Toniolo and Linder, Equation (10-)

Average Error: 47.4 → 1.1

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -1.6453940453160847 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \left(\left(\frac{\sqrt{\sqrt{2}}}{k} \cdot \frac{\sqrt{2}}{\tan k}\right) \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{elif}\;k \le 1.347321606507381 \cdot 10^{-87}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{\tan k} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\sqrt{2}}{\frac{t}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{k}} \cdot \frac{k}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;k \le 1.0217052900572645 \cdot 10^{+273}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \left(\left(\frac{\sqrt{\sqrt{2}}}{k} \cdot \frac{\sqrt{2}}{\tan k}\right) \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{k}}} \cdot \frac{\ell}{\sin k \cdot \tan k}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r3055030 = 2.0;
        double r3055031 = t;
        double r3055032 = 3.0;
        double r3055033 = pow(r3055031, r3055032);
        double r3055034 = l;
        double r3055035 = r3055034 * r3055034;
        double r3055036 = r3055033 / r3055035;
        double r3055037 = k;
        double r3055038 = sin(r3055037);
        double r3055039 = r3055036 * r3055038;
        double r3055040 = tan(r3055037);
        double r3055041 = r3055039 * r3055040;
        double r3055042 = 1.0;
        double r3055043 = r3055037 / r3055031;
        double r3055044 = pow(r3055043, r3055030);
        double r3055045 = r3055042 + r3055044;
        double r3055046 = r3055045 - r3055042;
        double r3055047 = r3055041 * r3055046;
        double r3055048 = r3055030 / r3055047;
        return r3055048;
}

↓

double f(double t, double l, double k) {
        double r3055049 = k;
        double r3055050 = -1.6453940453160847e-73;
        bool r3055051 = r3055049 <= r3055050;
        double r3055052 = 2.0;
        double r3055053 = sqrt(r3055052);
        double r3055054 = sqrt(r3055053);
        double r3055055 = t;
        double r3055056 = l;
        double r3055057 = r3055056 / r3055049;
        double r3055058 = r3055055 / r3055057;
        double r3055059 = r3055054 / r3055058;
        double r3055060 = r3055054 / r3055049;
        double r3055061 = tan(r3055049);
        double r3055062 = r3055053 / r3055061;
        double r3055063 = r3055060 * r3055062;
        double r3055064 = sin(r3055049);
        double r3055065 = r3055056 / r3055064;
        double r3055066 = r3055063 * r3055065;
        double r3055067 = r3055059 * r3055066;
        double r3055068 = 1.347321606507381e-87;
        bool r3055069 = r3055049 <= r3055068;
        double r3055070 = r3055062 * r3055065;
        double r3055071 = cbrt(r3055056);
        double r3055072 = r3055071 * r3055071;
        double r3055073 = r3055072 / r3055049;
        double r3055074 = r3055055 / r3055073;
        double r3055075 = r3055049 / r3055071;
        double r3055076 = r3055074 * r3055075;
        double r3055077 = r3055053 / r3055076;
        double r3055078 = r3055070 * r3055077;
        double r3055079 = 1.0217052900572645e+273;
        bool r3055080 = r3055049 <= r3055079;
        double r3055081 = r3055055 * r3055049;
        double r3055082 = r3055081 / r3055057;
        double r3055083 = r3055052 / r3055082;
        double r3055084 = r3055064 * r3055061;
        double r3055085 = r3055056 / r3055084;
        double r3055086 = r3055083 * r3055085;
        double r3055087 = r3055080 ? r3055067 : r3055086;
        double r3055088 = r3055069 ? r3055078 : r3055087;
        double r3055089 = r3055051 ? r3055067 : r3055088;
        return r3055089;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if k < -1.6453940453160847e-73 or 1.347321606507381e-87 < k < 1.0217052900572645e+273

   1. Initial program 46.2

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

   2. Simplified 13.0

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

   4. Applied div-inv 13.0

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

   5. Applied add-sqr-sqrt 13.2

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

   6. Applied times-frac 13.1

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

   7. Applied times-frac 13.1

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

   8. Simplified 5.4

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

   9. Simplified 5.4

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

   11. Applied associate-/r/ 5.4

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

   12. Applied times-frac 3.9

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

   13. Applied add-sqr-sqrt 3.9

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

   14. Applied sqrt-prod 3.8

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

   15. Applied times-frac 3.5

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

   16. Applied associate-*l* 0.6

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

   17. Simplified 0.6

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

   if -1.6453940453160847e-73 < k < 1.347321606507381e-87

   1. Initial program 61.8

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

   2. Simplified 44.0

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

   4. Applied div-inv 44.0

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

   5. Applied add-sqr-sqrt 44.1

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

   6. Applied times-frac 43.7

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

   7. Applied times-frac 29.3

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

   8. Simplified 19.7

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

   9. Simplified 3.8

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

   11. Applied div-inv 3.8

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

   12. Applied add-cube-cbrt 4.3

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

   13. Applied times-frac 4.3

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

   14. Applied times-frac 2.1

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

   15. Simplified 2.1

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

   if 1.0217052900572645e+273 < k

   1. Initial program 34.0

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

   2. Simplified 17.3

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

   4. Applied associate-/r/ 17.5

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

   5. Applied times-frac 17.5

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

   6. Simplified 8.6

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.6453940453160847 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \left(\left(\frac{\sqrt{\sqrt{2}}}{k} \cdot \frac{\sqrt{2}}{\tan k}\right) \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{elif}\;k \le 1.347321606507381 \cdot 10^{-87}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{\tan k} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\sqrt{2}}{\frac{t}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{k}} \cdot \frac{k}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;k \le 1.0217052900572645 \cdot 10^{+273}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \left(\left(\frac{\sqrt{\sqrt{2}}}{k} \cdot \frac{\sqrt{2}}{\tan k}\right) \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{k}}} \cdot \frac{\ell}{\sin k \cdot \tan k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(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))))