Jmat.Real.dawson

Average Error: 29.7 → 0.0

Time: 12.0s

Precision: binary64

Cost: 112201

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+22} \lor \neg \left(x \leq 40000000\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left({x}^{6}, 0.0072644182, \mathsf{fma}\left({x}^{8}, 0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right), \mathsf{fma}\left(0.0424060604, {x}^{4}, x \cdot \left(x \cdot 0.1049934947\right)\right)\right)\right), x\right)}{1 + \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{8} \cdot \left(x \cdot x\right), 0.0008327945, 0.0003579942 \cdot {x}^{12}\right)\right)\right)\right)\right)}\\ \end{array} \]

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
 (if (or (<= x -2e+22) (not (<= x 40000000.0)))
   (/ 0.5 x)
   (/
    (fma
     x
     (fma
      (pow x 6.0)
      0.0072644182
      (fma
       (pow x 8.0)
       (+ 0.0005064034 (* x (* x 0.0001789971)))
       (fma 0.0424060604 (pow x 4.0) (* x (* x 0.1049934947)))))
     x)
    (+
     1.0
     (fma
      x
      (* x 0.7715471019)
      (fma
       (pow x 4.0)
       0.2909738639
       (fma
        (pow x 6.0)
        0.0694555761
        (fma
         (pow x 8.0)
         0.0140005442
         (fma
          (* (pow x 8.0) (* x x))
          0.0008327945
          (* 0.0003579942 (pow x 12.0)))))))))))

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 tmp;
	if ((x <= -2e+22) || !(x <= 40000000.0)) {
		tmp = 0.5 / x;
	} else {
		tmp = fma(x, fma(pow(x, 6.0), 0.0072644182, fma(pow(x, 8.0), (0.0005064034 + (x * (x * 0.0001789971))), fma(0.0424060604, pow(x, 4.0), (x * (x * 0.1049934947))))), x) / (1.0 + fma(x, (x * 0.7715471019), fma(pow(x, 4.0), 0.2909738639, fma(pow(x, 6.0), 0.0694555761, fma(pow(x, 8.0), 0.0140005442, fma((pow(x, 8.0) * (x * x)), 0.0008327945, (0.0003579942 * pow(x, 12.0))))))));
	}
	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)
	tmp = 0.0
	if ((x <= -2e+22) || !(x <= 40000000.0))
		tmp = Float64(0.5 / x);
	else
		tmp = Float64(fma(x, fma((x ^ 6.0), 0.0072644182, fma((x ^ 8.0), Float64(0.0005064034 + Float64(x * Float64(x * 0.0001789971))), fma(0.0424060604, (x ^ 4.0), Float64(x * Float64(x * 0.1049934947))))), x) / Float64(1.0 + fma(x, Float64(x * 0.7715471019), fma((x ^ 4.0), 0.2909738639, fma((x ^ 6.0), 0.0694555761, fma((x ^ 8.0), 0.0140005442, fma(Float64((x ^ 8.0) * Float64(x * x)), 0.0008327945, Float64(0.0003579942 * (x ^ 12.0)))))))));
	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_] := If[Or[LessEqual[x, -2e+22], N[Not[LessEqual[x, 40000000.0]], $MachinePrecision]], N[(0.5 / x), $MachinePrecision], N[(N[(x * N[(N[Power[x, 6.0], $MachinePrecision] * 0.0072644182 + N[(N[Power[x, 8.0], $MachinePrecision] * N[(0.0005064034 + N[(x * N[(x * 0.0001789971), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * N[Power[x, 4.0], $MachinePrecision] + N[(x * N[(x * 0.1049934947), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + N[(x * N[(x * 0.7715471019), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * 0.2909738639 + N[(N[Power[x, 6.0], $MachinePrecision] * 0.0694555761 + N[(N[Power[x, 8.0], $MachinePrecision] * 0.0140005442 + N[(N[(N[Power[x, 8.0], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.0008327945 + N[(0.0003579942 * N[Power[x, 12.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e22 or 4e7 < x

