {
		tmp = fma(cbrt((0.5 / a)), cbrt((t_1 - g)), cbrt(((0.5 / -a) * (t_1 + g))));
	} else {
		tmp = cbrt((((0.25 * t_0) - g) / a)) + cbrt(((0.5 / a) * (-0.5 * t_0)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(h / g) * h)
	t_1 = sqrt(Float64(Float64(g - h) * Float64(h + g)))
	tmp = 0.0
	if (g <= -2.1e+152)
		tmp = Float64(cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25)) + cbrt(Float64(Float64(-g) / a)));
	elseif (g <= -6e-124)
		tmp = fma(cbrt(Float64(0.5 / a)), cbrt(Float64(t_1 - g)), cbrt(Float64(Float64(0.5 / Float64(-a)) * Float64(t_1 + g))));
	else
		tmp = Float64(cbrt(Float64(Float64(Float64(0.25 * t_0) - g) / a)) + cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * t_0))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[g, -2.1e+152], N[(N[Power[N[(N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, -6e-124], N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(t$95$1 - g), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / (-a)), $MachinePrecision] * N[(t$95$1 + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(0.25 * t$95$0), $MachinePrecision] - g), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if g < -2.1000000000000002e152

   1. Initial program 1.5%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 3.7

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

   5. Applied rewrites 3.7%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 61.9

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

   8. Applied rewrites 61.9%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower-cbrt.f64 N/A

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

       3. unpow2 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-cbrt.f64 N/A

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

       10. lower-cbrt.f64 63.4

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

   11. Applied rewrites 63.4%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 63.4

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

   13. Applied rewrites 63.4%

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

   if -2.1000000000000002e152 < g < -6e-124

   1. Initial program 76.1%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. cbrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. div-inv N/A

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

      15. metadata-eval N/A

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

   4. Applied rewrites 98.0%

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

   6. Applied rewrites 97.8%

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

   if -6e-124 < g

   1. Initial program 46.6%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 43.4

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

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.4

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

   8. Applied rewrites 77.4%

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

   9. Taylor expanded in h around 0

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

       1. +-commutative N/A

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

       2. associate-*r/ N/A

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

       3. times-frac N/A

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

       4. lower-fma.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)}} \]

       5. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{4}}{a}}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       6. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \color{blue}{\frac{{h}^{2}}{g}}, -1 \cdot \frac{g}{a}\right)} \]

       7. unpow2 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       9. associate-*r/ N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{-1 \cdot g}{a}}\right)} \]

       10. mul-1-neg N/A

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}\right)} \]

       11. lower-/.f64 N/A

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}\right)} \]

       12. lower-neg.f64 77.5

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{-g}}{a}\right)} \]

   11. Applied rewrites 77.5%

       \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{-g}{a}\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 81.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -2.1 \cdot 10^{+152}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}, \sqrt[3]{\frac{0.5}{-a} \cdot \left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + g\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.25 \cdot \left(\frac{h}{g} \cdot h\right) - g}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(\frac{h}{g} \cdot h\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{h}{g} \cdot h\\ \mathbf{if}\;g \leq -4.6 \cdot 10^{+145}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -1.55 \cdot 10^{-87}:\\ \;\;\;\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\left(-g\right) + g\right)} + \frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.25 \cdot t\_0 - g}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (/ h g) h)))
   (if (<= g -4.6e+145)
     (+ (cbrt (* (* (/ h a) (/ h g)) -0.25)) (cbrt (/ (- g) a)))
     (if (<= g -1.55e-87)
       (+
        (cbrt (* (/ -1.0 (* a 2.0)) (+ (- g) g)))
        (/ (cbrt (- (sqrt (* (- g h) (+ h g))) g)) (cbrt (* a 2.0))))
       (+
        (cbrt (/ (- (* 0.25 t_0) g) a))
        (cbrt (* (/ 0.5 a) (* -0.5 t_0))))))))

