Average Error: 16.1 → 12.1
Time: 25.2s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - 1 \cdot \frac{\frac{\tan \left(\sqrt{\sqrt{\pi}} \cdot \left(\sqrt{\sqrt{\pi}} \cdot \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \ell\right)\right)\right)}{F}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - 1 \cdot \frac{\frac{\tan \left(\sqrt{\sqrt{\pi}} \cdot \left(\sqrt{\sqrt{\pi}} \cdot \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \ell\right)\right)\right)}{F}}{F}
double f(double F, double l) {
        double r20664 = atan2(1.0, 0.0);
        double r20665 = l;
        double r20666 = r20664 * r20665;
        double r20667 = 1.0;
        double r20668 = F;
        double r20669 = r20668 * r20668;
        double r20670 = r20667 / r20669;
        double r20671 = tan(r20666);
        double r20672 = r20670 * r20671;
        double r20673 = r20666 - r20672;
        return r20673;
}
↓
double f(double F, double l) {
        double r20674 = atan2(1.0, 0.0);
        double r20675 = l;
        double r20676 = r20674 * r20675;
        double r20677 = 1.0;
        double r20678 = sqrt(r20674);
        double r20679 = sqrt(r20678);
        double r20680 = r20679 * r20679;
        double r20681 = r20680 * r20675;
        double r20682 = r20679 * r20681;
        double r20683 = r20679 * r20682;
        double r20684 = tan(r20683);
        double r20685 = F;
        double r20686 = r20684 / r20685;
        double r20687 = r20686 / r20685;
        double r20688 = r20677 * r20687;
        double r20689 = r20676 - r20688;
        return r20689;
}

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 16.1

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

  3. Applied div-inv16.1

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

  4. Applied associate-*l*16.1

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

  5. Simplified12.1

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

  7. Applied add-sqr-sqrt12.2

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

  8. Applied associate-*l*12.2

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

  10. Applied add-sqr-sqrt12.2

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

  11. Applied sqrt-prod12.1

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

  12. Applied associate-*l*12.1

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

  14. Applied add-sqr-sqrt12.1

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

  15. Applied sqrt-prod12.1

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

  16. Final simplification12.1

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

Reproduce

herbie shell --seed 2019291 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))