Toniolo and Linder, Equation (10+)

Average Error: 32.7 → 15.3

Time: 34.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -1.1912096216701534 \cdot 10^{+67}:\\ \;\;\;\;\left(\left(\frac{\ell}{t} \cdot \left(\cos k \cdot \frac{\ell}{t}\right)\right) \cdot 2\right) \cdot \frac{1}{\left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \sin k\right) \cdot \left(\sin k \cdot t\right)}\\ \mathbf{elif}\;\ell \le -1.7515234209579402 \cdot 10^{-149}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{\sin k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k\right)}{\left(\ell \cdot \ell\right) \cdot \cos k}, \frac{\left(\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)\right) \cdot t}{\left(\ell \cdot \ell\right) \cdot \cos k}\right)}\\ \mathbf{elif}\;\ell \le 3.438337439326428 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{\left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \sin k\right) \cdot \left(\sin k \cdot t\right)}{\frac{\ell}{t}}}{\frac{\ell}{t}}}{\cos k}}\\ \mathbf{elif}\;\ell \le 9.600966579055477 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{\sin k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k\right)}{\left(\ell \cdot \ell\right) \cdot \cos k}, \frac{\left(\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)\right) \cdot t}{\left(\ell \cdot \ell\right) \cdot \cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{\left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \sin k\right) \cdot \left(\sin k \cdot t\right)}{\frac{\ell}{t}}}{\frac{\ell}{t}}}{\cos k}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r1188271 = 2.0;
        double r1188272 = t;
        double r1188273 = 3.0;
        double r1188274 = pow(r1188272, r1188273);
        double r1188275 = l;
        double r1188276 = r1188275 * r1188275;
        double r1188277 = r1188274 / r1188276;
        double r1188278 = k;
        double r1188279 = sin(r1188278);
        double r1188280 = r1188277 * r1188279;
        double r1188281 = tan(r1188278);
        double r1188282 = r1188280 * r1188281;
        double r1188283 = 1.0;
        double r1188284 = r1188278 / r1188272;
        double r1188285 = pow(r1188284, r1188271);
        double r1188286 = r1188283 + r1188285;
        double r1188287 = r1188286 + r1188283;
        double r1188288 = r1188282 * r1188287;
        double r1188289 = r1188271 / r1188288;
        return r1188289;
}

↓

double f(double t, double l, double k) {
        double r1188290 = l;
        double r1188291 = -1.1912096216701534e+67;
        bool r1188292 = r1188290 <= r1188291;
        double r1188293 = t;
        double r1188294 = r1188290 / r1188293;
        double r1188295 = k;
        double r1188296 = cos(r1188295);
        double r1188297 = r1188296 * r1188294;
        double r1188298 = r1188294 * r1188297;
        double r1188299 = 2.0;
        double r1188300 = r1188298 * r1188299;
        double r1188301 = 1.0;
        double r1188302 = r1188295 / r1188293;
        double r1188303 = fma(r1188302, r1188302, r1188299);
        double r1188304 = sin(r1188295);
        double r1188305 = r1188303 * r1188304;
        double r1188306 = r1188304 * r1188293;
        double r1188307 = r1188305 * r1188306;
        double r1188308 = r1188301 / r1188307;
        double r1188309 = r1188300 * r1188308;
        double r1188310 = -1.7515234209579402e-149;
        bool r1188311 = r1188290 <= r1188310;
        double r1188312 = r1188293 * r1188293;
        double r1188313 = r1188293 * r1188312;
        double r1188314 = r1188313 * r1188304;
        double r1188315 = r1188304 * r1188314;
        double r1188316 = r1188290 * r1188290;
        double r1188317 = r1188316 * r1188296;
        double r1188318 = r1188315 / r1188317;
        double r1188319 = r1188295 * r1188304;
        double r1188320 = r1188319 * r1188319;
        double r1188321 = r1188320 * r1188293;
        double r1188322 = r1188321 / r1188317;
        double r1188323 = fma(r1188299, r1188318, r1188322);
        double r1188324 = r1188299 / r1188323;
        double r1188325 = 3.438337439326428e-85;
        bool r1188326 = r1188290 <= r1188325;
        double r1188327 = r1188307 / r1188294;
        double r1188328 = r1188327 / r1188294;
        double r1188329 = r1188328 / r1188296;
        double r1188330 = r1188299 / r1188329;
        double r1188331 = 9.600966579055477e+99;
        bool r1188332 = r1188290 <= r1188331;
        double r1188333 = r1188332 ? r1188324 : r1188330;
        double r1188334 = r1188326 ? r1188330 : r1188333;
        double r1188335 = r1188311 ? r1188324 : r1188334;
        double r1188336 = r1188292 ? r1188309 : r1188335;
        return r1188336;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if l < -1.1912096216701534e+67

   1. Initial program 51.2

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

   2. Simplified 28.8

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

   4. Applied tan-quot 28.9

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

   5. Applied associate-*l/ 24.8

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

   6. Applied frac-times 24.9

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

   7. Applied associate-*l/ 25.7

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

   9. Applied associate-*l* 25.7

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

   11. Applied div-inv 25.7

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

   12. Applied *-un-lft-identity 25.7

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

   13. Applied times-frac 25.7

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

   14. Simplified 25.7

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

   if -1.1912096216701534e+67 < l < -1.7515234209579402e-149 or 3.438337439326428e-85 < l < 9.600966579055477e+99

   1. Initial program 25.7

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

   2. Simplified 23.0

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

   3. Taylor expanded around inf 15.9

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

   4. Simplified 12.4

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

   if -1.7515234209579402e-149 < l < 3.438337439326428e-85 or 9.600966579055477e+99 < l

   1. Initial program 32.5

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

   2. Simplified 17.9

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

   4. Applied tan-quot 17.9

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

   5. Applied associate-*l/ 16.8

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

   6. Applied frac-times 18.8

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

   7. Applied associate-*l/ 17.1

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

   9. Applied associate-*l* 16.7

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

   11. Applied add-sqr-sqrt 16.8

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

   12. Applied associate-*r* 16.7

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

   14. Applied associate-/r* 16.7

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

   15. Simplified 14.3

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

4. Final simplification 15.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.1912096216701534 \cdot 10^{+67}:\\ \;\;\;\;\left(\left(\frac{\ell}{t} \cdot \left(\cos k \cdot \frac{\ell}{t}\right)\right) \cdot 2\right) \cdot \frac{1}{\left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \sin k\right) \cdot \left(\sin k \cdot t\right)}\\ \mathbf{elif}\;\ell \le -1.7515234209579402 \cdot 10^{-149}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{\sin k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k\right)}{\left(\ell \cdot \ell\right) \cdot \cos k}, \frac{\left(\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)\right) \cdot t}{\left(\ell \cdot \ell\right) \cdot \cos k}\right)}\\ \mathbf{elif}\;\ell \le 3.438337439326428 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{\left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \sin k\right) \cdot \left(\sin k \cdot t\right)}{\frac{\ell}{t}}}{\frac{\ell}{t}}}{\cos k}}\\ \mathbf{elif}\;\ell \le 9.600966579055477 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{\sin k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k\right)}{\left(\ell \cdot \ell\right) \cdot \cos k}, \frac{\left(\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)\right) \cdot t}{\left(\ell \cdot \ell\right) \cdot \cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{\left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \sin k\right) \cdot \left(\sin k \cdot t\right)}{\frac{\ell}{t}}}{\frac{\ell}{t}}}{\cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(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))))