Average Error: 16.5 → 12.3
Time: 8.8s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right) \cdot 1}{F}}{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right) \cdot 1}{F}}{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}
double f(double F, double l) {
        double r14992 = atan2(1.0, 0.0);
        double r14993 = l;
        double r14994 = r14992 * r14993;
        double r14995 = 1.0;
        double r14996 = F;
        double r14997 = r14996 * r14996;
        double r14998 = r14995 / r14997;
        double r14999 = tan(r14994);
        double r15000 = r14998 * r14999;
        double r15001 = r14994 - r15000;
        return r15001;
}
↓
double f(double F, double l) {
        double r15002 = atan2(1.0, 0.0);
        double r15003 = l;
        double r15004 = r15002 * r15003;
        double r15005 = 1.0;
        double r15006 = F;
        double r15007 = r15005 / r15006;
        double r15008 = sin(r15004);
        double r15009 = 1.0;
        double r15010 = r15008 * r15009;
        double r15011 = r15010 / r15006;
        double r15012 = sqrt(r15002);
        double r15013 = r15012 * r15003;
        double r15014 = r15012 * r15013;
        double r15015 = cos(r15014);
        double r15016 = r15011 / r15015;
        double r15017 = r15007 * r15016;
        double r15018 = r15004 - r15017;
        return r15018;
}

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. Initial program 16.5

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

  3. Applied *-un-lft-identity16.5

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

  4. Applied times-frac16.5

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

  5. Applied associate-*l*12.3

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

  7. Applied add-sqr-sqrt12.4

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

  8. Applied associate-*l*12.4

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

  10. Applied tan-quot12.4

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

  11. Applied associate-*r/12.4

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

  12. Simplified12.3

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

  13. Final simplification12.3

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

Reproduce

herbie shell --seed 2020057 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))