Average Error: 28.2 → 0.0
Time: 2.2m
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 -1.0210202014077706 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{0.2514179000665373}{x}}{x \cdot x} + \left(\left(\left(\frac{0.5}{x} + \frac{118.45460177846456}{{x}^{7}}\right) + \left(\frac{284585.51954496035}{{x}^{17}} + \frac{78254247.27596515}{{x}^{23}}\right)\right) + \left(\left(\frac{0.15298196345928972}{{x}^{5}} - \frac{388.67048156309664}{{x}^{9}}\right) - \left(\frac{2217.007530626692}{{x}^{11}} + \frac{96278.63573268967}{{x}^{15}}\right)\right)\right)\\ \mathbf{if}\;x \le 19300530.647801947:\\ \;\;\;\;\left((\left(\left({x}^{3} \cdot {x}^{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0001789971 + \left((0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{3} \cdot {x}^{3}\right)\right) + \left((0.0072644182 \cdot \left({x}^{3} \cdot {x}^{3}\right) + \left((0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left((0.1049934947 \cdot \left(x \cdot x\right) + 1)_*\right))_*\right))_*\right))_*\right))_* \cdot \frac{1}{(\left({\left({x}^{3}\right)}^{\left(1 + 3\right)}\right) \cdot \left(0.0001789971 + 0.0001789971\right) + \left(\left((\left(\left(0.2909738639 \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right) + \left(x \cdot \left(0.7715471019 \cdot x\right)\right))_* + (\left({x}^{3} \cdot {x}^{3}\right) \cdot 0.0694555761 + 1)_*\right) + (\left({x}^{\left(2 + 3\right)} \cdot {x}^{\left(2 + 3\right)}\right) \cdot 0.0008327945 + \left(0.0140005442 \cdot {\left(x \cdot x\right)}^{\left(1 + 3\right)}\right))_*\right))_*}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665373}{x}}{x \cdot x} + \left(\left(\left(\frac{0.5}{x} + \frac{118.45460177846456}{{x}^{7}}\right) + \left(\frac{284585.51954496035}{{x}^{17}} + \frac{78254247.27596515}{{x}^{23}}\right)\right) + \left(\left(\frac{0.15298196345928972}{{x}^{5}} - \frac{388.67048156309664}{{x}^{9}}\right) - \left(\frac{2217.007530626692}{{x}^{11}} + \frac{96278.63573268967}{{x}^{15}}\right)\right)\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -1.0210202014077706e+17 or 19300530.647801947 < x

    1. Initial program 60.2

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

    2. Applied simplify60.2

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

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.2514179000665373 \cdot \frac{1}{{x}^{3}} + \left(0.15298196345928972 \cdot \frac{1}{{x}^{5}} + \left(0.5 \cdot \frac{1}{x} + \left(284585.51954496035 \cdot \frac{1}{{x}^{17}} + \left(78254247.27596515 \cdot \frac{1}{{x}^{23}} + 118.45460177846456 \cdot \frac{1}{{x}^{7}}\right)\right)\right)\right)\right) - \left(2217.007530626692 \cdot \frac{1}{{x}^{11}} + \left(96278.63573268967 \cdot \frac{1}{{x}^{15}} + 388.67048156309664 \cdot \frac{1}{{x}^{9}}\right)\right)}\]

    4. Applied simplify0.0

      \[\leadsto \color{blue}{\frac{\frac{0.2514179000665373}{x}}{x \cdot x} + \left(\left(\left(\frac{0.5}{x} + \frac{118.45460177846456}{{x}^{7}}\right) + \left(\frac{284585.51954496035}{{x}^{17}} + \frac{78254247.27596515}{{x}^{23}}\right)\right) + \left(\left(\frac{0.15298196345928972}{{x}^{5}} - \frac{388.67048156309664}{{x}^{9}}\right) - \left(\frac{2217.007530626692}{{x}^{11}} + \frac{96278.63573268967}{{x}^{15}}\right)\right)\right)}\]

    if -1.0210202014077706e+17 < x < 19300530.647801947

    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

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

    4. Applied div-inv0.0

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

    5. Applied simplify0.0

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

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +o rules:numerics
(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))