Average Error: 42.6 → 8.6
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.6366721863158036 \cdot 10^{+110}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\sqrt{\frac{1}{8}}}{x} - \sqrt{\frac{1}{2}}\right) \cdot \frac{2 \cdot t}{x} - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le -2.501180277482156 \cdot 10^{-138}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(2 \cdot \ell\right) \cdot \frac{\ell}{x} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \le -1.0862198333401183 \cdot 10^{-221}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\sqrt{\frac{1}{8}}}{x} - \sqrt{\frac{1}{2}}\right) \cdot \frac{2 \cdot t}{x} - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 1.3253827230978788 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\sqrt{\left(\left(2 \cdot \ell\right) \cdot \ell\right) \cdot \left(\frac{4}{x} \cdot \frac{4}{x} + \left(\frac{-8}{x} + 4\right)\right) + \left(t \cdot \left(x \cdot t\right)\right) \cdot \left(8 + \frac{\frac{64}{x}}{x \cdot x}\right)}}{\sqrt{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right) \cdot \left(\left(\frac{4}{x} \cdot \frac{4}{x} - \frac{4}{x} \cdot 2\right) + 4\right)}}}\\ \mathbf{elif}\;t \le 3.6074033787200903 \cdot 10^{+47}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(2 \cdot \ell\right) \cdot \frac{\ell}{x} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t + \frac{\frac{\frac{2}{x}}{x}}{\sqrt{2}} \cdot \left(t - \frac{t}{2}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if t < -5.6366721863158036e+110 or -2.501180277482156e-138 < t < -1.0862198333401183e-221

    1. Initial program 52.9

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 47.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified44.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt44.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)} \cdot \sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)}\right) \cdot \sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]

    6. Taylor expanded around -inf 9.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \frac{t \cdot \sqrt{\frac{1}{8}}}{{x}^{2}} - \left(t \cdot \sqrt{2} + 2 \cdot \frac{t \cdot \sqrt{\frac{1}{2}}}{x}\right)}}\]

    7. Simplified9.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{2 \cdot t}{x} \cdot \left(\frac{\sqrt{\frac{1}{8}}}{x} - \sqrt{\frac{1}{2}}\right) - t \cdot \sqrt{2}}}\]

    if -5.6366721863158036e+110 < t < -2.501180277482156e-138 or 1.3253827230978788e-206 < t < 3.6074033787200903e+47

    1. Initial program 28.9

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 13.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified7.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt7.5

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot t}{\sqrt{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]

    6. Applied associate-*l*7.5

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}}{\sqrt{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]

    if -1.0862198333401183e-221 < t < 1.3253827230978788e-206

    1. Initial program 61.8

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 30.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified29.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt29.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)} \cdot \sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)}\right) \cdot \sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]
    6. Using strategy rm

    7. Applied flip3-+29.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\left(\sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)} \cdot \sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)}\right) \cdot \sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)} + \color{blue}{\frac{{2}^{3} + {\left(\frac{4}{x}\right)}^{3}}{2 \cdot 2 + \left(\frac{4}{x} \cdot \frac{4}{x} - 2 \cdot \frac{4}{x}\right)}} \cdot \left(t \cdot t\right)}}\]

    8. Applied associate-*l/29.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\left(\sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)} \cdot \sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)}\right) \cdot \sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)} + \color{blue}{\frac{\left({2}^{3} + {\left(\frac{4}{x}\right)}^{3}\right) \cdot \left(t \cdot t\right)}{2 \cdot 2 + \left(\frac{4}{x} \cdot \frac{4}{x} - 2 \cdot \frac{4}{x}\right)}}}}\]

    9. Applied associate-*l/30.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\left(\sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)} \cdot \sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)}\right) \cdot \sqrt[3]{\color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{x}}} + \frac{\left({2}^{3} + {\left(\frac{4}{x}\right)}^{3}\right) \cdot \left(t \cdot t\right)}{2 \cdot 2 + \left(\frac{4}{x} \cdot \frac{4}{x} - 2 \cdot \frac{4}{x}\right)}}}\]

    10. Applied cbrt-div30.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\left(\sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)} \cdot \sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)}\right) \cdot \color{blue}{\frac{\sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}}{\sqrt[3]{x}}} + \frac{\left({2}^{3} + {\left(\frac{4}{x}\right)}^{3}\right) \cdot \left(t \cdot t\right)}{2 \cdot 2 + \left(\frac{4}{x} \cdot \frac{4}{x} - 2 \cdot \frac{4}{x}\right)}}}\]

    11. Applied associate-*l/30.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\left(\sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)} \cdot \sqrt[3]{\color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{x}}}\right) \cdot \frac{\sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}}{\sqrt[3]{x}} + \frac{\left({2}^{3} + {\left(\frac{4}{x}\right)}^{3}\right) \cdot \left(t \cdot t\right)}{2 \cdot 2 + \left(\frac{4}{x} \cdot \frac{4}{x} - 2 \cdot \frac{4}{x}\right)}}}\]

    12. Applied cbrt-div30.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\left(\sqrt[3]{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right)} \cdot \color{blue}{\frac{\sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}}{\sqrt[3]{x}}}\right) \cdot \frac{\sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}}{\sqrt[3]{x}} + \frac{\left({2}^{3} + {\left(\frac{4}{x}\right)}^{3}\right) \cdot \left(t \cdot t\right)}{2 \cdot 2 + \left(\frac{4}{x} \cdot \frac{4}{x} - 2 \cdot \frac{4}{x}\right)}}}\]

