Average Error: 31.6 → 11.8
Time: 1.5m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.468498454543345 \cdot 10^{-161} \lor \neg \left(t \le 2.3614191920726883 \cdot 10^{-88}\right):\\ \;\;\;\;\frac{2}{\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t}{{\ell}^{2} \cdot \cos k} + \frac{{\left(\sin k\right)}^{2} \cdot {t}^{3}}{{\ell}^{2} \cdot \cos k} \cdot 2}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < -1.468498454543345e-161 or 2.3614191920726883e-88 < t

    1. Initial program 24.3

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

    3. Applied cube-mult24.3

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

    4. Applied times-frac16.9

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

    5. Applied associate-*l*14.9

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

    7. Applied pow114.9

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

    8. Applied pow114.9

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

    9. Applied pow114.9

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

    10. Applied pow-prod-down14.9

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

    11. Applied pow-prod-down14.9

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

    12. Simplified10.1

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

    14. Applied *-un-lft-identity10.1

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

    15. Applied associate-/r*10.1

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

    16. Simplified5.2

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

    18. Applied associate-*l*5.1

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

    if -1.468498454543345e-161 < t < 2.3614191920726883e-88

    1. Initial program 61.0

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

    2. Taylor expanded around inf 38.7

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

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.468498454543345 \cdot 10^{-161} \lor \neg \left(t \le 2.3614191920726883 \cdot 10^{-88}\right):\\ \;\;\;\;\frac{2}{\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t}{{\ell}^{2} \cdot \cos k} + \frac{{\left(\sin k\right)}^{2} \cdot {t}^{3}}{{\ell}^{2} \cdot \cos k} \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019030 +o rules:numerics
(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))))