C

double code(double g, double h, double a) {
	double t_0 = (h / g) * h;
	double tmp;
	if (g <= -4.6e+145) {
		tmp = cbrt((((h / a) * (h / g)) * -0.25)) + cbrt((-g / a));
	} else if (g <= -1.55e-87) {
		tmp = cbrt(((-1.0 / (a * 2.0)) * (-g + g))) + (cbrt((sqrt(((g - h) * (h + g))) - g)) / cbrt((a * 2.0)));
	} else {
		tmp = cbrt((((0.25 * t_0) - g) / a)) + cbrt(((0.5 / a) * (-0.5 * t_0)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = (h / g) * h;
	double tmp;
	if (g <= -4.6e+145) {
		tmp = Math.cbrt((((h / a) * (h / g)) * -0.25)) + Math.cbrt((-g / a));
	} else if (g <= -1.55e-87) {
		tmp = Math.cbrt(((-1.0 / (a * 2.0)) * (-g + g))) + (Math.cbrt((Math.sqrt(((g - h) * (h + g))) - g)) / Math.cbrt((a * 2.0)));
	} else {
		tmp = Math.cbrt((((0.25 * t_0) - g) / a)) + Math.cbrt(((0.5 / a) * (-0.5 * t_0)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(h / g) * h)
	tmp = 0.0
	if (g <= -4.6e+145)
		tmp = Float64(cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25)) + cbrt(Float64(Float64(-g) / a)));
	elseif (g <= -1.55e-87)
		tmp = Float64(cbrt(Float64(Float64(-1.0 / Float64(a * 2.0)) * Float64(Float64(-g) + g))) + Float64(cbrt(Float64(sqrt(Float64(Float64(g - h) * Float64(h + g))) - g)) / cbrt(Float64(a * 2.0))));
	else
		tmp = Float64(cbrt(Float64(Float64(Float64(0.25 * t_0) - g) / a)) + cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * t_0))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision]}, If[LessEqual[g, -4.6e+145], N[(N[Power[N[(N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, -1.55e-87], N[(N[Power[N[(N[(-1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[((-g) + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(0.25 * t$95$0), $MachinePrecision] - g), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if g < -4.6e145

   1. Initial program 4.4%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 4.1

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 63.0

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

   8. Applied rewrites 63.0%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower-cbrt.f64 N/A

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

       3. unpow2 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-cbrt.f64 N/A

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

       10. lower-cbrt.f64 64.5

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

   11. Applied rewrites 64.5%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 64.5

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

   13. Applied rewrites 64.5%

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

   if -4.6e145 < g < -1.54999999999999999e-87

   1. Initial program 73.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. pow1/3 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in g around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.6

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

   7. Applied rewrites 95.6%

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

   if -1.54999999999999999e-87 < g

   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. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 42.6

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

   5. Applied rewrites 42.6%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.7

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

   8. Applied rewrites 77.7%

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

   9. Taylor expanded in h around 0

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

       1. +-commutative N/A

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

       2. associate-*r/ N/A

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

       3. times-frac N/A

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

       4. lower-fma.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)}} \]

       5. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{4}}{a}}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       6. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \color{blue}{\frac{{h}^{2}}{g}}, -1 \cdot \frac{g}{a}\right)} \]

       7. unpow2 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       9. associate-*r/ N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{-1 \cdot g}{a}}\right)} \]

       10. mul-1-neg N/A

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}\right)} \]

       11. lower-/.f64 N/A

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}\right)} \]

       12. lower-neg.f64 77.9

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{-g}}{a}\right)} \]

   11. Applied rewrites 77.9%

       \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{-g}{a}\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 81.3%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -4.6 \cdot 10^{+145}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -1.55 \cdot 10^{-87}:\\ \;\;\;\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\left(-g\right) + g\right)} + \frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.25 \cdot \left(\frac{h}{g} \cdot h\right) - g}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(\frac{h}{g} \cdot h\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{h}{g} \cdot h\\ \mathbf{if}\;g \leq -4.6 \cdot 10^{+145}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -0.00021:\\ \;\;\;\;\sqrt[3]{\frac{\left(h \cdot h\right) \cdot -0.25}{a \cdot g}} + \frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.25 \cdot t\_0 - g}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (/ h g) h)))
   (if (<= g -4.6e+145)
     (+ (cbrt (* (* (/ h a) (/ h g)) -0.25)) (cbrt (/ (- g) a)))
     (if (<= g -0.00021)
       (+
        (cbrt (/ (* (* h h) -0.25) (* a g)))
        (/ (cbrt (- (sqrt (* (- g h) (+ h g))) g)) (cbrt (* a 2.0))))
       (+
        (cbrt (/ (- (* 0.25 t_0) g) a))
        (cbrt (* (/ 0.5 a) (* -0.5 t_0))))))))

