Average Error: 29.0 → 0.0
Time: 3.3s
Precision: binary64
\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
↓
\[\begin{array}{l} t_0 := x \cdot {x}^{3}\\ t_1 := {t_0}^{2}\\ \mathbf{if}\;x \leq -7.627463149032007 \cdot 10^{+24}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 20958266356.90171:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.0001789971, \left(x \cdot x\right) \cdot t_1, \mathsf{fma}\left(0.0005064034, t_1, \mathsf{fma}\left(0.0072644182, {\left(x \cdot x\right)}^{3}, \mathsf{fma}\left(0.0424060604, t_0, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0003579942, {t_0}^{3}, \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x \cdot \left(x \cdot {x}^{8}\right), 0.0008327945, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (/
   (+
    (+
     (+
      (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x))))
      (* 0.0072644182 (* (* (* x x) (* x x)) (* x x))))
     (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x))))
    (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x))))
   (+
    (+
     (+
      (+
       (+
        (+ 1.0 (* 0.7715471019 (* x x)))
        (* 0.2909738639 (* (* x x) (* x x))))
       (* 0.0694555761 (* (* (* x x) (* x x)) (* x x))))
      (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x))))
     (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x))))
    (*
     (* 2.0 0.0001789971)
     (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x)))))
  x))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (pow x 3.0))) (t_1 (pow t_0 2.0)))
   (if (<= x -7.627463149032007e+24)
     (/ 0.5 x)
     (if (<= x 20958266356.90171)
       (/
        (*
         x
         (fma
          0.0001789971
          (* (* x x) t_1)
          (fma
           0.0005064034
           t_1
           (fma
            0.0072644182
            (pow (* x x) 3.0)
            (fma 0.0424060604 t_0 (fma 0.1049934947 (* x x) 1.0))))))
        (fma
         0.0003579942
         (pow t_0 3.0)
         (expm1
          (log1p
           (fma
            (* x (* x (pow x 8.0)))
            0.0008327945
            (fma
             (pow x 8.0)
             0.0140005442
             (fma
              (pow x 6.0)
              0.0694555761
              (fma
               (pow x 4.0)
               0.2909738639
               (fma (* x x) 0.7715471019 1.0)))))))))
       (/ 0.5 x)))))
double code(double x) {
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * ((x * x) * (x * x)))) + (0.0072644182 * (((x * x) * (x * x)) * (x * x)))) + (0.0005064034 * ((((x * x) * (x * x)) * (x * x)) * (x * x)))) + (0.0001789971 * (((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)))) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * ((x * x) * (x * x)))) + (0.0694555761 * (((x * x) * (x * x)) * (x * x)))) + (0.0140005442 * ((((x * x) * (x * x)) * (x * x)) * (x * x)))) + (0.0008327945 * (((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)))) + ((2.0 * 0.0001789971) * ((((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)) * (x * x))))) * x;
}
↓
double code(double x) {
	double t_0 = x * pow(x, 3.0);
	double t_1 = pow(t_0, 2.0);
	double tmp;
	if (x <= -7.627463149032007e+24) {
		tmp = 0.5 / x;
	} else if (x <= 20958266356.90171) {
		tmp = (x * fma(0.0001789971, ((x * x) * t_1), fma(0.0005064034, t_1, fma(0.0072644182, pow((x * x), 3.0), fma(0.0424060604, t_0, fma(0.1049934947, (x * x), 1.0)))))) / fma(0.0003579942, pow(t_0, 3.0), expm1(log1p(fma((x * (x * pow(x, 8.0))), 0.0008327945, fma(pow(x, 8.0), 0.0140005442, fma(pow(x, 6.0), 0.0694555761, fma(pow(x, 4.0), 0.2909738639, fma((x * x), 0.7715471019, 1.0))))))));
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

