Average Error: 17.0 → 0.4
Time: 8.2s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(J \cdot \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)
double f(double J, double l, double K, double U) {
        double r118400 = J;
        double r118401 = l;
        double r118402 = exp(r118401);
        double r118403 = -r118401;
        double r118404 = exp(r118403);
        double r118405 = r118402 - r118404;
        double r118406 = r118400 * r118405;
        double r118407 = K;
        double r118408 = 2.0;
        double r118409 = r118407 / r118408;
        double r118410 = cos(r118409);
        double r118411 = r118406 * r118410;
        double r118412 = U;
        double r118413 = r118411 + r118412;
        return r118413;
}
↓
double f(double J, double l, double K, double U) {
        double r118414 = J;
        double r118415 = 0.3333333333333333;
        double r118416 = l;
        double r118417 = 3.0;
        double r118418 = pow(r118416, r118417);
        double r118419 = 0.016666666666666666;
        double r118420 = 5.0;
        double r118421 = pow(r118416, r118420);
        double r118422 = 2.0;
        double r118423 = r118422 * r118416;
        double r118424 = fma(r118419, r118421, r118423);
        double r118425 = fma(r118415, r118418, r118424);
        double r118426 = r118414 * r118425;
        double r118427 = K;
        double r118428 = 2.0;
        double r118429 = r118427 / r118428;
        double r118430 = cos(r118429);
        double r118431 = U;
        double r118432 = fma(r118426, r118430, r118431);
        return r118432;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.0

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

  2. Simplified17.0

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

  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2020065 +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))