VandenBroeck and Keller, Equation (6)

Average Error: 17.3 → 8.5

Time: 9.4s

Precision: 64

• could not determine a ground truth for program body

  1. F = 1.0925393300280745e+126
  2. l = -716357237560921.5

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -9.718573105215283439369659162476488255481 \cdot 10^{155}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 8.939014277543762652462397904252204810382 \cdot 10^{150}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{\mathsf{fma}\left(\frac{1}{24} \cdot {\pi}^{4}, {\ell}^{4}, 1 - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{\cos \left(\left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right) \cdot F}\\ \end{array}\]

C

double f(double F, double l) {
        double r15918 = atan2(1.0, 0.0);
        double r15919 = l;
        double r15920 = r15918 * r15919;
        double r15921 = 1.0;
        double r15922 = F;
        double r15923 = r15922 * r15922;
        double r15924 = r15921 / r15923;
        double r15925 = tan(r15920);
        double r15926 = r15924 * r15925;
        double r15927 = r15920 - r15926;
        return r15927;
}

↓

double f(double F, double l) {
        double r15928 = atan2(1.0, 0.0);
        double r15929 = l;
        double r15930 = r15928 * r15929;
        double r15931 = -9.718573105215283e+155;
        bool r15932 = r15930 <= r15931;
        double r15933 = 1.0;
        double r15934 = cbrt(r15933);
        double r15935 = r15934 * r15934;
        double r15936 = F;
        double r15937 = r15935 / r15936;
        double r15938 = r15934 / r15936;
        double r15939 = sqrt(r15928);
        double r15940 = r15939 * r15929;
        double r15941 = r15939 * r15940;
        double r15942 = tan(r15941);
        double r15943 = r15938 * r15942;
        double r15944 = r15937 * r15943;
        double r15945 = r15930 - r15944;
        double r15946 = 8.939014277543763e+150;
        bool r15947 = r15930 <= r15946;
        double r15948 = sin(r15930);
        double r15949 = r15948 * r15934;
        double r15950 = 0.041666666666666664;
        double r15951 = 4.0;
        double r15952 = pow(r15928, r15951);
        double r15953 = r15950 * r15952;
        double r15954 = pow(r15929, r15951);
        double r15955 = 1.0;
        double r15956 = 0.5;
        double r15957 = 2.0;
        double r15958 = pow(r15928, r15957);
        double r15959 = pow(r15929, r15957);
        double r15960 = r15958 * r15959;
        double r15961 = r15956 * r15960;
        double r15962 = r15955 - r15961;
        double r15963 = fma(r15953, r15954, r15962);
        double r15964 = r15963 * r15936;
        double r15965 = r15949 / r15964;
        double r15966 = r15937 * r15965;
        double r15967 = r15930 - r15966;
        double r15968 = cbrt(r15939);
        double r15969 = r15968 * r15968;
        double r15970 = r15969 * r15969;
        double r15971 = cbrt(r15928);
        double r15972 = r15971 * r15929;
        double r15973 = r15970 * r15972;
        double r15974 = cos(r15973);
        double r15975 = r15974 * r15936;
        double r15976 = r15949 / r15975;
        double r15977 = r15937 * r15976;
        double r15978 = r15930 - r15977;
        double r15979 = r15947 ? r15967 : r15978;
        double r15980 = r15932 ? r15945 : r15979;
        return r15980;
}

Error

Bits error versus F

Bits error versus l

Derivation

1. Split input into 3 regimes

2. if (* PI l) < -9.718573105215283e+155

   1. Initial program 21.5

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

   3. Applied add-cube-cbrt 21.5

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

   4. Applied times-frac 21.5

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

   5. Applied associate-*l* 21.5

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

   7. Applied add-sqr-sqrt 21.5

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

   8. Applied associate-*l* 21.6

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

   if -9.718573105215283e+155 < (* PI l) < 8.939014277543763e+150

   1. Initial program 15.8

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

   3. Applied add-cube-cbrt 15.8

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

   4. Applied times-frac 15.8

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

   5. Applied associate-*l* 9.8

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

   6. Taylor expanded around inf 9.7

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

   7. Taylor expanded around 0 3.9

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{\color{blue}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)} \cdot F}\]

   8. Simplified 4.0

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{\color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {\pi}^{4}, {\ell}^{4}, 1 - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)} \cdot F}\]

   if 8.939014277543763e+150 < (* PI l)

   1. Initial program 21.5

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

   3. Applied add-cube-cbrt 21.5

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

   4. Applied times-frac 21.5

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

   5. Applied associate-*l* 21.5

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

   6. Taylor expanded around inf 21.5

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

   8. Applied add-cube-cbrt 21.4

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

   9. Applied associate-*l* 21.4

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

   11. Applied add-sqr-sqrt 21.5

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

   12. Applied cbrt-prod 21.4

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

   13. Applied add-sqr-sqrt 21.4

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

   14. Applied cbrt-prod 21.4

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

   15. Applied swap-sqr 21.4

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

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -9.718573105215283439369659162476488255481 \cdot 10^{155}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 8.939014277543762652462397904252204810382 \cdot 10^{150}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{\mathsf{fma}\left(\frac{1}{24} \cdot {\pi}^{4}, {\ell}^{4}, 1 - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{\cos \left(\left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right) \cdot F}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))