2-ancestry mixing, positive discriminant

Percentage Accurate: 43.6% → 95.7%

Time: 16.1s

Alternatives: 8

Speedup: 2.1×

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: 95.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-1}{a}}\\ \mathbf{if}\;h\_m \leq 2 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, t\_0, \sqrt[3]{a \cdot \left(-0.25 \cdot \frac{{h\_m}^{2}}{g}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, t\_0, {\left(\sqrt[3]{h\_m}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{-0.25}{g}}{a}}\right)\\ \end{array} \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a)
 :precision binary64
 (let* ((t_0 (cbrt (/ -1.0 a))))
   (if (<= h_m 2e-229)
     (fma (cbrt g) t_0 (cbrt (* a (* -0.25 (/ (pow h_m 2.0) g)))))
     (fma (cbrt g) t_0 (* (pow (cbrt h_m) 2.0) (cbrt (/ (/ -0.25 g) a)))))))

C

h_m = fabs(h);
double code(double g, double h_m, double a) {
	double t_0 = cbrt((-1.0 / a));
	double tmp;
	if (h_m <= 2e-229) {
		tmp = fma(cbrt(g), t_0, cbrt((a * (-0.25 * (pow(h_m, 2.0) / g)))));
	} else {
		tmp = fma(cbrt(g), t_0, (pow(cbrt(h_m), 2.0) * cbrt(((-0.25 / g) / a))));
	}
	return tmp;
}

Julia

h_m = abs(h)
function code(g, h_m, a)
	t_0 = cbrt(Float64(-1.0 / a))
	tmp = 0.0
	if (h_m <= 2e-229)
		tmp = fma(cbrt(g), t_0, cbrt(Float64(a * Float64(-0.25 * Float64((h_m ^ 2.0) / g)))));
	else
		tmp = fma(cbrt(g), t_0, Float64((cbrt(h_m) ^ 2.0) * cbrt(Float64(Float64(-0.25 / g) / a))));
	end
	return tmp
end

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := Block[{t$95$0 = N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[h$95$m, 2e-229], N[(N[Power[g, 1/3], $MachinePrecision] * t$95$0 + N[Power[N[(a * N[(-0.25 * N[(N[Power[h$95$m, 2.0], $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[g, 1/3], $MachinePrecision] * t$95$0 + N[(N[Power[N[Power[h$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(N[(-0.25 / g), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < 2.00000000000000014e-229

