Average Error: 31.8 → 7.9
Time: 5.2m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.6331121973565042 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{\frac{1}{t}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(\frac{2}{\tan k} \cdot \frac{\frac{\ell}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\\ \mathbf{elif}\;t \le 3.0124796597905178 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{1}{\left(t \cdot t\right) \cdot 2 + k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{t}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(\frac{2}{\tan k} \cdot \frac{\frac{\ell}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < -1.6331121973565042e-53 or 3.0124796597905178e-30 < t

    1. Initial program 22.0

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

    2. Simplified11.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t}} \cdot \sin k}}{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity11.6

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

    5. Applied div-inv11.6

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

    6. Applied times-frac11.6

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

    7. Simplified11.6

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

    8. Simplified7.3

      \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\frac{\frac{\frac{\frac{\ell}{t}}{t}}{2 + \frac{k}{t} \cdot \frac{k}{t}}}{\frac{\sin k}{\frac{\ell}{t}}}}\]
    9. Using strategy rm

    10. Applied *-un-lft-identity7.3

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

    11. Applied add-sqr-sqrt7.4

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

    12. Applied div-inv7.4

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

    13. Applied times-frac7.3

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

    14. Applied times-frac5.9

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

    15. Applied associate-*r*2.7

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

    17. Applied add-sqr-sqrt2.6

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

    if -1.6331121973565042e-53 < t < 3.0124796597905178e-30

    1. Initial program 53.5

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

    2. Simplified39.5

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t}} \cdot \sin k}}{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity39.5

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

    5. Applied div-inv39.5

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

    6. Applied times-frac38.8

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

    7. Simplified38.8

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

    8. Simplified33.2

      \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\frac{\frac{\frac{\frac{\ell}{t}}{t}}{2 + \frac{k}{t} \cdot \frac{k}{t}}}{\frac{\sin k}{\frac{\ell}{t}}}}\]
    9. Using strategy rm

    10. Applied associate-/r/33.2

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

    11. Applied *-un-lft-identity33.2

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

    12. Applied div-inv33.2

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

    13. Applied times-frac28.7

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

    14. Applied times-frac30.4

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

    15. Simplified30.4

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

    16. Simplified19.5

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

  4. Final simplification7.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.6331121973565042 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{\frac{1}{t}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(\frac{2}{\tan k} \cdot \frac{\frac{\ell}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\\ \mathbf{elif}\;t \le 3.0124796597905178 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{1}{\left(t \cdot t\right) \cdot 2 + k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{t}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(\frac{2}{\tan k} \cdot \frac{\frac{\ell}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 
(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))))