Average Error: 17.2 → 0.3
Time: 8.9s
Precision: binary64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
↓
\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ \mathsf{fma}\left(J, 0.016666666666666666 \cdot \left({\ell}^{5} \cdot t_0\right) + \left(0.0003968253968253968 \cdot \left(t_0 \cdot {\ell}^{7}\right) + \left(0.3333333333333333 \cdot \left(t_0 \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot t_0\right)\right)\right), U\right) \end{array} \]
(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))
↓
(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K))))
   (fma
    J
    (+
     (* 0.016666666666666666 (* (pow l 5.0) t_0))
     (+
      (* 0.0003968253968253968 (* t_0 (pow l 7.0)))
      (+ (* 0.3333333333333333 (* t_0 (pow l 3.0))) (* 2.0 (* l t_0)))))
    U)))
double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}
↓
double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K));
	return fma(J, ((0.016666666666666666 * (pow(l, 5.0) * t_0)) + ((0.0003968253968253968 * (t_0 * pow(l, 7.0))) + ((0.3333333333333333 * (t_0 * pow(l, 3.0))) + (2.0 * (l * t_0))))), U);
}

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end
↓
function code(J, l, K, U)
	t_0 = cos(Float64(0.5 * K))
	return fma(J, Float64(Float64(0.016666666666666666 * Float64((l ^ 5.0) * t_0)) + Float64(Float64(0.0003968253968253968 * Float64(t_0 * (l ^ 7.0))) + Float64(Float64(0.3333333333333333 * Float64(t_0 * (l ^ 3.0))) + Float64(2.0 * Float64(l * t_0))))), U)
end

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]
↓
code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(J * N[(N[(0.016666666666666666 * N[(N[Power[l, 5.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0003968253968253968 * N[(t$95$0 * N[Power[l, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(t$95$0 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\begin{array}{l}
t_0 := \cos \left(0.5 \cdot K\right)\\
\mathsf{fma}\left(J, 0.016666666666666666 \cdot \left({\ell}^{5} \cdot t_0\right) + \left(0.0003968253968253968 \cdot \left(t_0 \cdot {\ell}^{7}\right) + \left(0.3333333333333333 \cdot \left(t_0 \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot t_0\right)\right)\right), U\right)
\end{array}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.2

    \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

  2. Simplified17.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]

  3. Taylor expanded in l around 0 0.3

    \[\leadsto \mathsf{fma}\left(J, \color{blue}{0.016666666666666666 \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right) + \left(0.0003968253968253968 \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right) + \left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right) + 2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)}, U\right) \]

  4. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(J, 0.016666666666666666 \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right) + \left(0.0003968253968253968 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{7}\right) + \left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right), U\right) \]

Reproduce

herbie shell --seed 2022162 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))