Average Error: 29.1 → 0.0
Time: 8.9m
Precision: 64
Internal Precision: 576
\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -42794565.356408514 \lor \neg \left(x \le 52046178.931661196\right):\\ \;\;\;\;\left(\frac{0.5}{x} - \frac{43733.204511252174}{{x}^{15}}\right) - \left(\frac{666.049723509856}{{x}^{9}} - \left(\frac{71.24974274308389}{{x}^{7}} + \frac{20.29199970278848}{{x}^{5}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0072644182 + 0.0424060604\right) + {\left(x \cdot x\right)}^{\left(3 + 1\right)} \cdot \left(\left(x \cdot x\right) \cdot 0.0001789971 + 0.0005064034\right)\right)}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(-0.2909738639\right) - \left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right)\right) - \left(\left({\left(x \cdot x\right)}^{3} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.0008327945 + \left(x \cdot x\right) \cdot \left(0.0001789971 \cdot 2\right)\right) + {\left(x \cdot x\right)}^{3} \cdot \left(0.0694555761 + \left(x \cdot x\right) \cdot 0.0140005442\right)\right)} \cdot \left(-x\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -42794565.356408514 or 52046178.931661196 < x

    1. Initial program 59.4

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

    2. Applied simplify59.3

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

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(71.24974274308389 \cdot \frac{1}{{x}^{7}} + \left(20.29199970278848 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right) - \left(666.049723509856 \cdot \frac{1}{{x}^{9}} + 43733.204511252174 \cdot \frac{1}{{x}^{15}}\right)}\]

    4. Applied simplify0.0

      \[\leadsto \color{blue}{\left(\frac{0.5}{x} - \frac{43733.204511252174}{{x}^{15}}\right) - \left(\frac{666.049723509856}{{x}^{9}} - \left(\frac{71.24974274308389}{{x}^{7}} + \frac{20.29199970278848}{{x}^{5}}\right)\right)}\]

    if -42794565.356408514 < x < 52046178.931661196

    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 simplify0.0

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

    4. Applied frac-2neg0.0

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

    5. Applied simplify0.0

      \[\leadsto \frac{-\left(\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.0424060604 + 0.0072644182 \cdot \left(x \cdot x\right)\right) + {\left(x \cdot x\right)}^{\left(3 + 1\right)} \cdot \left(\left(x \cdot x\right) \cdot 0.0001789971 + 0.0005064034\right)\right)\right)}{\color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(-0.2909738639\right) - \left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right)\right) - \left(\left(0.0694555761 + \left(x \cdot x\right) \cdot 0.0140005442\right) \cdot {\left(x \cdot x\right)}^{3} + \left({\left(x \cdot x\right)}^{3} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(0.0001789971 \cdot 2\right) \cdot \left(x \cdot x\right) + 0.0008327945\right)\right)}} \cdot x\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify0.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -42794565.356408514 \lor \neg \left(x \le 52046178.931661196\right):\\ \;\;\;\;\left(\frac{0.5}{x} - \frac{43733.204511252174}{{x}^{15}}\right) - \left(\frac{666.049723509856}{{x}^{9}} - \left(\frac{71.24974274308389}{{x}^{7}} + \frac{20.29199970278848}{{x}^{5}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0072644182 + 0.0424060604\right) + {\left(x \cdot x\right)}^{\left(3 + 1\right)} \cdot \left(\left(x \cdot x\right) \cdot 0.0001789971 + 0.0005064034\right)\right)}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(-0.2909738639\right) - \left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right)\right) - \left(\left({\left(x \cdot x\right)}^{3} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.0008327945 + \left(x \cdot x\right) \cdot \left(0.0001789971 \cdot 2\right)\right) + {\left(x \cdot x\right)}^{3} \cdot \left(0.0694555761 + \left(x \cdot x\right) \cdot 0.0140005442\right)\right)} \cdot \left(-x\right)\\ \end{array}}\]

Runtime

Time bar (total: 8.9m)Debug logProfile

herbie shell --seed 2018296 
(FPCore (x)
  :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))