Average Error: 46.8 → 11.2
Time: 4.2m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2 \le -5.102484297178739 \cdot 10^{+304}:\\ \;\;\;\;\left(\frac{\ell}{t \cdot \tan k} \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}\right) \cdot \frac{\ell}{t \cdot \left|\frac{k}{t}\right|}\\ \mathbf{elif}\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2 \le -1.7772874778501266 \cdot 10^{-209}:\\ \;\;\;\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t \cdot \tan k} \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}\right) \cdot \frac{\ell}{t \cdot \left|\frac{k}{t}\right|}\\ \end{array}\]

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 (* 2 (/ (* (pow l 2) (cos k)) (* t (* (pow k 2) (pow (sin k) 2))))) < -5.102484297178739e+304 or -1.7772874778501266e-209 < (* 2 (/ (* (pow l 2) (cos k)) (* t (* (pow k 2) (pow (sin k) 2)))))

    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 simplification29.6

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

    4. Applied add-sqr-sqrt29.6

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

    5. Applied times-frac29.5

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

    6. Applied times-frac26.6

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

    7. Simplified26.6

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

    8. Simplified14.0

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

    10. Applied associate-*r*12.7

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

    12. Applied associate-/l/12.7

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

    13. Taylor expanded around -inf 12.0

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

    if -5.102484297178739e+304 < (* 2 (/ (* (pow l 2) (cos k)) (* t (* (pow k 2) (pow (sin k) 2))))) < -1.7772874778501266e-209

    1. Initial program 50.3

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

    2. Initial simplification39.4

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

    3. Taylor expanded around -inf 1.8

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2 \le -5.102484297178739 \cdot 10^{+304}:\\ \;\;\;\;\left(\frac{\ell}{t \cdot \tan k} \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}\right) \cdot \frac{\ell}{t \cdot \left|\frac{k}{t}\right|}\\ \mathbf{elif}\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2 \le -1.7772874778501266 \cdot 10^{-209}:\\ \;\;\;\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t \cdot \tan k} \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}\right) \cdot \frac{\ell}{t \cdot \left|\frac{k}{t}\right|}\\ \end{array}\]

Runtime

Time bar (total: 4.2m)Debug logProfile

herbie shell --seed 2018225 +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))))