Average Error: 17.2 → 0.4
Time: 50.1s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), \cos \left(\frac{K}{2}\right), U\right)\]
\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 r82131 = J;
        double r82132 = l;
        double r82133 = exp(r82132);
        double r82134 = -r82132;
        double r82135 = exp(r82134);
        double r82136 = r82133 - r82135;
        double r82137 = r82131 * r82136;
        double r82138 = K;
        double r82139 = 2.0;
        double r82140 = r82138 / r82139;
        double r82141 = cos(r82140);
        double r82142 = r82137 * r82141;
        double r82143 = U;
        double r82144 = r82142 + r82143;
        return r82144;
}
↓
double f(double J, double l, double K, double U) {
        double r82145 = J;
        double r82146 = 0.3333333333333333;
        double r82147 = l;
        double r82148 = 3.0;
        double r82149 = pow(r82147, r82148);
        double r82150 = 0.016666666666666666;
        double r82151 = 5.0;
        double r82152 = pow(r82147, r82151);
        double r82153 = 2.0;
        double r82154 = r82153 * r82147;
        double r82155 = fma(r82150, r82152, r82154);
        double r82156 = fma(r82146, r82149, r82155);
        double r82157 = r82145 * r82156;
        double r82158 = K;
        double r82159 = 2.0;
        double r82160 = r82158 / r82159;
        double r82161 = cos(r82160);
        double r82162 = U;
        double r82163 = fma(r82157, r82161, r82162);
        return r82163;
}

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. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  5. Applied fma-def0.4

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

  6. 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 2019325 +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))