Average Error: 31.9 → 12.3
Time: 3.8m
Precision: 64
Internal Precision: 320
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left({1}^{3} + {\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}^{3}\right) \cdot \left(\tan k \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)}{\frac{\ell}{t} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 1\right) + \left(1 - \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)} \le -2.0323682233586974 \cdot 10^{-180}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot t\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{if}\;\frac{\left({1}^{3} + {\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}^{3}\right) \cdot \left(\tan k \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)}{\frac{\ell}{t} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 1\right) + \left(1 - \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)} \le 1.4641377692088345 \cdot 10^{-266}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)}\\ \mathbf{if}\;\frac{\left({1}^{3} + {\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}^{3}\right) \cdot \left(\tan k \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)}{\frac{\ell}{t} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 1\right) + \left(1 - \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)} \le +\infty:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot t\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\sin k \cdot t\right) \cdot \sin k\right)}{\frac{\ell}{t}}}{\frac{\ell}{t} \cdot \cos 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 (/ (* (* (* (* t (/ t l)) (sin k)) (tan k)) (+ (pow (+ 1 (pow (/ k t) 2)) 3) (pow 1 3))) (* (/ l t) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))) < -2.0323682233586974e-180 or 1.4641377692088345e-266 < (/ (* (* (* (* t (/ t l)) (sin k)) (tan k)) (+ (pow (+ 1 (pow (/ k t) 2)) 3) (pow 1 3))) (* (/ l t) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))) < +inf.0

    1. Initial program 22.7

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

    3. Applied add-cube-cbrt22.8

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

    4. Applied times-frac19.2

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

    5. Applied simplify19.1

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

    6. Applied simplify10.8

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

    8. Applied tan-quot10.8

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

    9. Applied associate-*l/10.8

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

    10. Applied associate-*l/7.2

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

    11. Applied frac-times6.4

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

    12. Applied associate-*l/6.4

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

    14. Applied associate-*l*1.3

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

    if -2.0323682233586974e-180 < (/ (* (* (* (* t (/ t l)) (sin k)) (tan k)) (+ (pow (+ 1 (pow (/ k t) 2)) 3) (pow 1 3))) (* (/ l t) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))) < 1.4641377692088345e-266

    1. Initial program 44.2

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

    3. Applied unpow344.2

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

    4. Applied times-frac16.9

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

    5. Applied associate-*l*9.6

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

    if +inf.0 < (/ (* (* (* (* t (/ t l)) (sin k)) (tan k)) (+ (pow (+ 1 (pow (/ k t) 2)) 3) (pow 1 3))) (* (/ l t) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1)))))

    1. Initial program 42.9

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

    3. Applied add-cube-cbrt42.9

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

    4. Applied times-frac41.5

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

    5. Applied simplify41.5

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

    6. Applied simplify31.3

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

    8. Applied tan-quot31.4

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

    9. Applied associate-*l/31.4

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

    10. Applied associate-*l/31.3

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

    11. Applied frac-times31.5

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

    12. Applied associate-*l/28.9

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

    14. Applied *-un-lft-identity28.9

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

    15. Applied associate-*r*28.9

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

    16. Applied simplify28.9

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

    18. Applied associate-*l/28.9

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

    19. Applied associate-*l/29.0

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

    20. Applied associate-*l/27.0

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

  4. Applied simplify12.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\left({1}^{3} + {\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}^{3}\right) \cdot \left(\tan k \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)}{\frac{\ell}{t} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 1\right) + \left(1 - \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)} \le -2.0323682233586974 \cdot 10^{-180}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot t\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{if}\;\frac{\left({1}^{3} + {\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}^{3}\right) \cdot \left(\tan k \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)}{\frac{\ell}{t} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 1\right) + \left(1 - \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)} \le 1.4641377692088345 \cdot 10^{-266}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)}\\ \mathbf{if}\;\frac{\left({1}^{3} + {\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}^{3}\right) \cdot \left(\tan k \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)}{\frac{\ell}{t} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 1\right) + \left(1 - \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)} \le +\infty:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot t\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\sin k \cdot t\right) \cdot \sin k\right)}{\frac{\ell}{t}}}{\frac{\ell}{t} \cdot \cos k}}\\ \end{array}}\]

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' 
(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))))