Average Error: 32.1 → 4.5
Time: 1.5m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right) \le -2.36471331288109 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{\sin k}{\cos k} \cdot (2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_*}\\ \mathbf{elif}\;\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right) \le 1.3463305703141285 \cdot 10^{+307}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t \cdot \left(\frac{\sin k}{\cos k} \cdot (2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_*\right)\right)}\\ \end{array}\]

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))))) < -2.36471331288109e+261

    1. Initial program 24.5

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

    2. Initial simplification16.9

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

    4. Applied times-frac16.9

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

    5. Applied associate-*l*16.0

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

    6. Taylor expanded around inf 8.3

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

    7. Simplified5.7

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

    9. Applied associate-/r*5.5

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

    if -2.36471331288109e+261 < (* (/ (sin k) (/ l t)) (* t (* (/ 1 (/ l t)) (fma (fma (/ k t) (/ k t) 1) (tan k) (tan k))))) < 1.3463305703141285e+307

    1. Initial program 46.4

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

    2. Initial simplification18.4

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

    4. Applied times-frac13.8

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

    5. Applied associate-*l*9.3

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

    7. Applied div-inv9.3

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

    8. Applied associate-*l*0.7

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

    if 1.3463305703141285e+307 < (* (/ (sin k) (/ l t)) (* t (* (/ 1 (/ l t)) (fma (fma (/ k t) (/ k t) 1) (tan k) (tan k)))))

    1. Initial program 27.3

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

    2. Initial simplification21.1

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

    4. Applied times-frac21.1

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

    5. Applied associate-*l*20.4

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

    6. Taylor expanded around inf 13.0

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

    7. Simplified11.3

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

    9. Applied associate-/r/10.8

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

    10. Applied associate-*l*6.8

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

  4. Final simplification4.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right) \le -2.36471331288109 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{\sin k}{\cos k} \cdot (2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_*}\\ \mathbf{elif}\;\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right) \le 1.3463305703141285 \cdot 10^{+307}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot \frac{1}{\frac{\ell}{t}}\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t \cdot \left(\frac{\sin k}{\cos k} \cdot (2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_*\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018216 +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))))