Toniolo and Linder, Equation (10+)

Average Error: 32.0 → 7.3

Time: 40.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.7347896894361278 \cdot 10^{+159}:\\ \;\;\;\;\frac{2}{\left(\left(8 + \left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)\right) \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \left(\cos k \cdot \left(\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right) + \left(4 - 2 \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)\right)\\ \mathbf{elif}\;t \le 1.9659319169181852 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{2}{\sin k}}{\left(\left(\left(\frac{2 \cdot \left(t \cdot t\right)}{\ell} + \frac{k}{\frac{\ell}{k}}\right) \cdot \frac{\sin k}{\cos k}\right) \cdot t\right) \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(8 + \left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)\right) \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \left(\cos k \cdot \left(\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right) + \left(4 - 2 \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r3293291 = 2.0;
        double r3293292 = t;
        double r3293293 = 3.0;
        double r3293294 = pow(r3293292, r3293293);
        double r3293295 = l;
        double r3293296 = r3293295 * r3293295;
        double r3293297 = r3293294 / r3293296;
        double r3293298 = k;
        double r3293299 = sin(r3293298);
        double r3293300 = r3293297 * r3293299;
        double r3293301 = tan(r3293298);
        double r3293302 = r3293300 * r3293301;
        double r3293303 = 1.0;
        double r3293304 = r3293298 / r3293292;
        double r3293305 = pow(r3293304, r3293291);
        double r3293306 = r3293303 + r3293305;
        double r3293307 = r3293306 + r3293303;
        double r3293308 = r3293302 * r3293307;
        double r3293309 = r3293291 / r3293308;
        return r3293309;
}

↓

double f(double t, double l, double k) {
        double r3293310 = t;
        double r3293311 = -2.7347896894361278e+159;
        bool r3293312 = r3293310 <= r3293311;
        double r3293313 = 2.0;
        double r3293314 = 8.0;
        double r3293315 = k;
        double r3293316 = r3293315 / r3293310;
        double r3293317 = r3293316 * r3293316;
        double r3293318 = r3293317 * r3293317;
        double r3293319 = r3293317 * r3293318;
        double r3293320 = r3293314 + r3293319;
        double r3293321 = sin(r3293315);
        double r3293322 = r3293321 * r3293310;
        double r3293323 = l;
        double r3293324 = r3293310 / r3293323;
        double r3293325 = r3293322 * r3293324;
        double r3293326 = r3293320 * r3293325;
        double r3293327 = r3293324 * r3293321;
        double r3293328 = r3293326 * r3293327;
        double r3293329 = r3293313 / r3293328;
        double r3293330 = cos(r3293315);
        double r3293331 = 4.0;
        double r3293332 = r3293313 * r3293317;
        double r3293333 = r3293331 - r3293332;
        double r3293334 = r3293318 + r3293333;
        double r3293335 = r3293330 * r3293334;
        double r3293336 = r3293329 * r3293335;
        double r3293337 = 1.9659319169181852e+157;
        bool r3293338 = r3293310 <= r3293337;
        double r3293339 = r3293313 / r3293321;
        double r3293340 = r3293310 * r3293310;
        double r3293341 = r3293313 * r3293340;
        double r3293342 = r3293341 / r3293323;
        double r3293343 = r3293323 / r3293315;
        double r3293344 = r3293315 / r3293343;
        double r3293345 = r3293342 + r3293344;
        double r3293346 = r3293321 / r3293330;
        double r3293347 = r3293345 * r3293346;
        double r3293348 = r3293347 * r3293310;
        double r3293349 = 1.0;
        double r3293350 = r3293349 / r3293323;
        double r3293351 = r3293348 * r3293350;
        double r3293352 = r3293339 / r3293351;
        double r3293353 = r3293338 ? r3293352 : r3293336;
        double r3293354 = r3293312 ? r3293336 : r3293353;
        return r3293354;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -2.7347896894361278e+159 or 1.9659319169181852e+157 < t

   1. Initial program 22.0

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

   2. Simplified 8.1

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

   4. Applied div-inv 8.1

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

   5. Applied times-frac 8.1

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

   6. Applied associate-/l* 8.1

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

   7. Simplified 5.1

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

   9. Applied tan-quot 5.1

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

   10. Applied flip3-+ 6.4

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

   11. Applied frac-times 6.4

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

   12. Applied associate-*l/ 6.8

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

   13. Applied associate-*l/ 6.8

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

   14. Applied associate-/r/ 6.8

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

   15. Simplified 2.7

       \[\leadsto \color{blue}{\frac{2}{\left(\left(8 + \left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)\right) \cdot \left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}} \cdot \left(\left(\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right) + \left(2 \cdot 2 - \left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot 2\right)\right) \cdot \cos k\right)\]

   if -2.7347896894361278e+159 < t < 1.9659319169181852e+157

   1. Initial program 37.0

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

   2. Simplified 26.7

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

   4. Applied div-inv 26.7

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

   5. Applied times-frac 26.7

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

   6. Applied associate-/l* 26.5

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

   7. Simplified 23.3

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

   9. Applied associate-*r* 19.3

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

   10. Taylor expanded around inf 15.0

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

   11. Simplified 13.9

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

   13. Applied div-inv 13.9

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

   14. Applied associate-*r* 14.0

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

   15. Simplified 9.7

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

4. Final simplification 7.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.7347896894361278 \cdot 10^{+159}:\\ \;\;\;\;\frac{2}{\left(\left(8 + \left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)\right) \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \left(\cos k \cdot \left(\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right) + \left(4 - 2 \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)\right)\\ \mathbf{elif}\;t \le 1.9659319169181852 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{2}{\sin k}}{\left(\left(\left(\frac{2 \cdot \left(t \cdot t\right)}{\ell} + \frac{k}{\frac{\ell}{k}}\right) \cdot \frac{\sin k}{\cos k}\right) \cdot t\right) \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(8 + \left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)\right) \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \left(\cos k \cdot \left(\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right) + \left(4 - 2 \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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))))