C

double code(double g, double h, double a) {
	double t_0 = (h / g) * h;
	double tmp;
	if (g <= -4.6e+145) {
		tmp = cbrt((((h / a) * (h / g)) * -0.25)) + cbrt((-g / a));
	} else if (g <= -0.00021) {
		tmp = cbrt((((h * h) * -0.25) / (a * g))) + (cbrt((sqrt(((g - h) * (h + g))) - g)) / cbrt((a * 2.0)));
	} else {
		tmp = cbrt((((0.25 * t_0) - g) / a)) + cbrt(((0.5 / a) * (-0.5 * t_0)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = (h / g) * h;
	double tmp;
	if (g <= -4.6e+145) {
		tmp = Math.cbrt((((h / a) * (h / g)) * -0.25)) + Math.cbrt((-g / a));
	} else if (g <= -0.00021) {
		tmp = Math.cbrt((((h * h) * -0.25) / (a * g))) + (Math.cbrt((Math.sqrt(((g - h) * (h + g))) - g)) / Math.cbrt((a * 2.0)));
	} else {
		tmp = Math.cbrt((((0.25 * t_0) - g) / a)) + Math.cbrt(((0.5 / a) * (-0.5 * t_0)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(h / g) * h)
	tmp = 0.0
	if (g <= -4.6e+145)
		tmp = Float64(cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25)) + cbrt(Float64(Float64(-g) / a)));
	elseif (g <= -0.00021)
		tmp = Float64(cbrt(Float64(Float64(Float64(h * h) * -0.25) / Float64(a * g))) + Float64(cbrt(Float64(sqrt(Float64(Float64(g - h) * Float64(h + g))) - g)) / cbrt(Float64(a * 2.0))));
	else
		tmp = Float64(cbrt(Float64(Float64(Float64(0.25 * t_0) - g) / a)) + cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * t_0))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision]}, If[LessEqual[g, -4.6e+145], N[(N[Power[N[(N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, -0.00021], N[(N[Power[N[(N[(N[(h * h), $MachinePrecision] * -0.25), $MachinePrecision] / N[(a * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(0.25 * t$95$0), $MachinePrecision] - g), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if g < -4.6e145

   1. Initial program 4.4%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 4.1

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 63.0

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

   8. Applied rewrites 63.0%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower-cbrt.f64 N/A

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

       3. unpow2 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-cbrt.f64 N/A

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

       10. lower-cbrt.f64 64.5

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

   11. Applied rewrites 64.5%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 64.5

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

   13. Applied rewrites 64.5%

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

   if -4.6e145 < g < -2.1000000000000001e-4

   1. Initial program 67.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. pow1/3 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in g around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 96.1

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

   7. Applied rewrites 96.1%

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

   if -2.1000000000000001e-4 < g

   1. Initial program 52.7%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 40.0

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 79.3

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

   8. Applied rewrites 79.3%

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

   9. Taylor expanded in h around 0

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

       1. +-commutative N/A

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

       2. associate-*r/ N/A

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

       3. times-frac N/A

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

       4. lower-fma.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)}} \]

       5. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{4}}{a}}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       6. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \color{blue}{\frac{{h}^{2}}{g}}, -1 \cdot \frac{g}{a}\right)} \]

       7. unpow2 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

       9. associate-*r/ N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{-1 \cdot g}{a}}\right)} \]

       10. mul-1-neg N/A

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}\right)} \]

       11. lower-/.f64 N/A

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}\right)} \]

       12. lower-neg.f64 79.5

           \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{-g}}{a}\right)} \]

   11. Applied rewrites 79.5%

       \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{-g}{a}\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 82.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -4.6 \cdot 10^{+145}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot \frac{h}{g}\right) \cdot -0.25} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -0.00021:\\ \;\;\;\;\sqrt[3]{\frac{\left(h \cdot h\right) \cdot -0.25}{a \cdot g}} + \frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.25 \cdot \left(\frac{h}{g} \cdot h\right) - g}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(\frac{h}{g} \cdot h\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{h}{g} \cdot h\\ \sqrt[3]{\frac{0.25 \cdot t\_0 - g}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (/ h g) h)))
   (+ (cbrt (/ (- (* 0.25 t_0) g) a)) (cbrt (* (/ 0.5 a) (* -0.5 t_0))))))

