Average Error: 46.9 → 2.1
Time: 6.8m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -8.954313681840263 \cdot 10^{-91} \lor \neg \left(k \le 1.5608918767812495 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \frac{\frac{\ell}{k}}{\sin k \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos k}{\sin k}}{\sin k \cdot t} \cdot \frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if k < -8.954313681840263e-91 or 1.5608918767812495e-87 < k

    1. Initial program 45.3

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

    3. Applied add-cbrt-cube47.1

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

    4. Applied simplify32.6

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

    5. Taylor expanded around inf 26.7

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

    6. Applied simplify6.6

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

    8. Applied div-inv6.7

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

    9. Applied times-frac0.6

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

    10. Applied simplify0.6

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

    if -8.954313681840263e-91 < k < 1.5608918767812495e-87

    1. Initial program 62.2

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

    3. Applied add-cbrt-cube62.2

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

    4. Applied simplify61.1

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

    5. Taylor expanded around inf 56.5

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

    6. Applied simplify31.7

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

    8. Applied div-inv32.0

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

    9. Applied simplify16.7

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

  4. Applied simplify2.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;k \le -8.954313681840263 \cdot 10^{-91} \lor \neg \left(k \le 1.5608918767812495 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \frac{\frac{\ell}{k}}{\sin k \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos k}{\sin k}}{\sin k \cdot t} \cdot \frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}\\ \end{array}}\]

Runtime

Time bar (total: 6.8m)Debug logProfile

herbie shell --seed 2018199 +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))))