Toniolo and Linder, Equation (10-)

Average Error: 48.6 → 8.9

Time: 26.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -4.57723594555655645 \cdot 10^{139} \lor \neg \left(k \le -2.7057282491360021 \cdot 10^{-140} \lor \neg \left(k \le 7.6001933799401753 \cdot 10^{-155} \lor \neg \left(k \le 1.15121578543094186 \cdot 10^{132}\right)\right)\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)}{\frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\sqrt{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)\right)}{\frac{\sin k}{\ell}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r102161 = 2.0;
        double r102162 = t;
        double r102163 = 3.0;
        double r102164 = pow(r102162, r102163);
        double r102165 = l;
        double r102166 = r102165 * r102165;
        double r102167 = r102164 / r102166;
        double r102168 = k;
        double r102169 = sin(r102168);
        double r102170 = r102167 * r102169;
        double r102171 = tan(r102168);
        double r102172 = r102170 * r102171;
        double r102173 = 1.0;
        double r102174 = r102168 / r102162;
        double r102175 = pow(r102174, r102161);
        double r102176 = r102173 + r102175;
        double r102177 = r102176 - r102173;
        double r102178 = r102172 * r102177;
        double r102179 = r102161 / r102178;
        return r102179;
}

↓

double f(double t, double l, double k) {
        double r102180 = k;
        double r102181 = -4.5772359455565565e+139;
        bool r102182 = r102180 <= r102181;
        double r102183 = -2.705728249136002e-140;
        bool r102184 = r102180 <= r102183;
        double r102185 = 7.600193379940175e-155;
        bool r102186 = r102180 <= r102185;
        double r102187 = 1.1512157854309419e+132;
        bool r102188 = r102180 <= r102187;
        double r102189 = !r102188;
        bool r102190 = r102186 || r102189;
        double r102191 = !r102190;
        bool r102192 = r102184 || r102191;
        double r102193 = !r102192;
        bool r102194 = r102182 || r102193;
        double r102195 = 2.0;
        double r102196 = 1.0;
        double r102197 = 2.0;
        double r102198 = r102195 / r102197;
        double r102199 = pow(r102180, r102198);
        double r102200 = t;
        double r102201 = 1.0;
        double r102202 = pow(r102200, r102201);
        double r102203 = r102199 * r102202;
        double r102204 = r102199 * r102203;
        double r102205 = r102196 / r102204;
        double r102206 = pow(r102205, r102201);
        double r102207 = cos(r102180);
        double r102208 = sin(r102180);
        double r102209 = r102207 / r102208;
        double r102210 = l;
        double r102211 = r102209 * r102210;
        double r102212 = r102206 * r102211;
        double r102213 = r102208 / r102210;
        double r102214 = r102212 / r102213;
        double r102215 = r102195 * r102214;
        double r102216 = sqrt(r102196);
        double r102217 = pow(r102180, r102195);
        double r102218 = r102216 / r102217;
        double r102219 = pow(r102218, r102201);
        double r102220 = r102196 / r102202;
        double r102221 = pow(r102220, r102201);
        double r102222 = r102221 * r102211;
        double r102223 = r102219 * r102222;
        double r102224 = r102223 / r102213;
        double r102225 = r102195 * r102224;
        double r102226 = r102194 ? r102215 : r102225;
        return r102226;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < -4.5772359455565565e+139 or -2.705728249136002e-140 < k < 7.600193379940175e-155 or 1.1512157854309419e+132 < k

   1. Initial program 42.0

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

   2. Simplified 36.9

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

   3. Taylor expanded around inf 26.4

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

   5. Applied add-sqr-sqrt 45.4

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

   6. Applied unpow-prod-down 45.4

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

   7. Applied times-frac 45.4

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

   8. Simplified 45.4

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

   9. Simplified 26.2

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

   11. Applied associate-*r/ 25.8

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

   12. Applied associate-*r/ 23.9

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

   14. Applied sqr-pow 23.9

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

   15. Applied associate-*l* 14.6

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

   if -4.5772359455565565e+139 < k < -2.705728249136002e-140 or 7.600193379940175e-155 < k < 1.1512157854309419e+132

   1. Initial program 54.8

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

   2. Simplified 43.8

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

   3. Taylor expanded around inf 18.5

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

   5. Applied add-sqr-sqrt 41.0

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

   6. Applied unpow-prod-down 41.0

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

   7. Applied times-frac 41.0

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

   8. Simplified 41.0

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

   9. Simplified 16.9

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

   11. Applied associate-*r/ 15.5

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

   12. Applied associate-*r/ 8.3

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

   14. Applied add-sqr-sqrt 8.3

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

   15. Applied times-frac 8.0

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

   16. Applied unpow-prod-down 8.0

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

   17. Applied associate-*l* 3.4

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

   18. Simplified 3.4

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -4.57723594555655645 \cdot 10^{139} \lor \neg \left(k \le -2.7057282491360021 \cdot 10^{-140} \lor \neg \left(k \le 7.6001933799401753 \cdot 10^{-155} \lor \neg \left(k \le 1.15121578543094186 \cdot 10^{132}\right)\right)\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)}{\frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\sqrt{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)\right)}{\frac{\sin k}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))