Toniolo and Linder, Equation (10-)

Average Error: 48.2 → 16.9

Time: 34.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le 1.0106088911309665 \cdot 10^{154}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\sin k}{{\ell}^{2}}}}{{\left(\sin k\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot {\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r99180 = 2.0;
        double r99181 = t;
        double r99182 = 3.0;
        double r99183 = pow(r99181, r99182);
        double r99184 = l;
        double r99185 = r99184 * r99184;
        double r99186 = r99183 / r99185;
        double r99187 = k;
        double r99188 = sin(r99187);
        double r99189 = r99186 * r99188;
        double r99190 = tan(r99187);
        double r99191 = r99189 * r99190;
        double r99192 = 1.0;
        double r99193 = r99187 / r99181;
        double r99194 = pow(r99193, r99180);
        double r99195 = r99192 + r99194;
        double r99196 = r99195 - r99192;
        double r99197 = r99191 * r99196;
        double r99198 = r99180 / r99197;
        return r99198;
}

↓

double f(double t, double l, double k) {
        double r99199 = l;
        double r99200 = 1.0106088911309665e+154;
        bool r99201 = r99199 <= r99200;
        double r99202 = 2.0;
        double r99203 = 1.0;
        double r99204 = k;
        double r99205 = 2.0;
        double r99206 = r99202 / r99205;
        double r99207 = pow(r99204, r99206);
        double r99208 = t;
        double r99209 = 1.0;
        double r99210 = pow(r99208, r99209);
        double r99211 = r99207 * r99210;
        double r99212 = r99203 / r99211;
        double r99213 = pow(r99212, r99209);
        double r99214 = cos(r99204);
        double r99215 = sin(r99204);
        double r99216 = pow(r99199, r99205);
        double r99217 = r99215 / r99216;
        double r99218 = r99214 / r99217;
        double r99219 = r99205 / r99205;
        double r99220 = pow(r99215, r99219);
        double r99221 = r99218 / r99220;
        double r99222 = r99213 * r99221;
        double r99223 = cbrt(r99203);
        double r99224 = r99223 * r99223;
        double r99225 = r99224 / r99207;
        double r99226 = pow(r99225, r99209);
        double r99227 = r99222 * r99226;
        double r99228 = r99202 * r99227;
        double r99229 = cbrt(r99208);
        double r99230 = r99229 * r99229;
        double r99231 = 3.0;
        double r99232 = pow(r99230, r99231);
        double r99233 = r99232 / r99199;
        double r99234 = pow(r99229, r99231);
        double r99235 = r99234 / r99199;
        double r99236 = r99233 * r99235;
        double r99237 = r99236 * r99215;
        double r99238 = tan(r99204);
        double r99239 = r99237 * r99238;
        double r99240 = r99202 / r99239;
        double r99241 = r99204 / r99208;
        double r99242 = pow(r99241, r99202);
        double r99243 = r99240 / r99242;
        double r99244 = r99201 ? r99228 : r99243;
        return r99244;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if l < 1.0106088911309665e+154

   1. Initial program 46.9

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

   2. Simplified 38.4

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

   3. Taylor expanded around inf 18.7

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

   5. Applied sqr-pow 18.7

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

   6. Applied associate-*l* 16.5

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

   8. Applied add-cube-cbrt 16.5

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

   9. Applied times-frac 16.3

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

   10. Applied unpow-prod-down 16.3

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

   11. Applied associate-*l* 14.5

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

   12. Simplified 14.5

       \[\leadsto 2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \color{blue}{\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\right)\]
   13. Using strategy rm

   14. Applied sqr-pow 14.5

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

   15. Applied associate-/r* 14.2

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

   16. Simplified 14.3

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

   if 1.0106088911309665e+154 < l

   1. Initial program 64.0

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

   2. Simplified 63.9

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

   4. Applied add-cube-cbrt 63.9

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

   5. Applied unpow-prod-down 63.9

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

   6. Applied times-frac 48.7

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

4. Final simplification 16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le 1.0106088911309665 \cdot 10^{154}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\sin k}{{\ell}^{2}}}}{{\left(\sin k\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot {\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))