Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 11.1

Time: 37.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 4.5430129409159728 \cdot 10^{185}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right)}{\frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right)\right)}{\frac{\left|\sin k\right|}{\ell}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r189 = 2.0;
        double r190 = t;
        double r191 = 3.0;
        double r192 = pow(r190, r191);
        double r193 = l;
        double r194 = r193 * r193;
        double r195 = r192 / r194;
        double r196 = k;
        double r197 = sin(r196);
        double r198 = r195 * r197;
        double r199 = tan(r196);
        double r200 = r198 * r199;
        double r201 = 1.0;
        double r202 = r196 / r190;
        double r203 = pow(r202, r189);
        double r204 = r201 + r203;
        double r205 = r204 - r201;
        double r206 = r200 * r205;
        double r207 = r189 / r206;
        return r207;
}

↓

double f(double t, double l, double k) {
        double r208 = l;
        double r209 = r208 * r208;
        double r210 = 4.543012940915973e+185;
        bool r211 = r209 <= r210;
        double r212 = 2.0;
        double r213 = 1.0;
        double r214 = k;
        double r215 = 2.0;
        double r216 = r212 / r215;
        double r217 = pow(r214, r216);
        double r218 = t;
        double r219 = 1.0;
        double r220 = pow(r218, r219);
        double r221 = r217 * r220;
        double r222 = r217 * r221;
        double r223 = r213 / r222;
        double r224 = pow(r223, r219);
        double r225 = cos(r214);
        double r226 = sin(r214);
        double r227 = fabs(r226);
        double r228 = r225 / r227;
        double r229 = r228 * r208;
        double r230 = r224 * r229;
        double r231 = r227 / r208;
        double r232 = r230 / r231;
        double r233 = r212 * r232;
        double r234 = pow(r214, r212);
        double r235 = r213 / r234;
        double r236 = pow(r235, r219);
        double r237 = r213 / r220;
        double r238 = pow(r237, r219);
        double r239 = r238 * r229;
        double r240 = r236 * r239;
        double r241 = r240 / r231;
        double r242 = r212 * r241;
        double r243 = r211 ? r233 : r242;
        return r243;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if (* l l) < 4.543012940915973e+185

   1. Initial program 44.7

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

   2. Simplified 35.3

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

   3. Taylor expanded around inf 12.6

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

   5. Applied add-sqr-sqrt 12.6

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

   6. Applied times-frac 12.6

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

   7. Simplified 12.6

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

   8. Simplified 11.3

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

   10. Applied associate-*r/ 10.4

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

   11. Applied associate-*r/ 8.5

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

   13. Applied sqr-pow 8.5

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

   14. Applied associate-*l* 4.3

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

   if 4.543012940915973e+185 < (* l l)

   1. Initial program 59.0

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

   2. Simplified 56.7

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

   3. Taylor expanded around inf 51.6

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

   5. Applied add-sqr-sqrt 51.6

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

   6. Applied times-frac 51.6

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

   7. Simplified 51.6

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

   8. Simplified 51.6

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

   10. Applied associate-*r/ 51.6

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

   11. Applied associate-*r/ 39.0

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

   13. Applied *-un-lft-identity 39.0

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

   14. Applied times-frac 38.5

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

   15. Applied unpow-prod-down 38.5

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

   16. Applied associate-*l* 33.0

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 4.5430129409159728 \cdot 10^{185}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right)}{\frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right)\right)}{\frac{\left|\sin k\right|}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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))))