Average Error: 30.9 → 4.9
Time: 1.7m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right) \le -2.6294014879752607 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\frac{\sin k}{\cos k} \cdot (2 \cdot \left(\frac{t}{\ell} \cdot t\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_*}\\ \mathbf{elif}\;\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right) \le 6.179793228191375 \cdot 10^{+296}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k}{\cos k} \cdot (2 \cdot \left(\frac{t}{\ell} \cdot t\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_*\right) \cdot t\right) \cdot \frac{\sin k}{\ell}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if (* (/ (sin k) (/ l t)) (* t (* (/ 1 (/ l t)) (fma (fma (/ k t) (/ k t) 1) (tan k) (tan k))))) < -2.6294014879752607e+307

    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 simplification15.2

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

    4. Applied times-frac15.1

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

    5. Applied associate-*l*14.3

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

    6. Taylor expanded around inf 7.2

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

    7. Simplified5.4

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

    9. Applied associate-/r*5.1

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

    10. Taylor expanded around inf 5.5

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

    if -2.6294014879752607e+307 < (* (/ (sin k) (/ l t)) (* t (* (/ 1 (/ l t)) (fma (fma (/ k t) (/ k t) 1) (tan k) (tan k))))) < 6.179793228191375e+296

    1. Initial program 46.1

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

    2. Initial simplification17.9

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

    4. Applied times-frac12.8

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

    5. Applied associate-*l*8.2

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

    7. Applied div-inv8.3

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

    8. Applied associate-*l*0.8

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

    if 6.179793228191375e+296 < (* (/ (sin k) (/ l t)) (* t (* (/ 1 (/ l t)) (fma (fma (/ k t) (/ k t) 1) (tan k) (tan k)))))

    1. Initial program 25.9

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

    2. Initial simplification20.3

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

    4. Applied times-frac20.3

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

    5. Applied associate-*l*19.7

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

    6. Taylor expanded around inf 13.8

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

    7. Simplified11.9

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

    9. Applied associate-/r/11.4

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

    10. Applied associate-*l*7.8

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

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right) \le -2.6294014879752607 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\frac{\sin k}{\cos k} \cdot (2 \cdot \left(\frac{t}{\ell} \cdot t\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_*}\\ \mathbf{elif}\;\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right) \le 6.179793228191375 \cdot 10^{+296}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k}{\cos k} \cdot (2 \cdot \left(\frac{t}{\ell} \cdot t\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_*\right) \cdot t\right) \cdot \frac{\sin k}{\ell}}\\ \end{array}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018225 +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))))