2-ancestry mixing, positive discriminant

Percentage Accurate: 43.6% → 79.7%

Time: 11.2s

Alternatives: 12

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 43.6% 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: 79.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-g}{a}}\\ t_1 := \sqrt{g - h}\\ t_2 := \left(h + g\right) \cdot \left(g - h\right)\\ t_3 := \sqrt{t\_2}\\ t_4 := t\_3 - g\\ t_5 := \sqrt{h + g}\\ \mathbf{if}\;g \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{h}{a} \cdot \left(-0.25 \cdot \frac{h}{g}\right)} + t\_0\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\sqrt[3]{\frac{\left(t\_2 - g \cdot g\right) \cdot \frac{0.5}{a}}{g - t\_3}} + \sqrt[3]{t\_4} \cdot \sqrt[3]{\frac{0.5}{a}}\\ \mathbf{elif}\;g \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;\sqrt[3]{\frac{-0.25}{a} \cdot \frac{h}{g}} \cdot \sqrt[3]{h} + t\_0\\ \mathbf{elif}\;g \leq 1.7 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{-1}{a}}, \sqrt[3]{\frac{\mathsf{fma}\left(t\_1, t\_5, g\right)}{2}}, \sqrt[3]{t\_4 \cdot \frac{0.5}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{\left(-2\right) \cdot a}{\mathsf{fma}\left(t\_5, t\_1, g\right)}}} + \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{\left(\left(\frac{\left(\frac{{\left(0 \cdot g\right)}^{2} \cdot 0.25}{g \cdot g} - -1\right) \cdot h}{g} - \frac{0 \cdot g}{g}\right) \cdot 0.5\right) \cdot h}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (/ (- g) a)))
        (t_1 (sqrt (- g h)))
        (t_2 (* (+ h g) (- g h)))
        (t_3 (sqrt t_2))
        (t_4 (- t_3 g))
        (t_5 (sqrt (+ h g))))
   (if (<= g -1.32e+154)
     (+ (cbrt (* (/ h a) (* -0.25 (/ h g)))) t_0)
     (if (<= g -6e-114)
       (+
        (cbrt (/ (* (- t_2 (* g g)) (/ 0.5 a)) (- g t_3)))
        (* (cbrt t_4) (cbrt (/ 0.5 a))))
       (if (<= g 1.05e-154)
         (+ (* (cbrt (* (/ -0.25 a) (/ h g))) (cbrt h)) t_0)
         (if (<= g 1.7e+153)
           (fma
            (cbrt (/ -1.0 a))
            (cbrt (/ (fma t_1 t_5 g) 2.0))
            (cbrt (* t_4 (/ 0.5 a))))
           (+
            (/ 1.0 (cbrt (/ (* (- 2.0) a) (fma t_5 t_1 g))))
            (/
             1.0
             (cbrt
              (/
               (* -2.0 a)
               (*
                (*
                 (-
                  (/ (* (- (/ (* (pow (* 0.0 g) 2.0) 0.25) (* g g)) -1.0) h) g)
                  (/ (* 0.0 g) g))
                 0.5)
                h)))))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((-g / a));
	double t_1 = sqrt((g - h));
	double t_2 = (h + g) * (g - h);
	double t_3 = sqrt(t_2);
	double t_4 = t_3 - g;
	double t_5 = sqrt((h + g));
	double tmp;
	if (g <= -1.32e+154) {
		tmp = cbrt(((h / a) * (-0.25 * (h / g)))) + t_0;
	} else if (g <= -6e-114) {
		tmp = cbrt((((t_2 - (g * g)) * (0.5 / a)) / (g - t_3))) + (cbrt(t_4) * cbrt((0.5 / a)));
	} else if (g <= 1.05e-154) {
		tmp = (cbrt(((-0.25 / a) * (h / g))) * cbrt(h)) + t_0;
	} else if (g <= 1.7e+153) {
		tmp = fma(cbrt((-1.0 / a)), cbrt((fma(t_1, t_5, g) / 2.0)), cbrt((t_4 * (0.5 / a))));
	} else {
		tmp = (1.0 / cbrt(((-2.0 * a) / fma(t_5, t_1, g)))) + (1.0 / cbrt(((-2.0 * a) / ((((((((pow((0.0 * g), 2.0) * 0.25) / (g * g)) - -1.0) * h) / g) - ((0.0 * g) / g)) * 0.5) * h))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(-g) / a))
	t_1 = sqrt(Float64(g - h))
	t_2 = Float64(Float64(h + g) * Float64(g - h))
	t_3 = sqrt(t_2)
	t_4 = Float64(t_3 - g)
	t_5 = sqrt(Float64(h + g))
	tmp = 0.0
	if (g <= -1.32e+154)
		tmp = Float64(cbrt(Float64(Float64(h / a) * Float64(-0.25 * Float64(h / g)))) + t_0);
	elseif (g <= -6e-114)
		tmp = Float64(cbrt(Float64(Float64(Float64(t_2 - Float64(g * g)) * Float64(0.5 / a)) / Float64(g - t_3))) + Float64(cbrt(t_4) * cbrt(Float64(0.5 / a))));
	elseif (g <= 1.05e-154)
		tmp = Float64(Float64(cbrt(Float64(Float64(-0.25 / a) * Float64(h / g))) * cbrt(h)) + t_0);
	elseif (g <= 1.7e+153)
		tmp = fma(cbrt(Float64(-1.0 / a)), cbrt(Float64(fma(t_1, t_5, g) / 2.0)), cbrt(Float64(t_4 * Float64(0.5 / a))));
	else
		tmp = Float64(Float64(1.0 / cbrt(Float64(Float64(Float64(-2.0) * a) / fma(t_5, t_1, g)))) + Float64(1.0 / cbrt(Float64(Float64(-2.0 * a) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64((Float64(0.0 * g) ^ 2.0) * 0.25) / Float64(g * g)) - -1.0) * h) / g) - Float64(Float64(0.0 * g) / g)) * 0.5) * h)))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - g), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[g, -1.32e+154], N[(N[Power[N[(N[(h / a), $MachinePrecision] * N[(-0.25 * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[g, -6e-114], N[(N[Power[N[(N[(N[(t$95$2 - N[(g * g), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(g - t$95$3), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[t$95$4, 1/3], $MachinePrecision] * N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 1.05e-154], N[(N[(N[Power[N[(N[(-0.25 / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[h, 1/3], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[g, 1.7e+153], N[(N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(t$95$1 * t$95$5 + g), $MachinePrecision] / 2.0), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$4 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Power[N[(N[((-2.0) * a), $MachinePrecision] / N[(t$95$5 * t$95$1 + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[N[(N[(-2.0 * a), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(N[(N[Power[N[(0.0 * g), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $MachinePrecision] / N[(g * g), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * h), $MachinePrecision] / g), $MachinePrecision] - N[(N[(0.0 * g), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if g < -1.31999999999999998e154