   1. Initial program 48.4%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in h around 0 70.3%

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

      1. pow1/3 32.2%

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

      2. div-inv 32.2%

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

      3. unpow-prod-down 22.6%

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

      4. pow1/3 40.1%

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

   7. Applied egg-rr 40.1%

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

      1. unpow1/3 84.2%

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

   9. Simplified 84.2%

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

       1. associate-*l* 84.1%

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

       2. fma-define 84.1%

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

       3. cbrt-unprod 84.6%

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

       4. metadata-eval 84.6%

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

       5. cbrt-unprod 84.6%

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

       6. associate-/l/ 90.4%

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

       7. cbrt-unprod 90.4%

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

       8. metadata-eval 90.4%

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

       9. cbrt-prod 90.4%

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

       10. associate-/l/ 84.6%

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

       11. *-commutative 84.6%

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

   11. Applied egg-rr 84.6%

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

       1. associate-*l/ 84.6%

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

       2. metadata-eval 84.6%

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

       3. associate-*l/ 84.6%

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

   13. Simplified 84.6%

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

   15. Applied egg-rr 82.4%

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

       1. *-lft-identity 82.4%

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

       2. *-commutative 82.4%

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

       3. associate-*l* 82.4%

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

       4. *-commutative 82.4%

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

   17. Simplified 82.4%

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

   if 2.00000000000000014e-229 < h

   1. Initial program 42.2%

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

   3. Simplified 42.2%

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

   5. Taylor expanded in h around 0 65.9%

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

      1. pow1/3 30.4%

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

      2. div-inv 30.4%

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

      3. unpow-prod-down 17.9%

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

      4. pow1/3 36.0%

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

   7. Applied egg-rr 36.0%

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

      1. unpow1/3 88.1%

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

   9. Simplified 88.1%

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

       1. associate-*l* 88.1%

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

       2. fma-define 88.1%

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

       3. cbrt-unprod 88.6%

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

       4. metadata-eval 88.6%

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

       5. cbrt-unprod 88.6%

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

       6. associate-/l/ 89.2%

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

       7. cbrt-unprod 89.2%

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

       8. metadata-eval 89.2%

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

       9. cbrt-prod 89.2%

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

       10. associate-/l/ 88.6%

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

       11. *-commutative 88.6%

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

   11. Applied egg-rr 88.6%

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

       1. associate-*l/ 88.6%

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

       2. metadata-eval 88.6%

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

       3. associate-*l/ 88.6%

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

   13. Simplified 88.6%

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

   15. Applied egg-rr 61.6%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{g}, \sqrt[3]{\frac{-1}{a}}, \color{blue}{{\left(\sqrt[3]{h}\right)}^{2} \cdot {\left(\frac{-0.25}{g \cdot a}\right)}^{0.3333333333333333}}\right) \]
   16. Step-by-step derivation

       1. unpow1/3 95.2%

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

       2. associate-/r* 95.2%

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

   17. Simplified 95.2%

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

Alternative 2: 92.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-1}{a}}\\ \mathbf{if}\;h\_m \leq 3.34 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, t\_0, \sqrt[3]{a \cdot \left(-0.25 \cdot \frac{{h\_m}^{2}}{g}\right)}\right)\\ \mathbf{elif}\;h\_m \leq 1.9 \cdot 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, t\_0, \sqrt[3]{\frac{-0.25 \cdot \left(h\_m \cdot h\_m\right)}{g \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \end{array} \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a)
 :precision binary64
 (let* ((t_0 (cbrt (/ -1.0 a))))
   (if (<= h_m 3.34e-237)
     (fma (cbrt g) t_0 (cbrt (* a (* -0.25 (/ (pow h_m 2.0) g)))))
     (if (<= h_m 1.9e+159)
       (fma (cbrt g) t_0 (cbrt (/ (* -0.25 (* h_m h_m)) (* g a))))
       (* (cbrt (/ g a)) (cbrt -1.0))))))

C

h_m = fabs(h);
double code(double g, double h_m, double a) {
	double t_0 = cbrt((-1.0 / a));
	double tmp;
	if (h_m <= 3.34e-237) {
		tmp = fma(cbrt(g), t_0, cbrt((a * (-0.25 * (pow(h_m, 2.0) / g)))));
	} else if (h_m <= 1.9e+159) {
		tmp = fma(cbrt(g), t_0, cbrt(((-0.25 * (h_m * h_m)) / (g * a))));
	} else {
		tmp = cbrt((g / a)) * cbrt(-1.0);
	}
	return tmp;
}

Julia

h_m = abs(h)
function code(g, h_m, a)
	t_0 = cbrt(Float64(-1.0 / a))
	tmp = 0.0
	if (h_m <= 3.34e-237)
		tmp = fma(cbrt(g), t_0, cbrt(Float64(a * Float64(-0.25 * Float64((h_m ^ 2.0) / g)))));
	elseif (h_m <= 1.9e+159)
		tmp = fma(cbrt(g), t_0, cbrt(Float64(Float64(-0.25 * Float64(h_m * h_m)) / Float64(g * a))));
	else
		tmp = Float64(cbrt(Float64(g / a)) * cbrt(-1.0));
	end
	return tmp
end

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := Block[{t$95$0 = N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[h$95$m, 3.34e-237], N[(N[Power[g, 1/3], $MachinePrecision] * t$95$0 + N[Power[N[(a * N[(-0.25 * N[(N[Power[h$95$m, 2.0], $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[h$95$m, 1.9e+159], N[(N[Power[g, 1/3], $MachinePrecision] * t$95$0 + N[Power[N[(N[(-0.25 * N[(h$95$m * h$95$m), $MachinePrecision]), $MachinePrecision] / N[(g * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < 3.3400000000000002e-237

