Average Error: 29.2 → 0.4
Time: 5.3s
Precision: binary64
Cost: 96456
\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
↓
\[\begin{array}{l} t_0 := {\left(x \cdot x\right)}^{3}\\ t_1 := \left(x \cdot x\right) \cdot t_0\\ t_2 := {\left(x \cdot x\right)}^{2}\\ t_3 := t_2 \cdot t_0\\ \mathbf{if}\;x \leq -217193238346.93753:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 2.6903770676420974 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + 0.0424060604 \cdot t_2\right)\right) + {x}^{6} \cdot 0.0072644182\right) + 0.0005064034 \cdot t_1\right) + 0.0001789971 \cdot t_3}{\left(\left(1 + \left(t_2 \cdot 0.2909738639 + \left(x \cdot x\right) \cdot 0.7715471019\right)\right) + \left({x}^{6} \cdot 0.0694555761 + t_1 \cdot 0.0140005442\right)\right) + \left(t_3 \cdot 0.0008327945 + 0.0003579942 \cdot \left(t_2 \cdot t_1\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (/
   (+
    (+
     (+
      (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x))))
      (* 0.0072644182 (* (* (* x x) (* x x)) (* x x))))
     (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x))))
    (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x))))
   (+
    (+
     (+
      (+
       (+
        (+ 1.0 (* 0.7715471019 (* x x)))
        (* 0.2909738639 (* (* x x) (* x x))))
       (* 0.0694555761 (* (* (* x x) (* x x)) (* x x))))
      (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x))))
     (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x))))
    (*
     (* 2.0 0.0001789971)
     (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x)))))
  x))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (* x x) 3.0))
        (t_1 (* (* x x) t_0))
        (t_2 (pow (* x x) 2.0))
        (t_3 (* t_2 t_0)))
   (if (<= x -217193238346.93753)
     (/ 0.5 x)
     (if (<= x 2.6903770676420974e-8)
       (log1p
        (expm1
         (*
          x
          (/
           (+
            (+
             (+
              (+ 1.0 (+ (* 0.1049934947 (* x x)) (* 0.0424060604 t_2)))
              (* (pow x 6.0) 0.0072644182))
             (* 0.0005064034 t_1))
            (* 0.0001789971 t_3))
           (+
            (+
             (+ 1.0 (+ (* t_2 0.2909738639) (* (* x x) 0.7715471019)))
             (+ (* (pow x 6.0) 0.0694555761) (* t_1 0.0140005442)))
            (+ (* t_3 0.0008327945) (* 0.0003579942 (* t_2 t_1))))))))
       (+ (/ 0.5 x) (/ 0.2514179000665374 (pow x 3.0)))))))
double code(double x) {
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * ((x * x) * (x * x)))) + (0.0072644182 * (((x * x) * (x * x)) * (x * x)))) + (0.0005064034 * ((((x * x) * (x * x)) * (x * x)) * (x * x)))) + (0.0001789971 * (((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)))) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * ((x * x) * (x * x)))) + (0.0694555761 * (((x * x) * (x * x)) * (x * x)))) + (0.0140005442 * ((((x * x) * (x * x)) * (x * x)) * (x * x)))) + (0.0008327945 * (((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)))) + ((2.0 * 0.0001789971) * ((((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)) * (x * x))))) * x;
}
↓
double code(double x) {
	double t_0 = pow((x * x), 3.0);
	double t_1 = (x * x) * t_0;
	double t_2 = pow((x * x), 2.0);
	double t_3 = t_2 * t_0;
	double tmp;
	if (x <= -217193238346.93753) {
		tmp = 0.5 / x;
	} else if (x <= 2.6903770676420974e-8) {
		tmp = log1p(expm1((x * (((((1.0 + ((0.1049934947 * (x * x)) + (0.0424060604 * t_2))) + (pow(x, 6.0) * 0.0072644182)) + (0.0005064034 * t_1)) + (0.0001789971 * t_3)) / (((1.0 + ((t_2 * 0.2909738639) + ((x * x) * 0.7715471019))) + ((pow(x, 6.0) * 0.0694555761) + (t_1 * 0.0140005442))) + ((t_3 * 0.0008327945) + (0.0003579942 * (t_2 * t_1))))))));
	} else {
		tmp = (0.5 / x) + (0.2514179000665374 / pow(x, 3.0));
	}
	return tmp;
}

