Average Error: 31.7 → 11.5
Time: 9.0m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.561216940282748 \cdot 10^{-153} \lor \neg \left(t \le 5.244282608629197 \cdot 10^{-259}\right):\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*}} \cdot \frac{\frac{\frac{2}{t}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\ell \cdot \ell}{t}\right) \cdot \left(\frac{2}{{k}^{4}} - \frac{\frac{5}{3}}{k \cdot k}\right) + \left(\frac{\ell \cdot t}{{k}^{4}} \cdot \left(-\ell \cdot \frac{10}{3}\right)\right))_*\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < -5.561216940282748e-153 or 5.244282608629197e-259 < t

    1. Initial program 27.9

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    2. Using strategy rm

    3. Applied unpow327.9

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    4. Applied times-frac20.4

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    5. Applied associate-*l*18.5

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    6. Taylor expanded around -inf 18.8

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(-1 \cdot \frac{\sin \left(-1 \cdot k\right) \cdot t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    7. Applied simplify12.8

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{t} \cdot \frac{\ell}{t}}{\frac{\sin k}{\frac{\ell}{t}}}}{\tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt12.9

      \[\leadsto \frac{\frac{\frac{\frac{2}{t} \cdot \frac{\ell}{t}}{\frac{\sin k}{\frac{\ell}{t}}}}{\tan k}}{\color{blue}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*}}}\]

    10. Applied *-un-lft-identity12.9

      \[\leadsto \frac{\frac{\frac{\frac{2}{t} \cdot \frac{\ell}{t}}{\frac{\sin k}{\frac{\ell}{t}}}}{\color{blue}{1 \cdot \tan k}}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*}}\]

    11. Applied add-cube-cbrt13.0

      \[\leadsto \frac{\frac{\frac{\frac{2}{t} \cdot \frac{\ell}{t}}{\color{blue}{\left(\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}} \cdot \sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}\right) \cdot \sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}}}{1 \cdot \tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*}}\]

    12. Applied times-frac12.2

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2}{t}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}} \cdot \sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}} \cdot \frac{\frac{\ell}{t}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}}}{1 \cdot \tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*}}\]

    13. Applied times-frac9.9

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{2}{t}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}} \cdot \sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}}{1} \cdot \frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}}{\tan k}}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*}}\]

    14. Applied times-frac8.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{t}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}} \cdot \sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}}{1}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*}} \cdot \frac{\frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*}}}\]

    15. Applied simplify8.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*} \cdot \sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}} \cdot \frac{\frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*}}\]

    if -5.561216940282748e-153 < t < 5.244282608629197e-259

    1. Initial program 62.7

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    2. Using strategy rm

    3. Applied unpow362.7

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    4. Applied times-frac61.4

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    5. Applied associate-*l*61.4

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    6. Taylor expanded around -inf 61.4

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(-1 \cdot \frac{\sin \left(-1 \cdot k\right) \cdot t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    7. Applied simplify51.8

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{t} \cdot \frac{\ell}{t}}{\frac{\sin k}{\frac{\ell}{t}}}}{\tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*}}\]

    8. Taylor expanded around 0 40.5

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} - \left(\frac{10}{3} \cdot \frac{t \cdot {\ell}^{2}}{{k}^{4}} + \frac{5}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}\]

    9. Applied simplify38.9

      \[\leadsto \color{blue}{(\left(\frac{\ell \cdot \ell}{t}\right) \cdot \left(\frac{2}{{k}^{4}} - \frac{\frac{5}{3}}{k \cdot k}\right) + \left(\frac{t \cdot \ell}{{k}^{4}} \cdot \left(\ell \cdot \left(-\frac{10}{3}\right)\right)\right))_*}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify11.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;t \le -5.561216940282748 \cdot 10^{-153} \lor \neg \left(t \le 5.244282608629197 \cdot 10^{-259}\right):\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*}} \cdot \frac{\frac{\frac{2}{t}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\ell \cdot \ell}{t}\right) \cdot \left(\frac{2}{{k}^{4}} - \frac{\frac{5}{3}}{k \cdot k}\right) + \left(\frac{\ell \cdot t}{{k}^{4}} \cdot \left(-\ell \cdot \frac{10}{3}\right)\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 9.0m)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))