Average Error: 16.3 → 0.8
Time: 32.0s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \left(F \cdot \log \left(e^{\pi \cdot \ell}\right)\right) \cdot \frac{1}{3}}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \left(F \cdot \log \left(e^{\pi \cdot \ell}\right)\right) \cdot \frac{1}{3}}}{F}
double f(double F, double l) {
        double r943272 = atan2(1.0, 0.0);
        double r943273 = l;
        double r943274 = r943272 * r943273;
        double r943275 = 1.0;
        double r943276 = F;
        double r943277 = r943276 * r943276;
        double r943278 = r943275 / r943277;
        double r943279 = tan(r943274);
        double r943280 = r943278 * r943279;
        double r943281 = r943274 - r943280;
        return r943281;
}
↓
double f(double F, double l) {
        double r943282 = atan2(1.0, 0.0);
        double r943283 = l;
        double r943284 = r943282 * r943283;
        double r943285 = 1.0;
        double r943286 = F;
        double r943287 = r943286 / r943284;
        double r943288 = exp(r943284);
        double r943289 = log(r943288);
        double r943290 = r943286 * r943289;
        double r943291 = 0.3333333333333333;
        double r943292 = r943290 * r943291;
        double r943293 = r943287 - r943292;
        double r943294 = r943285 / r943293;
        double r943295 = r943294 / r943286;
        double r943296 = r943284 - r943295;
        return r943296;
}

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 16.3

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

  2. Simplified16.1

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

  4. Applied associate-/r*12.3

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

  6. Applied clear-num12.3

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

  7. Taylor expanded around 0 8.1

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

  9. Applied add-log-exp0.8

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

  10. Final simplification0.8

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \left(F \cdot \log \left(e^{\pi \cdot \ell}\right)\right) \cdot \frac{1}{3}}}{F}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))