\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
Test:
Jmat.Real.dawson
Bits:
128 bits
Bits error versus x
Time: 18.8 s
Input Error: 28.9
Output Error: 28.9
Log: ⚲
Profile: 🕒
\(\frac{x \cdot \left((0.0005064034 * \left({x}^{8}\right) + \left(0.0072644182 \cdot {\left({x}^2\right)}^3\right))_* + (0.0001789971 * \left({\left({x}^2\right)}^3 \cdot {x}^{4}\right) + \left((0.0424060604 * \left({x}^{4}\right) + \left((\left(0.1049934947 \cdot x\right) * x + 1)_*\right))_*\right))_*\right)}{(0.0003579942 * \left({\left({x}^2\right)}^3 \cdot {x}^{6}\right) + \left((\left({x}^{8}\right) * 0.0140005442 + \left(0.0694555761 \cdot {\left({x}^2\right)}^3\right))_* + (0.0008327945 * \left({\left({x}^2\right)}^3 \cdot {x}^{4}\right) + \left((0.2909738639 * \left({x}^{4}\right) + \left((\left(0.7715471019 \cdot x\right) * x + 1)_*\right))_*\right))_*\right))_*}\)
  1. Started with
    \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
    28.9
  2. Applied simplify to get
    \[\color{red}{\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} \leadsto \color{blue}{\frac{x \cdot \left((0.0005064034 * \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \left({\left(x \cdot x\right)}^3 \cdot 0.0072644182\right))_* + (0.0001789971 * \left({\left(x \cdot x\right)}^3 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \left((0.0424060604 * \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left((\left(0.1049934947 \cdot x\right) * x + 1)_*\right))_*\right))_*\right)}{(\left(2 \cdot 0.0001789971\right) * \left(\left({\left(x \cdot x\right)}^3 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right) + \left((\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) * 0.0140005442 + \left(0.0694555761 \cdot {\left(x \cdot x\right)}^3\right))_* + (0.0008327945 * \left({\left(x \cdot x\right)}^3 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \left((0.2909738639 * \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left((\left(0.7715471019 \cdot x\right) * x + 1)_*\right))_*\right))_*\right))_*}}\]
    28.9
  3. Applied taylor to get
    \[\frac{x \cdot \left((0.0005064034 * \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \left({\left(x \cdot x\right)}^3 \cdot 0.0072644182\right))_* + (0.0001789971 * \left({\left(x \cdot x\right)}^3 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \left((0.0424060604 * \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left((\left(0.1049934947 \cdot x\right) * x + 1)_*\right))_*\right))_*\right)}{(\left(2 \cdot 0.0001789971\right) * \left(\left({\left(x \cdot x\right)}^3 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right) + \left((\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) * 0.0140005442 + \left(0.0694555761 \cdot {\left(x \cdot x\right)}^3\right))_* + (0.0008327945 * \left({\left(x \cdot x\right)}^3 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \left((0.2909738639 * \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left((\left(0.7715471019 \cdot x\right) * x + 1)_*\right))_*\right))_*\right))_*} \leadsto \frac{x \cdot \left((0.0005064034 * \left({x}^{8}\right) + \left(0.0072644182 \cdot {\left({x}^2\right)}^3\right))_* + (0.0001789971 * \left({\left({x}^2\right)}^3 \cdot {x}^{4}\right) + \left((0.0424060604 * \left({x}^{4}\right) + \left((\left(0.1049934947 \cdot x\right) * x + 1)_*\right))_*\right))_*\right)}{(0.0003579942 * \left({\left({x}^2\right)}^3 \cdot {x}^{6}\right) + \left((\left({x}^{8}\right) * 0.0140005442 + \left(0.0694555761 \cdot {\left({x}^2\right)}^3\right))_* + (0.0008327945 * \left({\left({x}^2\right)}^3 \cdot {x}^{4}\right) + \left((0.2909738639 * \left({x}^{4}\right) + \left((\left(0.7715471019 \cdot x\right) * x + 1)_*\right))_*\right))_*\right))_*}\]
    28.9
  4. Taylor expanded around 0 to get
    \[\color{red}{\frac{x \cdot \left((0.0005064034 * \left({x}^{8}\right) + \left(0.0072644182 \cdot {\left({x}^2\right)}^3\right))_* + (0.0001789971 * \left({\left({x}^2\right)}^3 \cdot {x}^{4}\right) + \left((0.0424060604 * \left({x}^{4}\right) + \left((\left(0.1049934947 \cdot x\right) * x + 1)_*\right))_*\right))_*\right)}{(0.0003579942 * \left({\left({x}^2\right)}^3 \cdot {x}^{6}\right) + \left((\left({x}^{8}\right) * 0.0140005442 + \left(0.0694555761 \cdot {\left({x}^2\right)}^3\right))_* + (0.0008327945 * \left({\left({x}^2\right)}^3 \cdot {x}^{4}\right) + \left((0.2909738639 * \left({x}^{4}\right) + \left((\left(0.7715471019 \cdot x\right) * x + 1)_*\right))_*\right))_*\right))_*}} \leadsto \color{blue}{\frac{x \cdot \left((0.0005064034 * \left({x}^{8}\right) + \left(0.0072644182 \cdot {\left({x}^2\right)}^3\right))_* + (0.0001789971 * \left({\left({x}^2\right)}^3 \cdot {x}^{4}\right) + \left((0.0424060604 * \left({x}^{4}\right) + \left((\left(0.1049934947 \cdot x\right) * x + 1)_*\right))_*\right))_*\right)}{(0.0003579942 * \left({\left({x}^2\right)}^3 \cdot {x}^{6}\right) + \left((\left({x}^{8}\right) * 0.0140005442 + \left(0.0694555761 \cdot {\left({x}^2\right)}^3\right))_* + (0.0008327945 * \left({\left({x}^2\right)}^3 \cdot {x}^{4}\right) + \left((0.2909738639 * \left({x}^{4}\right) + \left((\left(0.7715471019 \cdot x\right) * x + 1)_*\right))_*\right))_*\right))_*}}\]
    28.9

Original test:


(lambda ((x default))
  #:name "Jmat.Real.dawson"
  (* (/ (+ (+ (+ (+ (+ 1 (* 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 (* 0.7715471019 (* 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 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))