Jmat.Real.dawson

Average Accuracy: 54.0% → 100.0%

Time: 12.0s

Precision: binary64

Cost: 96200

Math

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

↓

\[\begin{array}{l} t_0 := {\left(x \cdot x\right)}^{2}\\ t_1 := {\left(x \cdot x\right)}^{3}\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ t_3 := t_1 \cdot t_0\\ \mathbf{if}\;x \leq -500000000:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 1000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot {x}^{4} + 0.0072644182 \cdot t_1\right)\right) + \left({x}^{8} \cdot 0.0005064034 + 0.0001789971 \cdot t_3\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + t_0 \cdot 0.2909738639\right)\right) + \left(t_1 \cdot 0.0694555761 + t_2 \cdot 0.0140005442\right)\right) + \left(t_3 \cdot 0.0008327945 + 0.0003579942 \cdot \left(t_0 \cdot t_2\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{\frac{0.2514179000665374}{x}}{x \cdot x} + \frac{0.15298196345929074}{{x}^{5}}\right)\\ \end{array} \]

FPCore

(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) 2.0))
        (t_1 (pow (* x x) 3.0))
        (t_2 (* (* x x) t_1))
        (t_3 (* t_1 t_0)))
   (if (<= x -500000000.0)
     (/ 0.5 x)
     (if (<= x 1000.0)
       (log1p
        (expm1
         (*
          x
          (/
           (+
            (+
             (+ 1.0 (* 0.1049934947 (* x x)))
             (+ (* 0.0424060604 (pow x 4.0)) (* 0.0072644182 t_1)))
            (+ (* (pow x 8.0) 0.0005064034) (* 0.0001789971 t_3)))
           (+
            (+
             (+ 1.0 (+ (* (* x x) 0.7715471019) (* t_0 0.2909738639)))
             (+ (* t_1 0.0694555761) (* t_2 0.0140005442)))
            (+ (* t_3 0.0008327945) (* 0.0003579942 (* t_0 t_2))))))))
       (+
        (/ 0.5 x)
        (+
         (/ (/ 0.2514179000665374 x) (* x x))
         (/ 0.15298196345929074 (pow x 5.0))))))))

C

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), 2.0);
	double t_1 = pow((x * x), 3.0);
	double t_2 = (x * x) * t_1;
	double t_3 = t_1 * t_0;
	double tmp;
	if (x <= -500000000.0) {
		tmp = 0.5 / x;
	} else if (x <= 1000.0) {
		tmp = log1p(expm1((x * ((((1.0 + (0.1049934947 * (x * x))) + ((0.0424060604 * pow(x, 4.0)) + (0.0072644182 * t_1))) + ((pow(x, 8.0) * 0.0005064034) + (0.0001789971 * t_3))) / (((1.0 + (((x * x) * 0.7715471019) + (t_0 * 0.2909738639))) + ((t_1 * 0.0694555761) + (t_2 * 0.0140005442))) + ((t_3 * 0.0008327945) + (0.0003579942 * (t_0 * t_2))))))));
	} else {
		tmp = (0.5 / x) + (((0.2514179000665374 / x) / (x * x)) + (0.15298196345929074 / pow(x, 5.0)));
	}
	return tmp;
}

Java

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), 2.0);
	double t_1 = Math.pow((x * x), 3.0);
	double t_2 = (x * x) * t_1;
	double t_3 = t_1 * t_0;
	double tmp;
	if (x <= -500000000.0) {
		tmp = 0.5 / x;
	} else if (x <= 1000.0) {
		tmp = Math.log1p(Math.expm1((x * ((((1.0 + (0.1049934947 * (x * x))) + ((0.0424060604 * Math.pow(x, 4.0)) + (0.0072644182 * t_1))) + ((Math.pow(x, 8.0) * 0.0005064034) + (0.0001789971 * t_3))) / (((1.0 + (((x * x) * 0.7715471019) + (t_0 * 0.2909738639))) + ((t_1 * 0.0694555761) + (t_2 * 0.0140005442))) + ((t_3 * 0.0008327945) + (0.0003579942 * (t_0 * t_2))))))));
	} else {
		tmp = (0.5 / x) + (((0.2514179000665374 / x) / (x * x)) + (0.15298196345929074 / Math.pow(x, 5.0)));
	}
	return tmp;
}

Python

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), 2.0)
	t_1 = math.pow((x * x), 3.0)
	t_2 = (x * x) * t_1
	t_3 = t_1 * t_0
	tmp = 0
	if x <= -500000000.0:
		tmp = 0.5 / x
	elif x <= 1000.0:
		tmp = math.log1p(math.expm1((x * ((((1.0 + (0.1049934947 * (x * x))) + ((0.0424060604 * math.pow(x, 4.0)) + (0.0072644182 * t_1))) + ((math.pow(x, 8.0) * 0.0005064034) + (0.0001789971 * t_3))) / (((1.0 + (((x * x) * 0.7715471019) + (t_0 * 0.2909738639))) + ((t_1 * 0.0694555761) + (t_2 * 0.0140005442))) + ((t_3 * 0.0008327945) + (0.0003579942 * (t_0 * t_2))))))))
	else:
		tmp = (0.5 / x) + (((0.2514179000665374 / x) / (x * x)) + (0.15298196345929074 / math.pow(x, 5.0)))
	return tmp

