Average Error: 47.7 → 8.0
Time: 33.6s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -1.1653754517657999 \cdot 10^{-138} \lor \neg \left(k \le 2.2473460020406468 \cdot 10^{-14}\right):\\ \;\;\;\;\ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}^{1} \cdot \left(\left(\ell \cdot \frac{\cos k}{{\left(\sin k\right)}^{2}}\right) \cdot {\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1} \cdot {\left({t}^{\left(-1\right)}\right)}^{1}\right)}^{1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1} \cdot \left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1} \cdot {\left({t}^{\left(-1\right)}\right)}^{1}\right)\right)}^{1} \cdot \left(\frac{1}{\sin k} \cdot \left(\ell \cdot \frac{\cos k}{\sin k}\right)\right)\right)\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if k < -1.1653754517657999e-138 or 2.2473460020406468e-14 < k

    1. Initial program 45.6

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

    2. Simplified35.7

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

    3. Taylor expanded around inf 52.9

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

    4. Simplified14.1

      \[\leadsto \ell \cdot \color{blue}{\left(2 \cdot \left({\left({\left({k}^{\left(-2\right)}\right)}^{1} \cdot {\left({t}^{\left(-1\right)}\right)}^{1}\right)}^{1} \cdot \left(\frac{\cos k}{{\left(\sin k\right)}^{2}} \cdot \ell\right)\right)\right)}\]
    5. Using strategy rm

    6. Applied sqr-pow14.1

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

    7. Applied unpow-prod-down14.1

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

    8. Applied associate-*l*10.5

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

    9. Simplified10.5

      \[\leadsto \ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1} \cdot \color{blue}{\left({\left({t}^{\left(-1\right)}\right)}^{1} \cdot {\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{{\left(\sin k\right)}^{2}} \cdot \ell\right)\right)\right)\]
    10. Using strategy rm

    11. Applied unpow-prod-down10.5

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

    12. Applied associate-*l*5.8

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

    13. Simplified6.5

      \[\leadsto \ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}^{1} \cdot \color{blue}{\left(\ell \cdot \left(\frac{\cos k}{{\left(\sin k\right)}^{2}} \cdot {\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1} \cdot {\left({t}^{\left(-1\right)}\right)}^{1}\right)}^{1}\right)\right)}\right)\right)\]
    14. Using strategy rm

    15. Applied associate-*r*5.8

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

    if -1.1653754517657999e-138 < k < 2.2473460020406468e-14

    1. Initial program 61.9

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

    2. Simplified57.7

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

    3. Taylor expanded around inf 47.8

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

    4. Simplified27.8

      \[\leadsto \ell \cdot \color{blue}{\left(2 \cdot \left({\left({\left({k}^{\left(-2\right)}\right)}^{1} \cdot {\left({t}^{\left(-1\right)}\right)}^{1}\right)}^{1} \cdot \left(\frac{\cos k}{{\left(\sin k\right)}^{2}} \cdot \ell\right)\right)\right)}\]
    5. Using strategy rm

    6. Applied sqr-pow27.9

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

    7. Applied unpow-prod-down27.9

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

    8. Applied associate-*l*27.9

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

    9. Simplified27.9

      \[\leadsto \ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1} \cdot \color{blue}{\left({\left({t}^{\left(-1\right)}\right)}^{1} \cdot {\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{{\left(\sin k\right)}^{2}} \cdot \ell\right)\right)\right)\]
    10. Using strategy rm

    11. Applied unpow227.9

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

    12. Applied *-un-lft-identity27.9

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

    13. Applied times-frac28.0

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

    14. Applied associate-*l*23.1

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

    15. Simplified23.1

      \[\leadsto \ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1} \cdot \left({\left({t}^{\left(-1\right)}\right)}^{1} \cdot {\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)\right)}^{1} \cdot \left(\frac{1}{\sin k} \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\sin k}\right)}\right)\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.1653754517657999 \cdot 10^{-138} \lor \neg \left(k \le 2.2473460020406468 \cdot 10^{-14}\right):\\ \;\;\;\;\ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}^{1} \cdot \left(\left(\ell \cdot \frac{\cos k}{{\left(\sin k\right)}^{2}}\right) \cdot {\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1} \cdot {\left({t}^{\left(-1\right)}\right)}^{1}\right)}^{1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1} \cdot \left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1} \cdot {\left({t}^{\left(-1\right)}\right)}^{1}\right)\right)}^{1} \cdot \left(\frac{1}{\sin k} \cdot \left(\ell \cdot \frac{\cos k}{\sin k}\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))