Average Error: 28.9 → 14.6
Time: 3.4m
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 -127015571187.22917:\\ \;\;\;\;(\left(\left(\left(\frac{78254247.27596515}{{x}^{24}} - \frac{388.67048156309664}{{x}^{10}}\right) + \left(\frac{118.45460177846456}{{x}^{8}} - \frac{96278.63573268967}{{x}^{16}}\right)\right) - \frac{2217.007530626692}{{x}^{12}}\right) \cdot x + \left(\left(\left(\frac{\frac{0.5}{x}}{x} + \frac{284585.51954496035}{{x}^{18}}\right) + \left(\frac{0.2514179000665373}{{x}^{4}} + \frac{0.15298196345928972}{{x}^{6}}\right)\right) \cdot x\right))_*\\ \mathbf{if}\;x \le 81516098.88895866:\\ \;\;\;\;\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\\ \mathbf{else}:\\ \;\;\;\;(\left(\left(\left(\frac{78254247.27596515}{{x}^{24}} - \frac{388.67048156309664}{{x}^{10}}\right) + \left(\frac{118.45460177846456}{{x}^{8}} - \frac{96278.63573268967}{{x}^{16}}\right)\right) - \frac{2217.007530626692}{{x}^{12}}\right) \cdot x + \left(\left(\left(\frac{\frac{0.5}{x}}{x} + \frac{284585.51954496035}{{x}^{18}}\right) + \left(\frac{0.2514179000665373}{{x}^{4}} + \frac{0.15298196345928972}{{x}^{6}}\right)\right) \cdot x\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -127015571187.22917 or 81516098.88895866 < x

    1. Initial program 59.5

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

    2. Applied simplify59.5

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

    3. Taylor expanded around inf 31.2

      \[\leadsto \color{blue}{\left(\left(118.45460177846456 \cdot \frac{1}{{x}^{8}} + \left(78254247.27596515 \cdot \frac{1}{{x}^{24}} + \left(284585.51954496035 \cdot \frac{1}{{x}^{18}} + \left(0.15298196345928972 \cdot \frac{1}{{x}^{6}} + \left(0.2514179000665373 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) - \left(2217.007530626692 \cdot \frac{1}{{x}^{12}} + \left(388.67048156309664 \cdot \frac{1}{{x}^{10}} + 96278.63573268967 \cdot \frac{1}{{x}^{16}}\right)\right)\right)} \cdot x\]

    4. Applied simplify31.2

      \[\leadsto \color{blue}{\left(\left(\left(\left(\frac{0.2514179000665373}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + \left(\frac{0.15298196345928972}{{x}^{6}} + \frac{284585.51954496035}{{x}^{18}}\right)\right) + \left(\left(\frac{78254247.27596515}{{x}^{24}} + \frac{118.45460177846456}{{x}^{8}}\right) - \frac{388.67048156309664}{{x}^{10}}\right)\right) - \left(\frac{2217.007530626692}{{x}^{12}} + \frac{96278.63573268967}{{x}^{16}}\right)\right) \cdot x}\]

    5. Taylor expanded around inf 31.2

      \[\leadsto \left(\left(\left(\left(\frac{0.2514179000665373}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + \left(\frac{0.15298196345928972}{{x}^{6}} + \frac{284585.51954496035}{{x}^{18}}\right)\right) + \color{blue}{\left(\left(118.45460177846456 \cdot \frac{1}{{x}^{8}} + 78254247.27596515 \cdot \frac{1}{{x}^{24}}\right) - 388.67048156309664 \cdot \frac{1}{{x}^{10}}\right)}\right) - \left(\frac{2217.007530626692}{{x}^{12}} + \frac{96278.63573268967}{{x}^{16}}\right)\right) \cdot x\]

    6. Applied simplify30.1

      \[\leadsto \color{blue}{(\left(\left(\left(\frac{78254247.27596515}{{x}^{24}} - \frac{388.67048156309664}{{x}^{10}}\right) + \left(\frac{118.45460177846456}{{x}^{8}} - \frac{96278.63573268967}{{x}^{16}}\right)\right) - \frac{2217.007530626692}{{x}^{12}}\right) \cdot x + \left(\left(\left(\frac{\frac{0.5}{x}}{x} + \frac{284585.51954496035}{{x}^{18}}\right) + \left(\frac{0.2514179000665373}{{x}^{4}} + \frac{0.15298196345928972}{{x}^{6}}\right)\right) \cdot x\right))_*}\]

    if -127015571187.22917 < x < 81516098.88895866

    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. Recombined 2 regimes into one program.

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' +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))