Average Error: 32.8 → 3.7
Time: 1.7m
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}\;\frac{\frac{2}{\frac{\sin k}{\frac{\cos k}{\sin k}}}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot 2\right) + \left(\frac{\frac{t}{\frac{\ell}{k}}}{\frac{\ell}{k}}\right))_*} \le -1.3415735269496097 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{2}{\frac{t \cdot \sin k}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{(\left(\tan k\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) + \left(\tan k\right))_*}\\ \mathbf{elif}\;\frac{\frac{2}{\frac{\sin k}{\frac{\cos k}{\sin k}}}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot 2\right) + \left(\frac{\frac{t}{\frac{\ell}{k}}}{\frac{\ell}{k}}\right))_*} \le -1.551048288954508 \cdot 10^{-276}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot 2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + \frac{\frac{t}{\frac{\ell}{k}}}{\frac{\ell}{k}}\right) \cdot \frac{\sin k}{\frac{\cos k}{\sin k}}}\\ \mathbf{elif}\;\frac{\frac{2}{\frac{\sin k}{\frac{\cos k}{\sin k}}}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot 2\right) + \left(\frac{\frac{t}{\frac{\ell}{k}}}{\frac{\ell}{k}}\right))_*} \le 4.4178162867776 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{2}{(\left(\tan k\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) + \left(\tan k\right))_* \cdot \left(t \cdot \sin k\right)}\\ \mathbf{elif}\;\frac{\frac{2}{\frac{\sin k}{\frac{\cos k}{\sin k}}}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot 2\right) + \left(\frac{\frac{t}{\frac{\ell}{k}}}{\frac{\ell}{k}}\right))_*} \le 1.7311082043702188 \cdot 10^{+288}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot 2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + \frac{\frac{t}{\frac{\ell}{k}}}{\frac{\ell}{k}}\right) \cdot \frac{\sin k}{\frac{\cos k}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}} \cdot \frac{\sqrt[3]{\sin k}}{\frac{1}{\sin k}} + \frac{\sin k \cdot \left(t \cdot 2\right)}{\frac{\ell}{t}} \cdot \frac{\sqrt[3]{\sin k}}{\frac{\ell}{t}}\right) \cdot \frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\cos k}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 4 regimes

  2. if (/ (/ 2 (/ (sin k) (/ (cos k) (sin k)))) (fma (* (/ t l) (/ t l)) (* t 2) (/ (/ t (/ l k)) (/ l k)))) < -1.3415735269496097e+305

    1. Initial program 42.6

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

    2. Initial simplification8.8

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

    4. Applied associate-/r*8.9

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

    if -1.3415735269496097e+305 < (/ (/ 2 (/ (sin k) (/ (cos k) (sin k)))) (fma (* (/ t l) (/ t l)) (* t 2) (/ (/ t (/ l k)) (/ l k)))) < -1.551048288954508e-276 or 4.4178162867776e-310 < (/ (/ 2 (/ (sin k) (/ (cos k) (sin k)))) (fma (* (/ t l) (/ t l)) (* t 2) (/ (/ t (/ l k)) (/ l k)))) < 1.7311082043702188e+288

    1. Initial program 51.1

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

    2. Initial simplification34.4

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

    3. Taylor expanded around inf 40.3

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

    4. Simplified9.1

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

    6. Applied associate-/r*1.0

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

    if -1.551048288954508e-276 < (/ (/ 2 (/ (sin k) (/ (cos k) (sin k)))) (fma (* (/ t l) (/ t l)) (* t 2) (/ (/ t (/ l k)) (/ l k)))) < 4.4178162867776e-310

    1. Initial program 17.7

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

    2. Initial simplification6.4

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

    4. Applied associate-*l/3.5

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

    5. Applied associate-/r/3.2

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

    if 1.7311082043702188e+288 < (/ (/ 2 (/ (sin k) (/ (cos k) (sin k)))) (fma (* (/ t l) (/ t l)) (* t 2) (/ (/ t (/ l k)) (/ l k))))

