Average Error: 36.7 → 32.6
Time: 12.5s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \sqrt{t_0}\\ \mathbf{if}\;g \leq 1.95 \cdot 10^{-223}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-2 \cdot g} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(t_0 - g\right)} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + t_1 \cdot t_1}\\ \end{array} \]
(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 (sqrt t_0)))
   (if (<= g 1.95e-223)
     (+ (* (cbrt (/ 0.5 a)) (cbrt (* -2.0 g))) (cbrt (* (/ -0.5 a) (+ g t_0))))
     (+
      (cbrt (* (/ 0.5 a) (- t_0 g)))
      (* (cbrt (/ -0.5 a)) (cbrt (+ g (* t_1 t_1))))))))
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 = sqrt(t_0);
	double tmp;
	if (g <= 1.95e-223) {
		tmp = (cbrt((0.5 / a)) * cbrt((-2.0 * g))) + cbrt(((-0.5 / a) * (g + t_0)));
	} else {
		tmp = cbrt(((0.5 / a) * (t_0 - g))) + (cbrt((-0.5 / a)) * cbrt((g + (t_1 * t_1))));
	}
	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.sqrt(t_0);
	double tmp;
	if (g <= 1.95e-223) {
		tmp = (Math.cbrt((0.5 / a)) * Math.cbrt((-2.0 * g))) + Math.cbrt(((-0.5 / a) * (g + t_0)));
	} else {
		tmp = Math.cbrt(((0.5 / a) * (t_0 - g))) + (Math.cbrt((-0.5 / a)) * Math.cbrt((g + (t_1 * t_1))));
	}
	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 = sqrt(t_0)
	tmp = 0.0
	if (g <= 1.95e-223)
		tmp = Float64(Float64(cbrt(Float64(0.5 / a)) * cbrt(Float64(-2.0 * g))) + cbrt(Float64(Float64(-0.5 / a) * Float64(g + t_0))));
	else
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(t_0 - g))) + Float64(cbrt(Float64(-0.5 / a)) * cbrt(Float64(g + Float64(t_1 * t_1)))));
	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[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[g, 1.95e-223], N[(N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(-2.0 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$0 - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], 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 := \sqrt{g \cdot g - h \cdot h}\\
t_1 := \sqrt{t_0}\\
\mathbf{if}\;g \leq 1.95 \cdot 10^{-223}:\\
\;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-2 \cdot g} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + t_0\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(t_0 - g\right)} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + t_1 \cdot t_1}\\


\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 < 1.94999999999999981e-223

    1. Initial program 37.0

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

    2. Simplified37.0

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

    3. Applied egg-rr33.3

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

    4. Taylor expanded in g around -inf 32.4

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

    if 1.94999999999999981e-223 < g

    1. Initial program 36.5

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

    2. Simplified36.5

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

    3. Applied egg-rr32.9

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

    4. Applied egg-rr32.9

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

  4. Final simplification32.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 1.95 \cdot 10^{-223}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-2 \cdot g} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \sqrt{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\sqrt{g \cdot g - h \cdot h}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022203 
(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))))))))