   1. Initial program 48.1%

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

   3. Simplified 48.7%

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

   5. Taylor expanded in h around 0 70.1%

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

      1. pow1/3 32.4%

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

      2. div-inv 32.4%

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

      3. unpow-prod-down 22.7%

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

      4. pow1/3 39.7%

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

   7. Applied egg-rr 39.7%

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

      1. unpow1/3 84.1%

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

   9. Simplified 84.1%

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

       1. associate-*l* 84.0%

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

       2. fma-define 84.1%

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

       3. cbrt-unprod 84.5%

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

       4. metadata-eval 84.5%

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

       5. cbrt-unprod 84.5%

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

       6. associate-/l/ 90.3%

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

       7. cbrt-unprod 90.3%

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

       8. metadata-eval 90.3%

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

       9. cbrt-prod 90.3%

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

       10. associate-/l/ 84.5%

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

       11. *-commutative 84.5%

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

   11. Applied egg-rr 84.5%

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

       1. associate-*l/ 84.5%

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

       2. metadata-eval 84.5%

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

       3. associate-*l/ 84.5%

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

   13. Simplified 84.5%

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

   15. Applied egg-rr 82.3%

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

       1. *-lft-identity 82.3%

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

       2. *-commutative 82.3%

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

       3. associate-*l* 82.3%

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

       4. *-commutative 82.3%

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

   17. Simplified 82.3%

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

   if 3.3400000000000002e-237 < h < 1.89999999999999983e159

   1. Initial program 46.1%

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

   3. Simplified 46.1%

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

   5. Taylor expanded in h around 0 71.2%

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

      1. pow1/3 32.3%

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

      2. div-inv 32.3%

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

      3. unpow-prod-down 19.1%

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

      4. pow1/3 39.3%

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

   7. Applied egg-rr 39.3%

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

      1. unpow1/3 94.8%

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

   9. Simplified 94.8%

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

       1. associate-*l* 94.9%

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

       2. fma-define 94.9%

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

       3. cbrt-unprod 95.3%

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

       4. metadata-eval 95.3%

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

       5. cbrt-unprod 95.3%

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

       6. associate-/l/ 95.9%

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

       7. cbrt-unprod 95.9%

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

       8. metadata-eval 95.9%

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

       9. cbrt-prod 95.9%

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

       10. associate-/l/ 95.3%

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

       11. *-commutative 95.3%

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

   11. Applied egg-rr 95.3%

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

       1. associate-*l/ 95.3%

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

       2. metadata-eval 95.3%

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

       3. associate-*l/ 95.3%

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

   13. Simplified 95.3%

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

   15. Applied egg-rr 95.3%

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

   if 1.89999999999999983e159 < h

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in h around 0 1.8%

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

      1. pow1/3 0.7%

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

      2. div-inv 0.7%

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

      3. unpow-prod-down 0.0%

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

      4. pow1/3 0.6%

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

   7. Applied egg-rr 0.6%

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

      1. unpow1/3 1.8%

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

   9. Simplified 1.8%

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

       1. associate-*l* 1.8%

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

       2. fma-define 1.8%

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

       3. cbrt-unprod 1.8%

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

       4. metadata-eval 1.8%

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

       5. cbrt-unprod 1.8%

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

       6. associate-/l/ 3.8%

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

       7. cbrt-unprod 3.8%

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

       8. metadata-eval 3.8%

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

       9. cbrt-prod 3.8%

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

       10. associate-/l/ 1.8%

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

       11. *-commutative 1.8%

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

   11. Applied egg-rr 1.8%

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

       1. associate-*l/ 1.8%

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

       2. metadata-eval 1.8%

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

       3. associate-*l/ 1.8%

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

   13. Simplified 1.8%

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

   14. Taylor expanded in g around inf 55.5%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 3.34 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, \sqrt[3]{\frac{-1}{a}}, \sqrt[3]{a \cdot \left(-0.25 \cdot \frac{{h}^{2}}{g}\right)}\right)\\ \mathbf{elif}\;h \leq 1.9 \cdot 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, \sqrt[3]{\frac{-1}{a}}, \sqrt[3]{\frac{-0.25 \cdot \left(h \cdot h\right)}{g \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ \begin{array}{l} \mathbf{if}\;h\_m \cdot h\_m \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, \sqrt[3]{\frac{-1}{a}}, \sqrt[3]{\frac{-0.25 \cdot \left(h\_m \cdot h\_m\right)}{g \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \end{array} \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a)
 :precision binary64
 (if (<= (* h_m h_m) 2e+298)
   (fma (cbrt g) (cbrt (/ -1.0 a)) (cbrt (/ (* -0.25 (* h_m h_m)) (* g a))))
   (* (cbrt (/ g a)) (cbrt -1.0))))

