Toniolo and Linder, Equation (10-)

Average Error: 48.2 → 6.9

Time: 39.4s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -4.36078991244498616 \cdot 10^{133} \lor \neg \left(t \le 4.29036885267607093 \cdot 10^{88}\right):\\ \;\;\;\;\left(2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot \ell}{{\left(\sin k\right)}^{2}}\right)\right)\right)\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)\right)\right)\right) \cdot \ell}{{\left(\sin k\right)}^{2}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r136960 = 2.0;
        double r136961 = t;
        double r136962 = 3.0;
        double r136963 = pow(r136961, r136962);
        double r136964 = l;
        double r136965 = r136964 * r136964;
        double r136966 = r136963 / r136965;
        double r136967 = k;
        double r136968 = sin(r136967);
        double r136969 = r136966 * r136968;
        double r136970 = tan(r136967);
        double r136971 = r136969 * r136970;
        double r136972 = 1.0;
        double r136973 = r136967 / r136961;
        double r136974 = pow(r136973, r136960);
        double r136975 = r136972 + r136974;
        double r136976 = r136975 - r136972;
        double r136977 = r136971 * r136976;
        double r136978 = r136960 / r136977;
        return r136978;
}

↓

double f(double t, double l, double k) {
        double r136979 = t;
        double r136980 = -4.360789912444986e+133;
        bool r136981 = r136979 <= r136980;
        double r136982 = 4.290368852676071e+88;
        bool r136983 = r136979 <= r136982;
        double r136984 = !r136983;
        bool r136985 = r136981 || r136984;
        double r136986 = 2.0;
        double r136987 = 1.0;
        double r136988 = cbrt(r136987);
        double r136989 = r136988 * r136988;
        double r136990 = k;
        double r136991 = 2.0;
        double r136992 = r136986 / r136991;
        double r136993 = pow(r136990, r136992);
        double r136994 = r136989 / r136993;
        double r136995 = 1.0;
        double r136996 = pow(r136994, r136995);
        double r136997 = r136987 / r136993;
        double r136998 = pow(r136997, r136995);
        double r136999 = pow(r136979, r136995);
        double r137000 = r136987 / r136999;
        double r137001 = pow(r137000, r136995);
        double r137002 = cos(r136990);
        double r137003 = l;
        double r137004 = r137002 * r137003;
        double r137005 = sin(r136990);
        double r137006 = pow(r137005, r136991);
        double r137007 = r137004 / r137006;
        double r137008 = r137001 * r137007;
        double r137009 = r136998 * r137008;
        double r137010 = r136996 * r137009;
        double r137011 = r136986 * r137010;
        double r137012 = r137011 * r137003;
        double r137013 = r136993 * r136999;
        double r137014 = r136987 / r137013;
        double r137015 = pow(r137014, r136995);
        double r137016 = r137015 * r137004;
        double r137017 = r136996 * r137016;
        double r137018 = r136986 * r137017;
        double r137019 = r137018 * r137003;
        double r137020 = r137019 / r137006;
        double r137021 = r136985 ? r137012 : r137020;
        return r137021;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -4.360789912444986e+133 or 4.290368852676071e+88 < t

   1. Initial program 52.1

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

   2. Simplified 37.0

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

   3. Taylor expanded around inf 14.2

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

   5. Applied sqr-pow 14.2

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

   6. Applied associate-*l* 14.1

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

   8. Applied add-cube-cbrt 14.1

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

   9. Applied times-frac 13.8

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

   10. Applied unpow-prod-down 13.8

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

   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 *-un-lft-identity 11.3

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

   15. Applied times-frac 10.9

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

   16. Applied unpow-prod-down 10.9

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

   17. Applied associate-*l* 8.7

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

   if -4.360789912444986e+133 < t < 4.290368852676071e+88

   1. Initial program 45.8

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

   2. Simplified 40.0

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

   3. Taylor expanded around inf 17.5

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

   5. Applied sqr-pow 17.5

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

   6. Applied associate-*l* 12.4

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

   8. Applied add-cube-cbrt 12.4

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

   9. Applied times-frac 12.1

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

   10. Applied unpow-prod-down 12.1

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

   11. Applied associate-*l* 6.6

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

   12. Simplified 6.6

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

   14. Applied associate-*r/ 6.6

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

   15. Applied associate-*r/ 6.6

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

   16. Applied associate-*r/ 6.6

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

   17. Applied associate-*l/ 5.8

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

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.36078991244498616 \cdot 10^{133} \lor \neg \left(t \le 4.29036885267607093 \cdot 10^{88}\right):\\ \;\;\;\;\left(2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot \ell}{{\left(\sin k\right)}^{2}}\right)\right)\right)\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)\right)\right)\right) \cdot \ell}{{\left(\sin k\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))