Jmat.Real.dawson

Average Error: 29.5 → 0.3

Time: 6.1s

Precision: binary64

Cost: 118344

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -17755209754.77356:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.0023813981322523565:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, 1 + \left(0.0072644182 \cdot {x}^{6} + \left(0.0424060604 \cdot {x}^{4} + 0.1049934947 \cdot {x}^{2}\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, 1\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

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 -17755209754.77356)
   (/ 0.5 x)
   (if (<= x 0.0023813981322523565)
     (*
      x
      (/
       (fma
        0.0001789971
        (pow x 10.0)
        (fma
         0.0005064034
         (pow x 8.0)
         (+
          1.0
          (+
           (* 0.0072644182 (pow x 6.0))
           (+ (* 0.0424060604 (pow x 4.0)) (* 0.1049934947 (pow x 2.0)))))))
       (fma
        (pow x 10.0)
        0.0008327945
        (fma
         0.0003579942
         (pow x 12.0)
         (fma
          (pow x 6.0)
          0.0694555761
          (fma
           (pow x 8.0)
           0.0140005442
           (fma x (* x 0.7715471019) (fma (pow x 4.0) 0.2909738639 1.0))))))))
     (/ 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 <= -17755209754.77356) {
		tmp = 0.5 / x;
	} else if (x <= 0.0023813981322523565) {
		tmp = x * (fma(0.0001789971, pow(x, 10.0), fma(0.0005064034, pow(x, 8.0), (1.0 + ((0.0072644182 * pow(x, 6.0)) + ((0.0424060604 * pow(x, 4.0)) + (0.1049934947 * pow(x, 2.0))))))) / fma(pow(x, 10.0), 0.0008327945, fma(0.0003579942, pow(x, 12.0), fma(pow(x, 6.0), 0.0694555761, fma(pow(x, 8.0), 0.0140005442, fma(x, (x * 0.7715471019), fma(pow(x, 4.0), 0.2909738639, 1.0)))))));
	} else {
		tmp = 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 <= -17755209754.77356)
		tmp = Float64(0.5 / x);
	elseif (x <= 0.0023813981322523565)
		tmp = Float64(x * Float64(fma(0.0001789971, (x ^ 10.0), fma(0.0005064034, (x ^ 8.0), Float64(1.0 + Float64(Float64(0.0072644182 * (x ^ 6.0)) + Float64(Float64(0.0424060604 * (x ^ 4.0)) + Float64(0.1049934947 * (x ^ 2.0))))))) / fma((x ^ 10.0), 0.0008327945, fma(0.0003579942, (x ^ 12.0), fma((x ^ 6.0), 0.0694555761, fma((x ^ 8.0), 0.0140005442, fma(x, Float64(x * 0.7715471019), fma((x ^ 4.0), 0.2909738639, 1.0))))))));
	else
		tmp = 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, -17755209754.77356], N[(0.5 / x), $MachinePrecision], If[LessEqual[x, 0.0023813981322523565], N[(x * N[(N[(0.0001789971 * N[Power[x, 10.0], $MachinePrecision] + N[(0.0005064034 * N[Power[x, 8.0], $MachinePrecision] + N[(1.0 + N[(N[(0.0072644182 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0424060604 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.1049934947 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, 10.0], $MachinePrecision] * 0.0008327945 + N[(0.0003579942 * N[Power[x, 12.0], $MachinePrecision] + N[(N[Power[x, 6.0], $MachinePrecision] * 0.0694555761 + N[(N[Power[x, 8.0], $MachinePrecision] * 0.0140005442 + N[(x * N[(x * 0.7715471019), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * 0.2909738639 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -17755209754.77356 or 0.0023813981322523565 < x

   1. Initial program 58.9

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

   2. Taylor expanded in x around inf 0.6

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

   if -17755209754.77356 < x < 0.0023813981322523565

   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

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

   3. Taylor expanded in x around inf 0.0

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, \color{blue}{1 + \left(0.0072644182 \cdot {x}^{6} + \left(0.0424060604 \cdot {x}^{4} + 0.1049934947 \cdot {x}^{2}\right)\right)}\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, 1\right)\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -17755209754.77356:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.0023813981322523565:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, 1 + \left(0.0072644182 \cdot {x}^{6} + \left(0.0424060604 \cdot {x}^{4} + 0.1049934947 \cdot {x}^{2}\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, 1\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	83784

\[\begin{array}{l} t_0 := {\left(x \cdot x\right)}^{3}\\ t_1 := \left(x \cdot x\right) \cdot t_0\\ t_2 := {\left(x \cdot x\right)}^{2}\\ t_3 := t_2 \cdot t_0\\ \mathbf{if}\;x \leq -2560198.7834962932:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \mathbf{elif}\;x \leq 0.0023813981322523565:\\ \;\;\;\;x \cdot \frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0072644182 \cdot {x}^{6} + 0.0424060604 \cdot t_2\right)\right) + \left(0.0005064034 \cdot t_1 + 0.0001789971 \cdot t_3\right)}{\left(\left(0.2909738639 \cdot t_2 + \left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right)\right) + \left(0.0694555761 \cdot t_0 + 0.0140005442 \cdot t_1\right)\right) + \left(0.0008327945 \cdot t_3 + 0.0003579942 \cdot \left(t_2 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	24008

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ t_3 := \left(x \cdot x\right) \cdot t_2\\ \mathbf{if}\;x \leq -2560198.7834962932:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \mathbf{elif}\;x \leq 0.0023813981322523565:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(\left(1 + \log \left(e^{0.1049934947 \cdot \left(x \cdot x\right)}\right)\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot t_1\right) + 0.0005064034 \cdot t_2\right) + 0.0001789971 \cdot t_3}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot t_0\right) + 0.0694555761 \cdot t_1\right) + 0.0140005442 \cdot t_2\right) + 0.0008327945 \cdot t_3\right) + 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	11208

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ t_3 := \left(x \cdot x\right) \cdot t_2\\ \mathbf{if}\;x \leq -2560198.7834962932:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \mathbf{elif}\;x \leq 0.0023813981322523565:\\ \;\;\;\;x \cdot \frac{0.0001789971 \cdot t_3 + \left(0.0005064034 \cdot t_2 + \left(0.0072644182 \cdot t_1 + \left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t_0\right)\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot t_0\right) + 0.0694555761 \cdot t_1\right) + 0.0140005442 \cdot t_2\right) + 0.0008327945 \cdot t_3\right) + 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 4
Error	0.6
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -11.9831158348553:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.0023813981322523565:\\ \;\;\;\;x + {x}^{3} \cdot -0.6665536072\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 5
Error	0.5
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -11.9831158348553:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \mathbf{elif}\;x \leq 0.0023813981322523565:\\ \;\;\;\;x + {x}^{3} \cdot -0.6665536072\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 6
Error	0.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -11.9831158348553:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.0023813981322523565:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 7
Error	31.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022225 
(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))