Average Error: 16.8 → 0.7
Time: 8.3s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
↓
\[\pi \cdot \ell - \frac{1}{F \cdot \left(\frac{F}{\pi \cdot \ell} - \left(0.3333333333333333 \cdot \left(\pi \cdot \left(\ell \cdot F\right)\right) + \left(0.0021164021164021165 \cdot \left({\pi}^{5} \cdot \left(F \cdot {\ell}^{5}\right)\right) + 0.022222222222222223 \cdot \left({\pi}^{3} \cdot \log \left({\left(e^{{\ell}^{3}}\right)}^{F}\right)\right)\right)\right)\right)} \]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F \cdot \left(\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 \log \left({\left(e^{{\ell}^{3}}\right)}^{F}\right)\right)\right)\right)\right)}

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))
↓
(FPCore (F l)
 :precision binary64
 (-
  (* PI l)
  (/
   1.0
   (*
    F
    (-
     (/ 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) (log (pow (exp (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 * ((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) * log(pow(exp(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 16.8

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

  2. Simplified16.6

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

  3. Applied egg-rr12.7

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

  4. Applied egg-rr12.7

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

  5. Taylor expanded in l around 0 2.2

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

  6. Applied egg-rr0.7

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

  7. Final simplification0.7

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

Reproduce

herbie shell --seed 2022125 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))