Average Error: 47.2 → 1.8
Time: 4.8m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{k}{\ell} \cdot (\frac{1}{6} \cdot \left(\frac{{k}^{5}}{\frac{\ell}{t}}\right) + \left(\frac{{k}^{3}}{\frac{\ell}{t}}\right))_* \le 3.6044206113176974 \cdot 10^{+180}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(\sin k\right)}^{2}}{\cos k} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{\sin k \cdot \left(t \cdot \sin k\right)}{\cos k}\right)}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if (* (/ k l) (fma 1/6 (/ (pow k 5) (/ l t)) (/ (pow k 3) (/ l t)))) < 3.6044206113176974e+180

    1. Initial program 47.8

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

    2. Initial simplification33.2

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

    3. Taylor expanded around inf 20.9

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

    5. Applied times-frac19.1

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

    7. Applied unpow219.1

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

    8. Applied unpow219.1

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

    9. Applied times-frac10.2

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

    10. Applied associate-*l*4.5

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

    12. Applied *-un-lft-identity4.5

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

    13. Applied times-frac4.4

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

    14. Applied associate-*r*2.5

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

    if 3.6044206113176974e+180 < (* (/ k l) (fma 1/6 (/ (pow k 5) (/ l t)) (/ (pow k 3) (/ l t))))

    1. Initial program 46.5

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

    2. Initial simplification35.7

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

    3. Taylor expanded around inf 24.7

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

    5. Applied times-frac24.8

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

    7. Applied unpow224.8

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

    8. Applied unpow224.8

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

    9. Applied times-frac9.2

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

    10. Applied associate-*l*4.2

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

    12. Applied unpow24.2

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

    13. Applied associate-*r*1.0

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{k}{\ell} \cdot (\frac{1}{6} \cdot \left(\frac{{k}^{5}}{\frac{\ell}{t}}\right) + \left(\frac{{k}^{3}}{\frac{\ell}{t}}\right))_* \le 3.6044206113176974 \cdot 10^{+180}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(\sin k\right)}^{2}}{\cos k} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{\sin k \cdot \left(t \cdot \sin k\right)}{\cos k}\right)}\\ \end{array}\]

Runtime

Time bar (total: 4.8m)Debug logProfile

herbie shell --seed 2018214 +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))))