Average Error: 16.7 → 0.8
Time: 28.2s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - 1 \cdot \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \log \left(e^{\pi \cdot \ell}\right)\right)}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - 1 \cdot \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \log \left(e^{\pi \cdot \ell}\right)\right)}}{F}
double f(double F, double l) {
        double r25522 = atan2(1.0, 0.0);
        double r25523 = l;
        double r25524 = r25522 * r25523;
        double r25525 = 1.0;
        double r25526 = F;
        double r25527 = r25526 * r25526;
        double r25528 = r25525 / r25527;
        double r25529 = tan(r25524);
        double r25530 = r25528 * r25529;
        double r25531 = r25524 - r25530;
        return r25531;
}
↓
double f(double F, double l) {
        double r25532 = atan2(1.0, 0.0);
        double r25533 = l;
        double r25534 = r25532 * r25533;
        double r25535 = 1.0;
        double r25536 = 1.0;
        double r25537 = F;
        double r25538 = r25537 / r25534;
        double r25539 = 0.3333333333333333;
        double r25540 = exp(r25534);
        double r25541 = log(r25540);
        double r25542 = r25537 * r25541;
        double r25543 = r25539 * r25542;
        double r25544 = r25538 - r25543;
        double r25545 = r25536 / r25544;
        double r25546 = r25545 / r25537;
        double r25547 = r25535 * r25546;
        double r25548 = r25534 - r25547;
        return r25548;
}

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 div-inv16.7

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

  4. Applied associate-*l*16.7

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

  5. Simplified12.5

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

  7. Applied clear-num12.6

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

  8. Taylor expanded around 0 8.4

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

  10. Applied add-log-exp0.8

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

  11. Final simplification0.8

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

Reproduce

herbie shell --seed 2019199 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))