Average Error: 32.1 → 4.0
Time: 1.2m
Precision: 64
Internal Precision: 320
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -8.8045644332429 \cdot 10^{+55}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right)}\\ \mathbf{elif}\;t \le 1.2554317064462554 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{\sin k}{\ell}\right) \cdot \left(\frac{{t}^{3}}{\frac{\ell}{2}} + \frac{k \cdot t}{\frac{\ell}{k}}\right)} \cdot \cos k\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{t}}}}{(2 \cdot \left(\frac{t}{\ell} \cdot t\right) + \left(\frac{k}{\ell} \cdot k\right))_* \cdot \frac{\sin k}{\cos k}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if t < -8.8045644332429e+55

    1. Initial program 23.1

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

    2. Initial simplification6.8

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

    4. Applied times-frac6.9

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

    5. Applied associate-*l*6.6

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

    7. Applied div-inv6.6

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

    8. Applied associate-*l*1.8

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

    if -8.8045644332429e+55 < t < 1.2554317064462554e-18

    1. Initial program 45.1

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

    2. Initial simplification33.8

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

    4. Applied times-frac31.2

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

    5. Applied associate-*l*27.3

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

    6. Taylor expanded around -inf 16.8

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

    7. Simplified12.3

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

    9. Applied associate-*r/12.3

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

    10. Applied associate-*r/12.3

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

    11. Applied associate-/r/12.3

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

    12. Taylor expanded around -inf 32.4

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

    13. Simplified4.3

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

    if 1.2554317064462554e-18 < t

    1. Initial program 22.8

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

    2. Initial simplification8.7

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

    4. Applied times-frac7.8

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

    5. Applied associate-*l*6.8

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

    6. Taylor expanded around -inf 12.6

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

    7. Simplified5.8

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

    9. Applied associate-/r*5.6

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

  4. Final simplification4.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.8045644332429 \cdot 10^{+55}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right)}\\ \mathbf{elif}\;t \le 1.2554317064462554 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{\sin k}{\ell}\right) \cdot \left(\frac{{t}^{3}}{\frac{\ell}{2}} + \frac{k \cdot t}{\frac{\ell}{k}}\right)} \cdot \cos k\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{t}}}}{(2 \cdot \left(\frac{t}{\ell} \cdot t\right) + \left(\frac{k}{\ell} \cdot k\right))_* \cdot \frac{\sin k}{\cos k}}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018227 +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))))