Average Error: 46.9 → 6.2
Time: 6.2m
Precision: 64
Internal Precision: 4224
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)\right)}{\ell}}{\ell \cdot \cos k}} = -\infty:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{\frac{\sin k \cdot t}{\frac{\cos k}{\sin k}}}\\ \mathbf{if}\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)\right)}{\ell}}{\ell \cdot \cos k}} \le -1.8180078768424565 \cdot 10^{-283}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)\right)}{\ell}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{\frac{\sin k \cdot t}{\frac{\cos k}{\sin k}}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if (/ 2 (/ (/ (* k (* k (* t (pow (sin k) 2)))) l) (* l (cos k)))) or -1.8180078768424565e-283 < (/ 2 (/ (/ (* k (* k (* t (pow (sin k) 2)))) l) (* l (cos k))))

    1. Initial program 45.2

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

    3. Applied add-cbrt-cube46.2

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

    4. Applied simplify34.9

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}}\]
    5. Using strategy rm

    6. Applied associate-*l/37.4

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\color{blue}{\frac{t \cdot \left(t \cdot t\right)}{\ell}} \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}\]

    7. Applied associate-*l/37.4

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}}\right)}^{3}}}\]

    8. Applied tan-quot37.4

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}\right)}^{3}}}\]

    9. Applied associate-*r/37.4

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\color{blue}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k}{\cos k}} \cdot \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}\right)}^{3}}}\]

    10. Applied frac-times37.2

      \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)}{\cos k \cdot \ell}\right)}}^{3}}}\]

    11. Applied cube-div40.2

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{\frac{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}{{\left(\cos k \cdot \ell\right)}^{3}}}}}\]

    12. Applied cbrt-div40.0

      \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}{\sqrt[3]{{\left(\cos k \cdot \ell\right)}^{3}}}}}\]

    13. Applied simplify32.9

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}}{\sqrt[3]{{\left(\cos k \cdot \ell\right)}^{3}}}}\]

    14. Applied simplify26.5

      \[\leadsto \frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}{\color{blue}{\ell \cdot \cos k}}}\]

    15. Taylor expanded around inf 19.4

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

    16. Taylor expanded around -inf 62.6

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

    17. Applied simplify7.1

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

    if (/ 2 (/ (/ (* k (* k (* t (pow (sin k) 2)))) l) (* l (cos k)))) < -1.8180078768424565e-283

    1. Initial program 56.5

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

    3. Applied add-cbrt-cube60.1

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

    4. Applied simplify56.6

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}}\]
    5. Using strategy rm

    6. Applied associate-*l/57.7

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\color{blue}{\frac{t \cdot \left(t \cdot t\right)}{\ell}} \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}\]

    7. Applied associate-*l/57.7

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}}\right)}^{3}}}\]

    8. Applied tan-quot57.7

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}\right)}^{3}}}\]

    9. Applied associate-*r/57.7

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\color{blue}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k}{\cos k}} \cdot \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}\right)}^{3}}}\]

    10. Applied frac-times57.3

      \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)}{\cos k \cdot \ell}\right)}}^{3}}}\]

    11. Applied cube-div59.8

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{\frac{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}{{\left(\cos k \cdot \ell\right)}^{3}}}}}\]

    12. Applied cbrt-div58.6

      \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}{\sqrt[3]{{\left(\cos k \cdot \ell\right)}^{3}}}}}\]

    13. Applied simplify50.7

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}}{\sqrt[3]{{\left(\cos k \cdot \ell\right)}^{3}}}}\]

    14. Applied simplify41.2

      \[\leadsto \frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}{\color{blue}{\ell \cdot \cos k}}}\]

    15. Taylor expanded around inf 10.0

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{\ell}}}{\ell \cdot \cos k}}\]
    16. Using strategy rm

    17. Applied unpow210.0

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

    18. Applied associate-*l*1.1

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

Runtime

Time bar (total: 6.2m)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' +o rules:numerics
(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))))