public static double code(double x) {
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * ((x * x) * (x * x)))) + (0.0072644182 * (((x * x) * (x * x)) * (x * x)))) + (0.0005064034 * ((((x * x) * (x * x)) * (x * x)) * (x * x)))) + (0.0001789971 * (((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)))) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * ((x * x) * (x * x)))) + (0.0694555761 * (((x * x) * (x * x)) * (x * x)))) + (0.0140005442 * ((((x * x) * (x * x)) * (x * x)) * (x * x)))) + (0.0008327945 * (((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)))) + ((2.0 * 0.0001789971) * ((((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)) * (x * x))))) * x;
}
↓
public static double code(double x) {
	double t_0 = Math.pow((x * x), 3.0);
	double t_1 = (x * x) * t_0;
	double t_2 = Math.pow((x * x), 2.0);
	double t_3 = t_2 * t_0;
	double tmp;
	if (x <= -217193238346.93753) {
		tmp = 0.5 / x;
	} else if (x <= 2.6903770676420974e-8) {
		tmp = Math.log1p(Math.expm1((x * (((((1.0 + ((0.1049934947 * (x * x)) + (0.0424060604 * t_2))) + (Math.pow(x, 6.0) * 0.0072644182)) + (0.0005064034 * t_1)) + (0.0001789971 * t_3)) / (((1.0 + ((t_2 * 0.2909738639) + ((x * x) * 0.7715471019))) + ((Math.pow(x, 6.0) * 0.0694555761) + (t_1 * 0.0140005442))) + ((t_3 * 0.0008327945) + (0.0003579942 * (t_2 * t_1))))))));
	} else {
		tmp = (0.5 / x) + (0.2514179000665374 / Math.pow(x, 3.0));
	}
	return tmp;
}

def code(x):
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * ((x * x) * (x * x)))) + (0.0072644182 * (((x * x) * (x * x)) * (x * x)))) + (0.0005064034 * ((((x * x) * (x * x)) * (x * x)) * (x * x)))) + (0.0001789971 * (((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)))) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * ((x * x) * (x * x)))) + (0.0694555761 * (((x * x) * (x * x)) * (x * x)))) + (0.0140005442 * ((((x * x) * (x * x)) * (x * x)) * (x * x)))) + (0.0008327945 * (((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)))) + ((2.0 * 0.0001789971) * ((((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)) * (x * x))))) * x
↓
def code(x):
	t_0 = math.pow((x * x), 3.0)
	t_1 = (x * x) * t_0
	t_2 = math.pow((x * x), 2.0)
	t_3 = t_2 * t_0
	tmp = 0
	if x <= -217193238346.93753:
		tmp = 0.5 / x
	elif x <= 2.6903770676420974e-8:
		tmp = math.log1p(math.expm1((x * (((((1.0 + ((0.1049934947 * (x * x)) + (0.0424060604 * t_2))) + (math.pow(x, 6.0) * 0.0072644182)) + (0.0005064034 * t_1)) + (0.0001789971 * t_3)) / (((1.0 + ((t_2 * 0.2909738639) + ((x * x) * 0.7715471019))) + ((math.pow(x, 6.0) * 0.0694555761) + (t_1 * 0.0140005442))) + ((t_3 * 0.0008327945) + (0.0003579942 * (t_2 * t_1))))))))
	else:
		tmp = (0.5 / x) + (0.2514179000665374 / math.pow(x, 3.0))
	return tmp