   1. Initial program 0.0%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 3.5

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

   5. Applied rewrites 3.5%

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

   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 61.8

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

   8. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   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 61.8

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

   10. Applied rewrites 61.8%

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

   if -1.31999999999999998e154 < g < -6.0000000000000003e-114

   1. Initial program 85.1%

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

      1. lift-cbrt.f64 N/A

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

      2. 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 97.7

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

      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 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. +-commutative N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-squares N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift--.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. sub-neg N/A

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

      16. lift--.f64 N/A

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

   6. Applied rewrites 97.8%

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

   if -6.0000000000000003e-114 < g < 1.04999999999999992e-154

   1. Initial program 25.8%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 16.2

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

   5. Applied rewrites 16.2%

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

   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 79.0

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

   8. Applied rewrites 79.0%

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

   10. Applied rewrites 80.3%

       \[\leadsto \sqrt[3]{\frac{h}{g} \cdot h} \cdot \color{blue}{\sqrt[3]{\frac{-0.25}{a}}} + \sqrt[3]{\frac{-g}{a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 80.4%

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

   if 1.04999999999999992e-154 < g < 1.6999999999999999e153

   1. Initial program 86.9%

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

   4. Applied rewrites 86.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. clear-num N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

      14. *-lft-identity N/A

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

   6. Applied rewrites 87.7%

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

   8. Applied rewrites 97.9%

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

   if 1.6999999999999999e153 < g

   1. Initial program 1.9%

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

   4. Applied rewrites 1.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. clear-num N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

      14. *-lft-identity N/A

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

   6. Applied rewrites 2.2%

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

   7. Taylor expanded in h around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-out-- N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

   9. Applied rewrites 65.1%

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{\color{blue}{h \cdot \left(0.5 \cdot \left(\frac{h \cdot \left(1 + \frac{0.25 \cdot {\left(0 \cdot g\right)}^{2}}{g \cdot g}\right)}{g} - \frac{0 \cdot g}{g}\right)\right)}}}} + \frac{1}{\sqrt[3]{\frac{a \cdot 2}{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{h}{a} \cdot \left(-0.25 \cdot \frac{h}{g}\right)} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\sqrt[3]{\frac{\left(\left(h + g\right) \cdot \left(g - h\right) - g \cdot g\right) \cdot \frac{0.5}{a}}{g - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}} + \sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g} \cdot \sqrt[3]{\frac{0.5}{a}}\\ \mathbf{elif}\;g \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;\sqrt[3]{\frac{-0.25}{a} \cdot \frac{h}{g}} \cdot \sqrt[3]{h} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq 1.7 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{-1}{a}}, \sqrt[3]{\frac{\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)}{2}}, \sqrt[3]{\left(\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g\right) \cdot \frac{0.5}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{\left(-2\right) \cdot a}{\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}}} + \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{\left(\left(\frac{\left(\frac{{\left(0 \cdot g\right)}^{2} \cdot 0.25}{g \cdot g} - -1\right) \cdot h}{g} - \frac{0 \cdot g}{g}\right) \cdot 0.5\right) \cdot h}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.6% accurate, 0.3× speedup

Math

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

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 (* (- g t_1) t_0))) (- INFINITY))
     (*
      (fma
       (* (cbrt -0.5) (cbrt (/ 1.0 (* (* g g) a))))
       (cbrt 2.0)
       (* (/ (pow (cbrt 0.5) 2.0) g) (cbrt (/ (* 0.0 h) a))))
      g)
     (+ (cbrt (/ (- g) a)) (* (cbrt (/ -0.25 a)) (cbrt (* (/ h g) h)))))))

