Average Error: 16.7 → 9.2
Time: 8.4s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.011948691715024543580909495142699086666 \cdot 10^{169}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \left(\left(\sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot F}} \cdot \sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot F}}\right) \cdot \sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot F}}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 2.691706907806316542182290270953314062387 \cdot 10^{133}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -1.011948691715024543580909495142699086666 \cdot 10^{169}:\\
\;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \left(\left(\sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot F}} \cdot \sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot F}}\right) \cdot \sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot F}}\right)\right)\\

\mathbf{elif}\;\pi \cdot \ell \le 2.691706907806316542182290270953314062387 \cdot 10^{133}:\\
\;\;\;\;\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 F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\

\end{array}
double f(double F, double l) {
        double r12970 = atan2(1.0, 0.0);
        double r12971 = l;
        double r12972 = r12970 * r12971;
        double r12973 = 1.0;
        double r12974 = F;
        double r12975 = r12974 * r12974;
        double r12976 = r12973 / r12975;
        double r12977 = tan(r12972);
        double r12978 = r12976 * r12977;
        double r12979 = r12972 - r12978;
        return r12979;
}
↓
double f(double F, double l) {
        double r12980 = atan2(1.0, 0.0);
        double r12981 = l;
        double r12982 = r12980 * r12981;
        double r12983 = -1.0119486917150245e+169;
        bool r12984 = r12982 <= r12983;
        double r12985 = 1.0;
        double r12986 = F;
        double r12987 = r12985 / r12986;
        double r12988 = 1.0;
        double r12989 = sin(r12982);
        double r12990 = cos(r12982);
        double r12991 = r12990 * r12986;
        double r12992 = r12989 / r12991;
        double r12993 = cbrt(r12992);
        double r12994 = r12993 * r12993;
        double r12995 = r12994 * r12993;
        double r12996 = r12988 * r12995;
        double r12997 = r12987 * r12996;
        double r12998 = r12982 - r12997;
        double r12999 = 2.6917069078063165e+133;
        bool r13000 = r12982 <= r12999;
        double r13001 = 0.041666666666666664;
        double r13002 = 4.0;
        double r13003 = pow(r12980, r13002);
        double r13004 = pow(r12981, r13002);
        double r13005 = r13003 * r13004;
        double r13006 = r13001 * r13005;
        double r13007 = r13006 + r12985;
        double r13008 = 0.5;
        double r13009 = 2.0;
        double r13010 = pow(r12980, r13009);
        double r13011 = pow(r12981, r13009);
        double r13012 = r13010 * r13011;
        double r13013 = r13008 * r13012;
        double r13014 = r13007 - r13013;
        double r13015 = r13014 * r12986;
        double r13016 = r12989 / r13015;
        double r13017 = r12988 * r13016;
        double r13018 = r12987 * r13017;
        double r13019 = r12982 - r13018;
        double r13020 = r12986 * r12986;
        double r13021 = r12988 / r13020;
        double r13022 = cbrt(r12982);
        double r13023 = r13022 * r13022;
        double r13024 = r13023 * r13022;
        double r13025 = tan(r13024);
        double r13026 = r13021 * r13025;
        double r13027 = r12982 - r13026;
        double r13028 = r13000 ? r13019 : r13027;
        double r13029 = r12984 ? r12998 : r13028;
        return r13029;
}

Error

Bits error versus F

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (* PI l) < -1.0119486917150245e+169

    1. Initial program 20.0

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

    3. Applied *-un-lft-identity20.0

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

    4. Applied times-frac20.0

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

    5. Applied associate-*l*20.0

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

    6. Taylor expanded around inf 20.0

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

    8. Applied add-cube-cbrt20.0

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

    if -1.0119486917150245e+169 < (* PI l) < 2.6917069078063165e+133

    1. Initial program 15.0

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

    3. Applied *-un-lft-identity15.0

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

    4. Applied times-frac15.1

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

    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 9.4

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

    7. Taylor expanded around 0 4.7

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

    if 2.6917069078063165e+133 < (* PI l)

    1. Initial program 21.6

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

    3. Applied add-cube-cbrt21.6

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.011948691715024543580909495142699086666 \cdot 10^{169}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \left(\left(\sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot F}} \cdot \sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot F}}\right) \cdot \sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot F}}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 2.691706907806316542182290270953314062387 \cdot 10^{133}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))