Jmat.Real.dawson

Average Error: 29.5 → 0.0

Time: 11.1s

Precision: binary64

Cost: 105928

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 20000000:\\ \;\;\;\;\frac{x}{\left(1 + \mathsf{fma}\left(x, x \cdot 0.7715471019, 0.2909738639 \cdot {x}^{4}\right)\right) + \left(\mathsf{fma}\left({x}^{6}, 0.0694555761, {x}^{8} \cdot 0.0140005442\right) + \mathsf{fma}\left(0.0008327945, {x}^{10}, 0.0003579942 \cdot {\left({x}^{4}\right)}^{3}\right)\right)} \cdot \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 1\right) + \left(\mathsf{fma}\left(0.0424060604, {x}^{4}, {x}^{6} \cdot 0.0072644182\right) + \mathsf{fma}\left(0.0005064034, {x}^{8}, {x}^{10} \cdot 0.0001789971\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

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 -1e+17)
   (/ 0.5 x)
   (if (<= x 20000000.0)
     (*
      (/
       x
       (+
        (+ 1.0 (fma x (* x 0.7715471019) (* 0.2909738639 (pow x 4.0))))
        (+
         (fma (pow x 6.0) 0.0694555761 (* (pow x 8.0) 0.0140005442))
         (fma
          0.0008327945
          (pow x 10.0)
          (* 0.0003579942 (pow (pow x 4.0) 3.0))))))
      (+
       (fma 0.1049934947 (* x x) 1.0)
       (+
        (fma 0.0424060604 (pow x 4.0) (* (pow x 6.0) 0.0072644182))
        (fma 0.0005064034 (pow x 8.0) (* (pow x 10.0) 0.0001789971)))))
     (/ 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 <= -1e+17) {
		tmp = 0.5 / x;
	} else if (x <= 20000000.0) {
		tmp = (x / ((1.0 + fma(x, (x * 0.7715471019), (0.2909738639 * pow(x, 4.0)))) + (fma(pow(x, 6.0), 0.0694555761, (pow(x, 8.0) * 0.0140005442)) + fma(0.0008327945, pow(x, 10.0), (0.0003579942 * pow(pow(x, 4.0), 3.0)))))) * (fma(0.1049934947, (x * x), 1.0) + (fma(0.0424060604, pow(x, 4.0), (pow(x, 6.0) * 0.0072644182)) + fma(0.0005064034, pow(x, 8.0), (pow(x, 10.0) * 0.0001789971))));
	} 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 <= -1e+17)
		tmp = Float64(0.5 / x);
	elseif (x <= 20000000.0)
		tmp = Float64(Float64(x / Float64(Float64(1.0 + fma(x, Float64(x * 0.7715471019), Float64(0.2909738639 * (x ^ 4.0)))) + Float64(fma((x ^ 6.0), 0.0694555761, Float64((x ^ 8.0) * 0.0140005442)) + fma(0.0008327945, (x ^ 10.0), Float64(0.0003579942 * ((x ^ 4.0) ^ 3.0)))))) * Float64(fma(0.1049934947, Float64(x * x), 1.0) + Float64(fma(0.0424060604, (x ^ 4.0), Float64((x ^ 6.0) * 0.0072644182)) + fma(0.0005064034, (x ^ 8.0), Float64((x ^ 10.0) * 0.0001789971)))));
	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, -1e+17], N[(0.5 / x), $MachinePrecision], If[LessEqual[x, 20000000.0], N[(N[(x / N[(N[(1.0 + N[(x * N[(x * 0.7715471019), $MachinePrecision] + N[(0.2909738639 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 6.0], $MachinePrecision] * 0.0694555761 + N[(N[Power[x, 8.0], $MachinePrecision] * 0.0140005442), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * N[Power[x, 10.0], $MachinePrecision] + N[(0.0003579942 * N[Power[N[Power[x, 4.0], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.1049934947 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(0.0424060604 * N[Power[x, 4.0], $MachinePrecision] + N[(N[Power[x, 6.0], $MachinePrecision] * 0.0072644182), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * N[Power[x, 8.0], $MachinePrecision] + N[(N[Power[x, 10.0], $MachinePrecision] * 0.0001789971), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1e17 or 2e7 < x

