Jmat.Real.dawson

Average Error: 29.1 → 0.0

Time: 3.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -19330359397.40189:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 6377.548293335222:\\ \;\;\;\;x \cdot {\left(\frac{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2514179000665374}{{x}^{3}} - \frac{-0.5}{x}\\ \end{array} \]

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
 (if (<= x -19330359397.40189)
   (/ 0.5 x)
   (if (<= x 6377.548293335222)
     (*
      x
      (pow
       (/
        (fma
         0.0003579942
         (pow x 12.0)
         (fma
          (pow x 10.0)
          0.0008327945
          (fma
           (pow x 8.0)
           0.0140005442
           (fma
            (pow x 6.0)
            0.0694555761
            (fma (pow x 4.0) 0.2909738639 (fma (* x x) 0.7715471019 1.0))))))
        (fma
         0.0001789971
         (pow x 10.0)
         (fma
          0.0005064034
          (pow x 8.0)
          (fma
           0.0072644182
           (pow x 6.0)
           (fma 0.0424060604 (pow x 4.0) (fma 0.1049934947 (* x x) 1.0))))))
       -1.0))
     (- (/ 0.2514179000665374 (pow x 3.0)) (/ -0.5 x)))))

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 tmp;
	if (x <= -19330359397.40189) {
		tmp = 0.5 / x;
	} else if (x <= 6377.548293335222) {
		tmp = x * pow((fma(0.0003579942, pow(x, 12.0), fma(pow(x, 10.0), 0.0008327945, fma(pow(x, 8.0), 0.0140005442, fma(pow(x, 6.0), 0.0694555761, fma(pow(x, 4.0), 0.2909738639, fma((x * x), 0.7715471019, 1.0)))))) / fma(0.0001789971, pow(x, 10.0), fma(0.0005064034, pow(x, 8.0), fma(0.0072644182, pow(x, 6.0), fma(0.0424060604, pow(x, 4.0), fma(0.1049934947, (x * x), 1.0)))))), -1.0);
	} else {
		tmp = (0.2514179000665374 / pow(x, 3.0)) - (-0.5 / x);
	}
	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)
	tmp = 0.0
	if (x <= -19330359397.40189)
		tmp = Float64(0.5 / x);
	elseif (x <= 6377.548293335222)
		tmp = Float64(x * (Float64(fma(0.0003579942, (x ^ 12.0), fma((x ^ 10.0), 0.0008327945, fma((x ^ 8.0), 0.0140005442, fma((x ^ 6.0), 0.0694555761, fma((x ^ 4.0), 0.2909738639, fma(Float64(x * x), 0.7715471019, 1.0)))))) / fma(0.0001789971, (x ^ 10.0), fma(0.0005064034, (x ^ 8.0), fma(0.0072644182, (x ^ 6.0), fma(0.0424060604, (x ^ 4.0), fma(0.1049934947, Float64(x * x), 1.0)))))) ^ -1.0));
	else
		tmp = Float64(Float64(0.2514179000665374 / (x ^ 3.0)) - Float64(-0.5 / x));
	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_] := If[LessEqual[x, -19330359397.40189], N[(0.5 / x), $MachinePrecision], If[LessEqual[x, 6377.548293335222], N[(x * N[Power[N[(N[(0.0003579942 * N[Power[x, 12.0], $MachinePrecision] + N[(N[Power[x, 10.0], $MachinePrecision] * 0.0008327945 + N[(N[Power[x, 8.0], $MachinePrecision] * 0.0140005442 + N[(N[Power[x, 6.0], $MachinePrecision] * 0.0694555761 + N[(N[Power[x, 4.0], $MachinePrecision] * 0.2909738639 + N[(N[(x * x), $MachinePrecision] * 0.7715471019 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0001789971 * N[Power[x, 10.0], $MachinePrecision] + N[(0.0005064034 * N[Power[x, 8.0], $MachinePrecision] + N[(0.0072644182 * N[Power[x, 6.0], $MachinePrecision] + N[(0.0424060604 * N[Power[x, 4.0], $MachinePrecision] + N[(0.1049934947 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(0.2514179000665374 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -19330359397.4018898

   1. Initial program 59.9

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

   2. Simplified 59.9

      \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)\right)\right)}} \]

   3. Applied egg-rr 60.0

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)\right)\right)}}\right)}^{3}} \]

   4. Taylor expanded in x around -inf 6.7

      \[\leadsto \color{blue}{-1 \cdot {\left(e^{0.3333333333333333 \cdot \left(\log \left(\frac{-1}{x}\right) + \log 0.5\right)}\right)}^{3}} \]

   5. Simplified 0

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

   if -19330359397.4018898 < x < 6377.5482933352223

   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. Simplified 0.0

      \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)\right)\right)}} \]

   3. Applied egg-rr 0.0

      \[\leadsto x \cdot \color{blue}{{\left(\frac{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}\right)}^{-1}} \]

   if 6377.5482933352223 < x

   1. Initial program 59.4

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

   2. Simplified 59.4

      \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)\right)\right)}} \]

   3. Taylor expanded in x around inf 0.0

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

   4. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19330359397.40189:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 6377.548293335222:\\ \;\;\;\;x \cdot {\left(\frac{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2514179000665374}{{x}^{3}} - \frac{-0.5}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022145 
(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))