Average Error: 28.1 → 0.0
Time: 2.3m
Precision: 64
Internal Precision: 576
\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -232196867.27439314 \lor \neg \left(x \le 704.7172246903538\right):\\ \;\;\;\;\frac{\frac{0.2514179000665375}{x}}{x \cdot x} + \left(\frac{0.5}{x} + \frac{0.15298196345929327}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{\sqrt{\left({x}^{4} \cdot \left(2 \cdot 0.0001789971\right) + \left(0.0140005442 + \left(0.0008327945 \cdot x\right) \cdot x\right)\right) \cdot \left({x}^{4} \cdot {x}^{4}\right) + \left(\left({x}^{4} \cdot 0.2909738639 + \left(x \cdot 0.0694555761\right) \cdot {x}^{5}\right) + \left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right)\right)}} \cdot \frac{\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0001789971 + 0.0005064034\right) + \left(\left(\left(x \cdot x\right) \cdot 0.1049934947 + 1\right) + \left(\left(x \cdot 0.0072644182\right) \cdot {x}^{5} + 0.0424060604 \cdot {x}^{4}\right)\right)}{\sqrt{\left(\left(1 + \left(x \cdot 0.7715471019\right) \cdot x\right) + \left({x}^{4} \cdot 0.2909738639 + \left(x \cdot 0.0694555761\right) \cdot {x}^{5}\right)\right) + \left({x}^{4} \cdot {x}^{4}\right) \cdot \left(\left(0.0140005442 + 0.0008327945 \cdot \left(x \cdot x\right)\right) + {x}^{4} \cdot \left(2 \cdot 0.0001789971\right)\right)}}\right)\\ \end{array}\]

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 < -232196867.27439314 or 704.7172246903538 < x

    1. Initial program 59.0

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

    2. Initial simplification58.9

      \[\leadsto \frac{\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + \left({x}^{4} \cdot \left(0.0424060604 + x \cdot \left(x \cdot 0.0072644182\right)\right) + \left({x}^{4} \cdot {x}^{4}\right) \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)}{\left(\left({x}^{4} \cdot \left(2 \cdot 0.0001789971\right)\right) \cdot \left({x}^{4} \cdot {x}^{4}\right) + \left(\left(x \cdot 0.7715471019\right) \cdot x + 1\right)\right) + \left(\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(0.0008327945 \cdot \left(x \cdot x\right) + 0.0140005442\right) + {x}^{4} \cdot \left(0.2909738639 + \left(x \cdot 0.0694555761\right) \cdot x\right)\right)} \cdot x\]

    3. Taylor expanded around inf 30.0

      \[\leadsto \color{blue}{\left(0.2514179000665375 \cdot \frac{1}{{x}^{4}} + \left(0.15298196345929327 \cdot \frac{1}{{x}^{6}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot x\]

    4. Simplified28.9

      \[\leadsto \color{blue}{\left(\frac{0.15298196345929327}{{x}^{6}} + \left(\frac{0.2514179000665375}{{x}^{4}} + \frac{\frac{0.5}{x}}{x}\right)\right)} \cdot x\]

    5. Taylor expanded around -inf 0.0

      \[\leadsto \color{blue}{0.15298196345929327 \cdot \frac{1}{{x}^{5}} + \left(0.2514179000665375 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)}\]

    6. Simplified0.0

      \[\leadsto \color{blue}{\frac{\frac{0.2514179000665375}{x}}{x \cdot x} + \left(\frac{0.15298196345929327}{{x}^{5}} + \frac{0.5}{x}\right)}\]

    if -232196867.27439314 < x < 704.7172246903538

    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

      \[\leadsto \frac{\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + \left({x}^{4} \cdot \left(0.0424060604 + x \cdot \left(x \cdot 0.0072644182\right)\right) + \left({x}^{4} \cdot {x}^{4}\right) \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)}{\left(\left({x}^{4} \cdot \left(2 \cdot 0.0001789971\right)\right) \cdot \left({x}^{4} \cdot {x}^{4}\right) + \left(\left(x \cdot 0.7715471019\right) \cdot x + 1\right)\right) + \left(\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(0.0008327945 \cdot \left(x \cdot x\right) + 0.0140005442\right) + {x}^{4} \cdot \left(0.2909738639 + \left(x \cdot 0.0694555761\right) \cdot x\right)\right)} \cdot x\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.0

