Toniolo and Linder, Equation (10+)

Average Error: 32.2 → 9.0

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.0373876915132293 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\sqrt{\sqrt{2}}}{\frac{t}{\ell}}}}{\frac{\frac{\sqrt{\sqrt{2}}}{\tan k}}{t}}}\\ \mathbf{elif}\;t \le 368905.46702557366:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\sin k \cdot {k}^{2}}{\left(\sqrt{2} \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{\sin k \cdot {t}^{2}}{\left(\cos k \cdot \ell\right) \cdot \sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sin k \cdot \frac{t}{\ell}}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{1}{\tan k}}{t}}} \cdot \frac{\sqrt{2}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\sqrt{2}}{\frac{t}{\ell}}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r2554017 = 2.0;
        double r2554018 = t;
        double r2554019 = 3.0;
        double r2554020 = pow(r2554018, r2554019);
        double r2554021 = l;
        double r2554022 = r2554021 * r2554021;
        double r2554023 = r2554020 / r2554022;
        double r2554024 = k;
        double r2554025 = sin(r2554024);
        double r2554026 = r2554023 * r2554025;
        double r2554027 = tan(r2554024);
        double r2554028 = r2554026 * r2554027;
        double r2554029 = 1.0;
        double r2554030 = r2554024 / r2554018;
        double r2554031 = pow(r2554030, r2554017);
        double r2554032 = r2554029 + r2554031;
        double r2554033 = r2554032 + r2554029;
        double r2554034 = r2554028 * r2554033;
        double r2554035 = r2554017 / r2554034;
        return r2554035;
}

↓

double f(double t, double l, double k) {
        double r2554036 = t;
        double r2554037 = -1.0373876915132293e+65;
        bool r2554038 = r2554036 <= r2554037;
        double r2554039 = 2.0;
        double r2554040 = sqrt(r2554039);
        double r2554041 = k;
        double r2554042 = sin(r2554041);
        double r2554043 = l;
        double r2554044 = r2554036 / r2554043;
        double r2554045 = r2554042 * r2554044;
        double r2554046 = r2554040 / r2554045;
        double r2554047 = r2554041 / r2554036;
        double r2554048 = r2554047 * r2554047;
        double r2554049 = r2554048 + r2554039;
        double r2554050 = sqrt(r2554040);
        double r2554051 = r2554050 / r2554044;
        double r2554052 = r2554049 / r2554051;
        double r2554053 = tan(r2554041);
        double r2554054 = r2554050 / r2554053;
        double r2554055 = r2554054 / r2554036;
        double r2554056 = r2554052 / r2554055;
        double r2554057 = r2554046 / r2554056;
        double r2554058 = 368905.46702557366;
        bool r2554059 = r2554036 <= r2554058;
        double r2554060 = pow(r2554041, r2554039);
        double r2554061 = r2554042 * r2554060;
        double r2554062 = r2554040 * r2554043;
        double r2554063 = cos(r2554041);
        double r2554064 = r2554062 * r2554063;
        double r2554065 = r2554061 / r2554064;
        double r2554066 = pow(r2554036, r2554039);
        double r2554067 = r2554042 * r2554066;
        double r2554068 = r2554063 * r2554043;
        double r2554069 = r2554068 * r2554040;
        double r2554070 = r2554067 / r2554069;
        double r2554071 = r2554039 * r2554070;
        double r2554072 = r2554065 + r2554071;
        double r2554073 = r2554046 / r2554072;
        double r2554074 = 1.0;
        double r2554075 = r2554074 / r2554045;
        double r2554076 = sqrt(r2554049);
        double r2554077 = r2554074 / r2554053;
        double r2554078 = r2554077 / r2554036;
        double r2554079 = r2554076 / r2554078;
        double r2554080 = r2554075 / r2554079;
        double r2554081 = r2554040 / r2554044;
        double r2554082 = r2554076 / r2554081;
        double r2554083 = r2554040 / r2554082;
        double r2554084 = r2554080 * r2554083;
        double r2554085 = r2554059 ? r2554073 : r2554084;
        double r2554086 = r2554038 ? r2554057 : r2554085;
        return r2554086;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -1.0373876915132293e+65

   1. Initial program 23.0

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

   2. Simplified 10.4

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

   4. Applied associate-*l* 9.8

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

   6. Applied associate-*r* 7.1

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

   8. Applied *-un-lft-identity 7.1

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

   9. Applied add-sqr-sqrt 7.2

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

   10. Applied times-frac 7.2

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

   11. Applied times-frac 6.6

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

   12. Applied associate-/l* 6.5

       \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\sqrt{2}}{\tan k}}{\frac{t}{\ell} \cdot t}}}}\]
   13. Using strategy rm

   14. Applied *-un-lft-identity 6.5

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

   15. Applied add-sqr-sqrt 6.5

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

   16. Applied sqrt-prod 6.4

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

   17. Applied times-frac 6.4

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

   18. Applied times-frac 1.2

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

   19. Applied associate-/r* 1.2

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

   if -1.0373876915132293e+65 < t < 368905.46702557366

   1. Initial program 44.3

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

   2. Simplified 34.5

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

   4. Applied associate-*l* 33.8

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

   6. Applied associate-*r* 31.8

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

   8. Applied *-un-lft-identity 31.8

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

   9. Applied add-sqr-sqrt 31.9

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

   10. Applied times-frac 31.9

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

   11. Applied times-frac 31.2

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

   12. Applied associate-/l* 27.8

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

   13. Taylor expanded around inf 17.7

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

   if 368905.46702557366 < t

   1. Initial program 21.7

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

   2. Simplified 10.6

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

   4. Applied associate-*l* 10.2

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

   6. Applied associate-*r* 7.3

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

   8. Applied *-un-lft-identity 7.3

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

   9. Applied add-sqr-sqrt 7.4

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

   10. Applied times-frac 7.4

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

   11. Applied times-frac 6.6

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

   12. Applied associate-/l* 6.2

       \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\sqrt{2}}{\tan k}}{\frac{t}{\ell} \cdot t}}}}\]
   13. Using strategy rm

   14. Applied div-inv 6.3

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

   15. Applied times-frac 2.1

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

   16. Applied add-sqr-sqrt 2.1

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

   17. Applied times-frac 2.0

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

   18. Applied div-inv 2.0

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

   19. Applied times-frac 2.4

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

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.0373876915132293 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\sqrt{\sqrt{2}}}{\frac{t}{\ell}}}}{\frac{\frac{\sqrt{\sqrt{2}}}{\tan k}}{t}}}\\ \mathbf{elif}\;t \le 368905.46702557366:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\sin k \cdot {k}^{2}}{\left(\sqrt{2} \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{\sin k \cdot {t}^{2}}{\left(\cos k \cdot \ell\right) \cdot \sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sin k \cdot \frac{t}{\ell}}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{1}{\tan k}}{t}}} \cdot \frac{\sqrt{2}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\sqrt{2}}{\frac{t}{\ell}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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))))