Average Error: 47.0 → 1.6
Time: 5.1m
Precision: 64
Internal Precision: 4224
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{\frac{\ell + \ell}{k}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\frac{\ell}{t}}{k}}{\sqrt[3]{\tan k}}\right) \cdot \frac{\frac{1}{\sin k}}{\sqrt[3]{\tan k}} \le -3.5776471143602304 \cdot 10^{+67}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\frac{2}{t}}{k}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;\left(\frac{\frac{\ell + \ell}{k}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\frac{\ell}{t}}{k}}{\sqrt[3]{\tan k}}\right) \cdot \frac{\frac{1}{\sin k}}{\sqrt[3]{\tan k}} \le -4.8667869991763 \cdot 10^{-315}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{t}\right)}{\frac{k}{1} \cdot \tan k}\\ \mathbf{if}\;\left(\frac{\frac{\ell + \ell}{k}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\frac{\ell}{t}}{k}}{\sqrt[3]{\tan k}}\right) \cdot \frac{\frac{1}{\sin k}}{\sqrt[3]{\tan k}} \le 0.0:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\frac{2}{t}}{k}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;\left(\frac{\frac{\ell + \ell}{k}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\frac{\ell}{t}}{k}}{\sqrt[3]{\tan k}}\right) \cdot \frac{\frac{1}{\sin k}}{\sqrt[3]{\tan k}} \le 1.0009407011640392 \cdot 10^{+196}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{t}\right)}{\frac{k}{1} \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\frac{\ell + \ell}{k}}{k \cdot t}} \cdot \sqrt{\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\frac{\ell + \ell}{k}}{k \cdot t}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if (* (* (/ (/ (+ l l) k) (cbrt (tan k))) (/ (/ (/ l t) k) (cbrt (tan k)))) (/ (/ 1 (sin k)) (cbrt (tan k)))) < -3.5776471143602304e+67 or -4.8667869991763e-315 < (* (* (/ (/ (+ l l) k) (cbrt (tan k))) (/ (/ (/ l t) k) (cbrt (tan k)))) (/ (/ 1 (sin k)) (cbrt (tan k)))) < 0.0

    1. Initial program 40.8

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

    2. Taylor expanded around -inf 63.1

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

    3. Applied simplify29.5

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

    5. Applied div-inv29.5

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

    6. Applied times-frac28.4

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

    7. Applied simplify7.1

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

    8. Applied simplify7.1

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

    10. Applied div-inv7.1

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

    11. Applied div-inv7.2

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

    12. Applied times-frac1.1

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

    13. Applied associate-*l*0.9

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

    14. Applied simplify0.9

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

    if -3.5776471143602304e+67 < (* (* (/ (/ (+ l l) k) (cbrt (tan k))) (/ (/ (/ l t) k) (cbrt (tan k)))) (/ (/ 1 (sin k)) (cbrt (tan k)))) < -4.8667869991763e-315 or 0.0 < (* (* (/ (/ (+ l l) k) (cbrt (tan k))) (/ (/ (/ l t) k) (cbrt (tan k)))) (/ (/ 1 (sin k)) (cbrt (tan k)))) < 1.0009407011640392e+196

    1. Initial program 55.4

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

    2. Taylor expanded around -inf 62.7

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

    3. Applied simplify49.6

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

    5. Applied div-inv49.6

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

    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 simplify9.7

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

    8. Applied simplify9.6

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

    10. Applied associate-*l/9.6

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

    11. Applied frac-times2.4

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

    12. Applied simplify2.4

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

    if 1.0009407011640392e+196 < (* (* (/ (/ (+ l l) k) (cbrt (tan k))) (/ (/ (/ l t) k) (cbrt (tan k)))) (/ (/ 1 (sin k)) (cbrt (tan k))))

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 59.9

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

    3. Applied simplify54.9

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

    5. Applied div-inv54.9

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

    6. Applied times-frac51.5

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

    7. Applied simplify30.7

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

    8. Applied simplify30.7

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

    10. Applied add-sqr-sqrt30.8

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

    11. Applied simplify32.3

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

    12. Applied simplify4.9

      \[\leadsto \sqrt{\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\frac{\ell + \ell}{k}}{k \cdot t}} \cdot \color{blue}{\sqrt{\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\frac{\ell + \ell}{k}}{k \cdot t}}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 5.1m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(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))))