Average Error: 16.6 → 2.2
Time: 10.7s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
↓
\[\pi \cdot \ell - {\left(F \cdot \left(\frac{F}{\pi \cdot \ell} - \mathsf{fma}\left(0.3333333333333333, \left(\pi \cdot \ell\right) \cdot F, \mathsf{fma}\left(0.0021164021164021165, F \cdot \left({\ell}^{5} \cdot {\pi}^{5}\right), 0.022222222222222223 \cdot \left(F \cdot \left({\ell}^{3} \cdot {\pi}^{3}\right)\right)\right)\right)\right)\right)}^{-1} \]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - {\left(F \cdot \left(\frac{F}{\pi \cdot \ell} - \mathsf{fma}\left(0.3333333333333333, \left(\pi \cdot \ell\right) \cdot F, \mathsf{fma}\left(0.0021164021164021165, F \cdot \left({\ell}^{5} \cdot {\pi}^{5}\right), 0.022222222222222223 \cdot \left(F \cdot \left({\ell}^{3} \cdot {\pi}^{3}\right)\right)\right)\right)\right)\right)}^{-1}

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

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 16.6

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

  2. Simplified16.3

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

  3. Applied clear-num_binary6416.3

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

  4. Simplified12.3

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

  5. Applied associate-/r/_binary6412.3

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

  6. Applied *-un-lft-identity_binary6412.3

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

  7. Applied times-frac_binary6412.3

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

  8. Taylor expanded in l around 0 2.2

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

  9. Applied inv-pow_binary642.2

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

  10. Applied inv-pow_binary642.2

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

  11. Applied pow-prod-down_binary642.2

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

  12. Simplified2.2

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

  13. Final simplification2.2

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

Reproduce

herbie shell --seed 2022068 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))