    1. Initial program 39.2

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

    2. Initial simplification25.9

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

    3. Taylor expanded around inf 60.8

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

    4. Simplified59.3

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

    6. Applied div-inv59.3

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

    7. Applied add-cube-cbrt59.3

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}}}{\cos k \cdot \frac{1}{\sin k}} \cdot \left(\frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}} + \frac{2 \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}\]

    8. Applied times-frac59.3

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\cos k} \cdot \frac{\sqrt[3]{\sin k}}{\frac{1}{\sin k}}\right)} \cdot \left(\frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}} + \frac{2 \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}\]

    9. Applied associate-*l*42.4

      \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\cos k} \cdot \left(\frac{\sqrt[3]{\sin k}}{\frac{1}{\sin k}} \cdot \left(\frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}} + \frac{2 \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)\right)}}\]
    10. Using strategy rm

    11. Applied distribute-lft-in42.4

      \[\leadsto \frac{2}{\frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\cos k} \cdot \color{blue}{\left(\frac{\sqrt[3]{\sin k}}{\frac{1}{\sin k}} \cdot \frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}} + \frac{\sqrt[3]{\sin k}}{\frac{1}{\sin k}} \cdot \frac{2 \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}}\]

    12. Simplified12.1

      \[\leadsto \frac{2}{\frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\cos k} \cdot \left(\frac{\sqrt[3]{\sin k}}{\frac{1}{\sin k}} \cdot \frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}} + \color{blue}{\frac{\sqrt[3]{\sin k}}{\frac{\ell}{t}} \cdot \frac{\sin k \cdot \left(t \cdot 2\right)}{\frac{\ell}{t}}}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{\frac{\sin k}{\frac{\cos k}{\sin k}}}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot 2\right) + \left(\frac{\frac{t}{\frac{\ell}{k}}}{\frac{\ell}{k}}\right))_*} \le -1.3415735269496097 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{2}{\frac{t \cdot \sin k}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{(\left(\tan k\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) + \left(\tan k\right))_*}\\ \mathbf{elif}\;\frac{\frac{2}{\frac{\sin k}{\frac{\cos k}{\sin k}}}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot 2\right) + \left(\frac{\frac{t}{\frac{\ell}{k}}}{\frac{\ell}{k}}\right))_*} \le -1.551048288954508 \cdot 10^{-276}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot 2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + \frac{\frac{t}{\frac{\ell}{k}}}{\frac{\ell}{k}}\right) \cdot \frac{\sin k}{\frac{\cos k}{\sin k}}}\\ \mathbf{elif}\;\frac{\frac{2}{\frac{\sin k}{\frac{\cos k}{\sin k}}}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot 2\right) + \left(\frac{\frac{t}{\frac{\ell}{k}}}{\frac{\ell}{k}}\right))_*} \le 4.4178162867776 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{2}{(\left(\tan k\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) + \left(\tan k\right))_* \cdot \left(t \cdot \sin k\right)}\\ \mathbf{elif}\;\frac{\frac{2}{\frac{\sin k}{\frac{\cos k}{\sin k}}}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot 2\right) + \left(\frac{\frac{t}{\frac{\ell}{k}}}{\frac{\ell}{k}}\right))_*} \le 1.7311082043702188 \cdot 10^{+288}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot 2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + \frac{\frac{t}{\frac{\ell}{k}}}{\frac{\ell}{k}}\right) \cdot \frac{\sin k}{\frac{\cos k}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}} \cdot \frac{\sqrt[3]{\sin k}}{\frac{1}{\sin k}} + \frac{\sin k \cdot \left(t \cdot 2\right)}{\frac{\ell}{t}} \cdot \frac{\sqrt[3]{\sin k}}{\frac{\ell}{t}}\right) \cdot \frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\cos k}}\\ \end{array}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018215 +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))))