2-ancestry mixing, positive discriminant

Percentage Accurate: 43.7% → 96.7%

Time: 19.9s

Alternatives: 7

Speedup: 2.0×

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: 43.7% 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: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{h}^{2}}{g}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(t_1 - g\right)} + \sqrt[3]{\left(g + t_1\right) \cdot \frac{-1}{2 \cdot a}} \leq 5 \cdot 10^{+70}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot t_0\right)} + \frac{\sqrt[3]{-0.5 \cdot \mathsf{fma}\left(-0.5, t_0, 2 \cdot g\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{\sqrt[3]{g}}{\sqrt[3]{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ (pow h 2.0) g)) (t_1 (sqrt (- (* g g) (* h h)))))
   (if (<=
        (+
         (cbrt (* (/ 1.0 (* 2.0 a)) (- t_1 g)))
         (cbrt (* (+ g t_1) (/ -1.0 (* 2.0 a)))))
        5e+70)
     (+
      (cbrt (* (/ 0.5 a) (* -0.5 t_0)))
      (/ (cbrt (* -0.5 (fma -0.5 t_0 (* 2.0 g)))) (cbrt a)))
     (+ (cbrt (* (/ 0.5 a) (- g g))) (/ (cbrt g) (cbrt (- a)))))))

C

double code(double g, double h, double a) {
	double t_0 = pow(h, 2.0) / g;
	double t_1 = sqrt(((g * g) - (h * h)));
	double tmp;
	if ((cbrt(((1.0 / (2.0 * a)) * (t_1 - g))) + cbrt(((g + t_1) * (-1.0 / (2.0 * a))))) <= 5e+70) {
		tmp = cbrt(((0.5 / a) * (-0.5 * t_0))) + (cbrt((-0.5 * fma(-0.5, t_0, (2.0 * g)))) / cbrt(a));
	} else {
		tmp = cbrt(((0.5 / a) * (g - g))) + (cbrt(g) / cbrt(-a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64((h ^ 2.0) / g)
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	tmp = 0.0
	if (Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(t_1 - g))) + cbrt(Float64(Float64(g + t_1) * Float64(-1.0 / Float64(2.0 * a))))) <= 5e+70)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * t_0))) + Float64(cbrt(Float64(-0.5 * fma(-0.5, t_0, Float64(2.0 * g)))) / cbrt(a)));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + Float64(cbrt(g) / cbrt(Float64(-a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[Power[h, 2.0], $MachinePrecision] / g), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(g + t$95$1), $MachinePrecision] * N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 5e+70], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 * N[(-0.5 * t$95$0 + N[(2.0 * g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[(-a), 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 5.0000000000000002e70

   1. Initial program 77.3%

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

   3. Simplified 77.3%

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

      1. add-cube-cbrt 77.2%

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

      2. pow3 77.2%

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

      3. pow2 77.2%

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

      4. pow2 77.2%

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

   5. Applied egg-rr 77.2%

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

   6. Taylor expanded in g around inf 35.7%

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

   7. Taylor expanded in g around inf 81.2%

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

      1. rem-cube-cbrt 81.4%

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

      2. associate-*r/ 81.4%

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

      3. cbrt-div 96.5%

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

      4. *-commutative 96.5%

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

      5. +-commutative 96.5%

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

      6. fma-def 96.5%

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

   9. Applied egg-rr 96.5%

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

       1. +-commutative 96.5%

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

       2. fma-udef 96.5%

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

       3. associate-+l+ 96.5%

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

       4. fma-def 96.5%

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

       5. *-lft-identity 96.5%

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

       6. *-lft-identity 96.5%

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

       7. distribute-rgt-out 96.5%

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

       8. metadata-eval 96.5%

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

   11. Simplified 96.5%

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

   if 5.0000000000000002e70 < (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 6.9%

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

   3. Simplified 6.9%

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

   4. Taylor expanded in g around inf 6.2%

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

   5. Taylor expanded in g around inf 66.7%

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

      1. associate-*r/ 66.7%

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

      2. neg-mul-1 66.7%

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

   7. Simplified 66.7%

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

      1. frac-2neg 66.7%

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

      2. cbrt-div 96.4%

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

      3. remove-double-neg 96.4%

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

   9. Applied egg-rr 96.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 5 \cdot 10^{+70}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)} + \frac{\sqrt[3]{-0.5 \cdot \mathsf{fma}\left(-0.5, \frac{{h}^{2}}{g}, 2 \cdot g\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{\sqrt[3]{g}}{\sqrt[3]{-a}}\\ \end{array} \]