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(((g - t_1) * t_0))) <= -((double) INFINITY)) {
		tmp = fma((cbrt(-0.5) * cbrt((1.0 / ((g * g) * a)))), cbrt(2.0), ((pow(cbrt(0.5), 2.0) / g) * cbrt(((0.0 * h) / a)))) * g;
	} else {
		tmp = cbrt((-g / a)) + (cbrt((-0.25 / a)) * cbrt(((h / g) * h)));
	}
	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(g - t_1) * t_0))) <= Float64(-Inf))
		tmp = Float64(fma(Float64(cbrt(-0.5) * cbrt(Float64(1.0 / Float64(Float64(g * g) * a)))), cbrt(2.0), Float64(Float64((cbrt(0.5) ^ 2.0) / g) * cbrt(Float64(Float64(0.0 * h) / a)))) * g);
	else
		tmp = Float64(cbrt(Float64(Float64(-g) / a)) + Float64(cbrt(Float64(-0.25 / a)) * cbrt(Float64(Float64(h / g) * h))));
	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[(g - t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[N[(1.0 / N[(N[(g * g), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision] + N[(N[(N[Power[N[Power[0.5, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / g), $MachinePrecision] * N[Power[N[(N[(0.0 * h), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * g), $MachinePrecision], N[(N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.25 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision], 1/3], $MachinePrecision]), $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. 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 67.3

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

      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 67.3

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

   4. Applied rewrites 67.3%

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

   5. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. lower-cbrt.f64 N/A

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

      12. lower-*.f64 N/A

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

   7. Applied rewrites 98.7%

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

   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 48.2%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 30.1

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   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 77.5

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

   8. Applied rewrites 77.5%

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

   10. Applied rewrites 77.7%

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

4. Final simplification 78.2%

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

Alternative 3: 75.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{a \cdot 2}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ t_2 := \sqrt[3]{t\_0 \cdot \left(t\_1 + g\right)}\\ \mathbf{if}\;t\_2 + \sqrt[3]{\left(g - t\_1\right) \cdot t\_0} \leq -1 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{a \cdot 2}} + t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{-0.25}{a}} \cdot \sqrt[3]{\frac{h}{g} \cdot h}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ -1.0 (* a 2.0)))
        (t_1 (sqrt (- (* g g) (* h h))))
        (t_2 (cbrt (* t_0 (+ t_1 g)))))
   (if (<= (+ t_2 (cbrt (* (- g t_1) t_0))) -1e+61)
     (+ (/ (cbrt (- (sqrt (* (+ h g) (- g h))) g)) (cbrt (* a 2.0))) t_2)
     (+ (cbrt (/ (- g) a)) (* (cbrt (/ -0.25 a)) (cbrt (* (/ h g) h)))))))

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 t_2 = cbrt((t_0 * (t_1 + g)));
	double tmp;
	if ((t_2 + cbrt(((g - t_1) * t_0))) <= -1e+61) {
		tmp = (cbrt((sqrt(((h + g) * (g - h))) - g)) / cbrt((a * 2.0))) + t_2;
	} else {
		tmp = cbrt((-g / a)) + (cbrt((-0.25 / a)) * cbrt(((h / g) * h)));
	}
	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 t_2 = Math.cbrt((t_0 * (t_1 + g)));
	double tmp;
	if ((t_2 + Math.cbrt(((g - t_1) * t_0))) <= -1e+61) {
		tmp = (Math.cbrt((Math.sqrt(((h + g) * (g - h))) - g)) / Math.cbrt((a * 2.0))) + t_2;
	} else {
		tmp = Math.cbrt((-g / a)) + (Math.cbrt((-0.25 / a)) * Math.cbrt(((h / g) * h)));
	}
	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)))
	t_2 = cbrt(Float64(t_0 * Float64(t_1 + g)))
	tmp = 0.0
	if (Float64(t_2 + cbrt(Float64(Float64(g - t_1) * t_0))) <= -1e+61)
		tmp = Float64(Float64(cbrt(Float64(sqrt(Float64(Float64(h + g) * Float64(g - h))) - g)) / cbrt(Float64(a * 2.0))) + t_2);
	else
		tmp = Float64(cbrt(Float64(Float64(-g) / a)) + Float64(cbrt(Float64(-0.25 / a)) * cbrt(Float64(Float64(h / g) * h))));
	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]}, Block[{t$95$2 = N[Power[N[(t$95$0 * N[(t$95$1 + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$2 + N[Power[N[(N[(g - t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], -1e+61], N[(N[(N[Power[N[(N[Sqrt[N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.25 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision], 1/3], $MachinePrecision]), $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))))))) < -9.99999999999999949e60

   1. Initial program 60.3%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 85.7%

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

   if -9.99999999999999949e60 < (+.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 46.4%

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

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

   5. Applied rewrites 29.2%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   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 76.9