   1. Initial program 61.0

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

   2. Simplified 61.0

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

      [Start] 61.0

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

      *-commutative [=>] 61.0

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

   3. Taylor expanded in x around inf 0.0

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

   if -1e17 < x < 2e7

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

   3. Simplified 0.0

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

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

   4. Applied egg-rr 0.0

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

   5. Simplified 0.0

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

      [Start] 0.0	\[ 1 \cdot \frac{x \cdot \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 1\right) + \left(\mathsf{fma}\left(0.0424060604, {x}^{4}, 0.0072644182 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.0005064034, {x}^{8}, {x}^{10} \cdot 0.0001789971\right)\right)\right)}{\left(1 + \mathsf{fma}\left(x, x \cdot 0.7715471019, {x}^{4} \cdot 0.2909738639\right)\right) + \left(\mathsf{fma}\left({x}^{6}, 0.0694555761, {x}^{8} \cdot 0.0140005442\right) + \mathsf{fma}\left(0.0008327945, {x}^{10}, {x}^{4} \cdot \left({x}^{8} \cdot 0.0003579942\right)\right)\right)} \]
      *-lft-identity [=>] 0.0	\[ \color{blue}{\frac{x \cdot \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 1\right) + \left(\mathsf{fma}\left(0.0424060604, {x}^{4}, 0.0072644182 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.0005064034, {x}^{8}, {x}^{10} \cdot 0.0001789971\right)\right)\right)}{\left(1 + \mathsf{fma}\left(x, x \cdot 0.7715471019, {x}^{4} \cdot 0.2909738639\right)\right) + \left(\mathsf{fma}\left({x}^{6}, 0.0694555761, {x}^{8} \cdot 0.0140005442\right) + \mathsf{fma}\left(0.0008327945, {x}^{10}, {x}^{4} \cdot \left({x}^{8} \cdot 0.0003579942\right)\right)\right)}} \]
      associate-*l/ [<=] 0.0	\[ \color{blue}{\frac{x}{\left(1 + \mathsf{fma}\left(x, x \cdot 0.7715471019, {x}^{4} \cdot 0.2909738639\right)\right) + \left(\mathsf{fma}\left({x}^{6}, 0.0694555761, {x}^{8} \cdot 0.0140005442\right) + \mathsf{fma}\left(0.0008327945, {x}^{10}, {x}^{4} \cdot \left({x}^{8} \cdot 0.0003579942\right)\right)\right)} \cdot \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 1\right) + \left(\mathsf{fma}\left(0.0424060604, {x}^{4}, 0.0072644182 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.0005064034, {x}^{8}, {x}^{10} \cdot 0.0001789971\right)\right)\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 20000000:\\ \;\;\;\;\frac{x}{\left(1 + \mathsf{fma}\left(x, x \cdot 0.7715471019, 0.2909738639 \cdot {x}^{4}\right)\right) + \left(\mathsf{fma}\left({x}^{6}, 0.0694555761, {x}^{8} \cdot 0.0140005442\right) + \mathsf{fma}\left(0.0008327945, {x}^{10}, 0.0003579942 \cdot {\left({x}^{4}\right)}^{3}\right)\right)} \cdot \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 1\right) + \left(\mathsf{fma}\left(0.0424060604, {x}^{4}, {x}^{6} \cdot 0.0072644182\right) + \mathsf{fma}\left(0.0005064034, {x}^{8}, {x}^{10} \cdot 0.0001789971\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	99464

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;\frac{x \cdot \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 1\right) + \left(\mathsf{fma}\left(0.0424060604, {x}^{4}, {x}^{6} \cdot 0.0072644182\right) + \mathsf{fma}\left(0.0005064034, {x}^{8}, {x}^{10} \cdot 0.0001789971\right)\right)\right)}{\left(\left(1 + \mathsf{fma}\left(x, x \cdot 0.7715471019, 0.2909738639 \cdot {x}^{4}\right)\right) + \mathsf{fma}\left({x}^{6}, 0.0694555761, {x}^{8} \cdot 0.0140005442\right)\right) + \mathsf{fma}\left(0.0008327945, {x}^{10}, 0.0003579942 \cdot {x}^{12}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 2
Error	0.0
Cost	86856

\[\begin{array}{l} \mathbf{if}\;x \leq -50000000000000:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(\mathsf{fma}\left(0.0005064034, {x}^{8}, {x}^{10} \cdot 0.0001789971\right) + \mathsf{fma}\left(0.1049934947, x \cdot x, \mathsf{fma}\left(0.0424060604, {x}^{4}, {x}^{6} \cdot 0.0072644182\right)\right)\right)\right)}{\left(\mathsf{fma}\left({x}^{6}, 0.0694555761, {x}^{8} \cdot 0.0140005442\right) + \left(1 + \mathsf{fma}\left(x \cdot x, 0.7715471019, 0.2909738639 \cdot {x}^{4}\right)\right)\right) + {x}^{10} \cdot \left(0.0008327945 + 0.0003579942 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 3
Error	0.0
Cost	70344

\[\begin{array}{l} t_0 := {\left(x \cdot x\right)}^{2}\\ t_1 := {\left(x \cdot x\right)}^{3}\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ \mathbf{if}\;x \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;\frac{x \cdot \left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot t_0 + 0.0072644182 \cdot t_1\right)\right) + \left(0.0005064034 \cdot t_2 + 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)\right)}{\left(\left(1 + \left(0.7715471019 \cdot \left(x \cdot x\right) + 0.2909738639 \cdot t_0\right)\right) + \left(0.0694555761 \cdot t_1 + 0.0140005442 \cdot t_2\right)\right) + \left(0.0008327945 \cdot {x}^{10} + 0.0003579942 \cdot \left(t_0 \cdot t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 4
Error	0.0
Cost	11208

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := t_0 \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ \mathbf{if}\;x \leq -50000000000000:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot t_0\right)\right) + \left(0.0005064034 \cdot t_1 + 0.0001789971 \cdot t_2\right)}{\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + \left(0.2909738639 \cdot t_0 + \left(x \cdot x\right) \cdot \left(0.0694555761 \cdot t_0\right)\right)\right) + 0.0140005442 \cdot t_1\right) + \left(0.0008327945 \cdot t_2 + 0.0003579942 \cdot \left(t_0 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 5
Error	0.0
Cost	11208

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ t_3 := \left(x \cdot x\right) \cdot t_2\\ \mathbf{if}\;x \leq -2 \cdot 10^{+23}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot t_1\right) + 0.0005064034 \cdot t_2\right) + 0.0001789971 \cdot t_3}{\left(\left(\left(\left(0.2909738639 \cdot t_0 + \left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot t_1\right) + 0.0140005442 \cdot t_2\right) + 0.0008327945 \cdot t_3\right) + 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 6
Error	0.4
Cost	7556

\[\begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;\frac{x \cdot \left(0.2514179000665374 \cdot {x}^{-2}\right) + x \cdot 0.5}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;x + x \cdot \left(\left(x \cdot x\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 7
Error	0.4
Cost	964

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

Alternative 8
Error	0.5
Cost	840

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

Alternative 9
Error	0.5
Cost	840

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

Alternative 10
Error	0.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -0.72:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 11
Error	31.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023027 
(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))