2-ancestry mixing, positive discriminant

Percentage Accurate: 44.1% → 80.4%

Time: 14.7s

Alternatives: 12

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 80.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(g \cdot g\right) \cdot a\\ t_1 := \sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}}\\ t_2 := \sqrt{g \cdot g - h \cdot h}\\ t_3 := \sqrt[3]{\left(t\_2 - g\right) \cdot \frac{1}{a \cdot 2}}\\ t_4 := \sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(t\_2 + g\right)} + t\_3\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{0.5} \cdot \sqrt[3]{\frac{1}{t\_0}}, \sqrt[3]{2}, \sqrt[3]{\frac{0}{t\_0}} \cdot \sqrt[3]{0.5}\right) \cdot \left(-g\right)\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-101}:\\ \;\;\;\;t\_1 + \sqrt[3]{\frac{0.25 \cdot {\left(\frac{h}{g}\right)}^{2} - 1}{a} \cdot g}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt[3]{-0.5} \cdot \left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \sqrt[3]{2}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{h}{g} \cdot \frac{h}{g}\right) \cdot \frac{0.25}{a} - \frac{1}{a}\right) \cdot g} + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (* g g) a))
        (t_1 (cbrt (* (* (/ h a) -0.25) (/ h g))))
        (t_2 (sqrt (- (* g g) (* h h))))
        (t_3 (cbrt (* (- t_2 g) (/ 1.0 (* a 2.0)))))
        (t_4 (+ (cbrt (* (/ -1.0 (* a 2.0)) (+ t_2 g))) t_3)))
   (if (<= t_4 (- INFINITY))
     (*
      (fma
       (* (cbrt 0.5) (cbrt (/ 1.0 t_0)))
       (cbrt 2.0)
       (* (cbrt (/ 0.0 t_0)) (cbrt 0.5)))
      (- g))
     (if (<= t_4 -2e-101)
       (+ t_1 (cbrt (* (/ (- (* 0.25 (pow (/ h g) 2.0)) 1.0) a) g)))
       (if (<= t_4 0.0)
         (+ (* (cbrt -0.5) (* (/ (cbrt g) (cbrt a)) (cbrt 2.0))) t_3)
         (+
          (cbrt (* (- (* (* (/ h g) (/ h g)) (/ 0.25 a)) (/ 1.0 a)) g))
          t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = (g * g) * a;
	double t_1 = cbrt((((h / a) * -0.25) * (h / g)));
	double t_2 = sqrt(((g * g) - (h * h)));
	double t_3 = cbrt(((t_2 - g) * (1.0 / (a * 2.0))));
	double t_4 = cbrt(((-1.0 / (a * 2.0)) * (t_2 + g))) + t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = fma((cbrt(0.5) * cbrt((1.0 / t_0))), cbrt(2.0), (cbrt((0.0 / t_0)) * cbrt(0.5))) * -g;
	} else if (t_4 <= -2e-101) {
		tmp = t_1 + cbrt(((((0.25 * pow((h / g), 2.0)) - 1.0) / a) * g));
	} else if (t_4 <= 0.0) {
		tmp = (cbrt(-0.5) * ((cbrt(g) / cbrt(a)) * cbrt(2.0))) + t_3;
	} else {
		tmp = cbrt((((((h / g) * (h / g)) * (0.25 / a)) - (1.0 / a)) * g)) + t_1;
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(g * g) * a)
	t_1 = cbrt(Float64(Float64(Float64(h / a) * -0.25) * Float64(h / g)))
	t_2 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_3 = cbrt(Float64(Float64(t_2 - g) * Float64(1.0 / Float64(a * 2.0))))
	t_4 = Float64(cbrt(Float64(Float64(-1.0 / Float64(a * 2.0)) * Float64(t_2 + g))) + t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(fma(Float64(cbrt(0.5) * cbrt(Float64(1.0 / t_0))), cbrt(2.0), Float64(cbrt(Float64(0.0 / t_0)) * cbrt(0.5))) * Float64(-g));
	elseif (t_4 <= -2e-101)
		tmp = Float64(t_1 + cbrt(Float64(Float64(Float64(Float64(0.25 * (Float64(h / g) ^ 2.0)) - 1.0) / a) * g)));
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(cbrt(-0.5) * Float64(Float64(cbrt(g) / cbrt(a)) * cbrt(2.0))) + t_3);
	else
		tmp = Float64(cbrt(Float64(Float64(Float64(Float64(Float64(h / g) * Float64(h / g)) * Float64(0.25 / a)) - Float64(1.0 / a)) * g)) + t_1);
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(g * g), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(N[(h / a), $MachinePrecision] * -0.25), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[(t$95$2 - g), $MachinePrecision] * N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(N[(-1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(N[Power[0.5, 1/3], $MachinePrecision] * N[Power[N[(1.0 / t$95$0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision] + N[(N[Power[N[(0.0 / t$95$0), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-g)), $MachinePrecision], If[LessEqual[t$95$4, -2e-101], N[(t$95$1 + N[Power[N[(N[(N[(N[(0.25 * N[Power[N[(h / g), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / a), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[(N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[Power[N[(N[(N[(N[(N[(h / g), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * N[(0.25 / a), $MachinePrecision]), $MachinePrecision] - N[(1.0 / a), $MachinePrecision]), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -inf.0

   1. Initial program 4.2%

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

   4. Applied rewrites 0.6%

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

   5. Taylor expanded in g around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\left(-g\right) \cdot \mathsf{fma}\left(\sqrt[3]{\frac{1}{a \cdot \left(g \cdot g\right)}} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{0}{a \cdot \left(g \cdot g\right)}} \cdot \sqrt[3]{0.5}\right)} \]

   if -inf.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -2.0000000000000001e-101

   1. Initial program 91.0%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. 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 97.2

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. associate-*r/ N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. unpow2 N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 97.5

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

   11. Applied rewrites 97.5%

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

   13. Applied rewrites 97.5%

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

   if -2.0000000000000001e-101 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 0.0

