Average Error: 16.7 → 2.3
Time: 9.1s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
↓
\[\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \left(0.3333333333333333 \cdot \left(\pi \cdot \left(\ell \cdot F\right)\right) + \left(0.0021164021164021165 \cdot \left({\pi}^{5} \cdot \left(F \cdot {\ell}^{5}\right)\right) + 0.022222222222222223 \cdot \left({\pi}^{3} \cdot \left(F \cdot {\ell}^{3}\right)\right)\right)\right)}}{F} \]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \left(0.3333333333333333 \cdot \left(\pi \cdot \left(\ell \cdot F\right)\right) + \left(0.0021164021164021165 \cdot \left({\pi}^{5} \cdot \left(F \cdot {\ell}^{5}\right)\right) + 0.022222222222222223 \cdot \left({\pi}^{3} \cdot \left(F \cdot {\ell}^{3}\right)\right)\right)\right)}}{F}

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))
↓
(FPCore (F l)
 :precision binary64
 (-
  (* PI l)
  (/
   (/
    1.0
    (-
     (/ F (* PI l))
     (+
      (* 0.3333333333333333 (* PI (* l F)))
      (+
       (* 0.0021164021164021165 (* (pow PI 5.0) (* F (pow l 5.0))))
       (* 0.022222222222222223 (* (pow PI 3.0) (* F (pow l 3.0))))))))
   F)))
double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan(((double) M_PI) * l));
}
↓
double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / ((F / (((double) M_PI) * l)) - ((0.3333333333333333 * (((double) M_PI) * (l * F))) + ((0.0021164021164021165 * (pow(((double) M_PI), 5.0) * (F * pow(l, 5.0)))) + (0.022222222222222223 * (pow(((double) M_PI), 3.0) * (F * pow(l, 3.0)))))))) / F);
}

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 16.7

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

  2. Simplified16.5

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

  3. Applied associate-/r*_binary6412.4

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

  4. Applied clear-num_binary6412.4

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

  5. Taylor expanded in l around 0 2.3

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\color{blue}{\frac{F}{\pi \cdot \ell} - \left(0.3333333333333333 \cdot \left(\pi \cdot \left(F \cdot \ell\right)\right) + \left(0.0021164021164021165 \cdot \left({\pi}^{5} \cdot \left({\ell}^{5} \cdot F\right)\right) + 0.022222222222222223 \cdot \left({\pi}^{3} \cdot \left({\ell}^{3} \cdot F\right)\right)\right)\right)}}}{F} \]

  6. Final simplification2.3

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \left(0.3333333333333333 \cdot \left(\pi \cdot \left(\ell \cdot F\right)\right) + \left(0.0021164021164021165 \cdot \left({\pi}^{5} \cdot \left(F \cdot {\ell}^{5}\right)\right) + 0.022222222222222223 \cdot \left({\pi}^{3} \cdot \left(F \cdot {\ell}^{3}\right)\right)\right)\right)}}{F} \]

Reproduce

herbie shell --seed 2021275 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))