2-ancestry mixing, positive discriminant

Percentage Accurate: 44.4% → 85.2%

Time: 14.9s

Alternatives: 8

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.4% 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: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq -5 \cdot 10^{-281}:\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + {\left(\sqrt[3]{a}\right)}^{-1} \cdot {\left(-0.25 \cdot \left(\frac{h}{g} \cdot h\right)\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}}{\sqrt[3]{a \cdot 2}} + \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{0 \cdot g}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= g -5e-281)
   (+
    (cbrt (/ (- g) a))
    (* (pow (cbrt a) -1.0) (pow (* -0.25 (* (/ h g) h)) 0.3333333333333333)))
   (+
    (/ (cbrt (- (fma (sqrt (+ h g)) (sqrt (- g h)) g))) (cbrt (* a 2.0)))
    (/ 1.0 (cbrt (/ (* -2.0 a) (* 0.0 g)))))))

C

double code(double g, double h, double a) {
	double tmp;
	if (g <= -5e-281) {
		tmp = cbrt((-g / a)) + (pow(cbrt(a), -1.0) * pow((-0.25 * ((h / g) * h)), 0.3333333333333333));
	} else {
		tmp = (cbrt(-fma(sqrt((h + g)), sqrt((g - h)), g)) / cbrt((a * 2.0))) + (1.0 / cbrt(((-2.0 * a) / (0.0 * g))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (g <= -5e-281)
		tmp = Float64(cbrt(Float64(Float64(-g) / a)) + Float64((cbrt(a) ^ -1.0) * (Float64(-0.25 * Float64(Float64(h / g) * h)) ^ 0.3333333333333333)));
	else
		tmp = Float64(Float64(cbrt(Float64(-fma(sqrt(Float64(h + g)), sqrt(Float64(g - h)), g))) / cbrt(Float64(a * 2.0))) + Float64(1.0 / cbrt(Float64(Float64(-2.0 * a) / Float64(0.0 * g)))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[g, -5e-281], N[(N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[Power[a, 1/3], $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(-0.25 * N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[(-N[(N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), 1/3], $MachinePrecision] / N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[N[(N[(-2.0 * a), $MachinePrecision] / N[(0.0 * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < -4.9999999999999998e-281

   1. Initial program 44.7%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 11.0

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

   5. Applied rewrites 11.0%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-cbrt.f64 71.4

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

   8. Applied rewrites 71.4%

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

   10. Applied rewrites 71.7%

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

   12. Applied rewrites 71.7%

       \[\leadsto {\left(\left(\frac{h}{g} \cdot h\right) \cdot -0.25\right)}^{0.3333333333333333} \cdot \color{blue}{{\left(\sqrt[3]{a}\right)}^{-1}} + \sqrt[3]{\frac{-g}{a}} \]

   if -4.9999999999999998e-281 < g

   1. Initial program 47.0%

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

   4. Applied rewrites 47.0%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 55.4%

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

   7. Taylor expanded in g around -inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval 96.6

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

   9. Applied rewrites 96.6%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -5 \cdot 10^{-281}:\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + {\left(\sqrt[3]{a}\right)}^{-1} \cdot {\left(-0.25 \cdot \left(\frac{h}{g} \cdot h\right)\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}}{\sqrt[3]{a \cdot 2}} + \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{0 \cdot g}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq 9.6 \cdot 10^{-279}:\\ \;\;\;\;\sqrt[3]{\frac{h}{g} \cdot h} \cdot \sqrt[3]{\frac{-0.25}{a}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}}{\sqrt[3]{a \cdot 2}} + \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{0 \cdot g}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= g 9.6e-279)
   (+ (* (cbrt (* (/ h g) h)) (cbrt (/ -0.25 a))) (cbrt (/ (- g) a)))
   (+
    (/ (cbrt (- (fma (sqrt (+ h g)) (sqrt (- g h)) g))) (cbrt (* a 2.0)))
    (/ 1.0 (cbrt (/ (* -2.0 a) (* 0.0 g)))))))

C

double code(double g, double h, double a) {
	double tmp;
	if (g <= 9.6e-279) {
		tmp = (cbrt(((h / g) * h)) * cbrt((-0.25 / a))) + cbrt((-g / a));
	} else {
		tmp = (cbrt(-fma(sqrt((h + g)), sqrt((g - h)), g)) / cbrt((a * 2.0))) + (1.0 / cbrt(((-2.0 * a) / (0.0 * g))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (g <= 9.6e-279)
		tmp = Float64(Float64(cbrt(Float64(Float64(h / g) * h)) * cbrt(Float64(-0.25 / a))) + cbrt(Float64(Float64(-g) / a)));
	else
		tmp = Float64(Float64(cbrt(Float64(-fma(sqrt(Float64(h + g)), sqrt(Float64(g - h)), g))) / cbrt(Float64(a * 2.0))) + Float64(1.0 / cbrt(Float64(Float64(-2.0 * a) / Float64(0.0 * g)))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[g, 9.6e-279], N[(N[(N[Power[N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(-0.25 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[(-N[(N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), 1/3], $MachinePrecision] / N[Power[N[(a * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[N[(N[(-2.0 * a), $MachinePrecision] / N[(0.0 * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < 9.59999999999999959e-279

