Average Error: 16.7 → 12.2
Time: 8.7s
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 r14231 = atan2(1.0, 0.0);
        double r14232 = l;
        double r14233 = r14231 * r14232;
        double r14234 = 1.0;
        double r14235 = F;
        double r14236 = r14235 * r14235;
        double r14237 = r14234 / r14236;
        double r14238 = tan(r14233);
        double r14239 = r14237 * r14238;
        double r14240 = r14233 - r14239;
        return r14240;
}
↓
double f(double F, double l) {
        double r14241 = atan2(1.0, 0.0);
        double r14242 = l;
        double r14243 = r14241 * r14242;
        double r14244 = 1.0;
        double r14245 = F;
        double r14246 = r14244 / r14245;
        double r14247 = sin(r14243);
        double r14248 = 1.0;
        double r14249 = r14247 * r14248;
        double r14250 = r14249 / r14245;
        double r14251 = sqrt(r14241);
        double r14252 = r14251 * r14242;
        double r14253 = r14251 * r14252;
        double r14254 = cos(r14253);
        double r14255 = r14250 / r14254;
        double r14256 = r14246 * r14255;
        double r14257 = r14243 - r14256;
        return r14257;
}

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 16.7

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

  3. Applied *-un-lft-identity16.7

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

  4. Applied times-frac16.8

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

  5. Applied associate-*l*12.2

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

  7. Applied add-sqr-sqrt12.3

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

  8. Applied associate-*l*12.3

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

  10. Applied tan-quot12.3

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

  11. Applied associate-*r/12.3

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

  12. Simplified12.2

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

  13. Final simplification12.2

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

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))