Average Error: 17.7 → 0.4
Time: 8.4s
Precision: binary64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
↓
\[\mathsf{fma}\left(J, \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]
(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))
↓
(FPCore (J l K U)
 :precision binary64
 (fma J (* (fma 0.3333333333333333 (pow l 3.0) (* l 2.0)) (cos (/ K 2.0))) 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) {
	return fma(J, (fma(0.3333333333333333, pow(l, 3.0), (l * 2.0)) * cos((K / 2.0))), 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)
	return fma(J, Float64(fma(0.3333333333333333, (l ^ 3.0), Float64(l * 2.0)) * cos(Float64(K / 2.0))), 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_] := N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(J, \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right), U\right)

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.7

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

  2. Simplified17.7

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

  3. Taylor expanded in l around 0 0.4

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

  4. Simplified0.4

    \[\leadsto \mathsf{fma}\left(J, \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)} \cdot \cos \left(\frac{K}{2}\right), U\right) \]

  5. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(J, \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]

Reproduce

herbie shell --seed 2022166 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))