Average Error: 47.5 → 1.2
Time: 4.2m
Precision: 64
Internal Precision: 4480
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\frac{2}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\ell}{\sin k}}{\frac{k}{1} \cdot \tan k} \le -4.009182609425654 \cdot 10^{+241}:\\ \;\;\;\;\left(\frac{\ell}{t \cdot k} \cdot \frac{2}{\frac{k}{1}}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;\frac{\left(\frac{2}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\ell}{\sin k}}{\frac{k}{1} \cdot \tan k} \le -4.8396724163879116 \cdot 10^{-300}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\ell + \ell}{\sin k}}{k}}{\frac{k}{1}}\\ \mathbf{if}\;\frac{\left(\frac{2}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\ell}{\sin k}}{\frac{k}{1} \cdot \tan k} \le -0.0:\\ \;\;\;\;\left(\sqrt[3]{\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{\frac{\ell + \ell}{k}}{t \cdot k}} \cdot \sqrt[3]{\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{\frac{\ell + \ell}{k}}{t \cdot k}}\right) \cdot \sqrt[3]{\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\frac{\ell + \ell}{k}}{t \cdot k}}\\ \mathbf{if}\;\frac{\left(\frac{2}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\ell}{\sin k}}{\frac{k}{1} \cdot \tan k} \le 2.841791613044812 \cdot 10^{+97}:\\ \;\;\;\;\left(\frac{\ell}{\tan k} \cdot \frac{\frac{2}{k}}{\sin k}\right) \cdot \frac{\frac{\ell}{t}}{\frac{k}{1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t \cdot k} \cdot \frac{2}{\frac{k}{1}}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 5 regimes

  2. if (/ (* (/ l (sin k)) (* (/ 2 k) (/ l t))) (* (/ k 1) (tan k))) < -4.009182609425654e+241

    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. Taylor expanded around -inf 63.8

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

    3. Applied simplify58.5

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

    5. Applied div-inv58.5

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

    6. Applied times-frac54.6

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

    7. Applied simplify30.0

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

    8. Applied simplify30.0

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

    10. Applied *-un-lft-identity30.0

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

    11. Applied *-un-lft-identity30.0

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

    12. Applied times-frac30.0

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

    13. Applied simplify30.0

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

    14. Applied simplify3.0

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

    if -4.009182609425654e+241 < (/ (* (/ l (sin k)) (* (/ 2 k) (/ l t))) (* (/ k 1) (tan k))) < -4.8396724163879116e-300

    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. 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 simplify50.2

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

    5. Applied div-inv50.3

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

    6. Applied times-frac48.3

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

    7. Applied simplify8.2

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

    8. Applied simplify8.1

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

    10. Applied associate-*l/8.2

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

    11. Applied associate-*l/0.8

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

    12. Applied simplify0.8

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

    if -4.8396724163879116e-300 < (/ (* (/ l (sin k)) (* (/ 2 k) (/ l t))) (* (/ k 1) (tan k))) < -0.0

    1. Initial program 36.5

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

    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 simplify23.2

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

    5. Applied div-inv23.2

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

    6. Applied times-frac22.6

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

    7. Applied simplify3.5

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

    8. Applied simplify3.5

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

    10. Applied add-cube-cbrt3.5

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

    11. Applied simplify3.5

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

    12. Applied simplify0.5

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

    if -0.0 < (/ (* (/ l (sin k)) (* (/ 2 k) (/ l t))) (* (/ k 1) (tan k))) < 2.841791613044812e+97

    1. Initial program 55.0

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

    2. 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-frac48.0

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

    7. Applied simplify10.4

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

    8. Applied simplify10.4

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

    10. Applied associate-*l*0.8

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

    11. Applied simplify0.9

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

    if 2.841791613044812e+97 < (/ (* (/ l (sin k)) (* (/ 2 k) (/ l t))) (* (/ k 1) (tan k)))

    1. Initial program 58.4

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

    2. Taylor expanded around -inf 60.7

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

    3. Applied simplify54.4

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

    5. Applied div-inv54.4

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

    6. Applied times-frac50.3

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

    7. Applied simplify25.2

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

    8. Applied simplify25.1

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

    10. Applied *-un-lft-identity25.1

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

    11. Applied *-un-lft-identity25.1

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

    12. Applied times-frac25.1

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

    13. Applied simplify25.1

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

    14. Applied simplify4.4

      \[\leadsto \left(\left(1 \cdot \color{blue}{\frac{\ell}{k \cdot t}}\right) \cdot \frac{2}{\frac{k}{1}}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\]
  3. Recombined 5 regimes into one program.

  4. Applied simplify1.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\left(\frac{2}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\ell}{\sin k}}{\frac{k}{1} \cdot \tan k} \le -4.009182609425654 \cdot 10^{+241}:\\ \;\;\;\;\left(\frac{\ell}{t \cdot k} \cdot \frac{2}{\frac{k}{1}}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;\frac{\left(\frac{2}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\ell}{\sin k}}{\frac{k}{1} \cdot \tan k} \le -4.8396724163879116 \cdot 10^{-300}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\ell + \ell}{\sin k}}{k}}{\frac{k}{1}}\\ \mathbf{if}\;\frac{\left(\frac{2}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\ell}{\sin k}}{\frac{k}{1} \cdot \tan k} \le -0.0:\\ \;\;\;\;\left(\sqrt[3]{\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{\frac{\ell + \ell}{k}}{t \cdot k}} \cdot \sqrt[3]{\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{\frac{\ell + \ell}{k}}{t \cdot k}}\right) \cdot \sqrt[3]{\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\frac{\ell + \ell}{k}}{t \cdot k}}\\ \mathbf{if}\;\frac{\left(\frac{2}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\ell}{\sin k}}{\frac{k}{1} \cdot \tan k} \le 2.841791613044812 \cdot 10^{+97}:\\ \;\;\;\;\left(\frac{\ell}{\tan k} \cdot \frac{\frac{2}{k}}{\sin k}\right) \cdot \frac{\frac{\ell}{t}}{\frac{k}{1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t \cdot k} \cdot \frac{2}{\frac{k}{1}}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array}}\]

Runtime

Time bar (total: 4.2m)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))))