Average Error: 16.8 → 12.7
Time: 22.5s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \left(1 \cdot \frac{\tan \left(\left(\sqrt{\pi} \cdot \ell\right) \cdot \left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)}{F}\right) \cdot \frac{1}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \left(1 \cdot \frac{\tan \left(\left(\sqrt{\pi} \cdot \ell\right) \cdot \left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)}{F}\right) \cdot \frac{1}{F}
double f(double F, double l) {
        double r22124 = atan2(1.0, 0.0);
        double r22125 = l;
        double r22126 = r22124 * r22125;
        double r22127 = 1.0;
        double r22128 = F;
        double r22129 = r22128 * r22128;
        double r22130 = r22127 / r22129;
        double r22131 = tan(r22126);
        double r22132 = r22130 * r22131;
        double r22133 = r22126 - r22132;
        return r22133;
}
↓
double f(double F, double l) {
        double r22134 = atan2(1.0, 0.0);
        double r22135 = l;
        double r22136 = r22134 * r22135;
        double r22137 = 1.0;
        double r22138 = sqrt(r22134);
        double r22139 = r22138 * r22135;
        double r22140 = sqrt(r22138);
        double r22141 = r22140 * r22140;
        double r22142 = r22139 * r22141;
        double r22143 = tan(r22142);
        double r22144 = F;
        double r22145 = r22143 / r22144;
        double r22146 = r22137 * r22145;
        double r22147 = 1.0;
        double r22148 = r22147 / r22144;
        double r22149 = r22146 * r22148;
        double r22150 = r22136 - r22149;
        return r22150;
}

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 16.8

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

  2. Simplified16.6

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

  4. Applied *-un-lft-identity16.6

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

  5. Applied times-frac16.6

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

  6. Applied *-un-lft-identity16.6

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

  7. Applied times-frac12.7

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

  8. Simplified12.7

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

  9. Simplified12.7

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

  11. Applied add-sqr-sqrt12.8

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

  12. Applied associate-*l*12.8

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

  13. Simplified12.8

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

  15. Applied add-sqr-sqrt12.8

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

  16. Applied sqrt-prod12.7

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

  17. Final simplification12.7

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

Reproduce

herbie shell --seed 2019174 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))