    13. Applied associate-*l/30.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\left(\sqrt[3]{\color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{x}}} \cdot \frac{\sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}}{\sqrt[3]{x}} + \frac{\left({2}^{3} + {\left(\frac{4}{x}\right)}^{3}\right) \cdot \left(t \cdot t\right)}{2 \cdot 2 + \left(\frac{4}{x} \cdot \frac{4}{x} - 2 \cdot \frac{4}{x}\right)}}}\]

    14. Applied cbrt-div30.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\left(\color{blue}{\frac{\sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}}{\sqrt[3]{x}} + \frac{\left({2}^{3} + {\left(\frac{4}{x}\right)}^{3}\right) \cdot \left(t \cdot t\right)}{2 \cdot 2 + \left(\frac{4}{x} \cdot \frac{4}{x} - 2 \cdot \frac{4}{x}\right)}}}\]

    15. Applied frac-times30.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)} \cdot \sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}}{\sqrt[3]{x}} + \frac{\left({2}^{3} + {\left(\frac{4}{x}\right)}^{3}\right) \cdot \left(t \cdot t\right)}{2 \cdot 2 + \left(\frac{4}{x} \cdot \frac{4}{x} - 2 \cdot \frac{4}{x}\right)}}}\]

    16. Applied frac-times30.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\left(\sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)} \cdot \sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}\right) \cdot \sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}} + \frac{\left({2}^{3} + {\left(\frac{4}{x}\right)}^{3}\right) \cdot \left(t \cdot t\right)}{2 \cdot 2 + \left(\frac{4}{x} \cdot \frac{4}{x} - 2 \cdot \frac{4}{x}\right)}}}\]

    17. Applied frac-add30.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\left(\left(\sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)} \cdot \sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}\right) \cdot \sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}\right) \cdot \left(2 \cdot 2 + \left(\frac{4}{x} \cdot \frac{4}{x} - 2 \cdot \frac{4}{x}\right)\right) + \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right) \cdot \left(\left({2}^{3} + {\left(\frac{4}{x}\right)}^{3}\right) \cdot \left(t \cdot t\right)\right)}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right) \cdot \left(2 \cdot 2 + \left(\frac{4}{x} \cdot \frac{4}{x} - 2 \cdot \frac{4}{x}\right)\right)}}}}\]

    18. Applied sqrt-div25.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{\sqrt{\left(\left(\sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)} \cdot \sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}\right) \cdot \sqrt[3]{\ell \cdot \left(\ell \cdot 2\right)}\right) \cdot \left(2 \cdot 2 + \left(\frac{4}{x} \cdot \frac{4}{x} - 2 \cdot \frac{4}{x}\right)\right) + \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right) \cdot \left(\left({2}^{3} + {\left(\frac{4}{x}\right)}^{3}\right) \cdot \left(t \cdot t\right)\right)}}{\sqrt{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right) \cdot \left(2 \cdot 2 + \left(\frac{4}{x} \cdot \frac{4}{x} - 2 \cdot \frac{4}{x}\right)\right)}}}}\]

    19. Simplified20.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\frac{\color{blue}{\sqrt{\left(\left(4 + \frac{-8}{x}\right) + \frac{4}{x} \cdot \frac{4}{x}\right) \cdot \left(\ell \cdot \left(\ell \cdot 2\right)\right) + \left(8 + \frac{\frac{64}{x}}{x \cdot x}\right) \cdot \left(\left(x \cdot t\right) \cdot t\right)}}}{\sqrt{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right) \cdot \left(2 \cdot 2 + \left(\frac{4}{x} \cdot \frac{4}{x} - 2 \cdot \frac{4}{x}\right)\right)}}}\]

    if 3.6074033787200903e+47 < t

    1. Initial program 44.7

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 4.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}} + \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}}\]

    3. Simplified4.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{\frac{\frac{2}{x}}{x}}{\sqrt{2}} \cdot \left(t - \frac{t}{2}\right) + t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.6366721863158036 \cdot 10^{+110}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\sqrt{\frac{1}{8}}}{x} - \sqrt{\frac{1}{2}}\right) \cdot \frac{2 \cdot t}{x} - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le -2.501180277482156 \cdot 10^{-138}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(2 \cdot \ell\right) \cdot \frac{\ell}{x} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \le -1.0862198333401183 \cdot 10^{-221}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\sqrt{\frac{1}{8}}}{x} - \sqrt{\frac{1}{2}}\right) \cdot \frac{2 \cdot t}{x} - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 1.3253827230978788 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\sqrt{\left(\left(2 \cdot \ell\right) \cdot \ell\right) \cdot \left(\frac{4}{x} \cdot \frac{4}{x} + \left(\frac{-8}{x} + 4\right)\right) + \left(t \cdot \left(x \cdot t\right)\right) \cdot \left(8 + \frac{\frac{64}{x}}{x \cdot x}\right)}}{\sqrt{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right) \cdot \left(\left(\frac{4}{x} \cdot \frac{4}{x} - \frac{4}{x} \cdot 2\right) + 4\right)}}}\\ \mathbf{elif}\;t \le 3.6074033787200903 \cdot 10^{+47}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(2 \cdot \ell\right) \cdot \frac{\ell}{x} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t + \frac{\frac{\frac{2}{x}}{x}}{\sqrt{2}} \cdot \left(t - \frac{t}{2}\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018252 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))