Average Error: 46.6 → 1.1
Time: 4.2m
Precision: 64
Internal Precision: 4224
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\ell}{\sin k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{t}\right)}{\frac{k}{1} \cdot \tan k} = -\infty:\\ \;\;\;\;\left(\left(\frac{\ell}{1} \cdot \frac{\frac{1}{k}}{t}\right) \cdot \frac{2}{\frac{k}{1}}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;\frac{\frac{\ell}{\sin k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{t}\right)}{\frac{k}{1} \cdot \tan k} \le -1.3810090760354852 \cdot 10^{-268}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{t}\right)}{\frac{k}{1} \cdot \tan k}\\ \mathbf{if}\;\frac{\frac{\ell}{\sin k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{t}\right)}{\frac{k}{1} \cdot \tan k} \le -0.0:\\ \;\;\;\;\frac{\frac{\frac{\ell + \ell}{k}}{k \cdot t}}{\sin k \cdot \frac{\sin k}{\ell}} \cdot \cos k\\ \mathbf{if}\;\frac{\frac{\ell}{\sin k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{t}\right)}{\frac{k}{1} \cdot \tan k} \le 3.596441304459706 \cdot 10^{+226}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{k}{1}} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\frac{2}{k}}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell + \ell}{k}}{k \cdot t}}{\sin k \cdot \frac{\sin k}{\ell}} \cdot \cos k\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 4 regimes

  2. if (/ (* (/ l (sin k)) (* (/ 2 k) (/ l t))) (* (/ k 1) (tan k))) < -inf.0

    1. Initial program 63.4

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

    2. Taylor expanded around -inf 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 simplify61.9

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

    5. Applied div-inv61.9

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

    6. Applied times-frac58.2

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

    7. Applied simplify42.0

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

    8. Applied simplify42.0

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

    10. Applied *-un-lft-identity42.0

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

    11. Applied div-inv42.0

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

    12. Applied times-frac0.8

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

    13. Applied simplify0.9

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

    if -inf.0 < (/ (* (/ l (sin k)) (* (/ 2 k) (/ l t))) (* (/ k 1) (tan k))) < -1.3810090760354852e-268

    1. Initial program 55.9

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

    2. Taylor expanded around -inf 63.6

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

    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-frac47.0

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

    7. Applied simplify5.1

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

    8. Applied simplify5.1

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

    10. Applied associate-*l/5.1

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

    11. Applied frac-times1.2

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

    12. Applied simplify1.2

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

    if -1.3810090760354852e-268 < (/ (* (/ l (sin k)) (* (/ 2 k) (/ l t))) (* (/ k 1) (tan k))) < -0.0 or 3.596441304459706e+226 < (/ (* (/ l (sin k)) (* (/ 2 k) (/ l t))) (* (/ k 1) (tan k)))

    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 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 simplify27.9

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

    5. Applied div-inv27.9

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

    6. Applied times-frac27.1

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

    7. Applied simplify9.1

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

    8. Applied simplify9.1

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

    10. Applied tan-quot9.1

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

    11. Applied associate-/r/9.1

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

    12. Applied associate-*r*9.1

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

    13. Applied simplify1.0

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

    if -0.0 < (/ (* (/ l (sin k)) (* (/ 2 k) (/ l t))) (* (/ k 1) (tan k))) < 3.596441304459706e+226

    1. Initial program 55.7

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

    2. Taylor expanded around -inf 62.1

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

    3. Applied simplify48.3

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

    5. Applied div-inv48.3

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

    6. Applied times-frac46.8

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

    7. Applied simplify7.7

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

    8. Applied simplify7.7

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

    10. Applied associate-*l*1.1

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

    11. Applied simplify1.4

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

Runtime

Time bar (total: 4.2m)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' 
(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))))