Average Error: 28.2 → 0.0
Time: 3.3m
Precision: 64
Internal Precision: 384
\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3724113583.442876:\\ \;\;\;\;\left(\left(\frac{20.29199970278848}{{x}^{6}} + \frac{71.24974274308389}{{x}^{8}}\right) - \left(\frac{666.049723509856}{{x}^{10}} + \frac{43733.204511252174}{{x}^{16}}\right)\right) \cdot x + \frac{0.5}{x}\\ \mathbf{if}\;x \le 34740955.142804675:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.0424060604 + \left(x \cdot x\right) \cdot 0.0072644182\right) + \left(\left(x \cdot \left(0.1049934947 \cdot x\right) + 1\right) + \left(0.0001789971 \cdot \left(x \cdot x\right) + 0.0005064034\right) \cdot {\left(x \cdot x\right)}^{\left(3 + 1\right)}\right)}{\left({x}^{3} \cdot {x}^{3}\right) \cdot \left(\left(x \cdot 0.0140005442\right) \cdot x + 0.0694555761\right) + \left(\left(1 + \left(0.7715471019 \cdot \left(x \cdot x\right) + \left(0.2909738639 \cdot x\right) \cdot {x}^{3}\right)\right) + \left(\left(\left(x \cdot x\right) \cdot {x}^{3}\right) \cdot \left(\left(x \cdot x\right) \cdot {x}^{3}\right)\right) \cdot \left(\left(0.0001789971 + 0.0001789971\right) \cdot \left(x \cdot x\right) + 0.0008327945\right)\right)}\right)}^{3}} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{20.29199970278848}{{x}^{6}} + \frac{71.24974274308389}{{x}^{8}}\right) - \left(\frac{666.049723509856}{{x}^{10}} + \frac{43733.204511252174}{{x}^{16}}\right)\right) \cdot x + \frac{0.5}{x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -3724113583.442876 or 34740955.142804675 < 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

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

    3. Taylor expanded around inf 31.1

      \[\leadsto \color{blue}{\left(\left(71.24974274308389 \cdot \frac{1}{{x}^{8}} + \left(20.29199970278848 \cdot \frac{1}{{x}^{6}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)\right) - \left(666.049723509856 \cdot \frac{1}{{x}^{10}} + 43733.204511252174 \cdot \frac{1}{{x}^{16}}\right)\right)} \cdot x\]

    4. Applied simplify0

      \[\leadsto \color{blue}{\left(\left(\frac{20.29199970278848}{{x}^{6}} + \frac{71.24974274308389}{{x}^{8}}\right) - \left(\frac{666.049723509856}{{x}^{10}} + \frac{43733.204511252174}{{x}^{16}}\right)\right) \cdot x + \frac{0.5}{x}}\]

    if -3724113583.442876 < x < 34740955.142804675

    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

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

    4. Applied add-cbrt-cube0.0

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

    5. Applied simplify0.0

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

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(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))