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

   8. Applied rewrites 76.9%

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

   10. Applied rewrites 77.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)} + \sqrt[3]{\left(g - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{a \cdot 2}} \leq -1 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{-0.25}{a}} \cdot \sqrt[3]{\frac{h}{g} \cdot h}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{a \cdot 2}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ t_2 := \sqrt[3]{t\_0 \cdot \left(t\_1 + g\right)}\\ \mathbf{if}\;t\_2 + \sqrt[3]{\left(g - t\_1\right) \cdot t\_0} \leq -4 \cdot 10^{+55}:\\ \;\;\;\;\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g} \cdot \sqrt[3]{\frac{0.5}{a}} + t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{-0.25}{a}} \cdot \sqrt[3]{\frac{h}{g} \cdot h}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ -1.0 (* a 2.0)))
        (t_1 (sqrt (- (* g g) (* h h))))
        (t_2 (cbrt (* t_0 (+ t_1 g)))))
   (if (<= (+ t_2 (cbrt (* (- g t_1) t_0))) -4e+55)
     (+ (* (cbrt (- (sqrt (* (+ h g) (- g h))) g)) (cbrt (/ 0.5 a))) t_2)
     (+ (cbrt (/ (- g) a)) (* (cbrt (/ -0.25 a)) (cbrt (* (/ h g) h)))))))

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 t_2 = cbrt((t_0 * (t_1 + g)));
	double tmp;
	if ((t_2 + cbrt(((g - t_1) * t_0))) <= -4e+55) {
		tmp = (cbrt((sqrt(((h + g) * (g - h))) - g)) * cbrt((0.5 / a))) + t_2;
	} else {
		tmp = cbrt((-g / a)) + (cbrt((-0.25 / a)) * cbrt(((h / g) * h)));
	}
	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 t_2 = Math.cbrt((t_0 * (t_1 + g)));
	double tmp;
	if ((t_2 + Math.cbrt(((g - t_1) * t_0))) <= -4e+55) {
		tmp = (Math.cbrt((Math.sqrt(((h + g) * (g - h))) - g)) * Math.cbrt((0.5 / a))) + t_2;
	} else {
		tmp = Math.cbrt((-g / a)) + (Math.cbrt((-0.25 / a)) * Math.cbrt(((h / g) * h)));
	}
	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)))
	t_2 = cbrt(Float64(t_0 * Float64(t_1 + g)))
	tmp = 0.0
	if (Float64(t_2 + cbrt(Float64(Float64(g - t_1) * t_0))) <= -4e+55)
		tmp = Float64(Float64(cbrt(Float64(sqrt(Float64(Float64(h + g) * Float64(g - h))) - g)) * cbrt(Float64(0.5 / a))) + t_2);
	else
		tmp = Float64(cbrt(Float64(Float64(-g) / a)) + Float64(cbrt(Float64(-0.25 / a)) * cbrt(Float64(Float64(h / g) * h))));
	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]}, Block[{t$95$2 = N[Power[N[(t$95$0 * N[(t$95$1 + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$2 + N[Power[N[(N[(g - t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], -4e+55], N[(N[(N[Power[N[(N[Sqrt[N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.25 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision], 1/3], $MachinePrecision]), $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))))))) < -4.00000000000000004e55

   1. Initial program 62.7%

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

      1. lift-cbrt.f64 N/A

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

      2. 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 86.3

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

      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 86.3

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

   4. Applied rewrites 86.3%

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

   if -4.00000000000000004e55 < (+.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 46.2%

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

   3. Taylor expanded in g around inf

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

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

   5. Applied rewrites 29.2%

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

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   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 76.8

