2-ancestry mixing, positive discriminant

Percentage Accurate: 44.0% → 97.0%

Time: 17.1s

Alternatives: 10

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.0% 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: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{0.5}\right) \cdot \mathsf{fma}\left(\sqrt[3]{g}, \sqrt[3]{-1} \cdot \sqrt[3]{2}, \sqrt[3]{\mathsf{fma}\left(-0.5, h \cdot \frac{h}{g}, \frac{0.5 \cdot \left(h \cdot \left(g - g\right)\right)}{g}\right)}\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (*
  (* (cbrt (/ 1.0 a)) (cbrt 0.5))
  (fma
   (cbrt g)
   (* (cbrt -1.0) (cbrt 2.0))
   (cbrt (fma -0.5 (* h (/ h g)) (/ (* 0.5 (* h (- g g))) g))))))

C

double code(double g, double h, double a) {
	return (cbrt((1.0 / a)) * cbrt(0.5)) * fma(cbrt(g), (cbrt(-1.0) * cbrt(2.0)), cbrt(fma(-0.5, (h * (h / g)), ((0.5 * (h * (g - g))) / g))));
}

Julia

function code(g, h, a)
	return Float64(Float64(cbrt(Float64(1.0 / a)) * cbrt(0.5)) * fma(cbrt(g), Float64(cbrt(-1.0) * cbrt(2.0)), cbrt(fma(-0.5, Float64(h * Float64(h / g)), Float64(Float64(0.5 * Float64(h * Float64(g - g))) / g)))))
end

Wolfram

code[g_, h_, a_] := N[(N[(N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[g, 1/3], $MachinePrecision] * N[(N[Power[-1.0, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(-0.5 * N[(h * N[(h / g), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[(h * N[(g - g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.3%

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

3. Taylor expanded in g around -inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 31.9

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

5. Applied rewrites 31.9%

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

   1. lift-+.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. cbrt-prod N/A

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

   5. pow1/3 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-*.f64 N/A

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

7. Applied rewrites 17.0%

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

8. Taylor expanded in h around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

10. Applied rewrites 89.0%

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{0.5}\right) \cdot \mathsf{fma}\left(\sqrt[3]{g}, \sqrt[3]{-1} \cdot \sqrt[3]{2}, \sqrt[3]{\mathsf{fma}\left(-0.5, \frac{\left(h \cdot h\right) \cdot \mathsf{fma}\left(0.25, \frac{\left(g + \left(-g\right)\right) \cdot \left(g + \left(-g\right)\right)}{g \cdot g}, 1\right)}{g}, \frac{0.5 \cdot \left(h \cdot \left(g + \left(-g\right)\right)\right)}{g}\right)}\right)} \]
11. Step-by-step derivation

12. Applied rewrites 97.3%

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

13. Final simplification 97.3%

    \[\leadsto \left(\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{0.5}\right) \cdot \mathsf{fma}\left(\sqrt[3]{g}, \sqrt[3]{-1} \cdot \sqrt[3]{2}, \sqrt[3]{\mathsf{fma}\left(-0.5, h \cdot \frac{h}{g}, \frac{0.5 \cdot \left(h \cdot \left(g - g\right)\right)}{g}\right)}\right) \]
14. Add Preprocessing

Alternative 2: 93.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \cdot h \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{-0.5 \cdot \left(h \cdot h\right)}{g}} + \frac{\sqrt[3]{g + g}}{\sqrt[3]{a \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \left(\sqrt[3]{g} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (* h h) 2e+300)
   (+
    (cbrt (* (/ 1.0 (* 2.0 a)) (/ (* -0.5 (* h h)) g)))
    (/ (cbrt (+ g g)) (cbrt (* a -2.0))))
   (*
    (pow (* 2.0 a) -0.3333333333333333)
    (* (cbrt g) (* (cbrt -1.0) (cbrt 2.0))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((h * h) <= 2e+300) {
		tmp = cbrt(((1.0 / (2.0 * a)) * ((-0.5 * (h * h)) / g))) + (cbrt((g + g)) / cbrt((a * -2.0)));
	} else {
		tmp = pow((2.0 * a), -0.3333333333333333) * (cbrt(g) * (cbrt(-1.0) * cbrt(2.0)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((h * h) <= 2e+300) {
		tmp = Math.cbrt(((1.0 / (2.0 * a)) * ((-0.5 * (h * h)) / g))) + (Math.cbrt((g + g)) / Math.cbrt((a * -2.0)));
	} else {
		tmp = Math.pow((2.0 * a), -0.3333333333333333) * (Math.cbrt(g) * (Math.cbrt(-1.0) * Math.cbrt(2.0)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(h * h) <= 2e+300)
		tmp = Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(Float64(-0.5 * Float64(h * h)) / g))) + Float64(cbrt(Float64(g + g)) / cbrt(Float64(a * -2.0))));
	else
		tmp = Float64((Float64(2.0 * a) ^ -0.3333333333333333) * Float64(cbrt(g) * Float64(cbrt(-1.0) * cbrt(2.0))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(h * h), $MachinePrecision], 2e+300], N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 * N[(h * h), $MachinePrecision]), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(g + g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(a * -2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[(N[Power[g, 1/3], $MachinePrecision] * N[(N[Power[-1.0, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 2.0000000000000001e300