      \[\leadsto \frac{\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + \left({x}^{4} \cdot \left(0.0424060604 + x \cdot \left(x \cdot 0.0072644182\right)\right) + \left({x}^{4} \cdot {x}^{4}\right) \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)}{\color{blue}{\sqrt{\left(\left({x}^{4} \cdot \left(2 \cdot 0.0001789971\right)\right) \cdot \left({x}^{4} \cdot {x}^{4}\right) + \left(\left(x \cdot 0.7715471019\right) \cdot x + 1\right)\right) + \left(\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(0.0008327945 \cdot \left(x \cdot x\right) + 0.0140005442\right) + {x}^{4} \cdot \left(0.2909738639 + \left(x \cdot 0.0694555761\right) \cdot x\right)\right)} \cdot \sqrt{\left(\left({x}^{4} \cdot \left(2 \cdot 0.0001789971\right)\right) \cdot \left({x}^{4} \cdot {x}^{4}\right) + \left(\left(x \cdot 0.7715471019\right) \cdot x + 1\right)\right) + \left(\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(0.0008327945 \cdot \left(x \cdot x\right) + 0.0140005442\right) + {x}^{4} \cdot \left(0.2909738639 + \left(x \cdot 0.0694555761\right) \cdot x\right)\right)}}} \cdot x\]

    5. Applied *-un-lft-identity0.0

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + \left({x}^{4} \cdot \left(0.0424060604 + x \cdot \left(x \cdot 0.0072644182\right)\right) + \left({x}^{4} \cdot {x}^{4}\right) \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)}}{\sqrt{\left(\left({x}^{4} \cdot \left(2 \cdot 0.0001789971\right)\right) \cdot \left({x}^{4} \cdot {x}^{4}\right) + \left(\left(x \cdot 0.7715471019\right) \cdot x + 1\right)\right) + \left(\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(0.0008327945 \cdot \left(x \cdot x\right) + 0.0140005442\right) + {x}^{4} \cdot \left(0.2909738639 + \left(x \cdot 0.0694555761\right) \cdot x\right)\right)} \cdot \sqrt{\left(\left({x}^{4} \cdot \left(2 \cdot 0.0001789971\right)\right) \cdot \left({x}^{4} \cdot {x}^{4}\right) + \left(\left(x \cdot 0.7715471019\right) \cdot x + 1\right)\right) + \left(\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(0.0008327945 \cdot \left(x \cdot x\right) + 0.0140005442\right) + {x}^{4} \cdot \left(0.2909738639 + \left(x \cdot 0.0694555761\right) \cdot x\right)\right)}} \cdot x\]

    6. Applied times-frac0.0

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\left(\left({x}^{4} \cdot \left(2 \cdot 0.0001789971\right)\right) \cdot \left({x}^{4} \cdot {x}^{4}\right) + \left(\left(x \cdot 0.7715471019\right) \cdot x + 1\right)\right) + \left(\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(0.0008327945 \cdot \left(x \cdot x\right) + 0.0140005442\right) + {x}^{4} \cdot \left(0.2909738639 + \left(x \cdot 0.0694555761\right) \cdot x\right)\right)}} \cdot \frac{\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + \left({x}^{4} \cdot \left(0.0424060604 + x \cdot \left(x \cdot 0.0072644182\right)\right) + \left({x}^{4} \cdot {x}^{4}\right) \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)}{\sqrt{\left(\left({x}^{4} \cdot \left(2 \cdot 0.0001789971\right)\right) \cdot \left({x}^{4} \cdot {x}^{4}\right) + \left(\left(x \cdot 0.7715471019\right) \cdot x + 1\right)\right) + \left(\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(0.0008327945 \cdot \left(x \cdot x\right) + 0.0140005442\right) + {x}^{4} \cdot \left(0.2909738639 + \left(x \cdot 0.0694555761\right) \cdot x\right)\right)}}\right)} \cdot x\]