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

   8. Applied rewrites 76.8%

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

   10. Applied rewrites 77.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)} + \sqrt[3]{\left(g - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{a \cdot 2}} \leq -4 \cdot 10^{+55}:\\ \;\;\;\;\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g} \cdot \sqrt[3]{\frac{0.5}{a}} + \sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{-0.25}{a}} \cdot \sqrt[3]{\frac{h}{g} \cdot h}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-g}{a}}\\ t_1 := \sqrt{g - h}\\ t_2 := \left(h + g\right) \cdot \left(g - h\right)\\ t_3 := \sqrt{t\_2}\\ t_4 := t\_3 - g\\ t_5 := \sqrt{h + g}\\ \mathbf{if}\;g \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{h}{a} \cdot \left(-0.25 \cdot \frac{h}{g}\right)} + t\_0\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\sqrt[3]{\frac{\left(t\_2 - g \cdot g\right) \cdot \frac{0.5}{a}}{g - t\_3}} + \sqrt[3]{t\_4} \cdot \sqrt[3]{\frac{0.5}{a}}\\ \mathbf{elif}\;g \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;\sqrt[3]{\frac{-0.25}{a} \cdot \frac{h}{g}} \cdot \sqrt[3]{h} + t\_0\\ \mathbf{elif}\;g \leq 1.7 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{-1}{a}}, \sqrt[3]{\frac{\mathsf{fma}\left(t\_1, t\_5, g\right)}{2}}, \sqrt[3]{t\_4 \cdot \frac{0.5}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{-2 \cdot a}{\left(0 \cdot h\right) \cdot -0.5}}} + \frac{1}{\sqrt[3]{\frac{\left(-2\right) \cdot a}{\mathsf{fma}\left(t\_5, t\_1, g\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (/ (- g) a)))
        (t_1 (sqrt (- g h)))
        (t_2 (* (+ h g) (- g h)))
        (t_3 (sqrt t_2))
        (t_4 (- t_3 g))
        (t_5 (sqrt (+ h g))))
   (if (<= g -1.32e+154)
     (+ (cbrt (* (/ h a) (* -0.25 (/ h g)))) t_0)
     (if (<= g -6e-114)
       (+
        (cbrt (/ (* (- t_2 (* g g)) (/ 0.5 a)) (- g t_3)))
        (* (cbrt t_4) (cbrt (/ 0.5 a))))
       (if (<= g 1.05e-154)
         (+ (* (cbrt (* (/ -0.25 a) (/ h g))) (cbrt h)) t_0)
         (if (<= g 1.7e+153)
           (fma
            (cbrt (/ -1.0 a))
            (cbrt (/ (fma t_1 t_5 g) 2.0))
            (cbrt (* t_4 (/ 0.5 a))))
           (+
            (/ 1.0 (cbrt (/ (* -2.0 a) (* (* 0.0 h) -0.5))))
            (/ 1.0 (cbrt (/ (* (- 2.0) a) (fma t_5 t_1 g)))))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((-g / a));
	double t_1 = sqrt((g - h));
	double t_2 = (h + g) * (g - h);
	double t_3 = sqrt(t_2);
	double t_4 = t_3 - g;
	double t_5 = sqrt((h + g));
	double tmp;
	if (g <= -1.32e+154) {
		tmp = cbrt(((h / a) * (-0.25 * (h / g)))) + t_0;
	} else if (g <= -6e-114) {
		tmp = cbrt((((t_2 - (g * g)) * (0.5 / a)) / (g - t_3))) + (cbrt(t_4) * cbrt((0.5 / a)));
	} else if (g <= 1.05e-154) {
		tmp = (cbrt(((-0.25 / a) * (h / g))) * cbrt(h)) + t_0;
	} else if (g <= 1.7e+153) {
		tmp = fma(cbrt((-1.0 / a)), cbrt((fma(t_1, t_5, g) / 2.0)), cbrt((t_4 * (0.5 / a))));
	} else {
		tmp = (1.0 / cbrt(((-2.0 * a) / ((0.0 * h) * -0.5)))) + (1.0 / cbrt(((-2.0 * a) / fma(t_5, t_1, g))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(-g) / a))
	t_1 = sqrt(Float64(g - h))
	t_2 = Float64(Float64(h + g) * Float64(g - h))
	t_3 = sqrt(t_2)
	t_4 = Float64(t_3 - g)
	t_5 = sqrt(Float64(h + g))
	tmp = 0.0
	if (g <= -1.32e+154)
		tmp = Float64(cbrt(Float64(Float64(h / a) * Float64(-0.25 * Float64(h / g)))) + t_0);
	elseif (g <= -6e-114)
		tmp = Float64(cbrt(Float64(Float64(Float64(t_2 - Float64(g * g)) * Float64(0.5 / a)) / Float64(g - t_3))) + Float64(cbrt(t_4) * cbrt(Float64(0.5 / a))));
	elseif (g <= 1.05e-154)
		tmp = Float64(Float64(cbrt(Float64(Float64(-0.25 / a) * Float64(h / g))) * cbrt(h)) + t_0);
	elseif (g <= 1.7e+153)
		tmp = fma(cbrt(Float64(-1.0 / a)), cbrt(Float64(fma(t_1, t_5, g) / 2.0)), cbrt(Float64(t_4 * Float64(0.5 / a))));
	else
		tmp = Float64(Float64(1.0 / cbrt(Float64(Float64(-2.0 * a) / Float64(Float64(0.0 * h) * -0.5)))) + Float64(1.0 / cbrt(Float64(Float64(Float64(-2.0) * a) / fma(t_5, t_1, g)))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - g), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[g, -1.32e+154], N[(N[Power[N[(N[(h / a), $MachinePrecision] * N[(-0.25 * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[g, -6e-114], N[(N[Power[N[(N[(N[(t$95$2 - N[(g * g), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(g - t$95$3), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[t$95$4, 1/3], $MachinePrecision] * N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 1.05e-154], N[(N[(N[Power[N[(N[(-0.25 / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[h, 1/3], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[g, 1.7e+153], N[(N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(t$95$1 * t$95$5 + g), $MachinePrecision] / 2.0), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$4 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Power[N[(N[(-2.0 * a), $MachinePrecision] / N[(N[(0.0 * h), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[N[(N[((-2.0) * a), $MachinePrecision] / N[(t$95$5 * t$95$1 + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if g < -1.31999999999999998e154

   1. Initial program 0.0%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 3.5

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

   5. Applied rewrites 3.5%

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

   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 61.8

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

   8. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   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 61.8

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

   10. Applied rewrites 61.8%

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

   if -1.31999999999999998e154 < g < -6.0000000000000003e-114

   1. Initial program 85.1%

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

      1. lift-cbrt.f64 N/A

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

      2. 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 97.7

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

      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 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. +-commutative N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-squares N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift--.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. sub-neg N/A

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

      16. lift--.f64 N/A

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

   6. Applied rewrites 97.8%

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

   if -6.0000000000000003e-114 < g < 1.04999999999999992e-154

   1. Initial program 25.8%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 16.2

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

   5. Applied rewrites 16.2%

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

   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 79.0

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

   8. Applied rewrites 79.0%

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

   10. Applied rewrites 80.3%

       \[\leadsto \sqrt[3]{\frac{h}{g} \cdot h} \cdot \color{blue}{\sqrt[3]{\frac{-0.25}{a}}} + \sqrt[3]{\frac{-g}{a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 80.4%