   1. Initial program 50.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. frac-2neg N/A

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

      6. *-lft-identity N/A

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

      7. cbrt-div N/A

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 26.1

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

   7. Applied rewrites 26.1%

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

   8. Taylor expanded in g around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 96.1

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

   10. Applied rewrites 96.1%

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

   if 2.0000000000000001e300 < (*.f64 h h)

   1. Initial program 0.0%

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

   3. Taylor expanded in g around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. cbrt-prod N/A

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

      5. pow1/3 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. lift-*.f64 N/A

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

   7. Applied rewrites 1.1%

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

   8. Taylor expanded in g around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-cbrt.f64 65.7

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

   10. Applied rewrites 65.7%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{-0.5 \cdot \left(h \cdot h\right)}{g}} + \frac{\sqrt[3]{g + g}}{\sqrt[3]{a \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \left(\sqrt[3]{g} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ \mathbf{if}\;h \cdot h \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\sqrt[3]{t\_0 \cdot \frac{-0.5 \cdot \left(h \cdot h\right)}{g}} + \frac{\sqrt[3]{g + g}}{\sqrt[3]{a \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{t\_0 \cdot \left(\frac{h}{g} \cdot \left(h \cdot -0.5\right)\right)} + \sqrt[3]{\frac{g}{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))))
   (if (<= (* h h) 2e+293)
     (+
      (cbrt (* t_0 (/ (* -0.5 (* h h)) g)))
      (/ (cbrt (+ g g)) (cbrt (* a -2.0))))
     (+ (cbrt (* t_0 (* (/ h g) (* h -0.5)))) (cbrt (/ g (- a)))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double tmp;
	if ((h * h) <= 2e+293) {
		tmp = cbrt((t_0 * ((-0.5 * (h * h)) / g))) + (cbrt((g + g)) / cbrt((a * -2.0)));
	} else {
		tmp = cbrt((t_0 * ((h / g) * (h * -0.5)))) + cbrt((g / -a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double tmp;
	if ((h * h) <= 2e+293) {
		tmp = Math.cbrt((t_0 * ((-0.5 * (h * h)) / g))) + (Math.cbrt((g + g)) / Math.cbrt((a * -2.0)));
	} else {
		tmp = Math.cbrt((t_0 * ((h / g) * (h * -0.5)))) + Math.cbrt((g / -a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	tmp = 0.0
	if (Float64(h * h) <= 2e+293)
		tmp = Float64(cbrt(Float64(t_0 * Float64(Float64(-0.5 * Float64(h * h)) / g))) + Float64(cbrt(Float64(g + g)) / cbrt(Float64(a * -2.0))));
	else
		tmp = Float64(cbrt(Float64(t_0 * Float64(Float64(h / g) * Float64(h * -0.5)))) + cbrt(Float64(g / Float64(-a))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h * h), $MachinePrecision], 2e+293], N[(N[Power[N[(t$95$0 * N[(N[(-0.5 * N[(h * h), $MachinePrecision]), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(g + g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(a * -2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t$95$0 * N[(N[(h / g), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(g / (-a)), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 h h) < 1.9999999999999998e293

   1. Initial program 50.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. frac-2neg N/A

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

      6. *-lft-identity N/A

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

      7. cbrt-div N/A

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

   4. Applied rewrites 53.4%

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

   5. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 26.3

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

   7. Applied rewrites 26.3%

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

   8. Taylor expanded in g around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 96.1

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

   10. Applied rewrites 96.1%

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

   if 1.9999999999999998e293 < (*.f64 h h)