   1. Initial program 44.7%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 11.0

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

   5. Applied rewrites 11.0%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-cbrt.f64 71.4

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

   8. Applied rewrites 71.4%

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

   10. Applied rewrites 71.7%

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

   if 9.59999999999999959e-279 < g

   1. Initial program 47.0%

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

   4. Applied rewrites 47.0%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 55.4%

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

   7. Taylor expanded in g around -inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval 96.6

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

   9. Applied rewrites 96.6%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 9.6 \cdot 10^{-279}:\\ \;\;\;\;\sqrt[3]{\frac{h}{g} \cdot h} \cdot \sqrt[3]{\frac{-0.25}{a}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}}{\sqrt[3]{a \cdot 2}} + \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{0 \cdot g}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(g \cdot g\right) \cdot a\\ \mathbf{if}\;\frac{1}{a \cdot 2} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;\sqrt[3]{\frac{h}{g} \cdot h} \cdot \sqrt[3]{\frac{-0.25}{a}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\sqrt[3]{2}} \cdot \left(\sqrt[3]{\frac{2}{t\_0}} + \sqrt[3]{\frac{0}{t\_0}}\right)\right) \cdot g\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (* (* g g) a)))
   (if (<= (/ 1.0 (* a 2.0)) 5e+176)
     (+ (* (cbrt (* (/ h g) h)) (cbrt (/ -0.25 a))) (cbrt (/ (- g) a)))
     (* (* (/ -1.0 (cbrt 2.0)) (+ (cbrt (/ 2.0 t_0)) (cbrt (/ 0.0 t_0)))) g))))

C

double code(double g, double h, double a) {
	double t_0 = (g * g) * a;
	double tmp;
	if ((1.0 / (a * 2.0)) <= 5e+176) {
		tmp = (cbrt(((h / g) * h)) * cbrt((-0.25 / a))) + cbrt((-g / a));
	} else {
		tmp = ((-1.0 / cbrt(2.0)) * (cbrt((2.0 / t_0)) + cbrt((0.0 / t_0)))) * g;
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = (g * g) * a;
	double tmp;
	if ((1.0 / (a * 2.0)) <= 5e+176) {
		tmp = (Math.cbrt(((h / g) * h)) * Math.cbrt((-0.25 / a))) + Math.cbrt((-g / a));
	} else {
		tmp = ((-1.0 / Math.cbrt(2.0)) * (Math.cbrt((2.0 / t_0)) + Math.cbrt((0.0 / t_0)))) * g;
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(g * g) * a)
	tmp = 0.0
	if (Float64(1.0 / Float64(a * 2.0)) <= 5e+176)
		tmp = Float64(Float64(cbrt(Float64(Float64(h / g) * h)) * cbrt(Float64(-0.25 / a))) + cbrt(Float64(Float64(-g) / a)));
	else
		tmp = Float64(Float64(Float64(-1.0 / cbrt(2.0)) * Float64(cbrt(Float64(2.0 / t_0)) + cbrt(Float64(0.0 / t_0)))) * g);
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(g * g), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 5e+176], N[(N[(N[Power[N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(-0.25 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(2.0 / t$95$0), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(0.0 / t$95$0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * g), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 5e176

   1. Initial program 47.6%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 30.1

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-cbrt.f64 78.0

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

   8. Applied rewrites 78.0%

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

   10. Applied rewrites 78.2%

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

   if 5e176 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 30.4%

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

   4. Applied rewrites 33.2%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 38.1%

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

   7. Taylor expanded in g around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

   9. Applied rewrites 55.2%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;\sqrt[3]{\frac{h}{g} \cdot h} \cdot \sqrt[3]{\frac{-0.25}{a}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\sqrt[3]{2}} \cdot \left(\sqrt[3]{\frac{2}{\left(g \cdot g\right) \cdot a}} + \sqrt[3]{\frac{0}{\left(g \cdot g\right) \cdot a}}\right)\right) \cdot g\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.9%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 29.4

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

5. Applied rewrites 29.4%

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

6. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 73.5

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

8. Applied rewrites 73.5%

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

10. Applied rewrites 73.7%

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

11. Final simplification 73.7%

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

Alternative 5: 75.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.9%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 29.4

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

5. Applied rewrites 29.4%

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

6. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 73.5

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

8. Applied rewrites 73.5%

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

10. Applied rewrites 73.7%

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

11. Final simplification 73.7%

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

Alternative 6: 75.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.9%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 29.4

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

5. Applied rewrites 29.4%

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

6. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 73.5

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

8. Applied rewrites 73.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 73.5

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

10. Applied rewrites 73.5%

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

11. Final simplification 73.5%

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

Alternative 7: 73.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{-1} \cdot \sqrt[3]{\frac{g}{a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.9%

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

4. Applied rewrites 46.1%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. cbrt-div N/A

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

   6. *-lft-identity N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 28.4%

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

7. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 72.3

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

9. Applied rewrites 72.3%

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

10. Final simplification 72.3%

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

Alternative 8: 3.0% accurate, 302.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[g_, h_, a_] := 0.0

Derivation

1. Initial program 45.9%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. cbrt-div N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 48.3%

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

5. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. rem-square-sqrt N/A

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

   10. metadata-eval N/A

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

   11. lower-cbrt.f64 3.0

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

7. Applied rewrites 3.0%

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

8. Taylor expanded in a around 0

   \[\leadsto 0 \]
9. Step-by-step derivation

10. Applied rewrites 3.0%

    \[\leadsto 0 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(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))))))))