   1. Initial program 6.9%

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

   3. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-cbrt.f64 6.9

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

   5. Applied rewrites 6.9%

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

   7. Applied rewrites 94.0%

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

   if 0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 36.8%

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

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   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 21.9

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

   5. Applied rewrites 21.9%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. 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 78.3

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. associate-*r/ N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. unpow2 N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 78.6

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

   11. Applied rewrites 78.6%

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

   13. Applied rewrites 78.6%

       \[\leadsto \color{blue}{\sqrt[3]{\left(-0.25 \cdot \frac{h}{a}\right) \cdot \frac{h}{g}}} + \sqrt[3]{g \cdot \left(\frac{0.25}{a} \cdot \left(\frac{h}{g} \cdot \frac{h}{g}\right) - \frac{1}{a}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)} + \sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{0.5} \cdot \sqrt[3]{\frac{1}{\left(g \cdot g\right) \cdot a}}, \sqrt[3]{2}, \sqrt[3]{\frac{0}{\left(g \cdot g\right) \cdot a}} \cdot \sqrt[3]{0.5}\right) \cdot \left(-g\right)\\ \mathbf{elif}\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)} + \sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} \leq -2 \cdot 10^{-101}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}} + \sqrt[3]{\frac{0.25 \cdot {\left(\frac{h}{g}\right)}^{2} - 1}{a} \cdot g}\\ \mathbf{elif}\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)} + \sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} \leq 0:\\ \;\;\;\;\sqrt[3]{-0.5} \cdot \left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \sqrt[3]{2}\right) + \sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{h}{g} \cdot \frac{h}{g}\right) \cdot \frac{0.25}{a} - \frac{1}{a}\right) \cdot g} + \sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(g \cdot g\right) \cdot a\\ t_1 := \sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}}\\ t_2 := \sqrt{g \cdot g - h \cdot h}\\ t_3 := \sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(t\_2 + g\right)} + \sqrt[3]{\left(t\_2 - g\right) \cdot \frac{1}{a \cdot 2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{0.5} \cdot \sqrt[3]{\frac{1}{t\_0}}, \sqrt[3]{2}, \sqrt[3]{\frac{0}{t\_0}} \cdot \sqrt[3]{0.5}\right) \cdot \left(-g\right)\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-101}:\\ \;\;\;\;t\_1 + \sqrt[3]{\frac{0.25 \cdot {\left(\frac{h}{g}\right)}^{2} - 1}{a} \cdot g}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\left({\left(\sqrt[3]{a}\right)}^{-1} \cdot \sqrt[3]{g}\right) \cdot \sqrt[3]{-1} + \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{h}{g} \cdot \frac{h}{g}\right) \cdot \frac{0.25}{a} - \frac{1}{a}\right) \cdot g} + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (* g g) a))
        (t_1 (cbrt (* (* (/ h a) -0.25) (/ h g))))
        (t_2 (sqrt (- (* g g) (* h h))))
        (t_3
         (+
          (cbrt (* (/ -1.0 (* a 2.0)) (+ t_2 g)))
          (cbrt (* (- t_2 g) (/ 1.0 (* a 2.0)))))))
   (if (<= t_3 (- INFINITY))
     (*
      (fma
       (* (cbrt 0.5) (cbrt (/ 1.0 t_0)))
       (cbrt 2.0)
       (* (cbrt (/ 0.0 t_0)) (cbrt 0.5)))
      (- g))
     (if (<= t_3 -2e-101)
       (+ t_1 (cbrt (* (/ (- (* 0.25 (pow (/ h g) 2.0)) 1.0) a) g)))
       (if (<= t_3 0.0)
         (+
          (* (* (pow (cbrt a) -1.0) (cbrt g)) (cbrt -1.0))
          (cbrt (* (/ 0.5 a) (- (sqrt (* (- g h) (+ h g))) g))))
         (+
          (cbrt (* (- (* (* (/ h g) (/ h g)) (/ 0.25 a)) (/ 1.0 a)) g))
          t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = (g * g) * a;
	double t_1 = cbrt((((h / a) * -0.25) * (h / g)));
	double t_2 = sqrt(((g * g) - (h * h)));
	double t_3 = cbrt(((-1.0 / (a * 2.0)) * (t_2 + g))) + cbrt(((t_2 - g) * (1.0 / (a * 2.0))));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma((cbrt(0.5) * cbrt((1.0 / t_0))), cbrt(2.0), (cbrt((0.0 / t_0)) * cbrt(0.5))) * -g;
	} else if (t_3 <= -2e-101) {
		tmp = t_1 + cbrt(((((0.25 * pow((h / g), 2.0)) - 1.0) / a) * g));
	} else if (t_3 <= 0.0) {
		tmp = ((pow(cbrt(a), -1.0) * cbrt(g)) * cbrt(-1.0)) + cbrt(((0.5 / a) * (sqrt(((g - h) * (h + g))) - g)));
	} else {
		tmp = cbrt((((((h / g) * (h / g)) * (0.25 / a)) - (1.0 / a)) * g)) + t_1;
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(g * g) * a)
	t_1 = cbrt(Float64(Float64(Float64(h / a) * -0.25) * Float64(h / g)))
	t_2 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_3 = Float64(cbrt(Float64(Float64(-1.0 / Float64(a * 2.0)) * Float64(t_2 + g))) + cbrt(Float64(Float64(t_2 - g) * Float64(1.0 / Float64(a * 2.0)))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(fma(Float64(cbrt(0.5) * cbrt(Float64(1.0 / t_0))), cbrt(2.0), Float64(cbrt(Float64(0.0 / t_0)) * cbrt(0.5))) * Float64(-g));
	elseif (t_3 <= -2e-101)
		tmp = Float64(t_1 + cbrt(Float64(Float64(Float64(Float64(0.25 * (Float64(h / g) ^ 2.0)) - 1.0) / a) * g)));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(Float64((cbrt(a) ^ -1.0) * cbrt(g)) * cbrt(-1.0)) + cbrt(Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(g - h) * Float64(h + g))) - g))));
	else
		tmp = Float64(cbrt(Float64(Float64(Float64(Float64(Float64(h / g) * Float64(h / g)) * Float64(0.25 / a)) - Float64(1.0 / a)) * g)) + t_1);
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(g * g), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(N[(h / a), $MachinePrecision] * -0.25), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[(-1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(t$95$2 - g), $MachinePrecision] * N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(N[Power[0.5, 1/3], $MachinePrecision] * N[Power[N[(1.0 / t$95$0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision] + N[(N[Power[N[(0.0 / t$95$0), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-g)), $MachinePrecision], If[LessEqual[t$95$3, -2e-101], N[(t$95$1 + N[Power[N[(N[(N[(N[(0.25 * N[Power[N[(h / g), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / a), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[(N[Power[N[Power[a, 1/3], $MachinePrecision], -1.0], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(N[(N[(h / g), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * N[(0.25 / a), $MachinePrecision]), $MachinePrecision] - N[(1.0 / a), $MachinePrecision]), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -inf.0