C

h_m = fabs(h);
double code(double g, double h_m, double a) {
	double tmp;
	if ((h_m * h_m) <= 2e+298) {
		tmp = fma(cbrt(g), cbrt((-1.0 / a)), cbrt(((-0.25 * (h_m * h_m)) / (g * a))));
	} else {
		tmp = cbrt((g / a)) * cbrt(-1.0);
	}
	return tmp;
}

Julia

h_m = abs(h)
function code(g, h_m, a)
	tmp = 0.0
	if (Float64(h_m * h_m) <= 2e+298)
		tmp = fma(cbrt(g), cbrt(Float64(-1.0 / a)), cbrt(Float64(Float64(-0.25 * Float64(h_m * h_m)) / Float64(g * a))));
	else
		tmp = Float64(cbrt(Float64(g / a)) * cbrt(-1.0));
	end
	return tmp
end

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := If[LessEqual[N[(h$95$m * h$95$m), $MachinePrecision], 2e+298], N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-0.25 * N[(h$95$m * h$95$m), $MachinePrecision]), $MachinePrecision] / N[(g * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 1.9999999999999999e298

   1. Initial program 49.8%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in h around 0 73.9%

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

      1. pow1/3 34.0%

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

      2. div-inv 34.0%

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

      3. unpow-prod-down 22.5%

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

      4. pow1/3 41.5%

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

   7. Applied egg-rr 41.5%

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

      1. unpow1/3 92.3%

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

   9. Simplified 92.3%

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

       1. associate-*l* 92.3%

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

       2. fma-define 92.3%

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

       3. cbrt-unprod 92.8%

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

       4. metadata-eval 92.8%

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

       5. cbrt-unprod 92.8%

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

       6. associate-/l/ 96.8%

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

       7. cbrt-unprod 96.8%

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

       8. metadata-eval 96.8%

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

       9. cbrt-prod 96.8%

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

       10. associate-/l/ 92.8%

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

       11. *-commutative 92.8%

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

   11. Applied egg-rr 92.8%

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

       1. associate-*l/ 92.8%

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

       2. metadata-eval 92.8%

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

       3. associate-*l/ 92.8%

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

   13. Simplified 92.8%

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

   15. Applied egg-rr 92.8%

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

   if 1.9999999999999999e298 < (*.f64 h h)

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in h around 0 2.6%

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

      1. pow1/3 0.7%

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

      2. div-inv 0.7%

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

      3. unpow-prod-down 0.0%

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

      4. pow1/3 1.0%

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

   7. Applied egg-rr 1.0%

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

      1. unpow1/3 2.6%

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

   9. Simplified 2.6%

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

       1. associate-*l* 2.6%

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

       2. fma-define 2.6%

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

       3. cbrt-unprod 2.6%

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

       4. metadata-eval 2.6%

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

       5. cbrt-unprod 2.6%

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

       6. associate-/l/ 4.0%

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

       7. cbrt-unprod 4.0%

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

       8. metadata-eval 4.0%

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

       9. cbrt-prod 4.0%

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

       10. associate-/l/ 2.6%

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

       11. *-commutative 2.6%

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

   11. Applied egg-rr 2.6%

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

       1. associate-*l/ 2.6%

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

       2. metadata-eval 2.6%

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

       3. associate-*l/ 2.6%

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

   13. Simplified 2.6%

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

   14. Taylor expanded in g around inf 53.5%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{g}, \sqrt[3]{\frac{-1}{a}}, \sqrt[3]{\frac{-0.25 \cdot \left(h \cdot h\right)}{g \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 9.2 \cdot 10^{-305}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \end{array} \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a)
 :precision binary64
 (if (<= a 9.2e-305)
   (* (cbrt (/ g a)) (cbrt -1.0))
   (+
    (* (cbrt (- g)) (pow (/ 1.0 a) 0.3333333333333333))
    (cbrt (* (- g g) (/ -0.5 a))))))