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

   if 1.04999999999999992e-154 < g < 1.6999999999999999e153

   1. Initial program 86.9%

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

   4. Applied rewrites 86.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. clear-num N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

      14. *-lft-identity N/A

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

   6. Applied rewrites 87.7%

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

   8. Applied rewrites 97.9%

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

   if 1.6999999999999999e153 < g

   1. Initial program 1.9%

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

   4. Applied rewrites 1.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. clear-num N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

      14. *-lft-identity N/A

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

   6. Applied rewrites 2.2%

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

   7. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 63.1

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

   9. Applied rewrites 63.1%

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{\color{blue}{-0.5 \cdot \left(0 \cdot h\right)}}}} + \frac{1}{\sqrt[3]{\frac{a \cdot 2}{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{h}{a} \cdot \left(-0.25 \cdot \frac{h}{g}\right)} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\sqrt[3]{\frac{\left(\left(h + g\right) \cdot \left(g - h\right) - g \cdot g\right) \cdot \frac{0.5}{a}}{g - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}} + \sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g} \cdot \sqrt[3]{\frac{0.5}{a}}\\ \mathbf{elif}\;g \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;\sqrt[3]{\frac{-0.25}{a} \cdot \frac{h}{g}} \cdot \sqrt[3]{h} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq 1.7 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{-1}{a}}, \sqrt[3]{\frac{\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)}{2}}, \sqrt[3]{\left(\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g\right) \cdot \frac{0.5}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{-2 \cdot a}{\left(0 \cdot h\right) \cdot -0.5}}} + \frac{1}{\sqrt[3]{\frac{\left(-2\right) \cdot a}{\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-g}{a}}\\ t_1 := \sqrt{g - h}\\ t_2 := \sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g\\ t_3 := \sqrt{h + g}\\ \mathbf{if}\;g \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{h}{a} \cdot \left(-0.25 \cdot \frac{h}{g}\right)} + t\_0\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt[3]{t\_2}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}}\\ \mathbf{elif}\;g \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;\sqrt[3]{\frac{-0.25}{a} \cdot \frac{h}{g}} \cdot \sqrt[3]{h} + t\_0\\ \mathbf{elif}\;g \leq 1.7 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{-1}{a}}, \sqrt[3]{\frac{\mathsf{fma}\left(t\_1, t\_3, g\right)}{2}}, \sqrt[3]{t\_2 \cdot \frac{0.5}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{-2 \cdot a}{\left(0 \cdot h\right) \cdot -0.5}}} + \frac{1}{\sqrt[3]{\frac{\left(-2\right) \cdot a}{\mathsf{fma}\left(t\_3, t\_1, g\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (/ (- g) a)))
        (t_1 (sqrt (- g h)))
        (t_2 (- (sqrt (* (+ h g) (- g h))) g))
        (t_3 (sqrt (+ h g))))
   (if (<= g -1.32e+154)
     (+ (cbrt (* (/ h a) (* -0.25 (/ h g)))) t_0)
     (if (<= g -6e-114)
       (+ (/ (cbrt t_2) (cbrt (* a 2.0))) (cbrt (* (/ (* h h) g) (/ -0.25 a))))
       (if (<= g 1.05e-154)
         (+ (* (cbrt (* (/ -0.25 a) (/ h g))) (cbrt h)) t_0)
         (if (<= g 1.7e+153)
           (fma
            (cbrt (/ -1.0 a))
            (cbrt (/ (fma t_1 t_3 g) 2.0))
            (cbrt (* t_2 (/ 0.5 a))))
           (+
            (/ 1.0 (cbrt (/ (* -2.0 a) (* (* 0.0 h) -0.5))))
            (/ 1.0 (cbrt (/ (* (- 2.0) a) (fma t_3 t_1 g)))))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((-g / a));
	double t_1 = sqrt((g - h));
	double t_2 = sqrt(((h + g) * (g - h))) - g;
	double t_3 = sqrt((h + g));
	double tmp;
	if (g <= -1.32e+154) {
		tmp = cbrt(((h / a) * (-0.25 * (h / g)))) + t_0;
	} else if (g <= -6e-114) {
		tmp = (cbrt(t_2) / cbrt((a * 2.0))) + cbrt((((h * h) / g) * (-0.25 / a)));
	} else if (g <= 1.05e-154) {
		tmp = (cbrt(((-0.25 / a) * (h / g))) * cbrt(h)) + t_0;
	} else if (g <= 1.7e+153) {
		tmp = fma(cbrt((-1.0 / a)), cbrt((fma(t_1, t_3, g) / 2.0)), cbrt((t_2 * (0.5 / a))));
	} else {
		tmp = (1.0 / cbrt(((-2.0 * a) / ((0.0 * h) * -0.5)))) + (1.0 / cbrt(((-2.0 * a) / fma(t_3, t_1, g))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(-g) / a))
	t_1 = sqrt(Float64(g - h))
	t_2 = Float64(sqrt(Float64(Float64(h + g) * Float64(g - h))) - g)
	t_3 = sqrt(Float64(h + g))
	tmp = 0.0
	if (g <= -1.32e+154)
		tmp = Float64(cbrt(Float64(Float64(h / a) * Float64(-0.25 * Float64(h / g)))) + t_0);
	elseif (g <= -6e-114)
		tmp = Float64(Float64(cbrt(t_2) / cbrt(Float64(a * 2.0))) + cbrt(Float64(Float64(Float64(h * h) / g) * Float64(-0.25 / a))));
	elseif (g <= 1.05e-154)
		tmp = Float64(Float64(cbrt(Float64(Float64(-0.25 / a) * Float64(h / g))) * cbrt(h)) + t_0);
	elseif (g <= 1.7e+153)
		tmp = fma(cbrt(Float64(-1.0 / a)), cbrt(Float64(fma(t_1, t_3, g) / 2.0)), cbrt(Float64(t_2 * Float64(0.5 / a))));
	else
		tmp = Float64(Float64(1.0 / cbrt(Float64(Float64(-2.0 * a) / Float64(Float64(0.0 * h) * -0.5)))) + Float64(1.0 / cbrt(Float64(Float64(Float64(-2.0) * a) / fma(t_3, t_1, g)))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[g, -1.32e+154], N[(N[Power[N[(N[(h / a), $MachinePrecision] * N[(-0.25 * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[g, -6e-114], N[(N[(N[Power[t$95$2, 1/3], $MachinePrecision] / N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[(h * h), $MachinePrecision] / g), $MachinePrecision] * N[(-0.25 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 1.05e-154], N[(N[(N[Power[N[(N[(-0.25 / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[h, 1/3], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[g, 1.7e+153], N[(N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(t$95$1 * t$95$3 + g), $MachinePrecision] / 2.0), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$2 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Power[N[(N[(-2.0 * a), $MachinePrecision] / N[(N[(0.0 * h), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[N[(N[((-2.0) * a), $MachinePrecision] / N[(t$95$3 * t$95$1 + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if g < -1.31999999999999998e154