Alternative 2: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{h}^{2}}{g}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(t_1 - g\right)} + \sqrt[3]{\left(g + t_1\right) \cdot \frac{-1}{2 \cdot a}} \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot t_0\right)} + \sqrt[3]{\mathsf{fma}\left(-0.5, t_0, 2 \cdot g\right)} \cdot \sqrt[3]{\frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{\sqrt[3]{g}}{\sqrt[3]{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ (pow h 2.0) g)) (t_1 (sqrt (- (* g g) (* h h)))))
   (if (<=
        (+
         (cbrt (* (/ 1.0 (* 2.0 a)) (- t_1 g)))
         (cbrt (* (+ g t_1) (/ -1.0 (* 2.0 a)))))
        5e+81)
     (+
      (cbrt (* (/ 0.5 a) (* -0.5 t_0)))
      (* (cbrt (fma -0.5 t_0 (* 2.0 g))) (cbrt (/ -0.5 a))))
     (+ (cbrt (* (/ 0.5 a) (- g g))) (/ (cbrt g) (cbrt (- a)))))))

C

double code(double g, double h, double a) {
	double t_0 = pow(h, 2.0) / g;
	double t_1 = sqrt(((g * g) - (h * h)));
	double tmp;
	if ((cbrt(((1.0 / (2.0 * a)) * (t_1 - g))) + cbrt(((g + t_1) * (-1.0 / (2.0 * a))))) <= 5e+81) {
		tmp = cbrt(((0.5 / a) * (-0.5 * t_0))) + (cbrt(fma(-0.5, t_0, (2.0 * g))) * cbrt((-0.5 / a)));
	} else {
		tmp = cbrt(((0.5 / a) * (g - g))) + (cbrt(g) / cbrt(-a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64((h ^ 2.0) / g)
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	tmp = 0.0
	if (Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(t_1 - g))) + cbrt(Float64(Float64(g + t_1) * Float64(-1.0 / Float64(2.0 * a))))) <= 5e+81)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * t_0))) + Float64(cbrt(fma(-0.5, t_0, Float64(2.0 * g))) * cbrt(Float64(-0.5 / a))));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + Float64(cbrt(g) / cbrt(Float64(-a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[Power[h, 2.0], $MachinePrecision] / g), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(g + t$95$1), $MachinePrecision] * N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 5e+81], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 * t$95$0 + N[(2.0 * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[(-a), 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 4.9999999999999998e81

   1. Initial program 77.7%

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

   3. Simplified 77.7%

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

      1. add-cube-cbrt 77.6%

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

      2. pow3 77.7%

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

      3. pow2 77.7%

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

      4. pow2 77.7%

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

   5. Applied egg-rr 77.7%

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

   6. Taylor expanded in g around inf 36.4%

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

   7. Taylor expanded in g around inf 81.6%

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

      1. cbrt-prod 96.2%

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

      2. rem-cbrt-cube 96.5%

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

      3. *-un-lft-identity 96.5%

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

      4. *-un-lft-identity 96.5%

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

      5. +-commutative 96.5%

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

      6. fma-def 96.5%

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

   9. Applied egg-rr 96.5%

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

       1. +-commutative 96.5%

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

       2. fma-udef 96.5%

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

       3. associate-+l+ 96.5%

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

       4. fma-def 96.5%

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

       5. *-lft-identity 96.5%

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

       6. *-lft-identity 96.5%

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

       7. distribute-rgt-out 96.5%

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

       8. metadata-eval 96.5%

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

   11. Simplified 96.5%

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

   if 4.9999999999999998e81 < (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 4.5%

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

   3. Simplified 4.5%

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

   4. Taylor expanded in g around inf 4.7%

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

   5. Taylor expanded in g around inf 65.9%

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

      1. associate-*r/ 65.9%

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

      2. neg-mul-1 65.9%

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

   7. Simplified 65.9%

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

      1. frac-2neg 65.9%

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

      2. cbrt-div 96.3%

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

      3. remove-double-neg 96.3%

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

   9. Applied egg-rr 96.3%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)} + \sqrt[3]{\mathsf{fma}\left(-0.5, \frac{{h}^{2}}{g}, 2 \cdot g\right)} \cdot \sqrt[3]{\frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{\sqrt[3]{g}}{\sqrt[3]{-a}}\\ \end{array} \]

