Toniolo and Linder, Equation (10-)

Average Error: 48.2 → 16.0

Time: 32.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 3.06321 \cdot 10^{-322}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \frac{\ell}{\frac{\left|\sin k\right|}{\ell}}\right)\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 5.8108209154164435 \cdot 10^{288}:\\ \;\;\;\;\left(\left(\frac{\frac{\cos k}{\frac{\sin k}{{\ell}^{2}}}}{{\left(\sin k\right)}^{\left(\frac{2}{2}\right)}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right) \cdot {\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}} \cdot \ell\right) \cdot \ell\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r115918 = 2.0;
        double r115919 = t;
        double r115920 = 3.0;
        double r115921 = pow(r115919, r115920);
        double r115922 = l;
        double r115923 = r115922 * r115922;
        double r115924 = r115921 / r115923;
        double r115925 = k;
        double r115926 = sin(r115925);
        double r115927 = r115924 * r115926;
        double r115928 = tan(r115925);
        double r115929 = r115927 * r115928;
        double r115930 = 1.0;
        double r115931 = r115925 / r115919;
        double r115932 = pow(r115931, r115918);
        double r115933 = r115930 + r115932;
        double r115934 = r115933 - r115930;
        double r115935 = r115929 * r115934;
        double r115936 = r115918 / r115935;
        return r115936;
}

↓

double f(double t, double l, double k) {
        double r115937 = l;
        double r115938 = r115937 * r115937;
        double r115939 = 3.0632070042157e-322;
        bool r115940 = r115938 <= r115939;
        double r115941 = 2.0;
        double r115942 = 1.0;
        double r115943 = k;
        double r115944 = pow(r115943, r115941);
        double r115945 = t;
        double r115946 = 1.0;
        double r115947 = pow(r115945, r115946);
        double r115948 = r115944 * r115947;
        double r115949 = r115942 / r115948;
        double r115950 = pow(r115949, r115946);
        double r115951 = cos(r115943);
        double r115952 = sin(r115943);
        double r115953 = fabs(r115952);
        double r115954 = r115951 / r115953;
        double r115955 = r115953 / r115937;
        double r115956 = r115937 / r115955;
        double r115957 = r115954 * r115956;
        double r115958 = r115950 * r115957;
        double r115959 = r115941 * r115958;
        double r115960 = 5.810820915416443e+288;
        bool r115961 = r115938 <= r115960;
        double r115962 = 2.0;
        double r115963 = pow(r115937, r115962);
        double r115964 = r115952 / r115963;
        double r115965 = r115951 / r115964;
        double r115966 = r115962 / r115962;
        double r115967 = pow(r115952, r115966);
        double r115968 = r115965 / r115967;
        double r115969 = r115941 / r115962;
        double r115970 = pow(r115943, r115969);
        double r115971 = r115970 * r115947;
        double r115972 = r115942 / r115971;
        double r115973 = pow(r115972, r115946);
        double r115974 = r115968 * r115973;
        double r115975 = cbrt(r115942);
        double r115976 = r115975 * r115975;
        double r115977 = r115976 / r115970;
        double r115978 = pow(r115977, r115946);
        double r115979 = r115974 * r115978;
        double r115980 = r115979 * r115941;
        double r115981 = 3.0;
        double r115982 = pow(r115945, r115981);
        double r115983 = r115941 / r115982;
        double r115984 = tan(r115943);
        double r115985 = r115952 * r115984;
        double r115986 = r115943 / r115945;
        double r115987 = pow(r115986, r115941);
        double r115988 = r115985 * r115987;
        double r115989 = r115983 / r115988;
        double r115990 = r115989 * r115937;
        double r115991 = r115990 * r115937;
        double r115992 = r115961 ? r115980 : r115991;
        double r115993 = r115940 ? r115959 : r115992;
        return r115993;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (* l l) < 3.0632070042157e-322

   1. Initial program 46.4

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

   2. Simplified 38.4

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

   3. Taylor expanded around inf 20.5

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

   5. Applied add-sqr-sqrt 20.5

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

   6. Applied times-frac 20.5

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

   7. Simplified 20.5

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

   8. Simplified 17.6

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

   if 3.0632070042157e-322 < (* l l) < 5.810820915416443e+288

   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 35.4

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

   3. Taylor expanded around inf 10.6

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

   5. Applied sqr-pow 10.6

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

   6. Applied associate-*l* 6.7

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

   8. Applied add-cube-cbrt 6.7

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

   9. Applied times-frac 6.5

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

   10. Applied unpow-prod-down 6.5

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

   11. Applied associate-*l* 3.4

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

   12. Simplified 3.4

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

   14. Applied sqr-pow 3.4

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

   15. Applied associate-/r* 3.0

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

   16. Simplified 3.1

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

   if 5.810820915416443e+288 < (* l l)

   1. Initial program 62.9

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

   2. Simplified 62.6

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

   4. Applied associate-*r* 53.0

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

4. Final simplification 16.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 3.06321 \cdot 10^{-322}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \frac{\ell}{\frac{\left|\sin k\right|}{\ell}}\right)\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 5.8108209154164435 \cdot 10^{288}:\\ \;\;\;\;\left(\left(\frac{\frac{\cos k}{\frac{\sin k}{{\ell}^{2}}}}{{\left(\sin k\right)}^{\left(\frac{2}{2}\right)}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right) \cdot {\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}} \cdot \ell\right) \cdot \ell\\ \end{array}\]

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))))