Average Error: 42.3 → 7.9
Time: 2.4m
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 -2.1869446493830736 \cdot 10^{+83}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{-\ell}{t \cdot \sqrt{2}}}{\frac{\frac{-1}{\ell}}{\frac{-1}{x}}} - t \cdot \left(\sqrt{2} + \frac{\sqrt{\frac{1}{2}}}{\frac{x}{2}}\right)}\\ \mathbf{if}\;t \le -1.2159894502301358 \cdot 10^{-169}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\sqrt{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}} \cdot \sqrt{\sqrt{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}}\\ \mathbf{if}\;t \le -1.497614897502176 \cdot 10^{-232}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{-\ell}{t \cdot \sqrt{2}}}{\frac{\frac{-1}{\ell}}{\frac{-1}{x}}} - t \cdot \left(\sqrt{2} + \frac{\sqrt{\frac{1}{2}}}{\frac{x}{2}}\right)}\\ \mathbf{if}\;t \le 5.606202260835721 \cdot 10^{-299}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\sqrt{\left(\left({\left(\frac{4}{x}\right)}^{3} + {2}^{3}\right) \cdot \left(t \cdot t\right)\right) \cdot \left(\sqrt[3]{\frac{x}{\ell}} \cdot \sqrt[3]{\frac{x}{\ell}}\right) + \left(\frac{4}{x} \cdot \frac{4}{x} + \left(2 \cdot 2 - \frac{4}{x} \cdot 2\right)\right) \cdot \left(2 \cdot \frac{\ell}{\sqrt[3]{\frac{x}{\ell}}}\right)}}{\sqrt{\left(2 \cdot 2 + \left(\frac{4}{x} - 2\right) \cdot \frac{4}{x}\right) \cdot \left(\sqrt[3]{\frac{x}{\ell}} \cdot \sqrt[3]{\frac{x}{\ell}}\right)}}}\\ \mathbf{if}\;t \le 6.075985997629053 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\frac{\frac{\ell}{t}}{\sqrt{2}}}{\frac{x}{\ell}} + t \cdot \left(\frac{\sqrt{\frac{1}{2}}}{\frac{x}{2}} + \sqrt{2}\right)}\\ \mathbf{if}\;t \le 8.928042441270439 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 6 regimes

  2. if t < -2.1869446493830736e+83 or -1.2159894502301358e-169 < t < -1.497614897502176e-232

    1. Initial program 49.7

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

    2. Taylor expanded around inf 45.9

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

    3. Applied simplify43.3

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

    5. Applied add-cube-cbrt43.3

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

    6. Applied times-frac43.3

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

    7. Taylor expanded around -inf 49.7

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

    8. Applied simplify6.5

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

    if -2.1869446493830736e+83 < t < -1.2159894502301358e-169

    1. Initial program 27.9

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

    2. Taylor expanded around inf 11.3

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

    3. Applied simplify6.0

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

    5. Applied add-sqr-sqrt6.0

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

    6. Applied sqrt-prod6.2

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

    if -1.497614897502176e-232 < t < 5.606202260835721e-299

    1. Initial program 61.9

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

    2. Taylor expanded around inf 29.5

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

    3. Applied simplify29.0

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

    5. Applied add-cube-cbrt29.1

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

    6. Applied times-frac29.1

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

    8. Applied associate-*l/29.1

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

    9. Applied flip3-+29.1

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

    10. Applied associate-*l/29.1

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

    11. Applied frac-add29.6

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

    12. Applied sqrt-div23.8

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

    13. Applied simplify23.8

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

    if 5.606202260835721e-299 < t < 6.075985997629053e-159

    1. Initial program 60.9

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

    2. Taylor expanded around inf 33.1

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

    3. Applied simplify30.7

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

    5. Applied add-cube-cbrt30.8

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

    6. Applied times-frac30.8

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

    7. Taylor expanded around inf 50.3

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

    8. Applied simplify25.3

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

    if 6.075985997629053e-159 < t < 8.928042441270439e+112

    1. Initial program 26.0

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

    2. Taylor expanded around inf 10.9

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

    3. Applied simplify5.0

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

    5. Applied add-sqr-sqrt5.2

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

    6. Applied associate-*r*5.1

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

    if 8.928042441270439e+112 < t

    1. Initial program 52.4

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

    2. Taylor expanded around inf 2.6

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

    3. Applied simplify2.6

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right)}}\]
  3. Recombined 6 regimes into one program.

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018207 
(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)))))