Average Error: 32.9 → 11.2
Time: 2.3m
Precision: 64
Internal Precision: 384
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.971341254147522 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{t}{\ell} \cdot \tan k}}{\frac{t}{\ell} \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\\ \mathbf{if}\;t \le 5.343108230184179 \cdot 10^{-188}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t + t\right) \cdot t + k \cdot k\right) \cdot \left(\left(t \cdot \sin k\right) \cdot \tan k\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{t}{\ell} \cdot \tan k}}{\frac{t}{\ell} \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < -1.971341254147522e-67 or 5.343108230184179e-188 < t

    1. Initial program 25.8

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

    3. Applied unpow325.8

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

    4. Applied times-frac18.5

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

    5. Applied associate-*l*16.2

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

    7. Applied *-un-lft-identity16.2

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

    8. Applied associate-*r*16.2

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

    9. Applied simplify7.9

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

    10. Taylor expanded around 0 48.7

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

    11. Applied simplify6.9

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

    if -1.971341254147522e-67 < t < 5.343108230184179e-188

    1. Initial program 59.0

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

    3. Applied unpow359.0

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

    4. Applied times-frac53.0

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

    5. Applied associate-*l*52.8

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

    7. Applied *-un-lft-identity52.8

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

    8. Applied associate-*r*52.8

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

    9. Applied simplify45.0

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

    11. Applied associate-*r/45.0

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

    12. Applied associate-*l/54.4

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

    13. Applied frac-times59.4

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

    14. Applied associate-*l/58.9

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

    15. Applied simplify27.2

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

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' 
(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))))