Average Error: 47.0 → 8.2
Time: 1.8m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -1.7160787418538492 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{t}{\sin k} \cdot \left(\left(\frac{\ell}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\frac{\cos k}{\sin k}}{k}\right)\right) \cdot 2\right)}{k}\\ \mathbf{elif}\;\ell \le -9.961281095666055 \cdot 10^{-190}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\cos k}{t \cdot k}\right)}{k}\\ \mathbf{elif}\;\ell \le -9.700569840782823 \cdot 10^{-293}:\\ \;\;\;\;\frac{t}{\sin k} \cdot \frac{\left(\frac{2}{\tan k} \cdot \ell\right) \cdot \frac{\ell}{t \cdot k}}{t \cdot k}\\ \mathbf{elif}\;\ell \le 8.080425029828749 \cdot 10^{+152}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\cos k}{t \cdot k}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t}{\sin k} \cdot \left(\left(\frac{\ell}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\frac{\cos k}{\sin k}}{k}\right)\right) \cdot 2\right)}{k}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if l < -1.7160787418538492e+130 or 8.080425029828749e+152 < l

    1. Initial program 60.5

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

    2. Simplified40.0

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

    4. Applied associate-*r/42.7

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

    5. Applied associate-/r/42.7

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

    6. Applied times-frac42.9

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{t} \cdot k}}{\tan k \cdot \frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}} \cdot \frac{t}{\sin k}}\]

    7. Simplified37.3

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

    8. Taylor expanded around inf 58.0

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

    9. Simplified26.9

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

    11. Applied associate-*l/22.7

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

    if -1.7160787418538492e+130 < l < -9.961281095666055e-190 or -9.700569840782823e-293 < l < 8.080425029828749e+152

    1. Initial program 43.8

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

    2. Simplified29.1

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

    4. Applied associate-*r/29.2

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

    5. Applied associate-/r/29.1

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

    6. Applied times-frac25.6

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{t} \cdot k}}{\tan k \cdot \frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}} \cdot \frac{t}{\sin k}}\]

    7. Simplified18.7

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

    8. Taylor expanded around inf 24.8

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

    9. Simplified13.8

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

    11. Applied associate-*l/12.7

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

    12. Taylor expanded around inf 9.4

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

    13. Simplified4.4

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

    if -9.961281095666055e-190 < l < -9.700569840782823e-293

    1. Initial program 46.2

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

    2. Simplified24.6

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

    4. Applied associate-*r/24.6

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

    5. Applied associate-/r/24.6

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

    6. Applied times-frac19.2

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{t} \cdot k}}{\tan k \cdot \frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}} \cdot \frac{t}{\sin k}}\]

    7. Simplified14.2

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

    9. Applied associate-*l/16.6

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

    10. Applied associate-*r/16.6

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

    11. Applied associate-*r/16.6

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

    12. Applied associate-/l/17.3

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

    13. Simplified10.3

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.7160787418538492 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{t}{\sin k} \cdot \left(\left(\frac{\ell}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\frac{\cos k}{\sin k}}{k}\right)\right) \cdot 2\right)}{k}\\ \mathbf{elif}\;\ell \le -9.961281095666055 \cdot 10^{-190}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\cos k}{t \cdot k}\right)}{k}\\ \mathbf{elif}\;\ell \le -9.700569840782823 \cdot 10^{-293}:\\ \;\;\;\;\frac{t}{\sin k} \cdot \frac{\left(\frac{2}{\tan k} \cdot \ell\right) \cdot \frac{\ell}{t \cdot k}}{t \cdot k}\\ \mathbf{elif}\;\ell \le 8.080425029828749 \cdot 10^{+152}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\cos k}{t \cdot k}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t}{\sin k} \cdot \left(\left(\frac{\ell}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\frac{\cos k}{\sin k}}{k}\right)\right) \cdot 2\right)}{k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 
(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))))