    7. Simplified0.0

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\left(\left(0.2909738639 \cdot {x}^{4} + \left(x \cdot 0.0694555761\right) \cdot {x}^{5}\right) + \left(\left(x \cdot x\right) \cdot 0.7715471019 + 1\right)\right) + \left({x}^{4} \cdot {x}^{4}\right) \cdot \left({x}^{4} \cdot \left(2 \cdot 0.0001789971\right) + \left(\left(0.0008327945 \cdot x\right) \cdot x + 0.0140005442\right)\right)}}} \cdot \frac{\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + \left({x}^{4} \cdot \left(0.0424060604 + x \cdot \left(x \cdot 0.0072644182\right)\right) + \left({x}^{4} \cdot {x}^{4}\right) \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)}{\sqrt{\left(\left({x}^{4} \cdot \left(2 \cdot 0.0001789971\right)\right) \cdot \left({x}^{4} \cdot {x}^{4}\right) + \left(\left(x \cdot 0.7715471019\right) \cdot x + 1\right)\right) + \left(\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(0.0008327945 \cdot \left(x \cdot x\right) + 0.0140005442\right) + {x}^{4} \cdot \left(0.2909738639 + \left(x \cdot 0.0694555761\right) \cdot x\right)\right)}}\right) \cdot x\]

    8. Simplified0.0

      \[\leadsto \left(\frac{1}{\sqrt{\left(\left(0.2909738639 \cdot {x}^{4} + \left(x \cdot 0.0694555761\right) \cdot {x}^{5}\right) + \left(\left(x \cdot x\right) \cdot 0.7715471019 + 1\right)\right) + \left({x}^{4} \cdot {x}^{4}\right) \cdot \left({x}^{4} \cdot \left(2 \cdot 0.0001789971\right) + \left(\left(0.0008327945 \cdot x\right) \cdot x + 0.0140005442\right)\right)}} \cdot \color{blue}{\frac{\left(\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + \left({x}^{4} \cdot 0.0424060604 + \left(x \cdot 0.0072644182\right) \cdot {x}^{5}\right)\right) + \left({x}^{4} \cdot {x}^{4}\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0001789971 + 0.0005064034\right)}{\sqrt{\left(\left(x \cdot \left(x \cdot 0.7715471019\right) + 1\right) + \left(\left(x \cdot 0.0694555761\right) \cdot {x}^{5} + 0.2909738639 \cdot {x}^{4}\right)\right) + \left({x}^{4} \cdot {x}^{4}\right) \cdot \left(\left(0.0001789971 \cdot 2\right) \cdot {x}^{4} + \left(0.0008327945 \cdot \left(x \cdot x\right) + 0.0140005442\right)\right)}}}\right) \cdot x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -232196867.27439314 \lor \neg \left(x \le 704.7172246903538\right):\\ \;\;\;\;\frac{\frac{0.2514179000665375}{x}}{x \cdot x} + \left(\frac{0.5}{x} + \frac{0.15298196345929327}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{\sqrt{\left({x}^{4} \cdot \left(2 \cdot 0.0001789971\right) + \left(0.0140005442 + \left(0.0008327945 \cdot x\right) \cdot x\right)\right) \cdot \left({x}^{4} \cdot {x}^{4}\right) + \left(\left({x}^{4} \cdot 0.2909738639 + \left(x \cdot 0.0694555761\right) \cdot {x}^{5}\right) + \left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right)\right)}} \cdot \frac{\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0001789971 + 0.0005064034\right) + \left(\left(\left(x \cdot x\right) \cdot 0.1049934947 + 1\right) + \left(\left(x \cdot 0.0072644182\right) \cdot {x}^{5} + 0.0424060604 \cdot {x}^{4}\right)\right)}{\sqrt{\left(\left(1 + \left(x \cdot 0.7715471019\right) \cdot x\right) + \left({x}^{4} \cdot 0.2909738639 + \left(x \cdot 0.0694555761\right) \cdot {x}^{5}\right)\right) + \left({x}^{4} \cdot {x}^{4}\right) \cdot \left(\left(0.0140005442 + 0.0008327945 \cdot \left(x \cdot x\right)\right) + {x}^{4} \cdot \left(2 \cdot 0.0001789971\right)\right)}}\right)\\ \end{array}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2018252 
(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))