Average Error: 42.3 → 8.2
Time: 2.3m
Precision: 64
Internal Precision: 1408
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.2830096441464198 \cdot 10^{+111}:\\ \;\;\;\;\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 -3.6850313831549756 \cdot 10^{-172}:\\ \;\;\;\;\frac{\left(\left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}{\sqrt{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}\\ \mathbf{if}\;t \le -2.279278951294948 \cdot 10^{-277}:\\ \;\;\;\;\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 2.2056957479059362 \cdot 10^{-212}:\\ \;\;\;\;\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 5.7963425764825095 \cdot 10^{+57}:\\ \;\;\;\;\frac{\left(\left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\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

Derivation

  1. Split input into 4 regimes

  2. if t < -2.2830096441464198e+111 or -3.6850313831549756e-172 < t < -2.279278951294948e-277

    1. Initial program 54.9

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

    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. Applied simplify45.4

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

    5. Applied add-cube-cbrt45.4

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

    6. Applied times-frac45.4

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

    7. Taylor expanded around -inf 51.1

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

    8. Applied simplify8.6

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

    if -2.2830096441464198e+111 < t < -3.6850313831549756e-172 or 2.2056957479059362e-212 < t < 5.7963425764825095e+57

    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 12.8

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

    3. Applied simplify7.4

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

    5. Applied add-cube-cbrt7.4

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

    6. Applied associate-*r*7.4

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

    8. Applied add-cube-cbrt7.4

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

    9. Applied associate-*r*7.5

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

    if -2.279278951294948e-277 < t < 2.2056957479059362e-212

    1. Initial program 61.2

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

    2. Taylor expanded around inf 29.9

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

    3. Applied simplify28.9

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

    5. Applied add-cube-cbrt29.0

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

    6. Applied times-frac29.0

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

    8. Applied associate-*l/29.0

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

    9. Applied flip3-+29.0

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

    10. Applied associate-*l/29.0

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

    11. Applied frac-add29.1

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

    12. Applied sqrt-div24.1

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

    13. Applied simplify24.1

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

    if 5.7963425764825095e+57 < t

    1. Initial program 45.1

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

    2. Taylor expanded around inf 3.9

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

    3. Applied simplify3.9

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right)}}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' 
(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)))))