Average Error: 31.7 → 6.3
Time: 1.4m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{t}}}}{(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_* \cdot \frac{\sin k}{\cos k}} \le -1.320886769440088 \cdot 10^{-203}:\\ \;\;\;\;\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{t}}}}{(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_* \cdot \frac{\sin k}{\cos k}}\\ \mathbf{elif}\;\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{t}}}}{(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_* \cdot \frac{\sin k}{\cos k}} \le 4.203719975492003 \cdot 10^{-271}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot (\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_*\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\sin k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{t}}}}{(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_* \cdot \frac{\sin k}{\cos k}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ 2 (/ (sin k) (/ l t))) (* (fma 2 (* (/ t l) t) (/ k (/ l k))) (/ (sin k) (cos k)))) < -1.320886769440088e-203 or 4.203719975492003e-271 < (/ (/ 2 (/ (sin k) (/ l t))) (* (fma 2 (* (/ t l) t) (/ k (/ l k))) (/ (sin k) (cos k))))

    1. Initial program 50.0

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

    2. Initial simplification30.4

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

    4. Applied times-frac24.9

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

    5. Applied associate-*l*21.4

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

    6. Taylor expanded around inf 24.2

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

    7. Simplified7.6

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

    9. Applied associate-/r*7.6

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

    if -1.320886769440088e-203 < (/ (/ 2 (/ (sin k) (/ l t))) (* (fma 2 (* (/ t l) t) (/ k (/ l k))) (/ (sin k) (cos k)))) < 4.203719975492003e-271

    1. Initial program 21.0

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

    2. Initial simplification11.6

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

    4. Applied times-frac12.5

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

    5. Applied associate-*l*11.2

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

    7. Applied associate-/r/11.2

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

    8. Applied associate-*l*5.5

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{t}}}}{(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_* \cdot \frac{\sin k}{\cos k}} \le -1.320886769440088 \cdot 10^{-203}:\\ \;\;\;\;\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{t}}}}{(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_* \cdot \frac{\sin k}{\cos k}}\\ \mathbf{elif}\;\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{t}}}}{(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_* \cdot \frac{\sin k}{\cos k}} \le 4.203719975492003 \cdot 10^{-271}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot (\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_*\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\sin k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{t}}}}{(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \left(\frac{k}{\frac{\ell}{k}}\right))_* \cdot \frac{\sin k}{\cos k}}\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018217 +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))))