Average Error: 29.2 → 0.1
Time: 29.4s
Precision: 64
Internal precision: 384
\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -4.248855072932943 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{0.10459085660078486}{{x}^{5}} + \frac{0.6105494012308682}{x}\right) + \frac{1.5116640102276828}{{x}^3}\\ \mathbf{if}\;x \le 3.0735832835760247 \cdot 10^{+26}:\\ \;\;\;\;\frac{\left(\left({x}^3 \cdot {x}^2\right) \cdot \left(0.0424060604 + 0.0072644182 \cdot {x}^2\right) + \left(x + \left(0.1049934947 \cdot x\right) \cdot {x}^2\right)\right) + \left({x}^2 \cdot 0.0001789971 + 0.0005064034\right) \cdot {\left({x}^3\right)}^3}{\left(\left(1 + \left(x \cdot 0.2909738639\right) \cdot {x}^3\right) + \left(\left(x \cdot \left(0.7715471019 \cdot x\right) + \left(0.0008327945 \cdot x\right) \cdot {\left({x}^3\right)}^3\right) + {\left(x \cdot x\right)}^3 \cdot \left(0.0694555761 + \left(x \cdot x\right) \cdot 0.0140005442\right)\right)\right) + {\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}^3 \cdot \left(0.0001789971 + 0.0001789971\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.10459085660078486}{{x}^{5}} + \frac{0.6105494012308682}{x}\right) + \frac{1.5116640102276828}{{x}^3}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes.

  2. if x < -4.248855072932943e+22 or 3.0735832835760247e+26 < x

    1. Initial program 62.9

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

    2. Applied simplify 62.9

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

    3. Applied taylor 62.9

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

    4. Taylor expanded around 0 62.9

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

    5. Applied simplify 62.9

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

    6. Applied taylor 0.3

      \[\leadsto 1.5116640102276828 \cdot \frac{1}{{x}^{3}} + \left(0.10459085660078486 \cdot \frac{1}{{x}^{5}} + 0.6105494012308682 \cdot \frac{1}{x}\right)\]

    7. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{1.5116640102276828 \cdot \frac{1}{{x}^{3}} + \left(0.10459085660078486 \cdot \frac{1}{{x}^{5}} + 0.6105494012308682 \cdot \frac{1}{x}\right)}\]

    8. Applied simplify 0

      \[\leadsto \color{blue}{\left(\frac{0.10459085660078486}{{x}^{5}} + \frac{0.6105494012308682}{x}\right) + \frac{1.5116640102276828}{{x}^3}}\]

    if -4.248855072932943e+22 < x < 3.0735832835760247e+26

    1. Initial program 0.1

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

    2. Applied simplify 0.1

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

    3. Applied simplify 0.1

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

    4. Applied simplify 0.2

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

Runtime

Time bar (total: 29.4s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(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))