Average Error: 17.0 → 2.4
Time: 12.2s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
↓
\[\pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\pi \cdot \ell} - \mathsf{fma}\left(F, \mathsf{fma}\left(0.3333333333333333, \pi \cdot \ell, 0.022222222222222223 \cdot \left({\pi}^{3} \cdot {\ell}^{3}\right)\right), 0.0021164021164021165 \cdot \left({\pi}^{5} \cdot \left(F \cdot {\ell}^{5}\right)\right)\right)} \]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\pi \cdot \ell} - \mathsf{fma}\left(F, \mathsf{fma}\left(0.3333333333333333, \pi \cdot \ell, 0.022222222222222223 \cdot \left({\pi}^{3} \cdot {\ell}^{3}\right)\right), 0.0021164021164021165 \cdot \left({\pi}^{5} \cdot \left(F \cdot {\ell}^{5}\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))
    (fma
     F
     (fma
      0.3333333333333333
      (* PI l)
      (* 0.022222222222222223 (* (pow PI 3.0) (pow l 3.0))))
     (* 0.0021164021164021165 (* (pow PI 5.0) (* F (pow l 5.0)))))))))
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) - ((1.0 / F) / ((F / (((double) M_PI) * l)) - fma(F, fma(0.3333333333333333, (((double) M_PI) * l), (0.022222222222222223 * (pow(((double) M_PI), 3.0) * pow(l, 3.0)))), (0.0021164021164021165 * (pow(((double) M_PI), 5.0) * (F * pow(l, 5.0)))))));
}

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 17.0

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

  2. Simplified16.7

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

  3. Applied clear-num_binary6416.8

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

  4. Simplified12.6

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

  5. Applied div-inv_binary6412.6

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

  6. Applied associate-/r*_binary6412.6

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

  7. Taylor expanded in l around 0 2.4

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

  8. Simplified2.4

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

  9. Final simplification2.4

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

Reproduce

herbie shell --seed 2022117 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))