Average Error: 17.2 → 2.7
Time: 10.3s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
↓
\[\pi \cdot \ell - \frac{1}{F \cdot \left(\frac{F}{\pi \cdot \ell} - \mathsf{fma}\left(\pi, \left(\ell \cdot F\right) \cdot 0.3333333333333333, \mathsf{log1p}\left(\mathsf{expm1}\left(F \cdot \mathsf{fma}\left(0.0021164021164021165, {\pi}^{5} \cdot {\ell}^{5}, 0.022222222222222223 \cdot {\left(\pi \cdot \ell\right)}^{3}\right)\right)\right)\right)\right)} \]
\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} - \mathsf{fma}\left(\pi, \left(\ell \cdot F\right) \cdot 0.3333333333333333, \mathsf{log1p}\left(\mathsf{expm1}\left(F \cdot \mathsf{fma}\left(0.0021164021164021165, {\pi}^{5} \cdot {\ell}^{5}, 0.022222222222222223 \cdot {\left(\pi \cdot \ell\right)}^{3}\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))
     (fma
      PI
      (* (* l F) 0.3333333333333333)
      (log1p
       (expm1
        (*
         F
         (fma
          0.0021164021164021165
          (* (pow PI 5.0) (pow l 5.0))
          (* 0.022222222222222223 (pow (* PI l) 3.0))))))))))))
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(((double) M_PI), ((l * F) * 0.3333333333333333), log1p(expm1(F * fma(0.0021164021164021165, (pow(((double) M_PI), 5.0) * pow(l, 5.0)), (0.022222222222222223 * pow((((double) M_PI) * l), 3.0)))))))));
}

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 17.2

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

  2. Simplified16.9

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

  3. Applied associate-/r*_binary6412.7

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

  4. Applied clear-num_binary6412.7

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

  5. Simplified12.7

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

  6. Taylor expanded in l around 0 2.4

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

  7. Simplified2.4

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

  8. Applied log1p-expm1-u_binary642.7

    \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot \left(\frac{F}{\pi \cdot \ell} - \mathsf{fma}\left(\pi, \left(\ell \cdot F\right) \cdot 0.3333333333333333, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(F \cdot \left(0.022222222222222223 \cdot \left({\pi}^{3} \cdot {\ell}^{3}\right) + 0.0021164021164021165 \cdot \left({\pi}^{5} \cdot {\ell}^{5}\right)\right)\right)\right)}\right)\right)} \]

  9. Simplified2.7

    \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot \left(\frac{F}{\pi \cdot \ell} - \mathsf{fma}\left(\pi, \left(\ell \cdot F\right) \cdot 0.3333333333333333, \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(F \cdot \mathsf{fma}\left(0.0021164021164021165, {\pi}^{5} \cdot {\ell}^{5}, 0.022222222222222223 \cdot {\left(\pi \cdot \ell\right)}^{3}\right)\right)}\right)\right)\right)} \]

  10. Final simplification2.7

    \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot \left(\frac{F}{\pi \cdot \ell} - \mathsf{fma}\left(\pi, \left(\ell \cdot F\right) \cdot 0.3333333333333333, \mathsf{log1p}\left(\mathsf{expm1}\left(F \cdot \mathsf{fma}\left(0.0021164021164021165, {\pi}^{5} \cdot {\ell}^{5}, 0.022222222222222223 \cdot {\left(\pi \cdot \ell\right)}^{3}\right)\right)\right)\right)\right)} \]

Reproduce

herbie shell --seed 2021339 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))