Average Error: 32.1 → 8.1
Time: 4.7m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -19796072.644097697:\\ \;\;\;\;\left(\frac{\frac{\ell}{t}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{2}{\tan k}\right) \cdot \frac{\frac{\frac{\frac{1}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}}{\frac{\sin k}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \le 3.628422635945199 \cdot 10^{-134}:\\ \;\;\;\;\left(\frac{2}{\tan k} \cdot \frac{\frac{\ell}{t}}{\frac{\sin k}{\ell}}\right) \cdot \frac{1}{k \cdot k + 2 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{t}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{2}{\tan k}\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if t < -19796072.644097697

    1. Initial program 23.0

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

    2. Simplified11.0

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

    4. Applied *-un-lft-identity11.0

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

    5. Applied div-inv11.0

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

    6. Applied times-frac11.0

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

    7. Simplified11.0

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

    8. Simplified7.5

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

    10. Applied *-un-lft-identity7.5

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

    11. Applied add-sqr-sqrt7.6

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

    12. Applied div-inv7.6

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

    13. Applied times-frac7.6

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

    14. Applied times-frac6.0

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

    15. Applied associate-*r*2.1

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

    17. Applied add-sqr-sqrt2.1

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

    18. Applied associate-/r*2.1

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

    if -19796072.644097697 < t < 3.628422635945199e-134

    1. Initial program 52.4

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

    2. Simplified39.3

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

    4. Applied *-un-lft-identity39.3

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

    5. Applied div-inv39.3

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

    6. Applied times-frac38.6

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

    7. Simplified38.6

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

    8. Simplified33.7

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

    10. Applied associate-/r/33.7

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

    11. Applied *-un-lft-identity33.7

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

    12. Applied div-inv33.7

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

    13. Applied times-frac29.4

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

    14. Applied times-frac30.5

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

    15. Applied associate-*r*30.5

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

    16. Simplified19.0

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

    if 3.628422635945199e-134 < t

    1. Initial program 24.8

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

    2. Simplified14.1

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

    4. Applied *-un-lft-identity14.1

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

    5. Applied div-inv14.1

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

    6. Applied times-frac14.1

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

    7. Simplified14.1

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

    8. Simplified9.1

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

    10. Applied *-un-lft-identity9.1

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

    11. Applied add-sqr-sqrt9.2

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

    12. Applied div-inv9.2

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

    13. Applied times-frac9.1

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

    14. Applied times-frac7.8

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

    15. Applied associate-*r*5.0

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

    17. Applied add-sqr-sqrt5.0

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

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -19796072.644097697:\\ \;\;\;\;\left(\frac{\frac{\ell}{t}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{2}{\tan k}\right) \cdot \frac{\frac{\frac{\frac{1}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}}{\frac{\sin k}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \le 3.628422635945199 \cdot 10^{-134}:\\ \;\;\;\;\left(\frac{2}{\tan k} \cdot \frac{\frac{\ell}{t}}{\frac{\sin k}{\ell}}\right) \cdot \frac{1}{k \cdot k + 2 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{t}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{2}{\tan k}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 
(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))))