Average Error: 35.7 → 31.6
Time: 11.4s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := g + \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;g \leq 10^{-155}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}{\sqrt[3]{a}} + \sqrt[3]{t_0 \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{t_0}\\ \end{array} \]
(FPCore (g h a)
 :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))))))))
↓
(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (+ g (sqrt (- (* g g) (* h h))))))
   (if (<= g 1e-155)
     (+ (/ (cbrt (* 0.5 (* g -2.0))) (cbrt a)) (cbrt (* t_0 (/ -0.5 a))))
     (+
      (cbrt (* (/ 0.5 a) (* -0.5 (/ (pow h 2.0) g))))
      (* (cbrt (/ -0.5 a)) (cbrt t_0))))))
double code(double g, double h, double a) {
	return cbrt(((1.0 / (2.0 * a)) * (-g + sqrt(((g * g) - (h * h)))))) + cbrt(((1.0 / (2.0 * a)) * (-g - sqrt(((g * g) - (h * h))))));
}
↓
double code(double g, double h, double a) {
	double t_0 = g + sqrt(((g * g) - (h * h)));
	double tmp;
	if (g <= 1e-155) {
		tmp = (cbrt((0.5 * (g * -2.0))) / cbrt(a)) + cbrt((t_0 * (-0.5 / a)));
	} else {
		tmp = cbrt(((0.5 / a) * (-0.5 * (pow(h, 2.0) / g)))) + (cbrt((-0.5 / a)) * cbrt(t_0));
	}
	return tmp;
}

public static double code(double g, double h, double a) {
	return Math.cbrt(((1.0 / (2.0 * a)) * (-g + Math.sqrt(((g * g) - (h * h)))))) + Math.cbrt(((1.0 / (2.0 * a)) * (-g - Math.sqrt(((g * g) - (h * h))))));
}
↓
public static double code(double g, double h, double a) {
	double t_0 = g + Math.sqrt(((g * g) - (h * h)));
	double tmp;
	if (g <= 1e-155) {
		tmp = (Math.cbrt((0.5 * (g * -2.0))) / Math.cbrt(a)) + Math.cbrt((t_0 * (-0.5 / a)));
	} else {
		tmp = Math.cbrt(((0.5 / a) * (-0.5 * (Math.pow(h, 2.0) / g)))) + (Math.cbrt((-0.5 / a)) * Math.cbrt(t_0));
	}
	return tmp;
}

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(Float64(-g) + sqrt(Float64(Float64(g * g) - Float64(h * h)))))) + cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(Float64(-g) - sqrt(Float64(Float64(g * g) - Float64(h * h)))))))
end
↓
function code(g, h, a)
	t_0 = Float64(g + sqrt(Float64(Float64(g * g) - Float64(h * h))))
	tmp = 0.0
	if (g <= 1e-155)
		tmp = Float64(Float64(cbrt(Float64(0.5 * Float64(g * -2.0))) / cbrt(a)) + cbrt(Float64(t_0 * Float64(-0.5 / a))));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * Float64((h ^ 2.0) / g)))) + Float64(cbrt(Float64(-0.5 / a)) * cbrt(t_0)));
	end
	return tmp
end

code[g_, h_, a_] := N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[((-g) + N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[((-g) - N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]
↓
code[g_, h_, a_] := Block[{t$95$0 = N[(g + N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[g, 1e-155], N[(N[(N[Power[N[(0.5 * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(t$95$0 * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * N[(N[Power[h, 2.0], $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t$95$0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\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)}
↓
\begin{array}{l}
t_0 := g + \sqrt{g \cdot g - h \cdot h}\\
\mathbf{if}\;g \leq 10^{-155}:\\
\;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}{\sqrt[3]{a}} + \sqrt[3]{t_0 \cdot \frac{-0.5}{a}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{t_0}\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if g < 1.00000000000000001e-155

    1. Initial program 37.3

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

    2. Simplified37.3

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

    3. Applied egg-rr33.9

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

    4. Taylor expanded in g around -inf 32.7

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

    5. Simplified32.7

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

    if 1.00000000000000001e-155 < g

    1. Initial program 33.9

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

    2. Simplified33.8

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

    3. Applied egg-rr30.4

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

    4. Taylor expanded in g around inf 30.3

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

  4. Final simplification31.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 10^{-155}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \sqrt{g \cdot g - h \cdot h}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022185 
(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))))))))