Average Error: 17.0 → 0.4
Time: 8.1s
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 r172046 = J;
        double r172047 = l;
        double r172048 = exp(r172047);
        double r172049 = -r172047;
        double r172050 = exp(r172049);
        double r172051 = r172048 - r172050;
        double r172052 = r172046 * r172051;
        double r172053 = K;
        double r172054 = 2.0;
        double r172055 = r172053 / r172054;
        double r172056 = cos(r172055);
        double r172057 = r172052 * r172056;
        double r172058 = U;
        double r172059 = r172057 + r172058;
        return r172059;
}
↓
double f(double J, double l, double K, double U) {
        double r172060 = J;
        double r172061 = 0.3333333333333333;
        double r172062 = l;
        double r172063 = 3.0;
        double r172064 = pow(r172062, r172063);
        double r172065 = r172061 * r172064;
        double r172066 = r172060 * r172065;
        double r172067 = 0.016666666666666666;
        double r172068 = 5.0;
        double r172069 = pow(r172062, r172068);
        double r172070 = 2.0;
        double r172071 = r172070 * r172062;
        double r172072 = fma(r172067, r172069, r172071);
        double r172073 = r172060 * r172072;
        double r172074 = r172066 + r172073;
        double r172075 = K;
        double r172076 = 2.0;
        double r172077 = r172075 / r172076;
        double r172078 = cos(r172077);
        double r172079 = U;
        double r172080 = fma(r172074, r172078, r172079);
        return r172080;
}

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. 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 2020039 +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))