Average Error: 32.3 → 10.7
Time: 4.9m
Precision: 64
Internal Precision: 384
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.457802529893899 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{t}{\ell} \cdot \tan k}} \cdot \sqrt[3]{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{t}{\ell} \cdot \tan k}}}{\frac{t}{\ell}} \cdot \frac{\sqrt[3]{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{t}{\ell} \cdot \tan k}}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\\ \mathbf{if}\;t \le 1.2519919771043373 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(t + t\right) \cdot t + k \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{t}{\ell} \cdot \tan k}}{\frac{\left(t + t\right) + \frac{k}{t} \cdot k}{\ell}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if t < -5.457802529893899e-89

    1. Initial program 23.2

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

    3. Applied unpow323.2

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

    4. Applied times-frac16.7

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

    5. Applied associate-*l*14.3

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

    7. Applied *-un-lft-identity14.3

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

    8. Applied associate-*r*14.3

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

    9. Applied simplify6.0

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

    10. Taylor expanded around 0 63.1

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

    11. Applied simplify5.2

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

    13. Applied add-cube-cbrt5.5

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

    14. Applied times-frac5.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{t}{\ell} \cdot \tan k}} \cdot \sqrt[3]{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{t}{\ell} \cdot \tan k}}}{\frac{t}{\ell}} \cdot \frac{\sqrt[3]{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{t}{\ell} \cdot \tan k}}}{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]

    if -5.457802529893899e-89 < t < 1.2519919771043373e-138

    1. Initial program 60.9

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

    3. Applied unpow360.9

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

    4. Applied times-frac52.9

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

    5. Applied associate-*l*52.9

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

    7. Applied *-un-lft-identity52.9

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

    8. Applied associate-*r*52.9

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

    9. Applied simplify45.4

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

    11. Applied associate-*r/45.4

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

    12. Applied associate-*l/55.6

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

    13. Applied frac-times61.2

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

    14. Applied associate-*l/61.0

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

    15. Applied simplify27.1

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

    if 1.2519919771043373e-138 < t

    1. Initial program 25.3

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

    3. Applied unpow325.3

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

    4. Applied times-frac17.2

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

    5. Applied associate-*l*15.1

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

    7. Applied *-un-lft-identity15.1

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

    8. Applied associate-*r*15.1

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

    9. Applied simplify7.2

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

    10. Taylor expanded around 0 35.5

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

    11. Applied simplify6.0

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

    13. Applied associate-*l/6.7

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

    14. Applied simplify6.3

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

Runtime

Time bar (total: 4.9m)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(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))))