Average Error: 32.2 → 5.8
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -669830280.6828679 \lor \neg \left(t \le 1.2807341793307659 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot t\right) \cdot \frac{\sin k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t \cdot \left(\frac{\sin k}{\cos k} \cdot (2 \cdot \left(\frac{t}{\ell} \cdot t\right) + \left(\frac{k}{\ell} \cdot k\right))_*\right)\right)}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < -669830280.6828679 or 1.2807341793307659e+70 < t

    1. Initial program 22.6

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

    2. Initial simplification7.2

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

    4. Applied times-frac6.8

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

    5. Applied associate-*l*6.1

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

    7. Applied associate-*l/2.7

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

    8. Applied associate-*r/3.1

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

    9. Applied associate-/r/3.0

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

    if -669830280.6828679 < t < 1.2807341793307659e+70

    1. Initial program 44.1

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

    2. Initial simplification33.3

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

    4. Applied times-frac30.2

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

    5. Applied associate-*l*26.0

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

    6. Taylor expanded around inf 16.7

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

    7. Simplified12.2

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

    9. Applied associate-/r/11.9

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

    10. Applied associate-*l*9.3

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

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -669830280.6828679 \lor \neg \left(t \le 1.2807341793307659 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot t\right) \cdot \frac{\sin k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t \cdot \left(\frac{\sin k}{\cos k} \cdot (2 \cdot \left(\frac{t}{\ell} \cdot t\right) + \left(\frac{k}{\ell} \cdot k\right))_*\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018234 +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))))