function code(x)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(0.0424060604 * Float64(Float64(x * x) * Float64(x * x)))) + Float64(0.0072644182 * Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0005064034 * Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0001789971 * Float64(Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) / Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.7715471019 * Float64(x * x))) + Float64(0.2909738639 * Float64(Float64(x * x) * Float64(x * x)))) + Float64(0.0694555761 * Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0140005442 * Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0008327945 * Float64(Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) + Float64(Float64(2.0 * 0.0001789971) * Float64(Float64(Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x))))) * x)
end
↓
function code(x)
	t_0 = Float64(x * (x ^ 3.0))
	t_1 = t_0 ^ 2.0
	tmp = 0.0
	if (x <= -7.627463149032007e+24)
		tmp = Float64(0.5 / x);
	elseif (x <= 20958266356.90171)
		tmp = Float64(Float64(x * fma(0.0001789971, Float64(Float64(x * x) * t_1), fma(0.0005064034, t_1, fma(0.0072644182, (Float64(x * x) ^ 3.0), fma(0.0424060604, t_0, fma(0.1049934947, Float64(x * x), 1.0)))))) / fma(0.0003579942, (t_0 ^ 3.0), expm1(log1p(fma(Float64(x * Float64(x * (x ^ 8.0))), 0.0008327945, fma((x ^ 8.0), 0.0140005442, fma((x ^ 6.0), 0.0694555761, fma((x ^ 4.0), 0.2909738639, fma(Float64(x * x), 0.7715471019, 1.0)))))))));
	else
		tmp = Float64(0.5 / x);
	end
	return tmp
end

code[x_] := N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(1.0 + N[(0.7715471019 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0694555761 * N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]
↓
code[x_] := Block[{t$95$0 = N[(x * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, If[LessEqual[x, -7.627463149032007e+24], N[(0.5 / x), $MachinePrecision], If[LessEqual[x, 20958266356.90171], N[(N[(x * N[(0.0001789971 * N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(0.0005064034 * t$95$1 + N[(0.0072644182 * N[Power[N[(x * x), $MachinePrecision], 3.0], $MachinePrecision] + N[(0.0424060604 * t$95$0 + N[(0.1049934947 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0003579942 * N[Power[t$95$0, 3.0], $MachinePrecision] + N[(Exp[N[Log[1 + N[(N[(x * N[(x * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.0008327945 + N[(N[Power[x, 8.0], $MachinePrecision] * 0.0140005442 + N[(N[Power[x, 6.0], $MachinePrecision] * 0.0694555761 + N[(N[Power[x, 4.0], $MachinePrecision] * 0.2909738639 + N[(N[(x * x), $MachinePrecision] * 0.7715471019 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]]]]

\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x
↓
\begin{array}{l}
t_0 := x \cdot {x}^{3}\\
t_1 := {t_0}^{2}\\
\mathbf{if}\;x \leq -7.627463149032007 \cdot 10^{+24}:\\
\;\;\;\;\frac{0.5}{x}\\

\mathbf{elif}\;x \leq 20958266356.90171:\\
\;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.0001789971, \left(x \cdot x\right) \cdot t_1, \mathsf{fma}\left(0.0005064034, t_1, \mathsf{fma}\left(0.0072644182, {\left(x \cdot x\right)}^{3}, \mathsf{fma}\left(0.0424060604, t_0, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0003579942, {t_0}^{3}, \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x \cdot \left(x \cdot {x}^{8}\right), 0.0008327945, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)\right)\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{0.5}{x}\\


\end{array}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -7.6274631490320069e24 or 20958266356.901711 < x

    1. Initial program 62.3

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

    2. Taylor expanded in x around inf 0

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

    if -7.6274631490320069e24 < x < 20958266356.901711

    1. Initial program 0.1

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

    2. Applied egg-rr0.1

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

    3. Applied egg-rr0.1

      \[\leadsto \frac{\mathsf{fma}\left(0.0001789971, \left(x \cdot x\right) \cdot {\left({x}^{3} \cdot x\right)}^{2}, \mathsf{fma}\left(0.0005064034, {\left({x}^{3} \cdot x\right)}^{2}, \mathsf{fma}\left(0.0072644182, {\left(x \cdot x\right)}^{3}, \mathsf{fma}\left(0.0424060604, {x}^{3} \cdot x, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right) \cdot x}{\mathsf{fma}\left(0.0003579942, {\left({x}^{3} \cdot x\right)}^{3}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x \cdot \left(x \cdot {x}^{8}\right), 0.0008327945, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)\right)\right)\right)}\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.627463149032007 \cdot 10^{+24}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 20958266356.90171:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.0001789971, \left(x \cdot x\right) \cdot {\left(x \cdot {x}^{3}\right)}^{2}, \mathsf{fma}\left(0.0005064034, {\left(x \cdot {x}^{3}\right)}^{2}, \mathsf{fma}\left(0.0072644182, {\left(x \cdot x\right)}^{3}, \mathsf{fma}\left(0.0424060604, x \cdot {x}^{3}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot {x}^{3}\right)}^{3}, \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x \cdot \left(x \cdot {x}^{8}\right), 0.0008327945, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022152 
(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))