   1. Initial program 4.2%

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

   4. Applied rewrites 0.6%

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

   5. Taylor expanded in g around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\left(-g\right) \cdot \mathsf{fma}\left(\sqrt[3]{\frac{1}{a \cdot \left(g \cdot g\right)}} \cdot \sqrt[3]{0.5}, \sqrt[3]{2}, \sqrt[3]{\frac{0}{a \cdot \left(g \cdot g\right)}} \cdot \sqrt[3]{0.5}\right)} \]

   if -inf.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -2.0000000000000001e-101

   1. Initial program 91.0%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. 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 97.2

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. associate-*r/ N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. unpow2 N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 97.5

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

   11. Applied rewrites 97.5%

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

   13. Applied rewrites 97.5%

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

   if -2.0000000000000001e-101 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 0.0

   1. Initial program 6.9%

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

   3. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-cbrt.f64 6.9

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

   5. Applied rewrites 6.9%

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

   7. Applied rewrites 4.4%

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

   9. Applied rewrites 93.6%

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

   if 0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 36.8%

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

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   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 21.9

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

   5. Applied rewrites 21.9%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. 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 78.3

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. associate-*r/ N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. unpow2 N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 78.6

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

   11. Applied rewrites 78.6%

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

   13. Applied rewrites 78.6%

       \[\leadsto \color{blue}{\sqrt[3]{\left(-0.25 \cdot \frac{h}{a}\right) \cdot \frac{h}{g}}} + \sqrt[3]{g \cdot \left(\frac{0.25}{a} \cdot \left(\frac{h}{g} \cdot \frac{h}{g}\right) - \frac{1}{a}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)} + \sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{0.5} \cdot \sqrt[3]{\frac{1}{\left(g \cdot g\right) \cdot a}}, \sqrt[3]{2}, \sqrt[3]{\frac{0}{\left(g \cdot g\right) \cdot a}} \cdot \sqrt[3]{0.5}\right) \cdot \left(-g\right)\\ \mathbf{elif}\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)} + \sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} \leq -2 \cdot 10^{-101}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}} + \sqrt[3]{\frac{0.25 \cdot {\left(\frac{h}{g}\right)}^{2} - 1}{a} \cdot g}\\ \mathbf{elif}\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)} + \sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} \leq 0:\\ \;\;\;\;\left({\left(\sqrt[3]{a}\right)}^{-1} \cdot \sqrt[3]{g}\right) \cdot \sqrt[3]{-1} + \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{h}{g} \cdot \frac{h}{g}\right) \cdot \frac{0.25}{a} - \frac{1}{a}\right) \cdot g} + \sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ t_2 := \sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(t\_1 + g\right)}\\ t_3 := t\_2 + \sqrt[3]{\left(t\_1 - g\right) \cdot \frac{1}{a \cdot 2}}\\ t_4 := \sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\sqrt[3]{t\_4} \cdot \sqrt[3]{\frac{0.5}{a}} + t\_2\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-101}:\\ \;\;\;\;t\_0 + \sqrt[3]{\frac{0.25 \cdot {\left(\frac{h}{g}\right)}^{2} - 1}{a} \cdot g}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\left({\left(\sqrt[3]{a}\right)}^{-1} \cdot \sqrt[3]{g}\right) \cdot \sqrt[3]{-1} + \sqrt[3]{\frac{0.5}{a} \cdot t\_4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{h}{g} \cdot \frac{h}{g}\right) \cdot \frac{0.25}{a} - \frac{1}{a}\right) \cdot g} + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (* (/ h a) -0.25) (/ h g))))
