Average Error: 47.0 → 2.1
Time: 4.3m
Precision: 64
Internal Precision: 4224
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\ell}{k}}{\sin k}}{\frac{t}{\ell} \cdot k} \cdot \frac{\cos k + \cos k}{\sin k} \le -7.490564257637328 \cdot 10^{-291}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{t}{\frac{\ell}{k}}} \cdot \frac{\cos k + \cos k}{\sin k \cdot \sin k}\\ \mathbf{if}\;\frac{\frac{\frac{\ell}{k}}{\sin k}}{\frac{t}{\ell} \cdot k} \cdot \frac{\cos k + \cos k}{\sin k} \le 8.026960324673611 \cdot 10^{-289}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\cos k}{\sin k} \cdot \frac{\frac{2}{t}}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{t}{\frac{\ell}{k}}} \cdot \frac{\cos k + \cos k}{\sin k \cdot \sin k}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if (* (/ (/ (/ l k) (sin k)) (* (/ t l) k)) (/ (+ (cos k) (cos k)) (sin k))) < -7.490564257637328e-291 or 8.026960324673611e-289 < (* (/ (/ (/ l k) (sin k)) (* (/ t l) k)) (/ (+ (cos k) (cos k)) (sin k)))

    1. Initial program 56.9

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

    3. Applied associate-*l*56.7

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

    4. Applied simplify53.9

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

    5. Taylor expanded around inf 37.9

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

    6. Taylor expanded around -inf 62.6

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

    7. Applied simplify13.0

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{\cos k + \cos k}{\sin k \cdot \sin k}}\]
    8. Using strategy rm

    9. Applied associate-/l*2.6

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{\frac{t}{\frac{\ell}{k}}}} \cdot \frac{\cos k + \cos k}{\sin k \cdot \sin k}\]

    if -7.490564257637328e-291 < (* (/ (/ (/ l k) (sin k)) (* (/ t l) k)) (/ (+ (cos k) (cos k)) (sin k))) < 8.026960324673611e-289

    1. Initial program 37.5

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

    3. Applied associate-*l*37.5

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

    4. Applied simplify26.6

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

    5. Taylor expanded around inf 8.0

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

    6. Taylor expanded around -inf 62.9

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

    7. Applied simplify6.0

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{\cos k + \cos k}{\sin k \cdot \sin k}}\]
    8. Using strategy rm

    9. Applied div-inv6.0

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

    10. Applied associate-*l*4.2

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

    11. Applied simplify1.5

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

Runtime

Time bar (total: 4.3m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' +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))))