Average Error: 35.9 → 34.4
Time: 16.8s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \sqrt[3]{\frac{0.5}{a} \cdot \left(t_0 - g\right)}\\ t_2 := t_1 + \sqrt[3]{\left(g + t_0\right) \cdot \frac{-0.5}{a}}\\ \mathbf{if}\;h \leq -1.85 \cdot 10^{-171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq 1.05 \cdot 10^{-162}:\\ \;\;\;\;t_1 + \frac{\sqrt[3]{-0.5 \cdot \left(g + \mathsf{hypot}\left(g, \sqrt{h \cdot \left(-h\right)}\right)\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
(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 (sqrt (- (* g g) (* h h))))
        (t_1 (cbrt (* (/ 0.5 a) (- t_0 g))))
        (t_2 (+ t_1 (cbrt (* (+ g t_0) (/ -0.5 a))))))
   (if (<= h -1.85e-171)
     t_2
     (if (<= h 1.05e-162)
       (+ t_1 (/ (cbrt (* -0.5 (+ g (hypot g (sqrt (* h (- h))))))) (cbrt a)))
       t_2))))
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 = sqrt(((g * g) - (h * h)));
	double t_1 = cbrt(((0.5 / a) * (t_0 - g)));
	double t_2 = t_1 + cbrt(((g + t_0) * (-0.5 / a)));
	double tmp;
	if (h <= -1.85e-171) {
		tmp = t_2;
	} else if (h <= 1.05e-162) {
		tmp = t_1 + (cbrt((-0.5 * (g + hypot(g, sqrt((h * -h)))))) / cbrt(a));
	} else {
		tmp = t_2;
	}
	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 = Math.sqrt(((g * g) - (h * h)));
	double t_1 = Math.cbrt(((0.5 / a) * (t_0 - g)));
	double t_2 = t_1 + Math.cbrt(((g + t_0) * (-0.5 / a)));
	double tmp;
	if (h <= -1.85e-171) {
		tmp = t_2;
	} else if (h <= 1.05e-162) {
		tmp = t_1 + (Math.cbrt((-0.5 * (g + Math.hypot(g, Math.sqrt((h * -h)))))) / Math.cbrt(a));
	} else {
		tmp = t_2;
	}
	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 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = cbrt(Float64(Float64(0.5 / a) * Float64(t_0 - g)))
	t_2 = Float64(t_1 + cbrt(Float64(Float64(g + t_0) * Float64(-0.5 / a))))
	tmp = 0.0
	if (h <= -1.85e-171)
		tmp = t_2;
	elseif (h <= 1.05e-162)
		tmp = Float64(t_1 + Float64(cbrt(Float64(-0.5 * Float64(g + hypot(g, sqrt(Float64(h * Float64(-h))))))) / cbrt(a)));
	else
		tmp = t_2;
	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[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$0 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Power[N[(N[(g + t$95$0), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.85e-171], t$95$2, If[LessEqual[h, 1.05e-162], N[(t$95$1 + N[(N[Power[N[(-0.5 * N[(g + N[Sqrt[g ^ 2 + N[Sqrt[N[(h * (-h)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}
↓
\begin{array}{l}
t_0 := \sqrt{g \cdot g - h \cdot h}\\
t_1 := \sqrt[3]{\frac{0.5}{a} \cdot \left(t_0 - g\right)}\\
t_2 := t_1 + \sqrt[3]{\left(g + t_0\right) \cdot \frac{-0.5}{a}}\\
\mathbf{if}\;h \leq -1.85 \cdot 10^{-171}:\\
\;\;\;\;t_2\\

\mathbf{elif}\;h \leq 1.05 \cdot 10^{-162}:\\
\;\;\;\;t_1 + \frac{\sqrt[3]{-0.5 \cdot \left(g + \mathsf{hypot}\left(g, \sqrt{h \cdot \left(-h\right)}\right)\right)}}{\sqrt[3]{a}}\\

\mathbf{else}:\\
\;\;\;\;t_2\\


\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 h < -1.85000000000000006e-171 or 1.05e-162 < h

    1. Initial program 39.5

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

    2. Simplified39.5

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

    if -1.85000000000000006e-171 < h < 1.05e-162

    1. Initial program 30.8

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

    2. Simplified30.7

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

    3. Applied egg-rr34.9

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \color{blue}{{\left({\left(g \cdot g - h \cdot h\right)}^{0.25}\right)}^{2}}\right) \cdot \frac{-0.5}{a}} \]

    4. Applied egg-rr27.3

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

  4. Final simplification34.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.85 \cdot 10^{-171}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{elif}\;h \leq 1.05 \cdot 10^{-162}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \mathsf{hypot}\left(g, \sqrt{h \cdot \left(-h\right)}\right)\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022209 
(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))))))))