Average Error: 8.4 → 0.8
Time: 1.3m
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{\frac{F}{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}{F}}}\]
double f(double F, double l) {
        double r1583486 = atan2(1.0, 0.0);
        double r1583487 = l;
        double r1583488 = r1583486 * r1583487;
        double r1583489 = 1.0;
        double r1583490 = F;
        double r1583491 = r1583490 * r1583490;
        double r1583492 = r1583489 / r1583491;
        double r1583493 = tan(r1583488);
        double r1583494 = r1583492 * r1583493;
        double r1583495 = r1583488 - r1583494;
        return r1583495;
}
↓
double f(double F, double l) {
        double r1583496 = atan2(1.0, 0.0);
        double r1583497 = l;
        double r1583498 = r1583496 * r1583497;
        double r1583499 = 1.0;
        double r1583500 = F;
        double r1583501 = sin(r1583498);
        double r1583502 = cos(r1583498);
        double r1583503 = r1583501 / r1583502;
        double r1583504 = r1583503 / r1583500;
        double r1583505 = r1583500 / r1583504;
        double r1583506 = r1583499 / r1583505;
        double r1583507 = r1583498 - r1583506;
        return r1583507;
}


\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{\frac{F}{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}{F}}}

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 8.4

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

  2. Simplified7.9

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

  4. Applied associate-/r*0.7

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

  6. Applied clear-num0.8

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

  7. Taylor expanded around -inf 0.8

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

  8. Final simplification0.8

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

Reproduce

herbie shell --seed 2019101 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))