Average Error: 17.3 → 0.4
Time: 8.0s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{3}\right) + J \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right), \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{3}\right) + J \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right), \cos \left(\frac{K}{2}\right), U\right)
double f(double J, double l, double K, double U) {
        double r144015 = J;
        double r144016 = l;
        double r144017 = exp(r144016);
        double r144018 = -r144016;
        double r144019 = exp(r144018);
        double r144020 = r144017 - r144019;
        double r144021 = r144015 * r144020;
        double r144022 = K;
        double r144023 = 2.0;
        double r144024 = r144022 / r144023;
        double r144025 = cos(r144024);
        double r144026 = r144021 * r144025;
        double r144027 = U;
        double r144028 = r144026 + r144027;
        return r144028;
}
↓
double f(double J, double l, double K, double U) {
        double r144029 = J;
        double r144030 = 0.3333333333333333;
        double r144031 = l;
        double r144032 = 3.0;
        double r144033 = pow(r144031, r144032);
        double r144034 = r144030 * r144033;
        double r144035 = r144029 * r144034;
        double r144036 = 0.016666666666666666;
        double r144037 = 5.0;
        double r144038 = pow(r144031, r144037);
        double r144039 = 2.0;
        double r144040 = r144039 * r144031;
        double r144041 = fma(r144036, r144038, r144040);
        double r144042 = r144029 * r144041;
        double r144043 = r144035 + r144042;
        double r144044 = K;
        double r144045 = 2.0;
        double r144046 = r144044 / r144045;
        double r144047 = cos(r144046);
        double r144048 = U;
        double r144049 = fma(r144043, r144047, r144048);
        return r144049;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.3

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

  2. Simplified17.3

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

  3. Taylor expanded around 0 0.4

    \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3} + \left(\frac{1}{60} \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}, \cos \left(\frac{K}{2}\right), U\right)\]

  4. Simplified0.4

    \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right)}, \cos \left(\frac{K}{2}\right), U\right)\]
  5. Using strategy rm

  6. Applied fma-udef0.4

    \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3} + \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right)}, \cos \left(\frac{K}{2}\right), U\right)\]

  7. Applied distribute-lft-in0.4

    \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(\frac{1}{3} \cdot {\ell}^{3}\right) + J \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)}, \cos \left(\frac{K}{2}\right), U\right)\]

  8. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{3}\right) + J \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right), \cos \left(\frac{K}{2}\right), U\right)\]

Reproduce

herbie shell --seed 2020059 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))