Average Error: 47.5 → 2.4
Time: 4.7m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -1.6816267635550583 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{2}{\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}}}{\frac{t}{\cos k}} \cdot \frac{\frac{\frac{\frac{\ell}{k}}{\sin k}}{\sqrt[3]{\frac{k}{\ell}}}}{\sin k}\\ \mathbf{if}\;k \le 2.010931763593779 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{k}}{\frac{t}{\cos k}} \cdot \left(\frac{\frac{\ell}{\sin k}}{\sin k} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}}}{\frac{t}{\cos k}} \cdot \frac{\frac{\frac{\frac{\ell}{k}}{\sin k}}{\sqrt[3]{\frac{k}{\ell}}}}{\sin k}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if k < -1.6816267635550583e+64 or 2.010931763593779e-157 < k

    1. Initial program 45.0

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

    3. Applied add-cbrt-cube46.5

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

    4. Applied simplify31.8

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\tan k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]

    5. Taylor expanded around inf 26.6

      \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}}^{3}}}\]

    6. Applied simplify6.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt6.4

      \[\leadsto \frac{\frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}\right) \cdot \sqrt[3]{\frac{k}{\ell}}}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]

    9. Applied div-inv6.4

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

    10. Applied times-frac6.4

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

    11. Applied times-frac2.3

      \[\leadsto \color{blue}{\frac{\frac{2}{\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}}}{\frac{t}{\cos k}} \cdot \frac{\frac{\frac{1}{\frac{k}{\ell}}}{\sqrt[3]{\frac{k}{\ell}}}}{\sin k \cdot \sin k}}\]

    12. Applied simplify2.2

      \[\leadsto \frac{\frac{2}{\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}}}{\frac{t}{\cos k}} \cdot \color{blue}{\frac{\frac{\frac{\frac{\ell}{k}}{\sin k}}{\sqrt[3]{\frac{k}{\ell}}}}{\sin k}}\]

    if -1.6816267635550583e+64 < k < 2.010931763593779e-157

    1. Initial program 57.1

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

    3. Applied add-cbrt-cube58.8

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

    4. Applied simplify49.6

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\tan k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]

    5. Taylor expanded around inf 39.4

      \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}}^{3}}}\]

    6. Applied simplify20.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}}\]
    7. Using strategy rm

    8. Applied div-inv20.1

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

    9. Applied add-cube-cbrt20.4

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

    10. Applied times-frac20.7

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

    11. Applied times-frac14.5

      \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{k}}{\frac{t}{\cos k}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{\frac{1}{\ell}}}{\sin k \cdot \sin k}}\]

    12. Applied simplify3.0

      \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{k}}{\frac{t}{\cos k}} \cdot \color{blue}{\left(\frac{\frac{\ell}{\sin k}}{\sin k} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 4.7m)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' +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))))