C

double code(double g, double h, double a) {
	double t_0 = (h / g) * h;
	return cbrt((((0.25 * t_0) - g) / a)) + cbrt(((0.5 / a) * (-0.5 * t_0)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = (h / g) * h;
	return Math.cbrt((((0.25 * t_0) - g) / a)) + Math.cbrt(((0.5 / a) * (-0.5 * t_0)));
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(h / g) * h)
	return Float64(cbrt(Float64(Float64(Float64(0.25 * t_0) - g) / a)) + cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * t_0))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision]}, N[(N[Power[N[(N[(N[(0.25 * t$95$0), $MachinePrecision] - g), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 42.3%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 26.1

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

5. Applied rewrites 26.1%

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

6. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 72.6

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

8. Applied rewrites 72.6%

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

9. Taylor expanded in h around 0

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

    1. +-commutative N/A

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

    2. associate-*r/ N/A

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

    3. times-frac N/A

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

    4. lower-fma.f64 N/A

       \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)}} \]

    5. lower-/.f64 N/A

       \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{4}}{a}}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)} \]

    6. lower-/.f64 N/A

       \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \color{blue}{\frac{{h}^{2}}{g}}, -1 \cdot \frac{g}{a}\right)} \]

    7. unpow2 N/A

       \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

    8. lower-*.f64 N/A

       \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

    9. associate-*r/ N/A

       \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{-1 \cdot g}{a}}\right)} \]

    10. mul-1-neg N/A

        \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}\right)} \]

    11. lower-/.f64 N/A

        \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{-1}{2} \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}\right)} \]

    12. lower-neg.f64 72.6

        \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{-g}}{a}\right)} \]

11. Applied rewrites 72.6%

    \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \frac{h \cdot h}{g}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{-g}{a}\right)}} \]
12. Step-by-step derivation

13. Applied rewrites 74.8%

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

14. Final simplification 74.8%

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

Alternative 8: 75.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(Float64(h / a) * Float64(h / g)) * -0.25)) + cbrt(Float64(Float64(-g) / a)))
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[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.3%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 26.1

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

5. Applied rewrites 26.1%

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

6. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 72.6

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

8. Applied rewrites 72.6%

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

9. Taylor expanded in g around inf

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

    1. lower-*.f64 N/A

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

    2. lower-cbrt.f64 N/A

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

    3. unpow2 N/A

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

    4. times-frac N/A

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

    5. lower-*.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. lower-/.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. lower-cbrt.f64 N/A

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

    10. lower-cbrt.f64 74.5

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

11. Applied rewrites 74.5%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-+.f64 74.5

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

13. Applied rewrites 74.5%

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

14. Final simplification 74.5%

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

Alternative 9: 71.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}} + \sqrt[3]{\frac{-g}{a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 26.1

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

5. Applied rewrites 26.1%

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

6. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 72.6

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

8. Applied rewrites 72.6%

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

9. Taylor expanded in g around inf

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

    1. associate-*r/ N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{-1}{4} \cdot {h}^{2}}{a \cdot g}}} + \sqrt[3]{\frac{-g}{a}} \]

    2. times-frac N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{-1}{4}}{a} \cdot \frac{{h}^{2}}{g}}} + \sqrt[3]{\frac{-g}{a}} \]

    3. lower-*.f64 N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{-1}{4}}{a} \cdot \frac{{h}^{2}}{g}}} + \sqrt[3]{\frac{-g}{a}} \]

    4. lower-/.f64 N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{-1}{4}}{a}} \cdot \frac{{h}^{2}}{g}} + \sqrt[3]{\frac{-g}{a}} \]

    5. lower-/.f64 N/A

       \[\leadsto \sqrt[3]{\frac{\frac{-1}{4}}{a} \cdot \color{blue}{\frac{{h}^{2}}{g}}} + \sqrt[3]{\frac{-g}{a}} \]

