Average Error: 47.0 → 3.1
Time: 8.3m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\cos k}{-\sin k} \cdot \frac{\frac{-2}{t}}{\sin k}}{\frac{\frac{-1}{\ell}}{\frac{-1}{k}} \cdot \frac{\frac{-1}{\ell}}{\frac{-1}{k}}} \le -7.324126507054615 \cdot 10^{+297}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}\right) \cdot \left(\left(\sin k \cdot \frac{k}{t}\right) \cdot \frac{k}{t}\right)}{\cos k}}\\ \mathbf{if}\;\frac{\frac{\cos k}{-\sin k} \cdot \frac{\frac{-2}{t}}{\sin k}}{\frac{\frac{-1}{\ell}}{\frac{-1}{k}} \cdot \frac{\frac{-1}{\ell}}{\frac{-1}{k}}} \le -4.989922658679945 \cdot 10^{-292}:\\ \;\;\;\;\frac{\frac{\cos k}{-\sin k} \cdot \frac{\frac{-2}{t}}{\sin k}}{\frac{\frac{-1}{\ell}}{\frac{-1}{k}} \cdot \frac{\frac{-1}{\ell}}{\frac{-1}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{\cos k \cdot \left(-2\right)}{-\sin k}} \cdot \sqrt[3]{\frac{\cos k \cdot \left(-2\right)}{-\sin k}}}{\sin k} \cdot \frac{\sqrt[3]{\frac{\cos k \cdot \left(-2\right)}{-\sin k}}}{\frac{k}{\ell}}\right) \cdot \frac{\frac{1}{t}}{\frac{\frac{-1}{\ell}}{\frac{-1}{k}}}\\ \end{array}\]

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 3 regimes

  2. if (/ (* (/ (cos k) (- (sin k))) (/ (/ (- 2) t) (sin k))) (* (/ (/ -1 l) (/ -1 k)) (/ (/ -1 l) (/ -1 k)))) < -7.324126507054615e+297

    1. Initial program 63.2

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

    3. Applied add-cbrt-cube63.2

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

    4. Applied simplify58.4

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

    6. Applied tan-quot58.4

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

    7. Applied associate-*r/58.4

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

    8. Applied associate-*l/58.4

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

    9. Applied cube-div58.4

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

    10. Applied cbrt-div58.4

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

    11. Applied simplify39.5

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

    12. Applied simplify39.4

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

    if -7.324126507054615e+297 < (/ (* (/ (cos k) (- (sin k))) (/ (/ (- 2) t) (sin k))) (* (/ (/ -1 l) (/ -1 k)) (/ (/ -1 l) (/ -1 k)))) < -4.989922658679945e-292

    1. Initial program 57.0

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

    3. Applied add-cbrt-cube60.7

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

    4. Applied simplify54.6

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

    5. Taylor expanded around -inf 63.6

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

    6. Applied simplify7.2

      \[\leadsto \color{blue}{\left(\frac{2}{-\sin k} \cdot \frac{\cos k}{\sin k}\right) \cdot \frac{\frac{-1}{t}}{\frac{\frac{-1}{\ell}}{\frac{-1}{k}} \cdot \frac{\frac{-1}{\ell}}{\frac{-1}{k}}}}\]
    7. Using strategy rm

    8. Applied associate-*r/4.9

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

    9. Applied simplify1.2

      \[\leadsto \frac{\color{blue}{\frac{\cos k}{-\sin k} \cdot \frac{\frac{-2}{t}}{\sin k}}}{\frac{\frac{-1}{\ell}}{\frac{-1}{k}} \cdot \frac{\frac{-1}{\ell}}{\frac{-1}{k}}}\]

    if -4.989922658679945e-292 < (/ (* (/ (cos k) (- (sin k))) (/ (/ (- 2) t) (sin k))) (* (/ (/ -1 l) (/ -1 k)) (/ (/ -1 l) (/ -1 k))))

    1. Initial program 43.7

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

    3. Applied add-cbrt-cube44.8

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

    4. Applied simplify29.3

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

    5. Taylor expanded around -inf 58.9

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

    6. Applied simplify9.6

      \[\leadsto \color{blue}{\left(\frac{2}{-\sin k} \cdot \frac{\cos k}{\sin k}\right) \cdot \frac{\frac{-1}{t}}{\frac{\frac{-1}{\ell}}{\frac{-1}{k}} \cdot \frac{\frac{-1}{\ell}}{\frac{-1}{k}}}}\]
    7. Using strategy rm

    8. Applied div-inv9.6

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

    9. Applied times-frac3.0

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

    10. Applied associate-*r*2.4

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

    11. Applied simplify2.1

      \[\leadsto \color{blue}{\frac{\frac{\cos k \cdot \left(-2\right)}{-\sin k}}{\sin k \cdot \frac{k}{\ell}}} \cdot \frac{\frac{1}{t}}{\frac{\frac{-1}{\ell}}{\frac{-1}{k}}}\]
    12. Using strategy rm

    13. Applied add-cube-cbrt2.3

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

    14. Applied times-frac2.4

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

Runtime

Time bar (total: 8.3m)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))))