        (t_1 (sqrt (- (* g g) (* h h))))
        (t_2 (cbrt (* (/ -1.0 (* a 2.0)) (+ t_1 g))))
        (t_3 (+ t_2 (cbrt (* (- t_1 g) (/ 1.0 (* a 2.0))))))
        (t_4 (- (sqrt (* (- g h) (+ h g))) g)))
   (if (<= t_3 (- INFINITY))
     (+ (* (cbrt t_4) (cbrt (/ 0.5 a))) t_2)
     (if (<= t_3 -2e-101)
       (+ t_0 (cbrt (* (/ (- (* 0.25 (pow (/ h g) 2.0)) 1.0) a) g)))
       (if (<= t_3 0.0)
         (+
          (* (* (pow (cbrt a) -1.0) (cbrt g)) (cbrt -1.0))
          (cbrt (* (/ 0.5 a) t_4)))
         (+
          (cbrt (* (- (* (* (/ h g) (/ h g)) (/ 0.25 a)) (/ 1.0 a)) g))
          t_0))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((((h / a) * -0.25) * (h / g)));
	double t_1 = sqrt(((g * g) - (h * h)));
	double t_2 = cbrt(((-1.0 / (a * 2.0)) * (t_1 + g)));
	double t_3 = t_2 + cbrt(((t_1 - g) * (1.0 / (a * 2.0))));
	double t_4 = sqrt(((g - h) * (h + g))) - g;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (cbrt(t_4) * cbrt((0.5 / a))) + t_2;
	} else if (t_3 <= -2e-101) {
		tmp = t_0 + cbrt(((((0.25 * pow((h / g), 2.0)) - 1.0) / a) * g));
	} else if (t_3 <= 0.0) {
		tmp = ((pow(cbrt(a), -1.0) * cbrt(g)) * cbrt(-1.0)) + cbrt(((0.5 / a) * t_4));
	} else {
		tmp = cbrt((((((h / g) * (h / g)) * (0.25 / a)) - (1.0 / a)) * g)) + t_0;
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt((((h / a) * -0.25) * (h / g)));
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	double t_2 = Math.cbrt(((-1.0 / (a * 2.0)) * (t_1 + g)));
	double t_3 = t_2 + Math.cbrt(((t_1 - g) * (1.0 / (a * 2.0))));
	double t_4 = Math.sqrt(((g - h) * (h + g))) - g;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.cbrt(t_4) * Math.cbrt((0.5 / a))) + t_2;
	} else if (t_3 <= -2e-101) {
		tmp = t_0 + Math.cbrt(((((0.25 * Math.pow((h / g), 2.0)) - 1.0) / a) * g));
	} else if (t_3 <= 0.0) {
		tmp = ((Math.pow(Math.cbrt(a), -1.0) * Math.cbrt(g)) * Math.cbrt(-1.0)) + Math.cbrt(((0.5 / a) * t_4));
	} else {
		tmp = Math.cbrt((((((h / g) * (h / g)) * (0.25 / a)) - (1.0 / a)) * g)) + t_0;
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(Float64(h / a) * -0.25) * Float64(h / g)))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_2 = cbrt(Float64(Float64(-1.0 / Float64(a * 2.0)) * Float64(t_1 + g)))
	t_3 = Float64(t_2 + cbrt(Float64(Float64(t_1 - g) * Float64(1.0 / Float64(a * 2.0)))))
	t_4 = Float64(sqrt(Float64(Float64(g - h) * Float64(h + g))) - g)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(cbrt(t_4) * cbrt(Float64(0.5 / a))) + t_2);
	elseif (t_3 <= -2e-101)
		tmp = Float64(t_0 + cbrt(Float64(Float64(Float64(Float64(0.25 * (Float64(h / g) ^ 2.0)) - 1.0) / a) * g)));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(Float64((cbrt(a) ^ -1.0) * cbrt(g)) * cbrt(-1.0)) + cbrt(Float64(Float64(0.5 / a) * t_4)));
	else
		tmp = Float64(cbrt(Float64(Float64(Float64(Float64(Float64(h / g) * Float64(h / g)) * Float64(0.25 / a)) - Float64(1.0 / a)) * g)) + t_0);
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(N[(h / a), $MachinePrecision] * -0.25), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[(-1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[N[(N[(t$95$1 - g), $MachinePrecision] * N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Power[t$95$4, 1/3], $MachinePrecision] * N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -2e-101], N[(t$95$0 + N[Power[N[(N[(N[(N[(0.25 * N[Power[N[(h / g), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / a), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[(N[Power[N[Power[a, 1/3], $MachinePrecision], -1.0], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * t$95$4), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(N[(N[(h / g), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * N[(0.25 / a), $MachinePrecision]), $MachinePrecision] - N[(1.0 / a), $MachinePrecision]), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision] + t$95$0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -inf.0

