Average Error: 46.9 → 12.8
Time: 3.7m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{\frac{2}{t}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\frac{\ell}{t}}{\sin k} \cdot \frac{\frac{\ell}{t}}{\sin k}\right)\right) \cdot \frac{\cos k}{\left|\frac{k}{t}\right|} \le -4.1330026057837656 \cdot 10^{+259}:\\ \;\;\;\;\left(\frac{\frac{\frac{\ell}{t}}{\frac{k}{t}}}{\tan k} \cdot \frac{\frac{\ell}{t}}{\frac{k}{t}}\right) \cdot \frac{\frac{2}{t}}{\sin k}\\ \mathbf{elif}\;\left(\frac{\frac{2}{t}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\frac{\ell}{t}}{\sin k} \cdot \frac{\frac{\ell}{t}}{\sin k}\right)\right) \cdot \frac{\cos k}{\left|\frac{k}{t}\right|} \le -7.9096411914411 \cdot 10^{-317}:\\ \;\;\;\;\left(\frac{\frac{2}{t}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\frac{\ell}{t}}{\sin k} \cdot \frac{\frac{\ell}{t}}{\sin k}\right)\right) \cdot \frac{\cos k}{\left|\frac{k}{t}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}\right) \cdot \frac{\ell}{t}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (* (* (* (/ (/ l t) (sin k)) (/ (/ l t) (sin k))) (/ (/ 2 t) (fabs (/ k t)))) (/ (cos k) (fabs (/ k t)))) < -4.1330026057837656e+259

    1. Initial program 62.8

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

    2. Initial simplification59.4

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 0}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity59.4

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 0\right)}}\]

    5. Applied times-frac59.0

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 0\right)}\]

    6. Applied times-frac57.5

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{1} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    7. Simplified57.5

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\sin k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 0}\]

    8. Simplified17.4

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

    if -4.1330026057837656e+259 < (* (* (* (/ (/ l t) (sin k)) (/ (/ l t) (sin k))) (/ (/ 2 t) (fabs (/ k t)))) (/ (cos k) (fabs (/ k t)))) < -7.9096411914411e-317

    1. Initial program 52.4

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

    2. Initial simplification28.4

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 0}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt28.4

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

    5. Applied tan-quot28.4

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \color{blue}{\frac{\sin k}{\cos k}}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    6. Applied associate-*r/28.4

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\color{blue}{\frac{\sin k \cdot \sin k}{\cos k}}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    7. Applied associate-/r/28.4

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \sin k} \cdot \cos k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    8. Applied times-frac28.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}} \cdot \frac{\cos k}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}}\]

    9. Simplified21.0

      \[\leadsto \color{blue}{\left(\left(\frac{\frac{\ell}{t}}{\sin k} \cdot \frac{\frac{\ell}{t}}{\sin k}\right) \cdot \frac{\frac{2}{t}}{\left|\frac{k}{t}\right|}\right)} \cdot \frac{\cos k}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    10. Simplified1.6

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

    if -7.9096411914411e-317 < (* (* (* (/ (/ l t) (sin k)) (/ (/ l t) (sin k))) (/ (/ 2 t) (fabs (/ k t)))) (/ (cos k) (fabs (/ k t))))

    1. Initial program 44.8

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

    2. Initial simplification28.4

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 0}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt28.4

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

    5. Applied times-frac28.4

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    6. Applied times-frac25.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}}\]

    7. Simplified25.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    8. Simplified14.3

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

    10. Applied associate-*r*14.3

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

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\frac{2}{t}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\frac{\ell}{t}}{\sin k} \cdot \frac{\frac{\ell}{t}}{\sin k}\right)\right) \cdot \frac{\cos k}{\left|\frac{k}{t}\right|} \le -4.1330026057837656 \cdot 10^{+259}:\\ \;\;\;\;\left(\frac{\frac{\frac{\ell}{t}}{\frac{k}{t}}}{\tan k} \cdot \frac{\frac{\ell}{t}}{\frac{k}{t}}\right) \cdot \frac{\frac{2}{t}}{\sin k}\\ \mathbf{elif}\;\left(\frac{\frac{2}{t}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\frac{\ell}{t}}{\sin k} \cdot \frac{\frac{\ell}{t}}{\sin k}\right)\right) \cdot \frac{\cos k}{\left|\frac{k}{t}\right|} \le -7.9096411914411 \cdot 10^{-317}:\\ \;\;\;\;\left(\frac{\frac{2}{t}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\frac{\ell}{t}}{\sin k} \cdot \frac{\frac{\ell}{t}}{\sin k}\right)\right) \cdot \frac{\cos k}{\left|\frac{k}{t}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}\right) \cdot \frac{\ell}{t}\\ \end{array}\]

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed 2018217 
(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))))