function code(x)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(0.0424060604 * Float64(Float64(x * x) * Float64(x * x)))) + Float64(0.0072644182 * Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0005064034 * Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0001789971 * Float64(Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) / Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.7715471019 * Float64(x * x))) + Float64(0.2909738639 * Float64(Float64(x * x) * Float64(x * x)))) + Float64(0.0694555761 * Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0140005442 * Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0008327945 * Float64(Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) + Float64(Float64(2.0 * 0.0001789971) * Float64(Float64(Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x))))) * x)
end
↓
function code(x)
	t_0 = Float64(x * x) ^ 3.0
	t_1 = Float64(Float64(x * x) * t_0)
	t_2 = Float64(x * x) ^ 2.0
	t_3 = Float64(t_2 * t_0)
	tmp = 0.0
	if (x <= -217193238346.93753)
		tmp = Float64(0.5 / x);
	elseif (x <= 2.6903770676420974e-8)
		tmp = log1p(expm1(Float64(x * Float64(Float64(Float64(Float64(Float64(1.0 + Float64(Float64(0.1049934947 * Float64(x * x)) + Float64(0.0424060604 * t_2))) + Float64((x ^ 6.0) * 0.0072644182)) + Float64(0.0005064034 * t_1)) + Float64(0.0001789971 * t_3)) / Float64(Float64(Float64(1.0 + Float64(Float64(t_2 * 0.2909738639) + Float64(Float64(x * x) * 0.7715471019))) + Float64(Float64((x ^ 6.0) * 0.0694555761) + Float64(t_1 * 0.0140005442))) + Float64(Float64(t_3 * 0.0008327945) + Float64(0.0003579942 * Float64(t_2 * t_1))))))));
	else
		tmp = Float64(Float64(0.5 / x) + Float64(0.2514179000665374 / (x ^ 3.0)));
	end
	return tmp
end

code[x_] := N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(1.0 + N[(0.7715471019 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0694555761 * N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]
↓
code[x_] := Block[{t$95$0 = N[Power[N[(x * x), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(x * x), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$0), $MachinePrecision]}, If[LessEqual[x, -217193238346.93753], N[(0.5 / x), $MachinePrecision], If[LessEqual[x, 2.6903770676420974e-8], N[Log[1 + N[(Exp[N[(x * N[(N[(N[(N[(N[(1.0 + N[(N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 6.0], $MachinePrecision] * 0.0072644182), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 + N[(N[(t$95$2 * 0.2909738639), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.7715471019), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 6.0], $MachinePrecision] * 0.0694555761), $MachinePrecision] + N[(t$95$1 * 0.0140005442), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 * 0.0008327945), $MachinePrecision] + N[(0.0003579942 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] + N[(0.2514179000665374 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x
↓
\begin{array}{l}
t_0 := {\left(x \cdot x\right)}^{3}\\
t_1 := \left(x \cdot x\right) \cdot t_0\\
t_2 := {\left(x \cdot x\right)}^{2}\\
t_3 := t_2 \cdot t_0\\
\mathbf{if}\;x \leq -217193238346.93753:\\
\;\;\;\;\frac{0.5}{x}\\

\mathbf{elif}\;x \leq 2.6903770676420974 \cdot 10^{-8}:\\
\;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + 0.0424060604 \cdot t_2\right)\right) + {x}^{6} \cdot 0.0072644182\right) + 0.0005064034 \cdot t_1\right) + 0.0001789971 \cdot t_3}{\left(\left(1 + \left(t_2 \cdot 0.2909738639 + \left(x \cdot x\right) \cdot 0.7715471019\right)\right) + \left({x}^{6} \cdot 0.0694555761 + t_1 \cdot 0.0140005442\right)\right) + \left(t_3 \cdot 0.0008327945 + 0.0003579942 \cdot \left(t_2 \cdot t_1\right)\right)}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -217193238346.937531

    1. Initial program 61.1

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

    2. Taylor expanded in x around inf 0

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]

    if -217193238346.937531 < x < 2.69037706764209737e-8

    1. Initial program 0.0

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

    2. Applied egg-rr0.0

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + 0.0424060604 \cdot {\left(x \cdot x\right)}^{2}\right)\right) + 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right)\right) + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left({\left(x \cdot x\right)}^{3} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot {\left(x \cdot x\right)}^{2}\right)\right)}\right)\right)} \]