   1. Initial program 0.0%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 0.1

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

   5. Applied rewrites 0.1%

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

   6. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 18.8

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

   8. Applied rewrites 18.8%

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

   10. Applied rewrites 55.4%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \cdot h \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{-0.5 \cdot \left(h \cdot h\right)}{g}} + \frac{\sqrt[3]{g + g}}{\sqrt[3]{a \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h}{g} \cdot \left(h \cdot -0.5\right)\right)} + \sqrt[3]{\frac{g}{-a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\sqrt[3]{t\_0 \cdot \left(\frac{h}{g} \cdot \left(h \cdot -0.5\right)\right)} + \sqrt[3]{\frac{g}{-a}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \left(\sqrt[3]{\mathsf{fma}\left(h \cdot h, \frac{-0.5}{g}, h \cdot 0\right)} + \sqrt[3]{g \cdot -2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))))
   (if (<= t_0 5e+61)
     (+ (cbrt (* t_0 (* (/ h g) (* h -0.5)))) (cbrt (/ g (- a))))
     (*
      (pow (* 2.0 a) -0.3333333333333333)
      (+ (cbrt (fma (* h h) (/ -0.5 g) (* h 0.0))) (cbrt (* g -2.0)))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double tmp;
	if (t_0 <= 5e+61) {
		tmp = cbrt((t_0 * ((h / g) * (h * -0.5)))) + cbrt((g / -a));
	} else {
		tmp = pow((2.0 * a), -0.3333333333333333) * (cbrt(fma((h * h), (-0.5 / g), (h * 0.0))) + cbrt((g * -2.0)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	tmp = 0.0
	if (t_0 <= 5e+61)
		tmp = Float64(cbrt(Float64(t_0 * Float64(Float64(h / g) * Float64(h * -0.5)))) + cbrt(Float64(g / Float64(-a))));
	else
		tmp = Float64((Float64(2.0 * a) ^ -0.3333333333333333) * Float64(cbrt(fma(Float64(h * h), Float64(-0.5 / g), Float64(h * 0.0))) + cbrt(Float64(g * -2.0))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+61], N[(N[Power[N[(t$95$0 * N[(N[(h / g), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(g / (-a)), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[(N[Power[N[(N[(h * h), $MachinePrecision] * N[(-0.5 / g), $MachinePrecision] + N[(h * 0.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(g * -2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 5.00000000000000018e61

   1. Initial program 50.9%

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

   3. Taylor expanded in g around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 27.3

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

   5. Applied rewrites 27.3%

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

   6. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.9

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

   8. Applied rewrites 81.9%

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

   10. Applied rewrites 84.6%

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

   if 5.00000000000000018e61 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 39.4%

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

   3. Taylor expanded in g around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.5

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

   5. Applied rewrites 27.5%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. cbrt-prod N/A

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

      5. pow1/3 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. lift-*.f64 N/A

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

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in h around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 89.6%

       \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{0.5}\right) \cdot \mathsf{fma}\left(\sqrt[3]{g}, \sqrt[3]{-1} \cdot \sqrt[3]{2}, \sqrt[3]{\mathsf{fma}\left(-0.5, \frac{\left(h \cdot h\right) \cdot \mathsf{fma}\left(0.25, \frac{\left(g + \left(-g\right)\right) \cdot \left(g + \left(-g\right)\right)}{g \cdot g}, 1\right)}{g}, \frac{0.5 \cdot \left(h \cdot \left(g + \left(-g\right)\right)\right)}{g}\right)}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 85.1%

       \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(h \cdot h, \frac{-0.5}{g}, h \cdot 0\right)} + \sqrt[3]{g \cdot -2}\right) \cdot {\left(2 \cdot a\right)}^{-0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{2 \cdot a} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h}{g} \cdot \left(h \cdot -0.5\right)\right)} + \sqrt[3]{\frac{g}{-a}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \left(\sqrt[3]{\mathsf{fma}\left(h \cdot h, \frac{-0.5}{g}, h \cdot 0\right)} + \sqrt[3]{g \cdot -2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (* (/ 1.0 (* 2.0 a)) (* (/ h g) (* h -0.5)))) (cbrt (/ g (- a)))))

C

double code(double g, double h, double a) {
	return cbrt(((1.0 / (2.0 * a)) * ((h / g) * (h * -0.5)))) + cbrt((g / -a));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((1.0 / (2.0 * a)) * ((h / g) * (h * -0.5)))) + Math.cbrt((g / -a));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(Float64(h / g) * Float64(h * -0.5)))) + cbrt(Float64(g / Float64(-a))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(N[(h / g), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(g / (-a)), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.3%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 25.6

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

5. Applied rewrites 25.6%

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

6. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 73.5

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

8. Applied rewrites 73.5%

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

10. Applied rewrites 75.7%

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

11. Final simplification 75.7%

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

Alternative 6: 72.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.3%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 25.6

