Toniolo and Linder, Equation (10+)

Average Error: 32.0 → 13.5

Time: 44.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.0530128908841045 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{t}{\ell} \cdot \left(\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \frac{\frac{\sqrt{2}}{\tan k}}{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}} \cdot t}\\ \mathbf{elif}\;t \le -2.883261033031283 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{t \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \sin k\right)}}{\frac{2 + \frac{k}{t} \cdot \frac{k}{t}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{\sqrt{\sqrt[3]{2 + \frac{k}{t} \cdot \frac{k}{t}} \cdot \sqrt[3]{2 + \frac{k}{t} \cdot \frac{k}{t}}}} \cdot \frac{\frac{\frac{1}{\tan k}}{t}}{\sqrt{\sqrt[3]{2 + \frac{k}{t} \cdot \frac{k}{t}}}}\right) \cdot \frac{\sqrt{2}}{\frac{t}{\ell} \cdot \left(\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r1637047 = 2.0;
        double r1637048 = t;
        double r1637049 = 3.0;
        double r1637050 = pow(r1637048, r1637049);
        double r1637051 = l;
        double r1637052 = r1637051 * r1637051;
        double r1637053 = r1637050 / r1637052;
        double r1637054 = k;
        double r1637055 = sin(r1637054);
        double r1637056 = r1637053 * r1637055;
        double r1637057 = tan(r1637054);
        double r1637058 = r1637056 * r1637057;
        double r1637059 = 1.0;
        double r1637060 = r1637054 / r1637048;
        double r1637061 = pow(r1637060, r1637047);
        double r1637062 = r1637059 + r1637061;
        double r1637063 = r1637062 + r1637059;
        double r1637064 = r1637058 * r1637063;
        double r1637065 = r1637047 / r1637064;
        return r1637065;
}

↓

double f(double t, double l, double k) {
        double r1637066 = t;
        double r1637067 = -2.0530128908841045e+39;
        bool r1637068 = r1637066 <= r1637067;
        double r1637069 = 2.0;
        double r1637070 = sqrt(r1637069);
        double r1637071 = l;
        double r1637072 = r1637066 / r1637071;
        double r1637073 = k;
        double r1637074 = r1637073 / r1637066;
        double r1637075 = r1637074 * r1637074;
        double r1637076 = r1637069 + r1637075;
        double r1637077 = sqrt(r1637076);
        double r1637078 = sin(r1637073);
        double r1637079 = r1637072 * r1637078;
        double r1637080 = r1637077 * r1637079;
        double r1637081 = r1637072 * r1637080;
        double r1637082 = r1637070 / r1637081;
        double r1637083 = tan(r1637073);
        double r1637084 = r1637070 / r1637083;
        double r1637085 = r1637077 * r1637066;
        double r1637086 = r1637084 / r1637085;
        double r1637087 = r1637082 * r1637086;
        double r1637088 = -2.883261033031283e-29;
        bool r1637089 = r1637066 <= r1637088;
        double r1637090 = r1637069 / r1637083;
        double r1637091 = r1637072 * r1637066;
        double r1637092 = r1637091 * r1637078;
        double r1637093 = r1637066 * r1637092;
        double r1637094 = r1637090 / r1637093;
        double r1637095 = r1637076 / r1637071;
        double r1637096 = r1637094 / r1637095;
        double r1637097 = cbrt(r1637076);
        double r1637098 = r1637097 * r1637097;
        double r1637099 = sqrt(r1637098);
        double r1637100 = r1637070 / r1637099;
        double r1637101 = 1.0;
        double r1637102 = r1637101 / r1637083;
        double r1637103 = r1637102 / r1637066;
        double r1637104 = sqrt(r1637097);
        double r1637105 = r1637103 / r1637104;
        double r1637106 = r1637100 * r1637105;
        double r1637107 = r1637106 * r1637082;
        double r1637108 = r1637089 ? r1637096 : r1637107;
        double r1637109 = r1637068 ? r1637087 : r1637108;
        return r1637109;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -2.0530128908841045e+39

   1. Initial program 22.6

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

   2. Simplified 10.2

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

   4. Applied associate-*r* 6.8

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

   6. Applied add-sqr-sqrt 6.9

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

   7. Applied *-un-lft-identity 6.9

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

   8. Applied add-sqr-sqrt 6.8

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

   9. Applied times-frac 6.9

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

   10. Applied times-frac 6.1

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

   11. Applied times-frac 6.1

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

   12. Simplified 2.9

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

   14. Applied associate-*r* 2.6

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

   16. Applied associate-/l/ 2.6

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

   if -2.0530128908841045e+39 < t < -2.883261033031283e-29

   1. Initial program 20.6

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

   2. Simplified 20.1

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

   4. Applied associate-*r* 19.9

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

   6. Applied associate-*r/ 19.9

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

   7. Applied associate-*r/ 14.5

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

   8. Applied associate-*l/ 14.0

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

   9. Applied associate-/r/ 13.8

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

   10. Applied associate-/l* 10.7

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

   if -2.883261033031283e-29 < t

   1. Initial program 36.8

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

   2. Simplified 25.6

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

   4. Applied associate-*r* 24.4

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

   6. Applied add-sqr-sqrt 24.4

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

   7. Applied *-un-lft-identity 24.4

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

   8. Applied add-sqr-sqrt 24.4

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

   9. Applied times-frac 24.4

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

   10. Applied times-frac 24.3

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

   11. Applied times-frac 21.5

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

   12. Simplified 19.0

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

   14. Applied associate-*r* 18.1

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

   16. Applied add-cube-cbrt 18.1

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

   17. Applied sqrt-prod 18.1

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

   18. Applied *-un-lft-identity 18.1

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

   19. Applied div-inv 18.1

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

   20. Applied times-frac 18.1

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

   21. Applied times-frac 18.2

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

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.0530128908841045 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{t}{\ell} \cdot \left(\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \frac{\frac{\sqrt{2}}{\tan k}}{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}} \cdot t}\\ \mathbf{elif}\;t \le -2.883261033031283 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{t \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \sin k\right)}}{\frac{2 + \frac{k}{t} \cdot \frac{k}{t}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{\sqrt{\sqrt[3]{2 + \frac{k}{t} \cdot \frac{k}{t}} \cdot \sqrt[3]{2 + \frac{k}{t} \cdot \frac{k}{t}}}} \cdot \frac{\frac{\frac{1}{\tan k}}{t}}{\sqrt{\sqrt[3]{2 + \frac{k}{t} \cdot \frac{k}{t}}}}\right) \cdot \frac{\sqrt{2}}{\frac{t}{\ell} \cdot \left(\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))