Toniolo and Linder, Equation (10-)

Average Error: 48.2 → 9.2

Time: 24.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -7.116427264207922396198059745161573311828 \cdot 10^{154} \lor \neg \left(k \le 1.596016421741683330355613183588958719568 \cdot 10^{147}\right):\\ \;\;\;\;\left(\frac{2 \cdot \ell}{{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)\right)}^{1}} \cdot \frac{\ell}{{\left(\sin k\right)}^{2}}\right) \cdot \cos k\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2 \cdot \ell}{{\left({k}^{2}\right)}^{1}}}{{\left({t}^{1}\right)}^{1}} \cdot \frac{\ell}{{\left(\sin k\right)}^{2}}\right) \cdot \cos k\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r77051 = 2.0;
        double r77052 = t;
        double r77053 = 3.0;
        double r77054 = pow(r77052, r77053);
        double r77055 = l;
        double r77056 = r77055 * r77055;
        double r77057 = r77054 / r77056;
        double r77058 = k;
        double r77059 = sin(r77058);
        double r77060 = r77057 * r77059;
        double r77061 = tan(r77058);
        double r77062 = r77060 * r77061;
        double r77063 = 1.0;
        double r77064 = r77058 / r77052;
        double r77065 = pow(r77064, r77051);
        double r77066 = r77063 + r77065;
        double r77067 = r77066 - r77063;
        double r77068 = r77062 * r77067;
        double r77069 = r77051 / r77068;
        return r77069;
}

↓

double f(double t, double l, double k) {
        double r77070 = k;
        double r77071 = -7.116427264207922e+154;
        bool r77072 = r77070 <= r77071;
        double r77073 = 1.5960164217416833e+147;
        bool r77074 = r77070 <= r77073;
        double r77075 = !r77074;
        bool r77076 = r77072 || r77075;
        double r77077 = 2.0;
        double r77078 = l;
        double r77079 = r77077 * r77078;
        double r77080 = 2.0;
        double r77081 = r77077 / r77080;
        double r77082 = pow(r77070, r77081);
        double r77083 = t;
        double r77084 = 1.0;
        double r77085 = pow(r77083, r77084);
        double r77086 = r77082 * r77085;
        double r77087 = r77082 * r77086;
        double r77088 = pow(r77087, r77084);
        double r77089 = r77079 / r77088;
        double r77090 = sin(r77070);
        double r77091 = pow(r77090, r77080);
        double r77092 = r77078 / r77091;
        double r77093 = r77089 * r77092;
        double r77094 = cos(r77070);
        double r77095 = r77093 * r77094;
        double r77096 = pow(r77070, r77077);
        double r77097 = pow(r77096, r77084);
        double r77098 = r77079 / r77097;
        double r77099 = pow(r77085, r77084);
        double r77100 = r77098 / r77099;
        double r77101 = r77100 * r77092;
        double r77102 = r77101 * r77094;
        double r77103 = r77076 ? r77095 : r77102;
        return r77103;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < -7.116427264207922e+154 or 1.5960164217416833e+147 < k

   1. Initial program 40.0

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

   2. Simplified 34.6

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

   3. Taylor expanded around inf 24.1

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

   5. Applied sqr-pow 24.1

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

   6. Applied associate-*l* 19.2

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

   8. Applied associate-*r/ 19.2

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

   9. Applied associate-*l/ 19.2

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

   10. Applied associate-/r/ 19.2

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

   11. Simplified 22.2

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

   13. Applied sqr-pow 22.2

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

   14. Applied associate-*l* 14.7

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

   if -7.116427264207922e+154 < k < 1.5960164217416833e+147

   1. Initial program 54.0

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

   2. Simplified 44.3

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

   3. Taylor expanded around inf 21.6

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

   5. Applied sqr-pow 21.6

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

   6. Applied associate-*l* 21.5

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

   8. Applied associate-*r/ 21.5

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

   9. Applied associate-*l/ 21.5

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

   10. Applied associate-/r/ 21.5

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

   11. Simplified 10.6

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

   13. Applied unpow-prod-down 10.6

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

   14. Applied associate-/r* 5.2

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -7.116427264207922396198059745161573311828 \cdot 10^{154} \lor \neg \left(k \le 1.596016421741683330355613183588958719568 \cdot 10^{147}\right):\\ \;\;\;\;\left(\frac{2 \cdot \ell}{{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)\right)}^{1}} \cdot \frac{\ell}{{\left(\sin k\right)}^{2}}\right) \cdot \cos k\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2 \cdot \ell}{{\left({k}^{2}\right)}^{1}}}{{\left({t}^{1}\right)}^{1}} \cdot \frac{\ell}{{\left(\sin k\right)}^{2}}\right) \cdot \cos k\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(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))))