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

5. Applied rewrites 25.6%

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

6. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 73.5

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

8. Applied rewrites 73.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 73.5

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

10. Applied rewrites 73.5%

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

11. Final simplification 73.5%

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

Alternative 7: 68.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.3%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 25.6

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

5. Applied rewrites 25.6%

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

6. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 73.5

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

8. Applied rewrites 73.5%

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

9. Taylor expanded in g around inf

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

    1. associate-*r/ N/A

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

    2. lower-/.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 N/A

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

    6. lower-*.f64 70.2

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

11. Applied rewrites 70.2%

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

12. Final simplification 70.2%

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

Alternative 8: 15.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{-a}}\\ t\_0 + t\_0 \end{array} \end{array} \]

FPCore

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

C

double code(double g, double h, double a) {
	double t_0 = cbrt((g / -a));
	return t_0 + t_0;
}

Java

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

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(g / Float64(-a)))
	return Float64(t_0 + t_0)
end

Wolfram

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

Derivation

1. Initial program 48.3%

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

3. Taylor expanded in g around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 25.6

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

5. Applied rewrites 25.6%

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

6. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 73.5

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

8. Applied rewrites 73.5%

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

9. Taylor expanded in g around -inf

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

    1. associate-*r/ N/A

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

    2. mul-1-neg N/A

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

    3. lower-/.f64 N/A

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

    4. lower-neg.f64 15.4

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

11. Applied rewrites 15.4%

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

12. Final simplification 15.4%

    \[\leadsto \sqrt[3]{\frac{g}{-a}} + \sqrt[3]{\frac{g}{-a}} \]
13. Add Preprocessing

Alternative 9: 2.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g \cdot 0}{a}} \cdot \left(-\sqrt[3]{0.5}\right) \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (* (cbrt (/ (* g 0.0) a)) (- (cbrt 0.5))))

C

double code(double g, double h, double a) {
	return cbrt(((g * 0.0) / a)) * -cbrt(0.5);
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((g * 0.0) / a)) * -Math.cbrt(0.5);
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(g * 0.0) / a)) * Float64(-cbrt(0.5)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(g * 0.0), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision] * (-N[Power[0.5, 1/3], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 48.3%

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

3. Taylor expanded in g around -inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 31.9

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

5. Applied rewrites 31.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. difference-of-squares N/A

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

   8. lift-+.f64 N/A

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

   9. lift--.f64 N/A

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

   10. *-commutative N/A

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

   11. sqrt-prod N/A

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

   12. pow1/2 N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. pow1/2 N/A

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

   16. lower-sqrt.f64 2.9

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

7. Applied rewrites 2.9%

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

8. Taylor expanded in g around -inf

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

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

   3. lower-*.f64 N/A

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

10. Applied rewrites 3.0%

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

11. Final simplification 3.0%

    \[\leadsto \sqrt[3]{\frac{g \cdot 0}{a}} \cdot \left(-\sqrt[3]{0.5}\right) \]
12. Add Preprocessing

Alternative 10: 1.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ {\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot 0 \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (* (pow (* 2.0 a) -0.3333333333333333) 0.0))

C

double code(double g, double h, double a) {
	return pow((2.0 * a), -0.3333333333333333) * 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 = ((2.0d0 * a) ** (-0.3333333333333333d0)) * 0.0d0
end function

Java

public static double code(double g, double h, double a) {
	return Math.pow((2.0 * a), -0.3333333333333333) * 0.0;
}

Python

def code(g, h, a):
	return math.pow((2.0 * a), -0.3333333333333333) * 0.0

Julia

function code(g, h, a)
	return Float64((Float64(2.0 * a) ^ -0.3333333333333333) * 0.0)
end

MATLAB

function tmp = code(g, h, a)
	tmp = ((2.0 * a) ^ -0.3333333333333333) * 0.0;
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(2.0 * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * 0.0), $MachinePrecision]

Derivation

1. Initial program 48.3%

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

3. Taylor expanded in g around -inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 31.9

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

5. Applied rewrites 31.9%

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

   1. lift-+.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. cbrt-prod N/A

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

   5. pow1/3 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-*.f64 N/A

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

7. Applied rewrites 17.0%

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

8. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. +-commutative N/A

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

   5. unpow2 N/A

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

   6. rem-square-sqrt N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 1.6

      \[\leadsto {\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \left(-\sqrt[3]{\color{blue}{g \cdot 0}}\right) \]

10. Applied rewrites 1.6%

    \[\leadsto {\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \color{blue}{\left(-\sqrt[3]{g \cdot 0}\right)} \]
11. Step-by-step derivation

12. Applied rewrites 1.6%

    \[\leadsto \color{blue}{{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot 0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024231 
(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))))))))