Average Error: 47.5 → 0.9
Time: 4.7m
Precision: 64
Internal Precision: 4480
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\ell + \ell}{\sin k}}{k}}{\frac{k}{1}} = -\infty:\\ \;\;\;\;\left(\frac{2}{\frac{k}{1}} \cdot \frac{\ell}{k \cdot t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;\frac{\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\ell + \ell}{\sin k}}{k}}{\frac{k}{1}} \le -7.743475866220657 \cdot 10^{-299}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\ell + \ell}{\sin k}}{k}}{\frac{k}{1}}\\ \mathbf{if}\;\frac{\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\ell + \ell}{\sin k}}{k}}{\frac{k}{1}} \le 0.0:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\frac{\ell + \ell}{k}}{k \cdot t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}\\ \mathbf{if}\;\frac{\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\ell + \ell}{\sin k}}{k}}{\frac{k}{1}} \le 2.934943614488963 \cdot 10^{+97}:\\ \;\;\;\;\left(\frac{\ell}{\tan k} \cdot \frac{\frac{2}{k}}{\sin k}\right) \cdot \frac{\frac{\ell}{t}}{\frac{k}{1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{\frac{k}{1}} \cdot \frac{\ell}{k \cdot t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 4 regimes

  2. if (/ (* (/ (/ (+ l l) (sin k)) k) (/ (/ l t) (tan k))) (/ k 1)) or 2.934943614488963e+97 < (/ (* (/ (/ (+ l l) (sin k)) k) (/ (/ l t) (tan k))) (/ k 1))

    1. Initial program 58.9

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

    2. Taylor expanded around -inf 61.1

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

    3. Applied simplify54.8

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell + \ell}{{t}^{3}}}{\frac{\sin k}{\ell}}}{\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \tan k}}\]
    4. Using strategy rm

    5. Applied div-inv54.8

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

    6. Applied times-frac50.4

      \[\leadsto \color{blue}{\frac{\frac{\ell + \ell}{{t}^{3}}}{\frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{\frac{1}{\frac{\sin k}{\ell}}}{\tan k}}\]

    7. Applied simplify23.7

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

    8. Applied simplify23.7

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

    10. Applied *-un-lft-identity23.7

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

    11. Applied *-un-lft-identity23.7

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

    12. Applied times-frac23.7

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

    13. Applied simplify23.7

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

    14. Applied simplify2.9

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

    if (/ (* (/ (/ (+ l l) (sin k)) k) (/ (/ l t) (tan k))) (/ k 1)) < -7.743475866220657e-299

    1. Initial program 57.5

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

    2. Taylor expanded around -inf 63.5

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

    3. Applied simplify50.5

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell + \ell}{{t}^{3}}}{\frac{\sin k}{\ell}}}{\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \tan k}}\]
    4. Using strategy rm

    5. Applied div-inv50.5

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

    6. Applied times-frac48.1

      \[\leadsto \color{blue}{\frac{\frac{\ell + \ell}{{t}^{3}}}{\frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{\frac{1}{\frac{\sin k}{\ell}}}{\tan k}}\]

    7. Applied simplify7.6

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

    8. Applied simplify7.5

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

    10. Applied associate-*l/7.5

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

    11. Applied associate-*l/1.0

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

    12. Applied simplify0.9

      \[\leadsto \frac{\color{blue}{\frac{\frac{\ell + \ell}{\sin k}}{k} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{\frac{k}{1}}\]

    if -7.743475866220657e-299 < (/ (* (/ (/ (+ l l) (sin k)) k) (/ (/ l t) (tan k))) (/ k 1)) < 0.0

    1. Initial program 37.5

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

    2. Taylor expanded around -inf 63.0

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

    3. Applied simplify24.8

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell + \ell}{{t}^{3}}}{\frac{\sin k}{\ell}}}{\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \tan k}}\]
    4. Using strategy rm

    5. Applied div-inv24.8

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

    6. Applied times-frac24.4

      \[\leadsto \color{blue}{\frac{\frac{\ell + \ell}{{t}^{3}}}{\frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{\frac{1}{\frac{\sin k}{\ell}}}{\tan k}}\]

    7. Applied simplify6.3

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

    8. Applied simplify6.3

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

    10. Applied add-cube-cbrt6.3

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

    11. Applied *-un-lft-identity6.3

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

    12. Applied times-frac6.3

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

    13. Applied associate-*r*6.3

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

    14. Applied simplify0.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell + \ell}{k}}{k \cdot t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}} \cdot \frac{\frac{\ell}{\sin k}}{\sqrt[3]{\tan k}}\]

    if 0.0 < (/ (* (/ (/ (+ l l) (sin k)) k) (/ (/ l t) (tan k))) (/ k 1)) < 2.934943614488963e+97

    1. Initial program 55.1

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

    2. Taylor expanded around -inf 62.4

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

    3. Applied simplify49.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell + \ell}{{t}^{3}}}{\frac{\sin k}{\ell}}}{\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \tan k}}\]
    4. Using strategy rm

    5. Applied div-inv49.1

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

    6. Applied times-frac47.9

      \[\leadsto \color{blue}{\frac{\frac{\ell + \ell}{{t}^{3}}}{\frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{\frac{1}{\frac{\sin k}{\ell}}}{\tan k}}\]

    7. Applied simplify10.3

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

    8. Applied simplify10.3

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

    10. Applied associate-*l*0.9

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

    11. Applied simplify0.9

      \[\leadsto \frac{\frac{\ell}{t}}{\frac{k}{1}} \cdot \color{blue}{\left(\frac{\ell}{\tan k} \cdot \frac{\frac{2}{k}}{\sin k}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify0.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\ell + \ell}{\sin k}}{k}}{\frac{k}{1}} = -\infty:\\ \;\;\;\;\left(\frac{2}{\frac{k}{1}} \cdot \frac{\ell}{k \cdot t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;\frac{\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\ell + \ell}{\sin k}}{k}}{\frac{k}{1}} \le -7.743475866220657 \cdot 10^{-299}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\ell + \ell}{\sin k}}{k}}{\frac{k}{1}}\\ \mathbf{if}\;\frac{\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\ell + \ell}{\sin k}}{k}}{\frac{k}{1}} \le 0.0:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\frac{\ell + \ell}{k}}{k \cdot t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}\\ \mathbf{if}\;\frac{\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\ell + \ell}{\sin k}}{k}}{\frac{k}{1}} \le 2.934943614488963 \cdot 10^{+97}:\\ \;\;\;\;\left(\frac{\ell}{\tan k} \cdot \frac{\frac{2}{k}}{\sin k}\right) \cdot \frac{\frac{\ell}{t}}{\frac{k}{1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{\frac{k}{1}} \cdot \frac{\ell}{k \cdot t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array}}\]

Runtime

Time bar (total: 4.7m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(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))))