Average Error: 42.4 → 9.0
Time: 2.3m
Precision: 64
Internal Precision: 1408
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.033787208186423 \cdot 10^{+42}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{-\ell}{t \cdot \sqrt{2}}}{\frac{\frac{-1}{\ell}}{\frac{-1}{x}}} - t \cdot \left(\sqrt{2} + \frac{\sqrt{\frac{1}{2}}}{\frac{x}{2}}\right)}\\ \mathbf{if}\;t \le -1.1747551287226268 \cdot 10^{-259}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{\frac{2 \cdot \ell}{\sqrt[3]{\frac{x}{\ell}} \cdot \sqrt[3]{\frac{x}{\ell}}}}{\sqrt[3]{\frac{x}{\ell}}}}}\\ \mathbf{if}\;t \le 2.770538867182025 \cdot 10^{-243}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\sqrt{\left(\left({\left(\frac{4}{x}\right)}^{3} + {2}^{3}\right) \cdot \left(t \cdot t\right)\right) \cdot \left(\sqrt[3]{\frac{x}{\ell}} \cdot \sqrt[3]{\frac{x}{\ell}}\right) + \left(\frac{4}{x} \cdot \frac{4}{x} + \left(2 \cdot 2 - \frac{4}{x} \cdot 2\right)\right) \cdot \left(2 \cdot \frac{\ell}{\sqrt[3]{\frac{x}{\ell}}}\right)}}{\sqrt{\left(2 \cdot 2 + \left(\frac{4}{x} - 2\right) \cdot \frac{4}{x}\right) \cdot \left(\sqrt[3]{\frac{x}{\ell}} \cdot \sqrt[3]{\frac{x}{\ell}}\right)}}}\\ \mathbf{if}\;t \le 3.399456339372293 \cdot 10^{-162}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{\ell}{t}}{\frac{x}{\ell} \cdot \sqrt{2}} + t \cdot \left(\frac{\sqrt{\frac{1}{2}}}{\frac{x}{2}} + \sqrt{2}\right)}\\ \mathbf{if}\;t \le 5.203050666686875 \cdot 10^{+81}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{\frac{2 \cdot \ell}{\sqrt[3]{\frac{x}{\ell}} \cdot \sqrt[3]{\frac{x}{\ell}}}}{\sqrt[3]{\frac{x}{\ell}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{\ell}{t}}{\frac{x}{\ell} \cdot \sqrt{2}} + t \cdot \left(\frac{\sqrt{\frac{1}{2}}}{\frac{x}{2}} + \sqrt{2}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 4 regimes

  2. if t < -7.033787208186423e+42

    1. Initial program 44.8

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

    2. Taylor expanded around inf 43.1

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

    3. Applied simplify39.8

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

    5. Applied add-cube-cbrt39.9

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

    6. Applied associate-/r*39.9

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

    7. Taylor expanded around -inf 48.6

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

    8. Applied simplify4.5

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

    if -7.033787208186423e+42 < t < -1.1747551287226268e-259 or 3.399456339372293e-162 < t < 5.203050666686875e+81

    1. Initial program 33.0

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

    2. Taylor expanded around inf 14.6

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

    3. Applied simplify10.0

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

    5. Applied add-cube-cbrt10.1

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

    6. Applied associate-/r*10.1

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

    if -1.1747551287226268e-259 < t < 2.770538867182025e-243

    1. Initial program 61.6

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

    2. Taylor expanded around inf 28.7

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

    3. Applied simplify28.4

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

    5. Applied add-cube-cbrt28.5

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

    6. Applied times-frac28.5

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

    8. Applied associate-*l/28.5

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

    9. Applied flip3-+28.5

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

    10. Applied associate-*l/28.5

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

    11. Applied frac-add28.7

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

    12. Applied sqrt-div24.2

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

    13. Applied simplify24.2

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

    if 2.770538867182025e-243 < t < 3.399456339372293e-162 or 5.203050666686875e+81 < t

    1. Initial program 50.2

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

    2. Taylor expanded around inf 44.1

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

    3. Applied simplify42.1

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

    5. Applied add-cube-cbrt42.1

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

    6. Applied associate-/r*42.1

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

    7. Taylor expanded around inf 48.1

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

    8. Applied simplify7.7

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

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' 
(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)))))