VandenBroeck and Keller, Equation (6)

Average Error: 16.4 → 8.7

Time: 8.6s

Precision: 64

• could not determine a ground truth for program body

  1. F = 4.508751374935895e+97
  2. l = -2192279693233314.5

Math

\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.94868822452690568 \cdot 10^{168}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{1}{\frac{\cos \left(\pi \cdot \ell\right) \cdot F}{\sin \left(\pi \cdot \ell\right)}}\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 5.3331432688450983 \cdot 10^{148}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\left(\pi \cdot \left({\ell}^{\frac{1}{3}} \cdot {\ell}^{\frac{1}{3}}\right)\right) \cdot \sqrt[3]{\ell}\right) \cdot F}\right)\\ \end{array}\]

C

double f(double F, double l) {
        double r16011 = atan2(1.0, 0.0);
        double r16012 = l;
        double r16013 = r16011 * r16012;
        double r16014 = 1.0;
        double r16015 = F;
        double r16016 = r16015 * r16015;
        double r16017 = r16014 / r16016;
        double r16018 = tan(r16013);
        double r16019 = r16017 * r16018;
        double r16020 = r16013 - r16019;
        return r16020;
}

↓

double f(double F, double l) {
        double r16021 = atan2(1.0, 0.0);
        double r16022 = l;
        double r16023 = r16021 * r16022;
        double r16024 = -1.9486882245269057e+168;
        bool r16025 = r16023 <= r16024;
        double r16026 = 1.0;
        double r16027 = F;
        double r16028 = r16026 / r16027;
        double r16029 = 1.0;
        double r16030 = cos(r16023);
        double r16031 = r16030 * r16027;
        double r16032 = sin(r16023);
        double r16033 = r16031 / r16032;
        double r16034 = r16026 / r16033;
        double r16035 = r16029 * r16034;
        double r16036 = r16028 * r16035;
        double r16037 = r16023 - r16036;
        double r16038 = 5.333143268845098e+148;
        bool r16039 = r16023 <= r16038;
        double r16040 = 0.041666666666666664;
        double r16041 = 4.0;
        double r16042 = pow(r16021, r16041);
        double r16043 = pow(r16022, r16041);
        double r16044 = r16042 * r16043;
        double r16045 = r16040 * r16044;
        double r16046 = r16045 + r16026;
        double r16047 = 0.5;
        double r16048 = 2.0;
        double r16049 = pow(r16021, r16048);
        double r16050 = pow(r16022, r16048);
        double r16051 = r16049 * r16050;
        double r16052 = r16047 * r16051;
        double r16053 = r16046 - r16052;
        double r16054 = r16053 * r16027;
        double r16055 = r16032 / r16054;
        double r16056 = r16029 * r16055;
        double r16057 = r16028 * r16056;
        double r16058 = r16023 - r16057;
        double r16059 = 0.3333333333333333;
        double r16060 = pow(r16022, r16059);
        double r16061 = r16060 * r16060;
        double r16062 = r16021 * r16061;
        double r16063 = cbrt(r16022);
        double r16064 = r16062 * r16063;
        double r16065 = cos(r16064);
        double r16066 = r16065 * r16027;
        double r16067 = r16032 / r16066;
        double r16068 = r16029 * r16067;
        double r16069 = r16028 * r16068;
        double r16070 = r16023 - r16069;
        double r16071 = r16039 ? r16058 : r16070;
        double r16072 = r16025 ? r16037 : r16071;
        return r16072;
}

Error

Bits error versus F

Bits error versus l

Derivation

1. Split input into 3 regimes

2. if (* PI l) < -1.9486882245269057e+168

   1. Initial program 20.4

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 20.4

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{1 \cdot 1}}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

   4. Applied times-frac 20.4

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{1}{F} \cdot \frac{1}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

   5. Applied associate-*l* 20.4

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]

   6. Taylor expanded around inf 20.4

      \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \color{blue}{\left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot F}\right)}\]
   7. Using strategy rm

   8. Applied clear-num 20.4

      \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \color{blue}{\frac{1}{\frac{\cos \left(\pi \cdot \ell\right) \cdot F}{\sin \left(\pi \cdot \ell\right)}}}\right)\]

   if -1.9486882245269057e+168 < (* PI l) < 5.333143268845098e+148

   1. Initial program 14.9

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 14.9

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{1 \cdot 1}}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

   4. Applied times-frac 15.0

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{1}{F} \cdot \frac{1}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

   5. Applied associate-*l* 9.5

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]

   6. Taylor expanded around inf 9.5

      \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \color{blue}{\left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot F}\right)}\]

   7. Taylor expanded around 0 4.5

      \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)} \cdot F}\right)\]

   if 5.333143268845098e+148 < (* PI l)

   1. Initial program 20.9

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 20.9

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{1 \cdot 1}}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

   4. Applied times-frac 20.9

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{1}{F} \cdot \frac{1}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

   5. Applied associate-*l* 20.9

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]

   6. Taylor expanded around inf 20.9

      \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \color{blue}{\left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot F}\right)}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 20.9

      \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \color{blue}{\left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}\right)}\right) \cdot F}\right)\]

   9. Applied associate-*r* 20.9

      \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\cos \color{blue}{\left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \sqrt[3]{\ell}\right)} \cdot F}\right)\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt 20.9

       \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell}}\right)}\right)\right) \cdot \sqrt[3]{\ell}\right) \cdot F}\right)\]

   12. Applied add-sqr-sqrt 20.9

       \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\left(\pi \cdot \left(\color{blue}{\left(\sqrt{\sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell}}\right)} \cdot \left(\sqrt{\sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell}}\right)\right)\right) \cdot \sqrt[3]{\ell}\right) \cdot F}\right)\]

   13. Applied swap-sqr 20.9

       \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\left(\pi \cdot \color{blue}{\left(\left(\sqrt{\sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell}}\right) \cdot \left(\sqrt{\sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell}}\right)\right)}\right) \cdot \sqrt[3]{\ell}\right) \cdot F}\right)\]

   14. Simplified 20.9

       \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\left(\pi \cdot \left(\color{blue}{{\ell}^{\frac{1}{3}}} \cdot \left(\sqrt{\sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell}}\right)\right)\right) \cdot \sqrt[3]{\ell}\right) \cdot F}\right)\]

   15. Simplified 20.9

       \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\left(\pi \cdot \left({\ell}^{\frac{1}{3}} \cdot \color{blue}{{\ell}^{\frac{1}{3}}}\right)\right) \cdot \sqrt[3]{\ell}\right) \cdot F}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.94868822452690568 \cdot 10^{168}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{1}{\frac{\cos \left(\pi \cdot \ell\right) \cdot F}{\sin \left(\pi \cdot \ell\right)}}\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 5.3331432688450983 \cdot 10^{148}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\left(\pi \cdot \left({\ell}^{\frac{1}{3}} \cdot {\ell}^{\frac{1}{3}}\right)\right) \cdot \sqrt[3]{\ell}\right) \cdot F}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))