   1. Initial program 0.0%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 3.5

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

   5. Applied rewrites 3.5%

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

   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 61.8

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

   8. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   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 61.8

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

   10. Applied rewrites 61.8%

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

   if -1.31999999999999998e154 < g < -6.0000000000000003e-114

   1. Initial program 85.1%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 16.8

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

   5. Applied rewrites 16.8%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. sub-neg N/A

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

      17. lift--.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

   7. Applied rewrites 22.5%

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

   8. Taylor expanded in g around -inf

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 97.7

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

   10. Applied rewrites 97.7%

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

   if -6.0000000000000003e-114 < g < 1.04999999999999992e-154

   1. Initial program 25.8%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 16.2

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

   5. Applied rewrites 16.2%

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

   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 79.0

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

   8. Applied rewrites 79.0%

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

   10. Applied rewrites 80.3%

       \[\leadsto \sqrt[3]{\frac{h}{g} \cdot h} \cdot \color{blue}{\sqrt[3]{\frac{-0.25}{a}}} + \sqrt[3]{\frac{-g}{a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 80.4%

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

   if 1.04999999999999992e-154 < g < 1.6999999999999999e153

   1. Initial program 86.9%

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

   4. Applied rewrites 86.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. clear-num N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

      14. *-lft-identity N/A

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

   6. Applied rewrites 87.7%

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

   8. Applied rewrites 97.9%

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

   if 1.6999999999999999e153 < g

   1. Initial program 1.9%

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

   4. Applied rewrites 1.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. clear-num N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