   1. Initial program 4.2%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing
   3. 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. pow1/3 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/3 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. pow1/3 N/A

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

      14. lower-cbrt.f64 78.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-neg.f64 N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 78.0

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

   4. Applied rewrites 78.0%

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

   if -inf.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -2.0000000000000001e-101

   1. Initial program 91.0%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. 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 97.2

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. associate-*r/ N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. unpow2 N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 97.5

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

   11. Applied rewrites 97.5%

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

   13. Applied rewrites 97.5%

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

   if -2.0000000000000001e-101 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 0.0

   1. Initial program 6.9%

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

   3. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-cbrt.f64 6.9

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

   5. Applied rewrites 6.9%

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

   7. Applied rewrites 4.4%

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

   9. Applied rewrites 93.6%

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

   if 0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 36.8%

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

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   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 21.9

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

   5. Applied rewrites 21.9%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. 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 78.3

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. associate-*r/ N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. unpow2 N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 78.6

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

   11. Applied rewrites 78.6%

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

   13. Applied rewrites 78.6%

       \[\leadsto \color{blue}{\sqrt[3]{\left(-0.25 \cdot \frac{h}{a}\right) \cdot \frac{h}{g}}} + \sqrt[3]{g \cdot \left(\frac{0.25}{a} \cdot \left(\frac{h}{g} \cdot \frac{h}{g}\right) - \frac{1}{a}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)} + \sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} \leq -\infty:\\ \;\;\;\;\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g} \cdot \sqrt[3]{\frac{0.5}{a}} + \sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)}\\ \mathbf{elif}\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)} + \sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} \leq -2 \cdot 10^{-101}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}} + \sqrt[3]{\frac{0.25 \cdot {\left(\frac{h}{g}\right)}^{2} - 1}{a} \cdot g}\\ \mathbf{elif}\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)} + \sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} \leq 0:\\ \;\;\;\;\left({\left(\sqrt[3]{a}\right)}^{-1} \cdot \sqrt[3]{g}\right) \cdot \sqrt[3]{-1} + \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{h}{g} \cdot \frac{h}{g}\right) \cdot \frac{0.25}{a} - \frac{1}{a}\right) \cdot g} + \sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(t\_0 + g\right)} + \sqrt[3]{\left(t\_0 - g\right) \cdot \frac{1}{a \cdot 2}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}, \sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}} + \sqrt[3]{\frac{0.25 \cdot {\left(\frac{h}{g}\right)}^{2} - 1}{a} \cdot g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h)))))
   (if (<=
        (+
         (cbrt (* (/ -1.0 (* a 2.0)) (+ t_0 g)))
         (cbrt (* (- t_0 g) (/ 1.0 (* a 2.0)))))
        (- INFINITY))
     (fma
      (cbrt (/ 0.5 a))
      (cbrt (- (sqrt (* (- g h) (+ h g))) g))
      (cbrt (* (/ (* h h) g) (/ -0.25 a))))
     (+
      (cbrt (* (* (/ h a) -0.25) (/ h g)))
      (cbrt (* (/ (- (* 0.25 (pow (/ h g) 2.0)) 1.0) a) g))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double tmp;
	if ((cbrt(((-1.0 / (a * 2.0)) * (t_0 + g))) + cbrt(((t_0 - g) * (1.0 / (a * 2.0))))) <= -((double) INFINITY)) {
		tmp = fma(cbrt((0.5 / a)), cbrt((sqrt(((g - h) * (h + g))) - g)), cbrt((((h * h) / g) * (-0.25 / a))));
	} else {
		tmp = cbrt((((h / a) * -0.25) * (h / g))) + cbrt(((((0.25 * pow((h / g), 2.0)) - 1.0) / a) * g));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	tmp = 0.0
	if (Float64(cbrt(Float64(Float64(-1.0 / Float64(a * 2.0)) * Float64(t_0 + g))) + cbrt(Float64(Float64(t_0 - g) * Float64(1.0 / Float64(a * 2.0))))) <= Float64(-Inf))
		tmp = fma(cbrt(Float64(0.5 / a)), cbrt(Float64(sqrt(Float64(Float64(g - h) * Float64(h + g))) - g)), cbrt(Float64(Float64(Float64(h * h) / g) * Float64(-0.25 / a))));
	else
		tmp = Float64(cbrt(Float64(Float64(Float64(h / a) * -0.25) * Float64(h / g))) + cbrt(Float64(Float64(Float64(Float64(0.25 * (Float64(h / g) ^ 2.0)) - 1.0) / a) * g)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -inf.0

   1. Initial program 4.2%

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

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   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 4.2

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

   5. Applied rewrites 4.2%

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

      1. lift-+.f64 N/A

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in g around -inf

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 77.9

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

   10. Applied rewrites 77.9%

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

   if -inf.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 49.4%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 29.9

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

   5. Applied rewrites 29.9%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. 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 79.2

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

   8. Applied rewrites 79.2%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. associate-*r/ N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. unpow2 N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 79.5

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

   11. Applied rewrites 79.5%

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

   13. Applied rewrites 79.5%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{{\left(\frac{h}{g}\right)}^{2} \cdot 0.25 - 1}{a} \cdot g} + \sqrt[3]{\left(-0.25 \cdot \frac{h}{a}\right) \cdot \frac{h}{g}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.5%

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

Alternative 5: 76.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ -1.0 (* a 2.0))) (t_1 (sqrt (- (* g g) (* h h)))))
   (if (<=
        (+ (cbrt (* t_0 (+ t_1 g))) (cbrt (* (- t_1 g) (/ 1.0 (* a 2.0)))))
        (- INFINITY))
     (+
      (cbrt (* t_0 (+ (- g) g)))
      (/ (cbrt (- (sqrt (* (- g h) (+ h g))) g)) (cbrt (* a 2.0))))
     (+
      (cbrt (* (* (/ h a) -0.25) (/ h g)))
      (cbrt (* (/ (- (* 0.25 (pow (/ h g) 2.0)) 1.0) a) g))))))

C

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

Java

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

Julia

function code(g, h, a)
	t_0 = Float64(-1.0 / Float64(a * 2.0))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	tmp = 0.0
	if (Float64(cbrt(Float64(t_0 * Float64(t_1 + g))) + cbrt(Float64(Float64(t_1 - g) * Float64(1.0 / Float64(a * 2.0))))) <= Float64(-Inf))
		tmp = Float64(cbrt(Float64(t_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(h / a) * -0.25) * Float64(h / g))) + cbrt(Float64(Float64(Float64(Float64(0.25 * (Float64(h / g) ^ 2.0)) - 1.0) / a) * g)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -inf.0

   1. Initial program 4.2%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing
   3. 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 77.7%

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

   5. Taylor expanded in g around -inf

      \[\leadsto \frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{-1 \cdot g}\right)} \]
   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 77.3

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

   7. Applied rewrites 77.3%

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

   if -inf.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 49.4%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 29.9

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

   5. Applied rewrites 29.9%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. 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 79.2

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

   8. Applied rewrites 79.2%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. associate-*r/ N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. unpow2 N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 79.5

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

   11. Applied rewrites 79.5%

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

   13. Applied rewrites 79.5%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{{\left(\frac{h}{g}\right)}^{2} \cdot 0.25 - 1}{a} \cdot g} + \sqrt[3]{\left(-0.25 \cdot \frac{h}{a}\right) \cdot \frac{h}{g}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.5%

