Toniolo and Linder, Equation (10+)

Average Error: 31.8 → 7.8

Time: 1.3m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -9.344040305863342 \cdot 10^{-120}:\\ \;\;\;\;\frac{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{t}} \cdot \frac{\ell}{t}\right) \cdot \frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{t}}}{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right) \cdot \frac{\sqrt[3]{t}}{\frac{\frac{\ell}{t}}{\sin k}}}\\ \mathbf{elif}\;t \le 4.3910634771150225 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\mathsf{fma}\left(2, \left(\frac{t \cdot \sin k}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right), \left(\frac{\left(k \cdot \left(t \cdot \sin k\right)\right) \cdot k}{\ell \cdot \ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right) \cdot \frac{\sqrt[3]{t}}{\frac{\frac{\ell}{t}}{\sin k}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r3145224 = 2.0;
        double r3145225 = t;
        double r3145226 = 3.0;
        double r3145227 = pow(r3145225, r3145226);
        double r3145228 = l;
        double r3145229 = r3145228 * r3145228;
        double r3145230 = r3145227 / r3145229;
        double r3145231 = k;
        double r3145232 = sin(r3145231);
        double r3145233 = r3145230 * r3145232;
        double r3145234 = tan(r3145231);
        double r3145235 = r3145233 * r3145234;
        double r3145236 = 1.0;
        double r3145237 = r3145231 / r3145225;
        double r3145238 = pow(r3145237, r3145224);
        double r3145239 = r3145236 + r3145238;
        double r3145240 = r3145239 + r3145236;
        double r3145241 = r3145235 * r3145240;
        double r3145242 = r3145224 / r3145241;
        return r3145242;
}

↓

double f(double t, double l, double k) {
        double r3145243 = t;
        double r3145244 = -9.344040305863342e-120;
        bool r3145245 = r3145243 <= r3145244;
        double r3145246 = 2.0;
        double r3145247 = k;
        double r3145248 = tan(r3145247);
        double r3145249 = r3145246 / r3145248;
        double r3145250 = cbrt(r3145249);
        double r3145251 = cbrt(r3145243);
        double r3145252 = r3145250 / r3145251;
        double r3145253 = l;
        double r3145254 = r3145253 / r3145243;
        double r3145255 = r3145252 * r3145254;
        double r3145256 = r3145250 * r3145250;
        double r3145257 = r3145256 / r3145251;
        double r3145258 = r3145255 * r3145257;
        double r3145259 = r3145247 / r3145243;
        double r3145260 = fma(r3145259, r3145259, r3145246);
        double r3145261 = sin(r3145247);
        double r3145262 = r3145254 / r3145261;
        double r3145263 = r3145251 / r3145262;
        double r3145264 = r3145260 * r3145263;
        double r3145265 = r3145258 / r3145264;
        double r3145266 = 4.3910634771150225e-67;
        bool r3145267 = r3145243 <= r3145266;
        double r3145268 = r3145243 * r3145261;
        double r3145269 = r3145254 * r3145254;
        double r3145270 = r3145268 / r3145269;
        double r3145271 = r3145247 * r3145268;
        double r3145272 = r3145271 * r3145247;
        double r3145273 = r3145253 * r3145253;
        double r3145274 = r3145272 / r3145273;
        double r3145275 = fma(r3145246, r3145270, r3145274);
        double r3145276 = r3145249 / r3145275;
        double r3145277 = r3145254 * r3145249;
        double r3145278 = r3145251 * r3145251;
        double r3145279 = r3145277 / r3145278;
        double r3145280 = r3145279 / r3145264;
        double r3145281 = r3145267 ? r3145276 : r3145280;
        double r3145282 = r3145245 ? r3145265 : r3145281;
        return r3145282;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -9.344040305863342e-120

   1. Initial program 23.2

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

   2. Simplified 11.3

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

   4. Applied *-un-lft-identity 11.3

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

   5. Applied times-frac 9.7

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

   6. Applied add-cube-cbrt 9.9

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

   7. Applied times-frac 9.3

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

   8. Applied associate-*l* 8.0

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\frac{\frac{\ell}{t}}{1}} \cdot \left(\frac{\sqrt[3]{t}}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)\right)}}\]
   9. Using strategy rm

   10. Applied associate-/r* 6.7

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

   11. Simplified 5.1

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

   13. Applied add-cube-cbrt 5.2

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

   14. Applied times-frac 5.2

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

   15. Applied associate-*l* 5.1

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

   if -9.344040305863342e-120 < t < 4.3910634771150225e-67

   1. Initial program 59.5

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

   2. Simplified 41.1

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

   3. Taylor expanded around -inf 38.8

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

   4. Simplified 18.6

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

   if 4.3910634771150225e-67 < t

   1. Initial program 22.6

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

   2. Simplified 10.1

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

   4. Applied *-un-lft-identity 10.1

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

   5. Applied times-frac 8.5

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

   6. Applied add-cube-cbrt 8.7

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

   7. Applied times-frac 7.9

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

   8. Applied associate-*l* 7.2

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\frac{\frac{\ell}{t}}{1}} \cdot \left(\frac{\sqrt[3]{t}}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)\right)}}\]
   9. Using strategy rm

   10. Applied associate-/r* 5.7

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

   11. Simplified 3.5

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

   13. Applied associate-*l/ 3.4

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

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.344040305863342 \cdot 10^{-120}:\\ \;\;\;\;\frac{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{t}} \cdot \frac{\ell}{t}\right) \cdot \frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{t}}}{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right) \cdot \frac{\sqrt[3]{t}}{\frac{\frac{\ell}{t}}{\sin k}}}\\ \mathbf{elif}\;t \le 4.3910634771150225 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\mathsf{fma}\left(2, \left(\frac{t \cdot \sin k}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right), \left(\frac{\left(k \cdot \left(t \cdot \sin k\right)\right) \cdot k}{\ell \cdot \ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right) \cdot \frac{\sqrt[3]{t}}{\frac{\frac{\ell}{t}}{\sin k}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +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))))