      14. *-lft-identity N/A

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

   6. Applied rewrites 2.2%

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

   7. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 63.1

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

   9. Applied rewrites 63.1%

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{\color{blue}{-0.5 \cdot \left(0 \cdot h\right)}}}} + \frac{1}{\sqrt[3]{\frac{a \cdot 2}{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{h}{a} \cdot \left(-0.25 \cdot \frac{h}{g}\right)} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}}\\ \mathbf{elif}\;g \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;\sqrt[3]{\frac{-0.25}{a} \cdot \frac{h}{g}} \cdot \sqrt[3]{h} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq 1.7 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{-1}{a}}, \sqrt[3]{\frac{\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)}{2}}, \sqrt[3]{\left(\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g\right) \cdot \frac{0.5}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{-2 \cdot a}{\left(0 \cdot h\right) \cdot -0.5}}} + \frac{1}{\sqrt[3]{\frac{\left(-2\right) \cdot a}{\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-g}{a}}\\ \mathbf{if}\;g \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{h}{a} \cdot \left(-0.25 \cdot \frac{h}{g}\right)} + t\_0\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-0.25}{a} \cdot \frac{h}{g}} \cdot \sqrt[3]{h} + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (/ (- g) a))))
   (if (<= g -1.32e+154)
     (+ (cbrt (* (/ h a) (* -0.25 (/ h g)))) t_0)
     (if (<= g -6e-114)
       (+
        (/ (cbrt (- (sqrt (* (+ h g) (- g h))) g)) (cbrt (* a 2.0)))
        (cbrt (* (/ (* h h) g) (/ -0.25 a))))
       (+ (* (cbrt (* (/ -0.25 a) (/ h g))) (cbrt h)) t_0)))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((-g / a));
	double tmp;
	if (g <= -1.32e+154) {
		tmp = cbrt(((h / a) * (-0.25 * (h / g)))) + t_0;
	} else if (g <= -6e-114) {
		tmp = (cbrt((sqrt(((h + g) * (g - h))) - g)) / cbrt((a * 2.0))) + cbrt((((h * h) / g) * (-0.25 / a)));
	} else {
		tmp = (cbrt(((-0.25 / a) * (h / g))) * cbrt(h)) + t_0;
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt((-g / a));
	double tmp;
	if (g <= -1.32e+154) {
		tmp = Math.cbrt(((h / a) * (-0.25 * (h / g)))) + t_0;
	} else if (g <= -6e-114) {
		tmp = (Math.cbrt((Math.sqrt(((h + g) * (g - h))) - g)) / Math.cbrt((a * 2.0))) + Math.cbrt((((h * h) / g) * (-0.25 / a)));
	} else {
		tmp = (Math.cbrt(((-0.25 / a) * (h / g))) * Math.cbrt(h)) + t_0;
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(-g) / a))
	tmp = 0.0
	if (g <= -1.32e+154)
		tmp = Float64(cbrt(Float64(Float64(h / a) * Float64(-0.25 * Float64(h / g)))) + t_0);
	elseif (g <= -6e-114)
		tmp = Float64(Float64(cbrt(Float64(sqrt(Float64(Float64(h + g) * Float64(g - h))) - g)) / cbrt(Float64(a * 2.0))) + cbrt(Float64(Float64(Float64(h * h) / g) * Float64(-0.25 / a))));
	else
		tmp = Float64(Float64(cbrt(Float64(Float64(-0.25 / a) * Float64(h / g))) * cbrt(h)) + t_0);
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[g, -1.32e+154], N[(N[Power[N[(N[(h / a), $MachinePrecision] * N[(-0.25 * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[g, -6e-114], N[(N[(N[Power[N[(N[Sqrt[N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[(h * h), $MachinePrecision] / g), $MachinePrecision] * N[(-0.25 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(-0.25 / a), $MachinePrecision] * N[(h / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[h, 1/3], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if g < -1.31999999999999998e154

   1. Initial program 0.0%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 3.5

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

   5. Applied rewrites 3.5%

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

   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 61.8

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

   8. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   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 61.8

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

   10. Applied rewrites 61.8%

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

   if -1.31999999999999998e154 < g < -6.0000000000000003e-114

   1. Initial program 85.1%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 16.8

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

   5. Applied rewrites 16.8%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. sub-neg N/A

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

      17. lift--.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

   7. Applied rewrites 22.5%

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

   8. Taylor expanded in g around -inf

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 97.7

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

   10. Applied rewrites 97.7%

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

   if -6.0000000000000003e-114 < g

   1. Initial program 45.5%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 44.2

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

   5. Applied rewrites 44.2%

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

   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 76.1

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

   8. Applied rewrites 76.1%

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

   10. Applied rewrites 76.4%

       \[\leadsto \sqrt[3]{\frac{h}{g} \cdot h} \cdot \color{blue}{\sqrt[3]{\frac{-0.25}{a}}} + \sqrt[3]{\frac{-g}{a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 76.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{h}{a} \cdot \left(-0.25 \cdot \frac{h}{g}\right)} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{h \cdot h}{g} \cdot \frac{-0.25}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-0.25}{a} \cdot \frac{h}{g}} \cdot \sqrt[3]{h} + \sqrt[3]{\frac{-g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.2%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 29.5

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

5. Applied rewrites 29.5%

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

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 75.8

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

8. Applied rewrites 75.8%

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

10. Applied rewrites 75.9%

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

11. Final simplification 75.9%

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

Alternative 9: 74.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.2%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 29.5

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

5. Applied rewrites 29.5%

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

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 75.8

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

8. Applied rewrites 75.8%

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

10. Applied rewrites 75.9%

    \[\leadsto \sqrt[3]{\frac{h}{g} \cdot h} \cdot \color{blue}{\sqrt[3]{\frac{-0.25}{a}}} + \sqrt[3]{\frac{-g}{a}} \]
11. Step-by-step derivation

12. Applied rewrites 75.9%

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

13. Final simplification 75.9%

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

Alternative 10: 74.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.2%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 29.5

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

5. Applied rewrites 29.5%

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

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 75.8

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

8. Applied rewrites 75.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
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 75.8

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

10. Applied rewrites 75.8%

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

11. Final simplification 75.8%

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

Alternative 11: 15.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.2%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 29.5

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

5. Applied rewrites 29.5%

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

6. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 15.3

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

8. Applied rewrites 15.3%

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

9. Final simplification 15.3%

   \[\leadsto \sqrt[3]{\left(\left(-g\right) - g\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{\frac{-g}{a}} \]
10. 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 47.2%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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 50.7

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

   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 50.7

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. rem-square-sqrt N/A

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

   10. metadata-eval N/A

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

   11. lower-cbrt.f64 3.0

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

7. Applied rewrites 3.0%

   \[\leadsto \color{blue}{-\sqrt[3]{\frac{g \cdot 0}{a}} \cdot \sqrt[3]{0.5}} \]
8. Step-by-step derivation

9. Applied rewrites 3.0%

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

Reproduce

herbie shell --seed 2024325 
(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))))))))