Average Error: 31.7 → 12.5
Time: 5.0m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{\sin k \cdot \left(\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \le -5.526145999018744 \cdot 10^{-276}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \mathbf{if}\;\frac{2}{\sin k \cdot \left(\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \le 7.72709538493361 \cdot 10^{-306}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \mathbf{if}\;\frac{2}{\sin k \cdot \left(\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \le 1.8279041662057353 \cdot 10^{+285}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\left(\frac{k \cdot t}{\ell} + \frac{1}{3} \cdot \frac{{k}^{3} \cdot t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array}\]

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 3 regimes

  2. if (/ 2 (* (sin k) (* (* (* (/ t l) (tan k)) (* (/ t l) t)) (+ (+ 1 (pow (/ k t) 2)) 1)))) < -5.526145999018744e-276 or 7.72709538493361e-306 < (/ 2 (* (sin k) (* (* (* (/ t l) (tan k)) (* (/ t l) t)) (+ (+ 1 (pow (/ k t) 2)) 1)))) < 1.8279041662057353e+285

    1. Initial program 42.8

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

    3. Applied unpow342.8

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

    4. Applied times-frac26.7

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

    5. Applied associate-*l*20.1

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity20.1

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

    8. Applied associate-*r*20.1

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

    9. Applied simplify3.7

      \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    10. Using strategy rm

    11. Applied associate-*l*2.8

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

    if -5.526145999018744e-276 < (/ 2 (* (sin k) (* (* (* (/ t l) (tan k)) (* (/ t l) t)) (+ (+ 1 (pow (/ k t) 2)) 1)))) < 7.72709538493361e-306

    1. Initial program 17.6

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

    3. Applied unpow317.6

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

    4. Applied times-frac12.7

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

    5. Applied associate-*l*12.7

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity12.7

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

    8. Applied associate-*r*12.7

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

    9. Applied simplify9.4

      \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    10. Using strategy rm

    11. Applied *-un-lft-identity9.4

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

    12. Applied associate-*r*9.4

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

    13. Applied simplify4.2

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    14. Using strategy rm

    15. Applied associate-*l*4.3

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

    if 1.8279041662057353e+285 < (/ 2 (* (sin k) (* (* (* (/ t l) (tan k)) (* (/ t l) t)) (+ (+ 1 (pow (/ k t) 2)) 1))))

    1. Initial program 61.9

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

    3. Applied unpow361.9

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

    4. Applied times-frac61.5

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

    5. Applied associate-*l*61.5

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity61.5

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

    8. Applied associate-*r*61.5

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

    9. Applied simplify61.2

      \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    10. Using strategy rm

    11. Applied *-un-lft-identity61.2

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

    12. Applied associate-*r*61.2

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

    13. Applied simplify61.9

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

    14. Taylor expanded around 0 52.9

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

Runtime

Time bar (total: 5.0m)Debug logProfile

herbie shell --seed 2020178 +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))))