    6. unpow2 N/A

       \[\leadsto \sqrt[3]{\frac{\frac{-1}{4}}{a} \cdot \frac{\color{blue}{h \cdot h}}{g}} + \sqrt[3]{\frac{-g}{a}} \]

    7. lower-*.f64 72.6

       \[\leadsto \sqrt[3]{\frac{-0.25}{a} \cdot \frac{\color{blue}{h \cdot h}}{g}} + \sqrt[3]{\frac{-g}{a}} \]

11. Applied rewrites 72.6%

    \[\leadsto \sqrt[3]{\color{blue}{\frac{-0.25}{a} \cdot \frac{h \cdot h}{g}}} + \sqrt[3]{\frac{-g}{a}} \]

12. Final simplification 72.6%

    \[\leadsto \sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}} + \sqrt[3]{\frac{-g}{a}} \]
13. Add Preprocessing

Alternative 10: 15.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{-2 \cdot g}{a} \cdot 0.5} + \sqrt[3]{\frac{-g}{a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 26.1

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

5. Applied rewrites 26.1%

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

6. Taylor expanded in g around -inf

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

   1. lower-*.f64 15.3

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

8. Applied rewrites 15.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 15.3

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

10. Applied rewrites 15.3%

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

11. Final simplification 15.3%

    \[\leadsto \sqrt[3]{\frac{-2 \cdot g}{a} \cdot 0.5} + \sqrt[3]{\frac{-g}{a}} \]
12. Add Preprocessing

Alternative 11: 15.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-g}{a}}\\ t\_0 + t\_0 \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 26.1

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

5. Applied rewrites 26.1%

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

6. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 72.6

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

8. Applied rewrites 72.6%

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

9. Taylor expanded in g around -inf

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

    1. associate-*r/ N/A

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

    2. mul-1-neg N/A

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

    3. lower-/.f64 N/A

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

    4. lower-neg.f64 15.3

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

11. Applied rewrites 15.3%

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

Alternative 12: 3.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(-\sqrt[3]{0.5}\right) \cdot \sqrt[3]{\frac{0}{a}} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (* (- (cbrt 0.5)) (cbrt (/ 0.0 a))))

C

double code(double g, double h, double a) {
	return -cbrt(0.5) * cbrt((0.0 / a));
}

Java

public static double code(double g, double h, double a) {
	return -Math.cbrt(0.5) * Math.cbrt((0.0 / a));
}

Julia

function code(g, h, a)
	return Float64(Float64(-cbrt(0.5)) * cbrt(Float64(0.0 / a)))
end

Wolfram

code[g_, h_, a_] := N[((-N[Power[0.5, 1/3], $MachinePrecision]) * N[Power[N[(0.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.3%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. cbrt-prod N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. inv-pow N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-cbrt.f64 N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

4. Applied rewrites 47.9%

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

5. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. metadata-eval N/A

      \[\leadsto -\sqrt[3]{\frac{\color{blue}{0} \cdot g}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   9. mul0-lft N/A

      \[\leadsto -\sqrt[3]{\frac{\color{blue}{0}}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   10. mul0-lft N/A

       \[\leadsto -\sqrt[3]{\frac{\color{blue}{0 \cdot h}}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   11. metadata-eval N/A

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

   12. distribute-rgt1-in N/A

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

   13. lower-cbrt.f64 N/A

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

   14. distribute-rgt1-in N/A

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

   15. metadata-eval N/A

       \[\leadsto -\sqrt[3]{\frac{\color{blue}{0} \cdot h}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   16. mul0-lft N/A

       \[\leadsto -\sqrt[3]{\frac{\color{blue}{0}}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   17. lower-/.f64 N/A

       \[\leadsto -\sqrt[3]{\color{blue}{\frac{0}{a}}} \cdot \sqrt[3]{\frac{1}{2}} \]

   18. lower-cbrt.f64 2.9

       \[\leadsto -\sqrt[3]{\frac{0}{a}} \cdot \color{blue}{\sqrt[3]{0.5}} \]

7. Applied rewrites 2.9%

   \[\leadsto \color{blue}{-\sqrt[3]{\frac{0}{a}} \cdot \sqrt[3]{0.5}} \]

8. Final simplification 2.9%

   \[\leadsto \left(-\sqrt[3]{0.5}\right) \cdot \sqrt[3]{\frac{0}{a}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024267 
(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))))))))