Average Error: 17.2 → 1.9
Time: 8.4s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
↓
\[\pi \cdot \ell - \frac{1}{F \cdot \left(\frac{F}{\pi \cdot \ell} + \mathsf{fma}\left(F, \mathsf{fma}\left(\pi \cdot \ell, -0.3333333333333333, {\ell}^{3} \cdot \left({\pi}^{3} \cdot -0.022222222222222223\right)\right), \mathsf{log1p}\left(\mathsf{expm1}\left({\ell}^{5} \cdot \left(F \cdot {\pi}^{5}\right)\right)\right) \cdot -0.0021164021164021165\right)\right)} \]
(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
       (* PI l)
       -0.3333333333333333
       (* (pow l 3.0) (* (pow PI 3.0) -0.022222222222222223)))
      (*
       (log1p (expm1 (* (pow l 5.0) (* F (pow PI 5.0)))))
       -0.0021164021164021165)))))))
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((((double) M_PI) * l), -0.3333333333333333, (pow(l, 3.0) * (pow(((double) M_PI), 3.0) * -0.022222222222222223))), (log1p(expm1((pow(l, 5.0) * (F * pow(((double) M_PI), 5.0))))) * -0.0021164021164021165)))));
}

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l))))
end
↓
function code(F, l)
	return Float64(Float64(pi * l) - Float64(1.0 / Float64(F * Float64(Float64(F / Float64(pi * l)) + fma(F, fma(Float64(pi * l), -0.3333333333333333, Float64((l ^ 3.0) * Float64((pi ^ 3.0) * -0.022222222222222223))), Float64(log1p(expm1(Float64((l ^ 5.0) * Float64(F * (pi ^ 5.0))))) * -0.0021164021164021165))))))
end

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(1.0 / N[(F * N[(N[(F / N[(Pi * l), $MachinePrecision]), $MachinePrecision] + N[(F * N[(N[(Pi * l), $MachinePrecision] * -0.3333333333333333 + N[(N[Power[l, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * -0.022222222222222223), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[1 + N[(Exp[N[(N[Power[l, 5.0], $MachinePrecision] * N[(F * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * -0.0021164021164021165), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

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 egg-rr12.8

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

  4. Applied egg-rr12.8

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

  5. Taylor expanded in l around 0 2.1

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

  6. Simplified2.3

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

  7. Applied egg-rr1.9

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

  8. Final simplification1.9

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

Reproduce

herbie shell --seed 2022153 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))