Average Error: 32.1 → 12.3
Time: 3.8m
Precision: 64
Internal Precision: 384
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right)}{\frac{\ell}{t}}} \le -8.80134732207218 \cdot 10^{-195}:\\ \;\;\;\;\frac{2}{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{if}\;\frac{2}{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right)}{\frac{\ell}{t}}} \le 2.8429283236850934 \cdot 10^{+19}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\frac{t}{k}} + \left(t + t\right)\right) \cdot \frac{\tan k \cdot \left(t \cdot \frac{\sin k}{\frac{\ell}{t}}\right)}{\ell}}\\ \mathbf{if}\;\frac{2}{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right)}{\frac{\ell}{t}}} \le +\infty:\\ \;\;\;\;\frac{2}{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{2 + \frac{k}{t} \cdot \frac{k}{t}}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k \cdot t}}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if (/ 2 (/ (* (* (* (/ (sin k) (/ l t)) t) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) (/ l t))) < -8.80134732207218e-195 or 2.8429283236850934e+19 < (/ 2 (/ (* (* (* (/ (sin k) (/ l t)) t) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) (/ l t))) < +inf.0

    1. Initial program 45.4

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

    3. Applied add-cube-cbrt45.6

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

    4. Applied times-frac39.9

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

    5. Applied simplify39.7

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

    6. Applied simplify20.4

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

    8. Applied associate-*l/20.4

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

    9. Applied associate-*l/12.8

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

    10. Applied associate-*l/17.0

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

    11. Applied associate-*l/13.1

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

    13. Applied *-un-lft-identity13.1

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

    14. Applied associate-*r*13.1

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

    15. Applied simplify7.8

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

    if -8.80134732207218e-195 < (/ 2 (/ (* (* (* (/ (sin k) (/ l t)) t) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) (/ l t))) < 2.8429283236850934e+19

    1. Initial program 22.2

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

    3. Applied add-cube-cbrt22.2

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

    4. Applied times-frac19.9

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

    5. Applied simplify19.9

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

    6. Applied simplify11.5

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

    8. Applied associate-*l/11.5

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

    9. Applied associate-*l/10.5

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

    10. Applied associate-*l/9.7

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

    11. Applied associate-*l/8.8

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

    13. Applied *-un-lft-identity8.8

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

    14. Applied associate-*r*8.8

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

    15. Applied simplify6.3

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

    17. Applied div-inv6.3

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

    18. Applied times-frac6.8

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

    19. Applied simplify5.5

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

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

    1. Initial program 62.5

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

    3. Applied add-cube-cbrt62.5

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

    4. Applied times-frac62.5

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

    5. Applied simplify62.5

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

    6. Applied simplify62.5

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

    8. Applied associate-*l/62.5

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

    9. Applied associate-*l/62.5

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

    10. Applied associate-*l/62.5

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

    11. Applied associate-*l/62.4

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

    13. Applied *-un-lft-identity62.4

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

    14. Applied associate-*r*62.4

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

    15. Applied simplify62.5

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

    16. Taylor expanded around 0 62.5

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

    17. Applied simplify53.9

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{2 + \frac{k}{t} \cdot \frac{k}{t}}{\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t \cdot k}}}}\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify12.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{2}{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right)}{\frac{\ell}{t}}} \le -8.80134732207218 \cdot 10^{-195}:\\ \;\;\;\;\frac{2}{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{if}\;\frac{2}{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right)}{\frac{\ell}{t}}} \le 2.8429283236850934 \cdot 10^{+19}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\frac{t}{k}} + \left(t + t\right)\right) \cdot \frac{\tan k \cdot \left(t \cdot \frac{\sin k}{\frac{\ell}{t}}\right)}{\ell}}\\ \mathbf{if}\;\frac{2}{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right)}{\frac{\ell}{t}}} \le +\infty:\\ \;\;\;\;\frac{2}{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{2 + \frac{k}{t} \cdot \frac{k}{t}}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k \cdot t}}}\\ \end{array}}\]

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' 
(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))))