Average Error: 32.1 → 10.4
Time: 4.5m
Precision: 64
Internal Precision: 384
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.8062231940298976 \cdot 10^{-149}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \mathbf{if}\;t \le 2.410588482727809 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(t + t\right) \cdot t + k \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < -1.8062231940298976e-149 or 2.410588482727809e-72 < t

    1. Initial program 24.1

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

    3. Applied unpow324.1

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

    4. Applied times-frac16.8

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

    5. Applied associate-*l*14.5

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

    7. Applied *-un-lft-identity14.5

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

    8. Applied associate-*r*14.5

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

    9. Applied simplify6.9

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

    11. Applied associate-*l*5.9

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

    if -1.8062231940298976e-149 < t < 2.410588482727809e-72

    1. Initial program 59.9

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

    3. Applied unpow359.9

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

    4. Applied times-frac53.5

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

    5. Applied associate-*l*53.3

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

    7. Applied *-un-lft-identity53.3

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

    8. Applied associate-*r*53.3

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

    9. Applied simplify45.6

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

    11. Applied associate-*r/45.6

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

    12. Applied associate-*l/55.5

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

    13. Applied frac-times60.3

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

    14. Applied associate-*l/60.1

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

    15. Applied simplify26.3

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

Runtime

Time bar (total: 4.5m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(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))))