Average Error: 46.6 → 3.2
Time: 3.0m
Precision: 64
Internal Precision: 4224
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\ell + \ell}{k \cdot k} \cdot \frac{\cos k}{\frac{t}{\ell}}}{\sin k \cdot \sin k} \le +\infty:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{\sin k}}{\frac{t}{\ell} \cdot k} \cdot \frac{\cos k + \cos k}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\cos k}{\sin k} \cdot \frac{\frac{2}{t}}{\sin k}\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (/ (+ l l) (* k k)) (/ (cos k) (/ t l))) (* (sin k) (sin k))) < +inf.0

    1. Initial program 45.5

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

    2. Taylor expanded around -inf 62.8

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

    3. Applied simplify35.0

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

    4. Taylor expanded around inf 18.9

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

    5. Taylor expanded around -inf 62.8

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

    6. Applied simplify8.1

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{\cos k + \cos k}{\sin k \cdot \sin k}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity8.1

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

    9. Applied times-frac8.1

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

    10. Applied associate-*r*7.5

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

    11. Applied simplify3.0

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

    if +inf.0 < (/ (* (/ (+ l l) (* k k)) (/ (cos k) (/ t l))) (* (sin k) (sin k)))

    1. Initial program 62.3

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

    2. Taylor expanded around -inf 62.3

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

    3. Applied simplify62.3

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

    4. Taylor expanded around inf 62.0

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

    5. Taylor expanded around -inf 62.3

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

    6. Applied simplify27.4

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{\cos k + \cos k}{\sin k \cdot \sin k}}\]
    7. Using strategy rm

    8. Applied div-inv27.4

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

    9. Applied associate-*l*27.4

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

    10. Applied simplify6.8

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

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' +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))))