Average Error: 47.2 → 5.1
Time: 9.1m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot \left(\frac{\frac{\sqrt{\cos k} \cdot \ell}{k}}{t} \cdot \frac{\frac{\sqrt{\cos k} \cdot \ell}{k}}{\sin k \cdot \sin k}\right) \le 5.930269536662363 \cdot 10^{+298}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\sqrt{\cos k} \cdot \ell}{k}}{t} \cdot \frac{\frac{\sqrt{\cos k} \cdot \ell}{k}}{\sin k \cdot \sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{\sin k \cdot \sin k} \cdot \left(\ell \cdot \left(\frac{1}{t} \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array}\]

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 (* (/ (/ (* (sqrt (cos k)) l) k) t) (/ (/ (* (sqrt (cos k)) l) k) (* (sin k) (sin k))))) < 5.930269536662363e+298

    1. Initial program 49.4

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

    2. Initial simplification35.9

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

    3. Taylor expanded around inf 23.8

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

    5. Applied associate-/r*22.3

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

    7. Applied add-sqr-sqrt22.3

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

    8. Applied times-frac22.6

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

    9. Simplified30.3

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

    10. Simplified3.3

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

    if 5.930269536662363e+298 < (* 2 (* (/ (/ (* (sqrt (cos k)) l) k) t) (/ (/ (* (sqrt (cos k)) l) k) (* (sin k) (sin k)))))

    1. Initial program 43.8

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

    2. Initial simplification31.1

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

    3. Taylor expanded around inf 20.7

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

    5. Applied associate-/r*20.5

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

    7. Applied add-sqr-sqrt20.5

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

    8. Applied times-frac20.5

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

    9. Applied times-frac20.4

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

    10. Simplified23.1

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

    11. Simplified9.0

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

    13. Applied div-inv9.0

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

    14. Applied associate-*l*7.7

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

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot \left(\frac{\frac{\sqrt{\cos k} \cdot \ell}{k}}{t} \cdot \frac{\frac{\sqrt{\cos k} \cdot \ell}{k}}{\sin k \cdot \sin k}\right) \le 5.930269536662363 \cdot 10^{+298}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\sqrt{\cos k} \cdot \ell}{k}}{t} \cdot \frac{\frac{\sqrt{\cos k} \cdot \ell}{k}}{\sin k \cdot \sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{\sin k \cdot \sin k} \cdot \left(\ell \cdot \left(\frac{1}{t} \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 9.1m)Debug logProfile

herbie shell --seed 2018214 
(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))))