Average Error: 16.8 → 8.1
Time: 9.0s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\sqrt{1} \cdot \frac{1}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \left(\pi \cdot \ell\right)\right)}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\sqrt{1} \cdot \frac{1}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \left(\pi \cdot \ell\right)\right)}\right)
double f(double F, double l) {
        double r16651 = atan2(1.0, 0.0);
        double r16652 = l;
        double r16653 = r16651 * r16652;
        double r16654 = 1.0;
        double r16655 = F;
        double r16656 = r16655 * r16655;
        double r16657 = r16654 / r16656;
        double r16658 = tan(r16653);
        double r16659 = r16657 * r16658;
        double r16660 = r16653 - r16659;
        return r16660;
}
↓
double f(double F, double l) {
        double r16661 = atan2(1.0, 0.0);
        double r16662 = l;
        double r16663 = r16661 * r16662;
        double r16664 = 1.0;
        double r16665 = sqrt(r16664);
        double r16666 = F;
        double r16667 = r16665 / r16666;
        double r16668 = 1.0;
        double r16669 = r16666 / r16663;
        double r16670 = 0.3333333333333333;
        double r16671 = r16666 * r16663;
        double r16672 = r16670 * r16671;
        double r16673 = r16669 - r16672;
        double r16674 = r16668 / r16673;
        double r16675 = r16665 * r16674;
        double r16676 = r16667 * r16675;
        double r16677 = r16663 - r16676;
        return r16677;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt16.8

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

  4. Applied times-frac16.8

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

  5. Applied associate-*l*12.2

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

  7. Applied div-inv12.2

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

  8. Applied associate-*l*12.2

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

  9. Simplified12.2

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

  11. Applied clear-num12.2

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

  12. Taylor expanded around 0 8.1

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

  13. Final simplification8.1

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

Reproduce

herbie shell --seed 2020083 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))