Average Error: 47.0 → 10.7
Time: 7.9m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(2 \cdot \frac{\cos k}{t}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\sin k}}{k \cdot k} \le -3.0768353072682817 \cdot 10^{+273}:\\ \;\;\;\;\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{t}{\frac{\ell}{t}} \cdot \left(\frac{\sin k}{\cos k} \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{if}\;\frac{\left(\left(2 \cdot \frac{\cos k}{t}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\sin k}}{k \cdot k} \le -1.0879876044475662 \cdot 10^{-289}:\\ \;\;\;\;\frac{\left(\left(2 \cdot \frac{\cos k}{t}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\sin k}}{k \cdot k}\\ \mathbf{if}\;\frac{\left(\left(2 \cdot \frac{\cos k}{t}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\sin k}}{k \cdot k} \le 1.516120556684992 \cdot 10^{-229}:\\ \;\;\;\;\frac{\left(2 \cdot \cos k\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \sin k} \cdot \frac{\ell}{\sin k}\\ \mathbf{if}\;\frac{\left(\left(2 \cdot \frac{\cos k}{t}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\sin k}}{k \cdot k} \le 2.9849685581566247 \cdot 10^{+293}:\\ \;\;\;\;\frac{\left(\left(2 \cdot \frac{\cos k}{t}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\sin k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\left(\frac{\ell}{\sin k} \cdot \frac{2}{k}\right) \cdot \frac{\cos k}{k \cdot t}\right)}^{3}} \cdot \frac{\ell}{\sin k}\\ \end{array}\]

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 4 regimes

  2. if (/ (* (* (* 2 (/ (cos k) t)) (/ l (sin k))) (/ l (sin k))) (* k k)) < -3.0768353072682817e+273

    1. Initial program 62.7

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

    3. Applied add-cbrt-cube62.8

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

    4. Applied simplify58.9

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]

    5. Taylor expanded around inf 60.3

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\color{blue}{\frac{{k}^{2} \cdot \sin k}{{t}^{2} \cdot \cos k}} \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}\]

    6. Applied simplify43.0

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

    if -3.0768353072682817e+273 < (/ (* (* (* 2 (/ (cos k) t)) (/ l (sin k))) (/ l (sin k))) (* k k)) < -1.0879876044475662e-289 or 1.516120556684992e-229 < (/ (* (* (* 2 (/ (cos k) t)) (/ l (sin k))) (/ l (sin k))) (* k k)) < 2.9849685581566247e+293

    1. Initial program 54.5

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

    3. Applied add-cbrt-cube58.8

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

    4. Applied simplify51.4

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]

    5. Taylor expanded around inf 44.7

      \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right)}}^{3}}}\]

    6. Applied simplify18.1

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

    8. Applied associate-*r*11.8

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

    10. Applied associate-*l/11.5

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

    11. Applied associate-*l/2.9

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

    12. Applied associate-*l/1.3

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

    if -1.0879876044475662e-289 < (/ (* (* (* 2 (/ (cos k) t)) (/ l (sin k))) (/ l (sin k))) (* k k)) < 1.516120556684992e-229

    1. Initial program 38.2

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

    3. Applied add-cbrt-cube39.0

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

    4. Applied simplify19.6

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]

    5. Taylor expanded around inf 11.9

      \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right)}}^{3}}}\]

    6. Applied simplify8.5

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

    8. Applied associate-*r*6.3

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

    10. Applied frac-times6.2

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

    11. Applied frac-times6.4

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

    if 2.9849685581566247e+293 < (/ (* (* (* 2 (/ (cos k) t)) (/ l (sin k))) (/ l (sin k))) (* k k))

    1. Initial program 61.8

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

    3. Applied add-cbrt-cube61.8

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

    4. Applied simplify58.3

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]

    5. Taylor expanded around inf 61.3

      \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right)}}^{3}}}\]

    6. Applied simplify57.5

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

    8. Applied associate-*r*53.0

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

    10. Applied add-cbrt-cube55.5

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left(\frac{2}{k \cdot k} \cdot \frac{\cos k}{t}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \left(\left(\frac{2}{k \cdot k} \cdot \frac{\cos k}{t}\right) \cdot \frac{\ell}{\sin k}\right)\right) \cdot \left(\left(\frac{2}{k \cdot k} \cdot \frac{\cos k}{t}\right) \cdot \frac{\ell}{\sin k}\right)}} \cdot \frac{\ell}{\sin k}\]

    11. Applied simplify38.8

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(\frac{\ell}{\sin k} \cdot \frac{2}{k}\right) \cdot \frac{\cos k}{k \cdot t}\right)}^{3}}} \cdot \frac{\ell}{\sin k}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 7.9m)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))))