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

Alternative 6: 78.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}}\\ t_1 := \sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\\ \mathbf{if}\;g \leq -5.7 \cdot 10^{+122}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{h}{g} \cdot \frac{h}{g}\right) \cdot \frac{0.25}{a} - \frac{1}{a}\right) \cdot g} + t\_0\\ \mathbf{elif}\;g \leq -4.4 \cdot 10^{-154}:\\ \;\;\;\;\sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}} + \frac{\sqrt[3]{t\_1}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{elif}\;g \leq 5.4 \cdot 10^{-218}:\\ \;\;\;\;t\_0 + \sqrt[3]{\frac{0.25 \cdot {\left(\frac{h}{g}\right)}^{2} - 1}{a} \cdot g}\\ \mathbf{elif}\;g \leq 1.1 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)}, \sqrt[3]{\frac{0.5}{a} \cdot t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{\left(\frac{h}{g} \cdot h\right) \cdot 0.25}{g \cdot a} - \frac{1}{a}\right) \cdot g} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (* (/ h a) -0.25) (/ h g))))
        (t_1 (- (sqrt (* (- g h) (+ h g))) g)))
   (if (<= g -5.7e+122)
     (+ (cbrt (* (- (* (* (/ h g) (/ h g)) (/ 0.25 a)) (/ 1.0 a)) g)) t_0)
     (if (<= g -4.4e-154)
       (+ (cbrt (* (/ (* h h) g) (/ -0.25 a))) (/ (cbrt t_1) (cbrt (* a 2.0))))
       (if (<= g 5.4e-218)
         (+ t_0 (cbrt (* (/ (- (* 0.25 (pow (/ h g) 2.0)) 1.0) a) g)))
         (if (<= g 1.1e+141)
           (fma
            (cbrt (/ 0.5 a))
            (cbrt (- (fma (sqrt (- g h)) (sqrt (+ h g)) g)))
            (cbrt (* (/ 0.5 a) t_1)))
           (+
            (cbrt (* (- (/ (* (* (/ h g) h) 0.25) (* g a)) (/ 1.0 a)) g))
            (* (* (cbrt -0.5) (cbrt 0.5)) (cbrt (* (/ h a) (/ h g)))))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((((h / a) * -0.25) * (h / g)));
	double t_1 = sqrt(((g - h) * (h + g))) - g;
	double tmp;
	if (g <= -5.7e+122) {
		tmp = cbrt((((((h / g) * (h / g)) * (0.25 / a)) - (1.0 / a)) * g)) + t_0;
	} else if (g <= -4.4e-154) {
		tmp = cbrt((((h * h) / g) * (-0.25 / a))) + (cbrt(t_1) / cbrt((a * 2.0)));
	} else if (g <= 5.4e-218) {
		tmp = t_0 + cbrt(((((0.25 * pow((h / g), 2.0)) - 1.0) / a) * g));
	} else if (g <= 1.1e+141) {
		tmp = fma(cbrt((0.5 / a)), cbrt(-fma(sqrt((g - h)), sqrt((h + g)), g)), cbrt(((0.5 / a) * t_1)));
	} else {
		tmp = cbrt(((((((h / g) * h) * 0.25) / (g * a)) - (1.0 / a)) * g)) + ((cbrt(-0.5) * cbrt(0.5)) * cbrt(((h / a) * (h / g))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(Float64(h / a) * -0.25) * Float64(h / g)))
	t_1 = Float64(sqrt(Float64(Float64(g - h) * Float64(h + g))) - g)
	tmp = 0.0
	if (g <= -5.7e+122)
		tmp = Float64(cbrt(Float64(Float64(Float64(Float64(Float64(h / g) * Float64(h / g)) * Float64(0.25 / a)) - Float64(1.0 / a)) * g)) + t_0);
	elseif (g <= -4.4e-154)
		tmp = Float64(cbrt(Float64(Float64(Float64(h * h) / g) * Float64(-0.25 / a))) + Float64(cbrt(t_1) / cbrt(Float64(a * 2.0))));
	elseif (g <= 5.4e-218)
		tmp = Float64(t_0 + cbrt(Float64(Float64(Float64(Float64(0.25 * (Float64(h / g) ^ 2.0)) - 1.0) / a) * g)));
	elseif (g <= 1.1e+141)
		tmp = fma(cbrt(Float64(0.5 / a)), cbrt(Float64(-fma(sqrt(Float64(g - h)), sqrt(Float64(h + g)), g))), cbrt(Float64(Float64(0.5 / a) * t_1)));
	else
		tmp = Float64(cbrt(Float64(Float64(Float64(Float64(Float64(Float64(h / g) * h) * 0.25) / Float64(g * a)) - Float64(1.0 / a)) * g)) + Float64(Float64(cbrt(-0.5) * cbrt(0.5)) * cbrt(Float64(Float64(h / a) * Float64(h / g)))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(N[(h / a), $MachinePrecision] * -0.25), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision]}, If[LessEqual[g, -5.7e+122], N[(N[Power[N[(N[(N[(N[(N[(h / g), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * N[(0.25 / a), $MachinePrecision]), $MachinePrecision] - N[(1.0 / a), $MachinePrecision]), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[g, -4.4e-154], N[(N[Power[N[(N[(N[(h * h), $MachinePrecision] / g), $MachinePrecision] * N[(-0.25 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[t$95$1, 1/3], $MachinePrecision] / N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 5.4e-218], N[(t$95$0 + N[Power[N[(N[(N[(N[(0.25 * N[Power[N[(h / g), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / a), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 1.1e+141], N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[(-N[(N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * t$95$1), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(N[(N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision] * 0.25), $MachinePrecision] / N[(g * a), $MachinePrecision]), $MachinePrecision] - N[(1.0 / a), $MachinePrecision]), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(h / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if g < -5.70000000000000006e122

   1. Initial program 14.8%

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

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   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 5.7

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

   5. Applied rewrites 5.7%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. 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 76.7

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

   8. Applied rewrites 76.7%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. associate-*r/ N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. unpow2 N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 76.9

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

   11. Applied rewrites 76.9%

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

   13. Applied rewrites 76.9%

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

   if -5.70000000000000006e122 < g < -4.40000000000000015e-154

   1. Initial program 80.3%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 96.1%

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

   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. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 96.5

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

   7. Applied rewrites 96.5%

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

   if -4.40000000000000015e-154 < g < 5.3999999999999999e-218

   1. Initial program 25.2%

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

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   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 15.7

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

   5. Applied rewrites 15.7%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. 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 71.1

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

   8. Applied rewrites 71.1%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. associate-*r/ N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. unpow2 N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 72.2

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

   11. Applied rewrites 72.2%

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

   13. Applied rewrites 72.3%

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

   if 5.3999999999999999e-218 < g < 1.1e141

   1. Initial program 79.5%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 80.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. sqrt-div N/A