Alternative 3: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(t_0 - g\right)} + \sqrt[3]{\left(g + t_0\right) \cdot \frac{-1}{2 \cdot a}} \leq 5 \cdot 10^{+70}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)} + \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{\sqrt[3]{g}}{\sqrt[3]{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h)))))
   (if (<=
        (+
         (cbrt (* (/ 1.0 (* 2.0 a)) (- t_0 g)))
         (cbrt (* (+ g t_0) (/ -1.0 (* 2.0 a)))))
        5e+70)
     (+
      (cbrt (* (/ 0.5 a) (* -0.5 (/ (pow h 2.0) g))))
      (* (* (cbrt g) (cbrt (/ 1.0 a))) (* (cbrt -0.5) (cbrt 2.0))))
     (+ (cbrt (* (/ 0.5 a) (- g g))) (/ (cbrt g) (cbrt (- a)))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double tmp;
	if ((cbrt(((1.0 / (2.0 * a)) * (t_0 - g))) + cbrt(((g + t_0) * (-1.0 / (2.0 * a))))) <= 5e+70) {
		tmp = cbrt(((0.5 / a) * (-0.5 * (pow(h, 2.0) / g)))) + ((cbrt(g) * cbrt((1.0 / a))) * (cbrt(-0.5) * cbrt(2.0)));
	} else {
		tmp = cbrt(((0.5 / a) * (g - g))) + (cbrt(g) / cbrt(-a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt(((g * g) - (h * h)));
	double tmp;
	if ((Math.cbrt(((1.0 / (2.0 * a)) * (t_0 - g))) + Math.cbrt(((g + t_0) * (-1.0 / (2.0 * a))))) <= 5e+70) {
		tmp = Math.cbrt(((0.5 / a) * (-0.5 * (Math.pow(h, 2.0) / g)))) + ((Math.cbrt(g) * Math.cbrt((1.0 / a))) * (Math.cbrt(-0.5) * Math.cbrt(2.0)));
	} else {
		tmp = Math.cbrt(((0.5 / a) * (g - g))) + (Math.cbrt(g) / Math.cbrt(-a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	tmp = 0.0
	if (Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(t_0 - g))) + cbrt(Float64(Float64(g + t_0) * Float64(-1.0 / Float64(2.0 * a))))) <= 5e+70)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * Float64((h ^ 2.0) / g)))) + Float64(Float64(cbrt(g) * cbrt(Float64(1.0 / a))) * Float64(cbrt(-0.5) * cbrt(2.0))));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + Float64(cbrt(g) / cbrt(Float64(-a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(g + t$95$0), $MachinePrecision] * N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 5e+70], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * N[(N[Power[h, 2.0], $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[(-a), 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 5.0000000000000002e70

   1. Initial program 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in h around 0 17.7%

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

   5. Taylor expanded in g around inf 38.1%

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

      1. *-commutative 38.1%

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

      2. associate-*r/ 38.1%

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

      3. unpow-prod-down 19.6%

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

      4. pow1/3 42.3%

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

   7. Applied egg-rr 42.3%

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

      1. unpow1/3 95.9%

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

   9. Simplified 95.9%

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

   if 5.0000000000000002e70 < (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 6.9%

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

   3. Simplified 6.9%

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

   4. Taylor expanded in g around inf 6.2%

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

   5. Taylor expanded in g around inf 66.7%

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

      1. associate-*r/ 66.7%

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

      2. neg-mul-1 66.7%

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

   7. Simplified 66.7%

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

      1. frac-2neg 66.7%

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

      2. cbrt-div 96.4%

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

      3. remove-double-neg 96.4%

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

   9. Applied egg-rr 96.4%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 5 \cdot 10^{+70}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)} + \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{\sqrt[3]{g}}{\sqrt[3]{-a}}\\ \end{array} \]

Alternative 4: 96.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.3%

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

3. Simplified 44.3%

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

4. Taylor expanded in g around inf 20.1%

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

5. Taylor expanded in g around inf 72.7%

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

   1. associate-*r/ 72.7%

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

   2. neg-mul-1 72.7%

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

7. Simplified 72.7%

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

   1. frac-2neg 72.7%

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

   2. cbrt-div 94.5%

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

   3. remove-double-neg 94.5%

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

9. Applied egg-rr 94.5%

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

10. Final simplification 94.5%

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

Alternative 5: 73.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.3%

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

3. Simplified 44.3%

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

4. Taylor expanded in g around inf 20.1%

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

5. Taylor expanded in g around inf 72.7%

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

6. Final simplification 72.7%

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

Alternative 6: 73.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.3%

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

3. Simplified 44.3%

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

4. Taylor expanded in g around inf 20.1%

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

5. Taylor expanded in g around inf 72.7%

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

   1. associate-*r/ 72.7%

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

   2. neg-mul-1 72.7%

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

7. Simplified 72.7%

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

8. Final simplification 72.7%

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

Alternative 7: 1.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.3%

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

3. Simplified 44.3%

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

4. Taylor expanded in g around inf 20.1%

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

5. Taylor expanded in g around inf 72.7%

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

   1. associate-*r/ 72.7%

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

   2. neg-mul-1 72.7%

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

7. Simplified 72.7%

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

   1. expm1-log1p-u 49.1%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{-g}{a}}\right)\right)} \]

   2. expm1-udef 28.2%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{-g}{a}}\right)} - 1\right)} \]

   3. add-sqr-sqrt 12.2%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{\sqrt{-g} \cdot \sqrt{-g}}}{a}}\right)} - 1\right) \]

   4. sqrt-unprod 7.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{\sqrt{\left(-g\right) \cdot \left(-g\right)}}}{a}}\right)} - 1\right) \]

   5. sqr-neg 7.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\sqrt{\color{blue}{g \cdot g}}}{a}}\right)} - 1\right) \]

   6. sqrt-unprod 0.6%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{\sqrt{g} \cdot \sqrt{g}}}{a}}\right)} - 1\right) \]

   7. add-sqr-sqrt 1.4%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{g}}{a}}\right)} - 1\right) \]

9. Applied egg-rr 1.4%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{g}{a}}\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 1.1%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{g}{a}}\right)\right)} \]

    2. expm1-log1p 1.4%

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

11. Simplified 1.4%

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

12. Final simplification 1.4%

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

Reproduce

herbie shell --seed 2023305 
(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))))))))