Average Error: 47.6 → 1.5
Time: 3.8m
Precision: 64
Internal Precision: 4224
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos k \cdot \frac{\frac{\frac{\ell + \ell}{k}}{k \cdot t}}{\frac{\sin k}{\ell} \cdot \sin k} \le -1.4574511191016394 \cdot 10^{-205}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(\frac{2}{\frac{k}{1}} \cdot \frac{1}{\frac{t}{\frac{\ell}{k}}}\right)\\ \mathbf{if}\;\cos k \cdot \frac{\frac{\frac{\ell + \ell}{k}}{k \cdot t}}{\frac{\sin k}{\ell} \cdot \sin k} \le 4.823938800371537 \cdot 10^{-292}:\\ \;\;\;\;\frac{\left(\frac{2}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\ell}{\sin k}}{\frac{k}{1} \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{\sin k} \cdot \frac{\frac{\ell + \ell}{k}}{k \cdot t}\\ \end{array}\]

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) (* k t)) (* (sin k) (/ (sin k) l))) (cos k)) < -1.4574511191016394e-205

    1. Initial program 58.7

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

    2. Taylor expanded around -inf 63.7

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

    3. Applied simplify53.1

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

    5. Applied div-inv53.1

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

    6. Applied times-frac49.7

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

    7. Applied simplify14.1

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

    8. Applied simplify14.1

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

    10. Applied clear-num14.2

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

    11. Applied simplify1.6

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

    if -1.4574511191016394e-205 < (* (/ (/ (/ (+ l l) k) (* k t)) (* (sin k) (/ (sin k) l))) (cos k)) < 4.823938800371537e-292

    1. Initial program 39.2

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

    2. 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 simplify27.1

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

    5. Applied div-inv27.1

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

    6. Applied times-frac26.9

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

    7. Applied simplify6.0

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

    8. Applied simplify6.0

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

    10. Applied associate-*l/6.0

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

    11. Applied frac-times1.3

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

    12. Applied simplify1.3

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

    if 4.823938800371537e-292 < (* (/ (/ (/ (+ l l) k) (* k t)) (* (sin k) (/ (sin k) l))) (cos k))

    1. Initial program 56.9

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

    2. Taylor expanded around -inf 61.4

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

    3. Applied simplify51.3

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

    5. Applied div-inv51.3

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

    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 simplify13.8

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

    8. Applied simplify13.8

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

    10. Applied pow113.8

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

    11. Applied pow113.8

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

    12. Applied pow113.8

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

    13. Applied pow-prod-down13.8

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

    14. Applied pow-prod-down13.8

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

    15. Applied simplify2.1

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

  4. Applied simplify1.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\cos k \cdot \frac{\frac{\frac{\ell + \ell}{k}}{k \cdot t}}{\frac{\sin k}{\ell} \cdot \sin k} \le -1.4574511191016394 \cdot 10^{-205}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(\frac{2}{\frac{k}{1}} \cdot \frac{1}{\frac{t}{\frac{\ell}{k}}}\right)\\ \mathbf{if}\;\cos k \cdot \frac{\frac{\frac{\ell + \ell}{k}}{k \cdot t}}{\frac{\sin k}{\ell} \cdot \sin k} \le 4.823938800371537 \cdot 10^{-292}:\\ \;\;\;\;\frac{\left(\frac{2}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\ell}{\sin k}}{\frac{k}{1} \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{\sin k} \cdot \frac{\frac{\ell + \ell}{k}}{k \cdot t}\\ \end{array}}\]

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' 
(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))))