    3. Taylor expanded in x around 0 0.0

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + 0.0424060604 \cdot {\left(x \cdot x\right)}^{2}\right)\right) + \color{blue}{0.0072644182 \cdot {x}^{6}}\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right)\right) + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left({\left(x \cdot x\right)}^{3} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot {\left(x \cdot x\right)}^{2}\right)\right)}\right)\right) \]

    4. Simplified0.0

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + 0.0424060604 \cdot {\left(x \cdot x\right)}^{2}\right)\right) + \color{blue}{{x}^{6} \cdot 0.0072644182}\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right)\right) + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left({\left(x \cdot x\right)}^{3} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot {\left(x \cdot x\right)}^{2}\right)\right)}\right)\right) \]

    5. Taylor expanded in x around 0 0.0

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + 0.0424060604 \cdot {\left(x \cdot x\right)}^{2}\right)\right) + {x}^{6} \cdot 0.0072644182\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right)\right) + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left(\color{blue}{0.0694555761 \cdot {x}^{6}} + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot {\left(x \cdot x\right)}^{2}\right)\right)}\right)\right) \]

    6. Simplified0.0

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + 0.0424060604 \cdot {\left(x \cdot x\right)}^{2}\right)\right) + {x}^{6} \cdot 0.0072644182\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right)\right) + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left(\color{blue}{{x}^{6} \cdot 0.0694555761} + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot {\left(x \cdot x\right)}^{2}\right)\right)}\right)\right) \]

    if 2.69037706764209737e-8 < x

    1. Initial program 56.9

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

    2. Taylor expanded in x around inf 1.6

      \[\leadsto \color{blue}{0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}} \]

    3. Simplified1.6

      \[\leadsto \color{blue}{\frac{0.2514179000665374}{{x}^{3}} + \frac{0.5}{x}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -217193238346.93753:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 2.6903770676420974 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + 0.0424060604 \cdot {\left(x \cdot x\right)}^{2}\right)\right) + {x}^{6} \cdot 0.0072644182\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right)\right) + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{2} \cdot {\left(x \cdot x\right)}^{3}\right)}{\left(\left(1 + \left({\left(x \cdot x\right)}^{2} \cdot 0.2909738639 + \left(x \cdot x\right) \cdot 0.7715471019\right)\right) + \left({x}^{6} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{2} \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left({\left(x \cdot x\right)}^{2} \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \end{array} \]

Alternatives

Alternative 1
Error0.4
Cost11208
\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ t_3 := \left(x \cdot x\right) \cdot t_2\\ \mathbf{if}\;x \leq -217193238346.93753:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 2.6903770676420974 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot t_1\right) + 0.0005064034 \cdot t_2\right) + 0.0001789971 \cdot t_3}{\left(\left(\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + 0.2909738639 \cdot t_0\right) + 0.0694555761 \cdot t_1\right) + 0.0140005442 \cdot t_2\right) + 0.0008327945 \cdot t_3\right) + 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \end{array} \]
Alternative 2
Error1.1
Cost7176
\[\begin{array}{l} \mathbf{if}\;x \leq -217193238346.93753:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 2.6903770676420974 \cdot 10^{-8}:\\ \;\;\;\;x + {x}^{3} \cdot -0.6665536072\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \end{array} \]
Alternative 3
Error1.2
Cost7048
\[\begin{array}{l} \mathbf{if}\;x \leq -217193238346.93753:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 2.6903770676420974 \cdot 10^{-8}:\\ \;\;\;\;x + {x}^{3} \cdot -0.6665536072\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
Alternative 4
Error1.2
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -217193238346.93753:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 2.6903770676420974 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
Alternative 5
Error31.1
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022217 
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
  (* (/ (+ (+ (+ (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x)))) (* 0.0072644182 (* (* (* x x) (* x x)) (* x x)))) (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (+ (+ (+ (+ (+ (+ 1.0 (* 0.7715471019 (* x x))) (* 0.2909738639 (* (* x x) (* x x)))) (* 0.0694555761 (* (* (* x x) (* x x)) (* x x)))) (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (* (* 2.0 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))