Julia

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) ^ 2.0
	t_1 = Float64(x * x) ^ 3.0
	t_2 = Float64(Float64(x * x) * t_1)
	t_3 = Float64(t_1 * t_0)
	tmp = 0.0
	if (x <= -500000000.0)
		tmp = Float64(0.5 / x);
	elseif (x <= 1000.0)
		tmp = log1p(expm1(Float64(x * Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(Float64(0.0424060604 * (x ^ 4.0)) + Float64(0.0072644182 * t_1))) + Float64(Float64((x ^ 8.0) * 0.0005064034) + Float64(0.0001789971 * t_3))) / Float64(Float64(Float64(1.0 + Float64(Float64(Float64(x * x) * 0.7715471019) + Float64(t_0 * 0.2909738639))) + Float64(Float64(t_1 * 0.0694555761) + Float64(t_2 * 0.0140005442))) + Float64(Float64(t_3 * 0.0008327945) + Float64(0.0003579942 * Float64(t_0 * t_2))))))));
	else
		tmp = Float64(Float64(0.5 / x) + Float64(Float64(Float64(0.2514179000665374 / x) / Float64(x * x)) + Float64(0.15298196345929074 / (x ^ 5.0))));
	end
	return tmp
end

Wolfram

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], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x * x), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$0), $MachinePrecision]}, If[LessEqual[x, -500000000.0], N[(0.5 / x), $MachinePrecision], If[LessEqual[x, 1000.0], N[Log[1 + N[(Exp[N[(x * N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0424060604 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 8.0], $MachinePrecision] * 0.0005064034), $MachinePrecision] + N[(0.0001789971 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * 0.7715471019), $MachinePrecision] + N[(t$95$0 * 0.2909738639), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * 0.0694555761), $MachinePrecision] + N[(t$95$2 * 0.0140005442), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 * 0.0008327945), $MachinePrecision] + N[(0.0003579942 * N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] + N[(N[(N[(0.2514179000665374 / x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.15298196345929074 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5e8

   1. Initial program 5.9%

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

   2. Simplified 5.9%

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

      [Start] 5.9

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

      *-commutative [=>] 5.9

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

   3. Taylor expanded in x around inf 100.0%

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

   if -5e8 < x < 1e3

   1. Initial program 100.0%

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

   2. Applied egg-rr 100.0%

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

      [Start] 100.0

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

      log1p-expm1-u [=>] 100.0

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot {\left(x \cdot x\right)}^{2} + 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + \left(\color{blue}{0.0005064034 \cdot {x}^{8}} + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right)\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left({\left(x \cdot x\right)}^{3} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot {\left(x \cdot x\right)}^{2}\right)\right)}\right)\right) \]

   4. Simplified 100.0%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot {\left(x \cdot x\right)}^{2} + 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + \left(\color{blue}{{x}^{8} \cdot 0.0005064034} + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right)\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left({\left(x \cdot x\right)}^{3} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot {\left(x \cdot x\right)}^{2}\right)\right)}\right)\right) \]
      Proof

      [Start] 100.0

      \[ \mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot {\left(x \cdot x\right)}^{2} + 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right)\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left({\left(x \cdot x\right)}^{3} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot {\left(x \cdot x\right)}^{2}\right)\right)}\right)\right) \]

      *-commutative [=>] 100.0

      \[ \mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot {\left(x \cdot x\right)}^{2} + 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + \left(\color{blue}{{x}^{8} \cdot 0.0005064034} + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right)\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left({\left(x \cdot x\right)}^{3} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot {\left(x \cdot x\right)}^{2}\right)\right)}\right)\right) \]

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(\color{blue}{0.0424060604 \cdot {x}^{4}} + 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + \left({x}^{8} \cdot 0.0005064034 + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right)\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left({\left(x \cdot x\right)}^{3} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot {\left(x \cdot x\right)}^{2}\right)\right)}\right)\right) \]

   if 1e3 < x

   1. Initial program 6.7%

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

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{0.2514179000665374 \cdot \frac{1}{{x}^{3}} + \left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.5}{x} + \left(\frac{0.2514179000665374}{{x}^{3}} + \frac{0.15298196345929074}{{x}^{5}}\right)} \]
      Proof

