Average Error: 28.5 → 0.0
Time: 6.2m
Precision: 64
Internal Precision: 320
\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -74985928.13026966 \lor \neg \left(x \le 45521744.94755249\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}:\\ \;\;\;\;x \cdot \left(\sqrt{\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({x}^{3} \cdot \left(0.2909738639 \cdot x\right) + \left(x \cdot \left(0.7715471019 \cdot x\right) + 1\right)\right) + \left(\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(x \cdot x\right) \cdot \left(0.0001789971 \cdot 2\right) + 0.0008327945\right) + \left(\left(x \cdot x\right) \cdot 0.0140005442 + 0.0694555761\right) \cdot \left({x}^{3} \cdot {x}^{3}\right)\right)}} \cdot \sqrt{\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({x}^{3} \cdot \left(0.2909738639 \cdot x\right) + \left(x \cdot \left(0.7715471019 \cdot x\right) + 1\right)\right) + \left(\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(x \cdot x\right) \cdot \left(0.0001789971 \cdot 2\right) + 0.0008327945\right) + \left(\left(x \cdot x\right) \cdot 0.0140005442 + 0.0694555761\right) \cdot \left({x}^{3} \cdot {x}^{3}\right)\right)}}\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 < -74985928.13026966 or 45521744.94755249 < x

    1. Initial program 58.9

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

    2. Initial simplification58.9

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

    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. Simplified0.0

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

    if -74985928.13026966 < x < 45521744.94755249

    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. Initial simplification0.0

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

    4. Applied add-sqr-sqrt0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -74985928.13026966 \lor \neg \left(x \le 45521744.94755249\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}:\\ \;\;\;\;x \cdot \left(\sqrt{\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({x}^{3} \cdot \left(0.2909738639 \cdot x\right) + \left(x \cdot \left(0.7715471019 \cdot x\right) + 1\right)\right) + \left(\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(x \cdot x\right) \cdot \left(0.0001789971 \cdot 2\right) + 0.0008327945\right) + \left(\left(x \cdot x\right) \cdot 0.0140005442 + 0.0694555761\right) \cdot \left({x}^{3} \cdot {x}^{3}\right)\right)}} \cdot \sqrt{\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({x}^{3} \cdot \left(0.2909738639 \cdot x\right) + \left(x \cdot \left(0.7715471019 \cdot x\right) + 1\right)\right) + \left(\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(x \cdot x\right) \cdot \left(0.0001789971 \cdot 2\right) + 0.0008327945\right) + \left(\left(x \cdot x\right) \cdot 0.0140005442 + 0.0694555761\right) \cdot \left({x}^{3} \cdot {x}^{3}\right)\right)}}\right)\\ \end{array}\]

Runtime

Time bar (total: 6.2m)Debug logProfile

herbie shell --seed 2018220 
(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))