Average Error: 46.9 → 4.4
Time: 1.6m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 5515486.226576313:\\ \;\;\;\;\frac{\frac{2}{k} \cdot \ell}{\left(\frac{\sin k}{\ell} \cdot \left(t \cdot k\right)\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{\tan k \cdot \sin k} \cdot \frac{\frac{2}{k}}{\frac{1}{\frac{\ell}{t}}}\right) \cdot \frac{\frac{\ell}{t}}{k}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if k < 5515486.226576313

    1. Initial program 49.2

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

    2. Simplified33.2

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

    4. Applied associate-*r/33.6

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

    5. Applied associate-/r/33.6

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

    6. Applied times-frac29.4

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

    7. Simplified17.3

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{k}}{\frac{\frac{k}{1}}{\frac{\ell}{t}}}} \cdot \frac{t}{\sin k \cdot \tan k}\]
    8. Using strategy rm

    9. Applied frac-times9.1

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

    10. Simplified9.0

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

    11. Simplified8.4

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

    13. Applied associate-*r*6.0

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

    15. Applied associate-*r*4.5

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

    if 5515486.226576313 < k

    1. Initial program 43.1

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

    2. Simplified25.2

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

    4. Applied associate-*r/25.9

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

    5. Applied associate-/r/25.8

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

    6. Applied times-frac23.2

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

    7. Simplified9.4

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{k}}{\frac{\frac{k}{1}}{\frac{\ell}{t}}}} \cdot \frac{t}{\sin k \cdot \tan k}\]
    8. Using strategy rm

    9. Applied div-inv9.4

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

    10. Applied times-frac9.4

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

    11. Applied associate-*l*4.3

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

    12. Simplified4.3

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k}} \cdot \left(\frac{\frac{2}{k}}{\frac{1}{\frac{\ell}{t}}} \cdot \frac{t}{\sin k \cdot \tan k}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 5515486.226576313:\\ \;\;\;\;\frac{\frac{2}{k} \cdot \ell}{\left(\frac{\sin k}{\ell} \cdot \left(t \cdot k\right)\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{\tan k \cdot \sin k} \cdot \frac{\frac{2}{k}}{\frac{1}{\frac{\ell}{t}}}\right) \cdot \frac{\frac{\ell}{t}}{k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +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))))