Average Error: 32.1 → 5.5
Time: 1.4m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.674970086705644 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right)}\\ \mathbf{elif}\;t \le 3.680141818671534 \cdot 10^{+19}:\\ \;\;\;\;\frac{2}{\left(\frac{\sqrt[3]{\sin k}}{\frac{1}{t}} \cdot \left(\frac{\sin k}{\cos k} \cdot (2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \left(\frac{k}{\ell} \cdot k\right))_*\right)\right) \cdot \frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot (\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_*}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if t < -3.674970086705644e-14

    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 simplification9.1

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

    4. Applied times-frac8.3

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

    5. Applied associate-*l*7.3

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

    7. Applied div-inv7.3

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

    8. Applied associate-*l*2.7

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

    if -3.674970086705644e-14 < t < 3.680141818671534e+19

    1. Initial program 47.5

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

    2. Initial simplification34.8

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

    4. Applied times-frac31.7

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

    5. Applied associate-*l*27.8

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

    6. Taylor expanded around inf 18.0

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

    7. Simplified12.8

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

    9. Applied div-inv12.8

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

    10. Applied add-cube-cbrt13.1

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

    11. Applied times-frac12.9

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

    12. Applied associate-*l*10.4

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

    if 3.680141818671534e+19 < t

    1. Initial program 22.2

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

    2. Initial simplification7.8

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

    4. Applied times-frac7.4

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

    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 associate-*l/2.4

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

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.674970086705644 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right)}\\ \mathbf{elif}\;t \le 3.680141818671534 \cdot 10^{+19}:\\ \;\;\;\;\frac{2}{\left(\frac{\sqrt[3]{\sin k}}{\frac{1}{t}} \cdot \left(\frac{\sin k}{\cos k} \cdot (2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \left(\frac{k}{\ell} \cdot k\right))_*\right)\right) \cdot \frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot (\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_*}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}}\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018221 +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))))