Toniolo and Linder, Equation (10-)

Average Error: 48.0 → 7.9

Time: 40.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1689046141557.58716:\\ \;\;\;\;\left(2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot \ell}{{\left(\sin k\right)}^{2}}\right)\right)\right)\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)\right)\right)\right) \cdot \ell}{{\left(\sin k\right)}^{2}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r140186 = 2.0;
        double r140187 = t;
        double r140188 = 3.0;
        double r140189 = pow(r140187, r140188);
        double r140190 = l;
        double r140191 = r140190 * r140190;
        double r140192 = r140189 / r140191;
        double r140193 = k;
        double r140194 = sin(r140193);
        double r140195 = r140192 * r140194;
        double r140196 = tan(r140193);
        double r140197 = r140195 * r140196;
        double r140198 = 1.0;
        double r140199 = r140193 / r140187;
        double r140200 = pow(r140199, r140186);
        double r140201 = r140198 + r140200;
        double r140202 = r140201 - r140198;
        double r140203 = r140197 * r140202;
        double r140204 = r140186 / r140203;
        return r140204;
}

↓

double f(double t, double l, double k) {
        double r140205 = t;
        double r140206 = -1689046141557.5872;
        bool r140207 = r140205 <= r140206;
        double r140208 = 2.0;
        double r140209 = 1.0;
        double r140210 = cbrt(r140209);
        double r140211 = r140210 * r140210;
        double r140212 = k;
        double r140213 = 2.0;
        double r140214 = r140208 / r140213;
        double r140215 = pow(r140212, r140214);
        double r140216 = r140211 / r140215;
        double r140217 = 1.0;
        double r140218 = pow(r140216, r140217);
        double r140219 = pow(r140205, r140217);
        double r140220 = r140210 / r140219;
        double r140221 = pow(r140220, r140217);
        double r140222 = cos(r140212);
        double r140223 = l;
        double r140224 = r140222 * r140223;
        double r140225 = sin(r140212);
        double r140226 = pow(r140225, r140213);
        double r140227 = r140224 / r140226;
        double r140228 = r140221 * r140227;
        double r140229 = r140218 * r140228;
        double r140230 = r140218 * r140229;
        double r140231 = r140208 * r140230;
        double r140232 = r140231 * r140223;
        double r140233 = r140215 * r140219;
        double r140234 = r140209 / r140233;
        double r140235 = pow(r140234, r140217);
        double r140236 = r140235 * r140224;
        double r140237 = r140218 * r140236;
        double r140238 = r140208 * r140237;
        double r140239 = r140238 * r140223;
        double r140240 = r140239 / r140226;
        double r140241 = r140207 ? r140232 : r140240;
        return r140241;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -1689046141557.5872

   1. Initial program 47.7

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

   2. Simplified 32.8

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

   3. Taylor expanded around inf 15.0

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

   5. Applied sqr-pow 15.0

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

   6. Applied associate-*l* 15.0

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

   8. Applied add-cube-cbrt 15.0

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

   9. Applied times-frac 14.6

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

   10. Applied unpow-prod-down 14.6

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

   11. Applied associate-*l* 11.3

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

   12. Simplified 11.3

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

   14. Applied add-cube-cbrt 11.3

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

   15. Applied times-frac 11.0

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

   16. Applied unpow-prod-down 11.0

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

   17. Applied associate-*l* 9.5

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

   if -1689046141557.5872 < t

   1. Initial program 48.2

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

   2. Simplified 40.8

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

   3. Taylor expanded around inf 16.8

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

   5. Applied sqr-pow 16.8

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

   6. Applied associate-*l* 12.7

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

   8. Applied add-cube-cbrt 12.7

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

   9. Applied times-frac 12.2

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

   10. Applied unpow-prod-down 12.2

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

   11. Applied associate-*l* 7.5

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

   12. Simplified 7.5

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

   14. Applied associate-*r/ 7.7

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

   15. Applied associate-*r/ 7.7

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

   16. Applied associate-*r/ 7.7

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

   17. Applied associate-*l/ 7.3

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

4. Final simplification 7.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1689046141557.58716:\\ \;\;\;\;\left(2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot \ell}{{\left(\sin k\right)}^{2}}\right)\right)\right)\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)\right)\right)\right) \cdot \ell}{{\left(\sin k\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(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))))