C

h_m = fabs(h);
double code(double g, double h_m, double a) {
	double tmp;
	if (a <= 9.2e-305) {
		tmp = cbrt((g / a)) * cbrt(-1.0);
	} else {
		tmp = (cbrt(-g) * pow((1.0 / a), 0.3333333333333333)) + cbrt(((g - g) * (-0.5 / a)));
	}
	return tmp;
}

Java

h_m = Math.abs(h);
public static double code(double g, double h_m, double a) {
	double tmp;
	if (a <= 9.2e-305) {
		tmp = Math.cbrt((g / a)) * Math.cbrt(-1.0);
	} else {
		tmp = (Math.cbrt(-g) * Math.pow((1.0 / a), 0.3333333333333333)) + Math.cbrt(((g - g) * (-0.5 / a)));
	}
	return tmp;
}

Julia

h_m = abs(h)
function code(g, h_m, a)
	tmp = 0.0
	if (a <= 9.2e-305)
		tmp = Float64(cbrt(Float64(g / a)) * cbrt(-1.0));
	else
		tmp = Float64(Float64(cbrt(Float64(-g)) * (Float64(1.0 / a) ^ 0.3333333333333333)) + cbrt(Float64(Float64(g - g) * Float64(-0.5 / a))));
	end
	return tmp
end

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := If[LessEqual[a, 9.2e-305], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[(-g), 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.1999999999999998e-305

   1. Initial program 44.0%

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

   3. Simplified 44.7%

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

   5. Taylor expanded in h around 0 72.0%

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

      1. pow1/3 29.5%

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

      2. div-inv 29.5%

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

      3. unpow-prod-down 0.0%

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

      4. pow1/3 0.0%

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

   7. Applied egg-rr 0.0%

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

      1. unpow1/3 88.7%

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

   9. Simplified 88.7%

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

       1. associate-*l* 88.6%

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

       2. fma-define 88.6%

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

       3. cbrt-unprod 89.2%

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

       4. metadata-eval 89.2%

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

       5. cbrt-unprod 89.2%

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

       6. associate-/l/ 92.1%

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

       7. cbrt-unprod 92.1%

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

       8. metadata-eval 92.1%

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

       9. cbrt-prod 92.1%

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

       10. associate-/l/ 89.2%

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

       11. *-commutative 89.2%

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

   11. Applied egg-rr 89.2%

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

       1. associate-*l/ 89.2%

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

       2. metadata-eval 89.2%

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

       3. associate-*l/ 89.2%

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

   13. Simplified 89.2%

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

   14. Taylor expanded in g around inf 76.7%

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

   if 9.1999999999999998e-305 < a

   1. Initial program 48.2%

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

   3. Simplified 48.2%

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

   5. Taylor expanded in g around -inf 27.0%

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

      1. associate-*r/ 27.0%

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

      2. neg-mul-1 27.0%

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

   7. Simplified 27.0%

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

      1. pow1/3 19.0%

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

      2. div-inv 19.0%

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

      3. unpow-prod-down 20.2%

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

      4. pow1/3 30.3%

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

   9. Applied egg-rr 30.3%

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

   10. Taylor expanded in g around -inf 88.2%

       \[\leadsto \sqrt[3]{-g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} + \sqrt[3]{\left(g + \color{blue}{-1 \cdot g}\right) \cdot \frac{-0.5}{a}} \]
   11. Step-by-step derivation

       1. neg-mul-1 88.2%

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

   12. Simplified 88.2%

       \[\leadsto \sqrt[3]{-g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} + \sqrt[3]{\left(g + \color{blue}{\left(-g\right)}\right) \cdot \frac{-0.5}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.2 \cdot 10^{-305}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1} \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a) :precision binary64 (* (cbrt (/ g a)) (cbrt -1.0)))

