Average Error: 8.4 → 0.8
Time: 1.4m
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 r1648214 = atan2(1.0, 0.0);
        double r1648215 = l;
        double r1648216 = r1648214 * r1648215;
        double r1648217 = 1.0;
        double r1648218 = F;
        double r1648219 = r1648218 * r1648218;
        double r1648220 = r1648217 / r1648219;
        double r1648221 = tan(r1648216);
        double r1648222 = r1648220 * r1648221;
        double r1648223 = r1648216 - r1648222;
        return r1648223;
}
↓
double f(double F, double l) {
        double r1648224 = atan2(1.0, 0.0);
        double r1648225 = l;
        double r1648226 = r1648224 * r1648225;
        double r1648227 = 1.0;
        double r1648228 = F;
        double r1648229 = sin(r1648226);
        double r1648230 = cos(r1648226);
        double r1648231 = r1648229 / r1648230;
        double r1648232 = r1648231 / r1648228;
        double r1648233 = r1648228 / r1648232;
        double r1648234 = r1648227 / r1648233;
        double r1648235 = r1648226 - r1648234;
        return r1648235;
}


\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 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))