Average Error: 28.8 → 0.0
Time: 3.8m
Precision: 64
Internal Precision: 384
\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -727361566.8424039:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.2514179000665375}{{x}^{4}} + \frac{0.15298196345929327}{{x}^{6}}\right) \cdot x\\ \mathbf{if}\;x \le 42729.05365024598:\\ \;\;\;\;\frac{\left(-\left(0.0005064034 \cdot {\left(x \cdot x\right)}^{\left(3 + 1\right)} + \left(\left(x \cdot x\right) \cdot 0.0001789971\right) \cdot {\left(x \cdot x\right)}^{\left(3 + 1\right)}\right)\right) + \left(\left(\left(-\left(x \cdot 0.0424060604\right) \cdot {x}^{3}\right) + \left(-0.0072644182\right) \cdot \left({x}^{3} \cdot {x}^{3}\right)\right) + \left(-\left(\left(x \cdot x\right) \cdot 0.1049934947 + 1\right)\right)\right)}{-\left(\left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{\left(1 + 3\right)}\right) \cdot \left(\left(0.0001789971 + 0.0001789971\right) \cdot \left(x \cdot x\right) + 0.0008327945\right) + \left(\left(\left(x \cdot 0.0694555761\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(\left(0.2909738639 \cdot x\right) \cdot x\right)\right)\right) + \left(0.0140005442 \cdot {\left(x \cdot x\right)}^{\left(1 + 3\right)} + \left(\left(x \cdot x\right) \cdot 0.7715471019 + 1\right)\right)\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.2514179000665375}{{x}^{4}} + \frac{0.15298196345929327}{{x}^{6}}\right) \cdot x\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -727361566.8424039 or 42729.05365024598 < x

    1. Initial program 58.7

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

    2. Taylor expanded around inf 31.6

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

    3. Applied simplify0.0

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

    if -727361566.8424039 < x < 42729.05365024598

    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. Using strategy rm

    3. Applied frac-2neg0.0

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

    4. Applied simplify0.0

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

    5. Applied simplify0.0

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

    6. Applied simplify0.0

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

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(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))