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

      13. sqrt-unprod N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-/.f64 N/A

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

   6. Applied rewrites 56.6%

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

   8. Applied rewrites 95.4%

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

   if 1.1e141 < g

   1. Initial program 10.1%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 10.4

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

   5. Applied rewrites 10.4%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. 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 69.2

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

   8. Applied rewrites 69.2%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. associate-*r/ N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. unpow2 N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 69.4

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

   11. Applied rewrites 69.4%

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

   13. Applied rewrites 69.4%

       \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) + \sqrt[3]{g \cdot \left(\frac{\left(\frac{h}{g} \cdot h\right) \cdot 0.25}{a \cdot g} - \frac{\color{blue}{1}}{a}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -5.7 \cdot 10^{+122}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{h}{g} \cdot \frac{h}{g}\right) \cdot \frac{0.25}{a} - \frac{1}{a}\right) \cdot g} + \sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}}\\ \mathbf{elif}\;g \leq -4.4 \cdot 10^{-154}:\\ \;\;\;\;\sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}} + \frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{elif}\;g \leq 5.4 \cdot 10^{-218}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}} + \sqrt[3]{\frac{0.25 \cdot {\left(\frac{h}{g}\right)}^{2} - 1}{a} \cdot g}\\ \mathbf{elif}\;g \leq 1.1 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)}, \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{\left(\frac{h}{g} \cdot h\right) \cdot 0.25}{g \cdot a} - \frac{1}{a}\right) \cdot g} + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}}\\ t_1 := \sqrt[3]{\left(\left(\frac{h}{g} \cdot \frac{h}{g}\right) \cdot \frac{0.25}{a} - \frac{1}{a}\right) \cdot g} + t\_0\\ t_2 := \sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\\ \mathbf{if}\;g \leq -5.7 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;g \leq -4.4 \cdot 10^{-154}:\\ \;\;\;\;\sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}} + \frac{\sqrt[3]{t\_2}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{elif}\;g \leq 5.4 \cdot 10^{-218}:\\ \;\;\;\;t\_0 + \sqrt[3]{\frac{0.25 \cdot {\left(\frac{h}{g}\right)}^{2} - 1}{a} \cdot g}\\ \mathbf{elif}\;g \leq 1.1 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)}, \sqrt[3]{\frac{0.5}{a} \cdot t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (* (/ h a) -0.25) (/ h g))))
        (t_1
         (+ (cbrt (* (- (* (* (/ h g) (/ h g)) (/ 0.25 a)) (/ 1.0 a)) g)) t_0))
        (t_2 (- (sqrt (* (- g h) (+ h g))) g)))
   (if (<= g -5.7e+122)
     t_1
     (if (<= g -4.4e-154)
       (+ (cbrt (* (/ (* h h) g) (/ -0.25 a))) (/ (cbrt t_2) (cbrt (* a 2.0))))
       (if (<= g 5.4e-218)
         (+ t_0 (cbrt (* (/ (- (* 0.25 (pow (/ h g) 2.0)) 1.0) a) g)))
         (if (<= g 1.1e+141)
           (fma
            (cbrt (/ 0.5 a))
            (cbrt (- (fma (sqrt (- g h)) (sqrt (+ h g)) g)))
            (cbrt (* (/ 0.5 a) t_2)))
           t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((((h / a) * -0.25) * (h / g)));
	double t_1 = cbrt((((((h / g) * (h / g)) * (0.25 / a)) - (1.0 / a)) * g)) + t_0;
	double t_2 = sqrt(((g - h) * (h + g))) - g;
	double tmp;
	if (g <= -5.7e+122) {
		tmp = t_1;
	} else if (g <= -4.4e-154) {
		tmp = cbrt((((h * h) / g) * (-0.25 / a))) + (cbrt(t_2) / cbrt((a * 2.0)));
	} else if (g <= 5.4e-218) {
		tmp = t_0 + cbrt(((((0.25 * pow((h / g), 2.0)) - 1.0) / a) * g));
	} else if (g <= 1.1e+141) {
		tmp = fma(cbrt((0.5 / a)), cbrt(-fma(sqrt((g - h)), sqrt((h + g)), g)), cbrt(((0.5 / a) * t_2)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(Float64(h / a) * -0.25) * Float64(h / g)))
	t_1 = Float64(cbrt(Float64(Float64(Float64(Float64(Float64(h / g) * Float64(h / g)) * Float64(0.25 / a)) - Float64(1.0 / a)) * g)) + t_0)
	t_2 = Float64(sqrt(Float64(Float64(g - h) * Float64(h + g))) - g)
	tmp = 0.0
	if (g <= -5.7e+122)
		tmp = t_1;
	elseif (g <= -4.4e-154)
		tmp = Float64(cbrt(Float64(Float64(Float64(h * h) / g) * Float64(-0.25 / a))) + Float64(cbrt(t_2) / cbrt(Float64(a * 2.0))));
	elseif (g <= 5.4e-218)
		tmp = Float64(t_0 + cbrt(Float64(Float64(Float64(Float64(0.25 * (Float64(h / g) ^ 2.0)) - 1.0) / a) * g)));
	elseif (g <= 1.1e+141)
		tmp = fma(cbrt(Float64(0.5 / a)), cbrt(Float64(-fma(sqrt(Float64(g - h)), sqrt(Float64(h + g)), g))), cbrt(Float64(Float64(0.5 / a) * t_2)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(N[(h / a), $MachinePrecision] * -0.25), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[(N[(N[(N[(h / g), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision] * N[(0.25 / a), $MachinePrecision]), $MachinePrecision] - N[(1.0 / a), $MachinePrecision]), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(g - h), $MachinePrecision] * N[(h + g), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision]}, If[LessEqual[g, -5.7e+122], t$95$1, If[LessEqual[g, -4.4e-154], N[(N[Power[N[(N[(N[(h * h), $MachinePrecision] / g), $MachinePrecision] * N[(-0.25 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[t$95$2, 1/3], $MachinePrecision] / N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 5.4e-218], N[(t$95$0 + N[Power[N[(N[(N[(N[(0.25 * N[Power[N[(h / g), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / a), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 1.1e+141], N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[(-N[(N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * t$95$2), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if g < -5.70000000000000006e122 or 1.1e141 < g