   1. Initial program 61.9

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

   2. Simplified 61.9

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

      [Start] 61.9

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

      *-commutative [=>] 61.9

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

   3. Taylor expanded in x around inf 0.0

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

   if -2e22 < x < 4e7

   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

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

      [Start] 0.0

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

      *-commutative [=>] 0.0

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

   3. Applied egg-rr 0.0

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

   4. Simplified 0.0

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

      [Start] 0.0

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

      associate-*r/ [<=] 0.0

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

   5. Applied egg-rr 0.0

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

   6. Simplified 0.0

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

      [Start] 0.0	\[ 1 \cdot \frac{x}{\frac{\left(1 + x \cdot \left(x \cdot 0.7715471019\right)\right) + \left(\mathsf{fma}\left(x \cdot {x}^{3}, 0.2909738639, {x}^{6} \cdot 0.0694555761\right) + \left({\left(x \cdot {x}^{3}\right)}^{2} \cdot 0.0140005442 + \mathsf{fma}\left(x \cdot \left(x \cdot {\left(x \cdot {x}^{3}\right)}^{2}\right), 0.0008327945, 0.0003579942 \cdot {\left(x \cdot {x}^{3}\right)}^{3}\right)\right)\right)}{1 + \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 0.0424060604 \cdot \left(x \cdot {x}^{3}\right)\right) + \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0005064034, {\left(x \cdot {x}^{3}\right)}^{2}, {\left(x \cdot {x}^{3}\right)}^{2} \cdot \left(x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)\right)}} \]
      *-lft-identity [=>] 0.0	\[ \color{blue}{\frac{x}{\frac{\left(1 + x \cdot \left(x \cdot 0.7715471019\right)\right) + \left(\mathsf{fma}\left(x \cdot {x}^{3}, 0.2909738639, {x}^{6} \cdot 0.0694555761\right) + \left({\left(x \cdot {x}^{3}\right)}^{2} \cdot 0.0140005442 + \mathsf{fma}\left(x \cdot \left(x \cdot {\left(x \cdot {x}^{3}\right)}^{2}\right), 0.0008327945, 0.0003579942 \cdot {\left(x \cdot {x}^{3}\right)}^{3}\right)\right)\right)}{1 + \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 0.0424060604 \cdot \left(x \cdot {x}^{3}\right)\right) + \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0005064034, {\left(x \cdot {x}^{3}\right)}^{2}, {\left(x \cdot {x}^{3}\right)}^{2} \cdot \left(x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)\right)}}} \]
      associate-/l* [<=] 0.0	\[ \color{blue}{\frac{x \cdot \left(1 + \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 0.0424060604 \cdot \left(x \cdot {x}^{3}\right)\right) + \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0005064034, {\left(x \cdot {x}^{3}\right)}^{2}, {\left(x \cdot {x}^{3}\right)}^{2} \cdot \left(x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)\right)\right)}{\left(1 + x \cdot \left(x \cdot 0.7715471019\right)\right) + \left(\mathsf{fma}\left(x \cdot {x}^{3}, 0.2909738639, {x}^{6} \cdot 0.0694555761\right) + \left({\left(x \cdot {x}^{3}\right)}^{2} \cdot 0.0140005442 + \mathsf{fma}\left(x \cdot \left(x \cdot {\left(x \cdot {x}^{3}\right)}^{2}\right), 0.0008327945, 0.0003579942 \cdot {\left(x \cdot {x}^{3}\right)}^{3}\right)\right)\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+22} \lor \neg \left(x \leq 40000000\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left({x}^{6}, 0.0072644182, \mathsf{fma}\left({x}^{8}, 0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right), \mathsf{fma}\left(0.0424060604, {x}^{4}, x \cdot \left(x \cdot 0.1049934947\right)\right)\right)\right), x\right)}{1 + \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{8} \cdot \left(x \cdot x\right), 0.0008327945, 0.0003579942 \cdot {x}^{12}\right)\right)\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	93129

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+22} \lor \neg \left(x \leq 40000000\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(\mathsf{fma}\left(0.0424060604, {x}^{4}, x \cdot \left(x \cdot 0.1049934947\right)\right) + \mathsf{fma}\left({x}^{6}, 0.0072644182, {x}^{8} \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)\right)}{1 + \left(\mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, {x}^{6} \cdot 0.0694555761\right)\right) + \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{10}, 0.0008327945, 0.0003579942 \cdot {x}^{12}\right)\right)\right)}\\ \end{array} \]

Alternative 2
Error	0.0
Cost	23369

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ t_3 := \left(x \cdot x\right) \cdot t_2\\ \mathbf{if}\;x \leq -5000000000 \lor \neg \left(x \leq 50000000\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot t_1\right) + 0.0005064034 \cdot t_2\right) + 0.0001789971 \cdot t_3}{\left(\left(\left(\left(0.2909738639 \cdot t_0 + \left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot t_1\right) + 0.0140005442 \cdot t_2\right) + 0.0008327945 \cdot t_3\right) + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot {\left(x \cdot x\right)}^{2}\right)}\\ \end{array} \]

Alternative 3
Error	0.0
Cost	11209

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := t_0 \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ \mathbf{if}\;x \leq -5000000000 \lor \neg \left(x \leq 40000000\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot t_0\right)\right) + \left(0.0005064034 \cdot t_1 + 0.0001789971 \cdot t_2\right)}{\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + \left(0.2909738639 \cdot t_0 + \left(x \cdot x\right) \cdot \left(0.0694555761 \cdot t_0\right)\right)\right) + 0.0140005442 \cdot t_1\right) + \left(0.0008327945 \cdot t_2 + 0.0003579942 \cdot \left(t_0 \cdot t_1\right)\right)}\\ \end{array} \]

Alternative 4
Error	0.0
Cost	11209

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ t_3 := \left(x \cdot x\right) \cdot t_2\\ \mathbf{if}\;x \leq -5000000000 \lor \neg \left(x \leq 50000000\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot t_1\right) + 0.0005064034 \cdot t_2\right) + 0.0001789971 \cdot t_3}{\left(\left(\left(\left(0.2909738639 \cdot t_0 + \left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot t_1\right) + 0.0140005442 \cdot t_2\right) + 0.0008327945 \cdot t_3\right) + 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot t_3\right)}\\ \end{array} \]

Alternative 5
Error	0.3
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{0.5}{x} + \frac{\frac{0.2514179000665374}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot 0.6665536072, 1\right)}\\ \end{array} \]

Alternative 6
Error	0.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{0.5}{x} + \frac{\frac{0.2514179000665374}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot \left(x \cdot 0.6665536072\right)}\\ \end{array} \]

Alternative 7
Error	0.4
Cost	841

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

Alternative 8
Error	0.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x}{1 + x \cdot \left(x \cdot 0.6665536072\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 9
Error	0.6
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 10
Error	31.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023041 
(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))