C

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

Java

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

Julia

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

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.1%

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

3. Simplified 46.4%

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

5. Taylor expanded in h around 0 68.6%

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

   1. pow1/3 31.5%

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

   2. div-inv 31.5%

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

   3. unpow-prod-down 20.8%

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

   4. pow1/3 38.5%

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

7. Applied egg-rr 38.5%

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

   1. unpow1/3 85.7%

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

9. Simplified 85.7%

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

    1. associate-*l* 85.6%

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

    2. fma-define 85.7%

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

    3. cbrt-unprod 86.1%

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

    4. metadata-eval 86.1%

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

    5. cbrt-unprod 86.1%

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

    6. associate-/l/ 89.9%

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

    7. cbrt-unprod 89.9%

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

    8. metadata-eval 89.9%

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

    9. cbrt-prod 89.9%

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

    10. associate-/l/ 86.1%

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

    11. *-commutative 86.1%

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

11. Applied egg-rr 86.1%

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

    1. associate-*l/ 86.1%

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

    2. metadata-eval 86.1%

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

    3. associate-*l/ 86.1%

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

13. Simplified 86.1%

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

14. Taylor expanded in g around inf 75.5%

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

Alternative 6: 15.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ \frac{1}{\sqrt[3]{\frac{a}{g}}} \cdot -2 \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a) :precision binary64 (* (/ 1.0 (cbrt (/ a g))) -2.0))

C

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

Java

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

Julia

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

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := N[(N[(1.0 / N[Power[N[(a / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 46.1%

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

3. Simplified 46.4%

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

5. Taylor expanded in g around -inf 27.5%

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

   1. associate-*r/ 27.5%

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

   2. neg-mul-1 27.5%

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

7. Simplified 27.5%

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

8. Taylor expanded in g around inf 15.6%

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

9. Taylor expanded in g around -inf 15.6%

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

    1. *-commutative 15.6%

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

11. Simplified 15.6%

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

    1. clear-num 15.6%

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

    2. cbrt-div 15.9%

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

    3. metadata-eval 15.9%

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

13. Applied egg-rr 15.9%

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

Alternative 7: 15.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ \sqrt[3]{\frac{g}{a}} \cdot -2 \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a) :precision binary64 (* (cbrt (/ g a)) -2.0))

C

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

Java

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

Julia

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

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 46.1%

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

3. Simplified 46.4%

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

5. Taylor expanded in g around -inf 27.5%

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

   1. associate-*r/ 27.5%

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

   2. neg-mul-1 27.5%

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

7. Simplified 27.5%

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

8. Taylor expanded in g around inf 15.6%

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

9. Taylor expanded in g around -inf 15.6%

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

    1. *-commutative 15.6%

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

11. Simplified 15.6%

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

Alternative 8: 5.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} h_m = \left|h\right| \\ -2 \cdot \sqrt[3]{g \cdot a} \end{array} \]

FPCore

h_m = (fabs.f64 h)
(FPCore (g h_m a) :precision binary64 (* -2.0 (cbrt (* g a))))

C

h_m = fabs(h);
double code(double g, double h_m, double a) {
	return -2.0 * cbrt((g * a));
}

Java

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

Julia

h_m = abs(h)
function code(g, h_m, a)
	return Float64(-2.0 * cbrt(Float64(g * a)))
end

Wolfram

h_m = N[Abs[h], $MachinePrecision]
code[g_, h$95$m_, a_] := N[(-2.0 * N[Power[N[(g * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.1%

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

3. Simplified 46.4%

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

5. Taylor expanded in g around -inf 27.5%

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

   1. associate-*r/ 27.5%

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

   2. neg-mul-1 27.5%

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

7. Simplified 27.5%

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

8. Taylor expanded in g around inf 15.6%

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

9. Taylor expanded in g around -inf 15.6%

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

    1. *-commutative 15.6%

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

11. Simplified 15.6%

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

13. Applied egg-rr 5.8%

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

    1. unpow1 5.8%

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

    2. *-commutative 5.8%

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

    3. *-commutative 5.8%

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

15. Simplified 5.8%

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

16. Final simplification 5.8%

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

Reproduce

herbie shell --seed 2024137 
(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))))))))