   1. Initial program 12.3%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 8.2

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

   5. Applied rewrites 8.2%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. 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 72.7

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

   8. Applied rewrites 72.7%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. associate-*r/ N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. unpow2 N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 72.9

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

   11. Applied rewrites 72.9%

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

   13. Applied rewrites 72.9%

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

   if -5.70000000000000006e122 < g < -4.40000000000000015e-154

   1. Initial program 80.3%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 96.1%

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

   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. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 96.5

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

   7. Applied rewrites 96.5%

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

   if -4.40000000000000015e-154 < g < 5.3999999999999999e-218

   1. Initial program 25.2%

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

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   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 15.7

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

   5. Applied rewrites 15.7%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. 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 71.1

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

   8. Applied rewrites 71.1%

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

   9. Taylor expanded in g around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. associate-*r/ N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. unpow2 N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 72.2

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

   11. Applied rewrites 72.2%

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

   13. Applied rewrites 72.3%

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

   if 5.3999999999999999e-218 < g < 1.1e141

   1. Initial program 79.5%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 80.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. sqrt-div N/A

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

      13. sqrt-unprod N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-/.f64 N/A

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

   6. Applied rewrites 56.6%

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

   8. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)}, \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -5.7 \cdot 10^{+122}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{h}{g} \cdot \frac{h}{g}\right) \cdot \frac{0.25}{a} - \frac{1}{a}\right) \cdot g} + \sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}}\\ \mathbf{elif}\;g \leq -4.4 \cdot 10^{-154}:\\ \;\;\;\;\sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}} + \frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{elif}\;g \leq 5.4 \cdot 10^{-218}:\\ \;\;\;\;\sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}} + \sqrt[3]{\frac{0.25 \cdot {\left(\frac{h}{g}\right)}^{2} - 1}{a} \cdot g}\\ \mathbf{elif}\;g \leq 1.1 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)}, \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{h}{g} \cdot \frac{h}{g}\right) \cdot \frac{0.25}{a} - \frac{1}{a}\right) \cdot g} + \sqrt[3]{\left(\frac{h}{a} \cdot -0.25\right) \cdot \frac{h}{g}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.8%

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

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
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 29.0

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

5. Applied rewrites 29.0%

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

6. Taylor expanded in g around inf

   \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
7. 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 76.6

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

8. Applied rewrites 76.6%

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

9. Taylor expanded in g around inf

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

    1. lower-*.f64 N/A

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

    2. lower--.f64 N/A

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

    3. associate-*r/ N/A

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

    4. times-frac N/A

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

    5. lower-*.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. unpow2 N/A

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

    8. unpow2 N/A

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

    9. times-frac N/A

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

    10. lower-*.f64 N/A

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

    11. lower-/.f64 N/A

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

    12. lower-/.f64 N/A

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

    13. lower-/.f64 76.9

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

11. Applied rewrites 76.9%

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

13. Applied rewrites 76.9%

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

14. Final simplification 76.9%

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

Alternative 9: 75.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.8%

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

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
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 29.0

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

5. Applied rewrites 29.0%

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

6. Taylor expanded in g around inf

   \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
7. 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 76.6

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

8. Applied rewrites 76.6%

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

9. Taylor expanded in g around inf

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

    1. lower-*.f64 N/A

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

    2. lower--.f64 N/A

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

    3. associate-*r/ N/A

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

    4. times-frac N/A

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

    5. lower-*.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. unpow2 N/A

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

    8. unpow2 N/A

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

    9. times-frac N/A

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

    10. lower-*.f64 N/A

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

    11. lower-/.f64 N/A

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

    12. lower-/.f64 N/A

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

    13. lower-/.f64 76.9

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

11. Applied rewrites 76.9%

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

13. Applied rewrites 76.9%

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

14. Final simplification 76.9%

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

Alternative 10: 75.2% 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 47.8%

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

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
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 29.0

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

5. Applied rewrites 29.0%

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

6. Taylor expanded in g around inf

   \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
7. 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 76.6

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

8. Applied rewrites 76.6%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
9. 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 76.6

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

10. Applied rewrites 76.6%

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

11. Final simplification 76.6%

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

Alternative 11: 73.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.8%

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

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
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 29.0

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

5. Applied rewrites 29.0%

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

6. Taylor expanded in g around inf

   \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
7. 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 76.6

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

8. Applied rewrites 76.6%

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

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

10. Applied rewrites 37.7%

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

11. Taylor expanded in g around inf

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

    1. lower-*.f64 N/A

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

    2. lower-cbrt.f64 N/A

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

    3. lower-/.f64 N/A

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

    4. unpow2 N/A

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

    5. rem-square-sqrt N/A

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

    6. lower-*.f64 N/A

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

    7. lower-cbrt.f64 73.6

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

13. Applied rewrites 73.6%

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

14. Final simplification 73.6%

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

Alternative 12: 3.0% accurate, 302.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 0.0)

C

double code(double g, double h, double a) {
	return 0.0;
}

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    code = 0.0d0
end function

Java

public static double code(double g, double h, double a) {
	return 0.0;
}

Python

def code(g, h, a):
	return 0.0

Julia

function code(g, h, a)
	return 0.0
end

MATLAB

function tmp = code(g, h, a)
	tmp = 0.0;
end

Wolfram

code[g_, h_, a_] := 0.0

Derivation

1. Initial program 47.8%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing
3. 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 52.2%

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

5. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-cbrt.f64 3.0

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

7. Applied rewrites 3.0%

   \[\leadsto \color{blue}{-\sqrt[3]{\frac{g \cdot 0}{a}} \cdot \frac{1}{\sqrt[3]{2}}} \]
8. Step-by-step derivation

9. Applied rewrites 3.0%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024251 
(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))))))))