Average Error: 41.8 → 9.7
Time: 2.4m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.185833878878676 \cdot 10^{+45}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\sqrt{2}} \cdot \left(\frac{1}{x \cdot x} - \frac{2}{x}\right) - t \cdot \sqrt{2}}\\ \mathbf{if}\;t \le -3.5152694293520474 \cdot 10^{-160}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}{\sqrt{2}}}\\ \mathbf{if}\;t \le -2.85640140794466 \cdot 10^{-244}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\sqrt{2}} \cdot \left(\frac{1}{x \cdot x} - \frac{2}{x}\right) - t \cdot \sqrt{2}}\\ \mathbf{if}\;t \le 1.6833506141016324 \cdot 10^{-215}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}{\sqrt{2}}}\\ \mathbf{if}\;t \le 1.4208460157490442 \cdot 10^{-168} \lor \neg \left(t \le 8.866934073187334 \cdot 10^{+33}\right):\\ \;\;\;\;\frac{t}{\left(t + \frac{t}{x}\right) - \frac{\frac{2}{x} \cdot \frac{t}{x}}{{\left(\sqrt{2}\right)}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{2}} \cdot \left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}\\ \end{array}\]

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 5 regimes

  2. if t < -3.185833878878676e+45 or -3.5152694293520474e-160 < t < -2.85640140794466e-244

    1. Initial program 46.5

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

    2. Taylor expanded around inf 40.1

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

    3. Applied simplify37.4

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

    4. Taylor expanded around -inf 10.1

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

    5. Applied simplify10.1

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

    if -3.185833878878676e+45 < t < -3.5152694293520474e-160 or -2.85640140794466e-244 < t < 1.6833506141016324e-215

    1. Initial program 40.0

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

    2. Taylor expanded around inf 17.5

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

    3. Applied simplify13.3

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

    5. Applied associate-/l*13.3

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

    if 1.6833506141016324e-215 < t < 1.4208460157490442e-168

    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 35.0

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

    3. Applied simplify31.4

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

    5. Applied associate-/l*31.4

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

    6. Taylor expanded around inf 34.7

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

    7. Applied simplify34.7

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

    if 1.4208460157490442e-168 < t < 8.866934073187334e+33

    1. Initial program 30.0

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

    2. Taylor expanded around inf 9.9

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

    3. Applied simplify5.3

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

    5. Applied add-cube-cbrt5.3

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

    6. Applied associate-*r*5.3

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

    if 8.866934073187334e+33 < t

    1. Initial program 42.9

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

    2. Taylor expanded around inf 40.6

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

    3. Applied simplify38.2

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

    5. Applied associate-/l*38.2

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

    6. Taylor expanded around inf 5.4

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

    7. Applied simplify5.4

      \[\leadsto \color{blue}{\frac{t}{\left(t + \frac{t}{x}\right) - \frac{\frac{t}{x} \cdot \frac{2}{x}}{{\left(\sqrt{2}\right)}^{4}}}}\]
  3. Recombined 5 regimes into one program.

  4. Applied simplify9.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;t \le -3.185833878878676 \cdot 10^{+45}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\sqrt{2}} \cdot \left(\frac{1}{x \cdot x} - \frac{2}{x}\right) - t \cdot \sqrt{2}}\\ \mathbf{if}\;t \le -3.5152694293520474 \cdot 10^{-160}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}{\sqrt{2}}}\\ \mathbf{if}\;t \le -2.85640140794466 \cdot 10^{-244}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\sqrt{2}} \cdot \left(\frac{1}{x \cdot x} - \frac{2}{x}\right) - t \cdot \sqrt{2}}\\ \mathbf{if}\;t \le 1.6833506141016324 \cdot 10^{-215}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}{\sqrt{2}}}\\ \mathbf{if}\;t \le 1.4208460157490442 \cdot 10^{-168} \lor \neg \left(t \le 8.866934073187334 \cdot 10^{+33}\right):\\ \;\;\;\;\frac{t}{\left(t + \frac{t}{x}\right) - \frac{\frac{2}{x} \cdot \frac{t}{x}}{{\left(\sqrt{2}\right)}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{2}} \cdot \left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}\\ \end{array}}\]

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018167 
(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)))))