      [Start] 100.0	\[ 0.2514179000665374 \cdot \frac{1}{{x}^{3}} + \left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right) \]
      associate-+r+ [=>] 100.0	\[ \color{blue}{\left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right) + 0.5 \cdot \frac{1}{x}} \]
      +-commutative [=>] 100.0	\[ \color{blue}{0.5 \cdot \frac{1}{x} + \left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right)} \]
      associate-*r/ [=>] 100.0	\[ \color{blue}{\frac{0.5 \cdot 1}{x}} + \left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right) \]
      metadata-eval [=>] 100.0	\[ \frac{\color{blue}{0.5}}{x} + \left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right) \]
      associate-*r/ [=>] 100.0	\[ \frac{0.5}{x} + \left(\color{blue}{\frac{0.2514179000665374 \cdot 1}{{x}^{3}}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right) \]
      metadata-eval [=>] 100.0	\[ \frac{0.5}{x} + \left(\frac{\color{blue}{0.2514179000665374}}{{x}^{3}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right) \]
      associate-*r/ [=>] 100.0	\[ \frac{0.5}{x} + \left(\frac{0.2514179000665374}{{x}^{3}} + \color{blue}{\frac{0.15298196345929074 \cdot 1}{{x}^{5}}}\right) \]
      metadata-eval [=>] 100.0	\[ \frac{0.5}{x} + \left(\frac{0.2514179000665374}{{x}^{3}} + \frac{\color{blue}{0.15298196345929074}}{{x}^{5}}\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{0.5}{x} + \left(\color{blue}{\frac{1}{x \cdot x} \cdot \frac{0.2514179000665374}{x}} + \frac{0.15298196345929074}{{x}^{5}}\right) \]
      Proof

      [Start] 100.0	\[ \frac{0.5}{x} + \left(\frac{0.2514179000665374}{{x}^{3}} + \frac{0.15298196345929074}{{x}^{5}}\right) \]
      clear-num [=>] 100.0	\[ \frac{0.5}{x} + \left(\color{blue}{\frac{1}{\frac{{x}^{3}}{0.2514179000665374}}} + \frac{0.15298196345929074}{{x}^{5}}\right) \]
      unpow3 [=>] 100.0	\[ \frac{0.5}{x} + \left(\frac{1}{\frac{\color{blue}{\left(x \cdot x\right) \cdot x}}{0.2514179000665374}} + \frac{0.15298196345929074}{{x}^{5}}\right) \]
      associate-/l* [=>] 100.0	\[ \frac{0.5}{x} + \left(\frac{1}{\color{blue}{\frac{x \cdot x}{\frac{0.2514179000665374}{x}}}} + \frac{0.15298196345929074}{{x}^{5}}\right) \]
      associate-/r/ [=>] 100.0	\[ \frac{0.5}{x} + \left(\color{blue}{\frac{1}{x \cdot x} \cdot \frac{0.2514179000665374}{x}} + \frac{0.15298196345929074}{{x}^{5}}\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{0.5}{x} + \left(\color{blue}{\frac{\frac{0.2514179000665374}{x}}{x \cdot x}} + \frac{0.15298196345929074}{{x}^{5}}\right) \]
      Proof

      [Start] 100.0	\[ \frac{0.5}{x} + \left(\frac{1}{x \cdot x} \cdot \frac{0.2514179000665374}{x} + \frac{0.15298196345929074}{{x}^{5}}\right) \]
      associate-*l/ [=>] 100.0	\[ \frac{0.5}{x} + \left(\color{blue}{\frac{1 \cdot \frac{0.2514179000665374}{x}}{x \cdot x}} + \frac{0.15298196345929074}{{x}^{5}}\right) \]
      *-un-lft-identity [<=] 100.0	\[ \frac{0.5}{x} + \left(\frac{\color{blue}{\frac{0.2514179000665374}{x}}}{x \cdot x} + \frac{0.15298196345929074}{{x}^{5}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -500000000:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 1000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot {x}^{4} + 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + \left({x}^{8} \cdot 0.0005064034 + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right)\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left({\left(x \cdot x\right)}^{3} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left({\left(x \cdot x\right)}^{2} \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{\frac{0.2514179000665374}{x}}{x \cdot x} + \frac{0.15298196345929074}{{x}^{5}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	11208

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := t_0 \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ \mathbf{if}\;x \leq -100000000:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 1000:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot t_0\right)\right) + \left(0.0005064034 \cdot t_1 + 0.0001789971 \cdot t_2\right)}{\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + \left(0.2909738639 \cdot t_0 + \left(x \cdot x\right) \cdot \left(0.0694555761 \cdot t_0\right)\right)\right) + 0.0140005442 \cdot t_1\right) + \left(0.0008327945 \cdot t_2 + 0.0003579942 \cdot \left(t_0 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{\frac{0.2514179000665374}{x}}{x \cdot x} + \frac{0.15298196345929074}{{x}^{5}}\right)\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	7688

\[\begin{array}{l} \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{\frac{0.2514179000665374}{x}}{x \cdot x} + \frac{0.15298196345929074}{{x}^{5}}\right)\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \frac{\frac{0.2514179000665374}{x}}{x \cdot x}\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -0.8 \lor \neg \left(x \leq 0.78\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.6665536072\right)\\ \end{array} \]

Alternative 5
Accuracy	99.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -0.7:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 6
Accuracy	51.3%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023131 
(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))