Average Error: 32.1 → 16.0
Time: 2.1m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -2.782015650547342 \cdot 10^{+50}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(1 + {\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}^{3}\right)}{\left(\frac{\ell}{t} \cdot \cos k\right) \cdot \left(1 - \left(-{\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}}\\ \mathbf{elif}\;\ell \le 4.313367124248792 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\left(\frac{t \cdot k}{\ell} \cdot t\right) \cdot \sin k\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{elif}\;\ell \le 3.48848675235572 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\sin k \cdot \left(\left(\sin k \cdot t\right) \cdot t\right)\right)}{\ell}}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{elif}\;\ell \le 7.056352983768264 \cdot 10^{+130}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(1 + {\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}^{3}\right)}{\left(\frac{\ell}{t} \cdot \cos k\right) \cdot \left(1 - \left(-{\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \sqrt{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\right) \cdot \sqrt{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if l < -2.782015650547342e+50 or 3.48848675235572e+96 < l < 7.056352983768264e+130

    1. Initial program 47.9

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

    3. Applied unpow347.9

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

    4. Applied times-frac36.2

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

    5. Applied associate-*l*36.2

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

    7. Applied associate-/l*25.9

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

    9. Applied flip3-+32.7

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

    10. Applied tan-quot32.7

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

    11. Applied associate-*l/30.7

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

    12. Applied frac-times28.6

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

    13. Applied frac-times29.4

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

    14. Simplified29.4

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

    15. Simplified29.4

      \[\leadsto \frac{2}{\frac{\left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot t\right)\right) \cdot \left({\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + 1\right)}{\color{blue}{\left(1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(0 - {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\cos k \cdot \frac{\ell}{t}\right)}}}\]

    if -2.782015650547342e+50 < l < 4.313367124248792e-143

    1. Initial program 24.1

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

    3. Applied unpow324.1

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

    4. Applied times-frac18.8

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

    5. Applied associate-*l*16.2

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

    7. Applied associate-/l*12.0

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

    9. Applied tan-quot12.0

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

    10. Applied associate-*l/11.7

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

    11. Applied frac-times10.9

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

    12. Applied associate-*l/10.0

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

    13. Taylor expanded around 0 9.9

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

    if 4.313367124248792e-143 < l < 3.48848675235572e+96

    1. Initial program 24.3

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

    3. Applied unpow324.3

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

    4. Applied times-frac21.7

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

    5. Applied associate-*l*19.5

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

    7. Applied associate-/l*18.7

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

    9. Applied tan-quot18.7

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

    10. Applied associate-*l/17.9

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

    11. Applied frac-times15.0

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

    12. Applied associate-*l/13.4

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

    14. Applied associate-*l/13.4

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

    15. Applied associate-*r/14.6

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

    16. Applied associate-*l/15.7

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

    17. Applied associate-*l/15.9

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

    if 7.056352983768264e+130 < l

    1. Initial program 58.7

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

    3. Applied unpow358.7

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

    4. Applied times-frac41.1

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

    5. Applied associate-*l*41.1

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

    7. Applied associate-/l*23.2

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

    9. Applied add-sqr-sqrt23.4

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

    10. Applied associate-*r*23.3

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}\right) \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification16.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -2.782015650547342 \cdot 10^{+50}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(1 + {\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}^{3}\right)}{\left(\frac{\ell}{t} \cdot \cos k\right) \cdot \left(1 - \left(-{\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}}\\ \mathbf{elif}\;\ell \le 4.313367124248792 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\left(\frac{t \cdot k}{\ell} \cdot t\right) \cdot \sin k\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{elif}\;\ell \le 3.48848675235572 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\sin k \cdot \left(\left(\sin k \cdot t\right) \cdot t\right)\right)}{\ell}}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{elif}\;\ell \le 7.056352983768264 \cdot 10^{+130}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(1 + {\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}^{3}\right)}{\left(\frac{\ell}{t} \cdot \cos k\right) \cdot \left(1 - \left(-{\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \sqrt{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\right) \cdot \sqrt{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}}\\ \end{array}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018221 
(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))))