Average Error: 17.6 → 0.5
Time: 6.6s
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 r129087 = J;
        double r129088 = l;
        double r129089 = exp(r129088);
        double r129090 = -r129088;
        double r129091 = exp(r129090);
        double r129092 = r129089 - r129091;
        double r129093 = r129087 * r129092;
        double r129094 = K;
        double r129095 = 2.0;
        double r129096 = r129094 / r129095;
        double r129097 = cos(r129096);
        double r129098 = r129093 * r129097;
        double r129099 = U;
        double r129100 = r129098 + r129099;
        return r129100;
}
↓
double f(double J, double l, double K, double U) {
        double r129101 = J;
        double r129102 = 0.3333333333333333;
        double r129103 = l;
        double r129104 = 3.0;
        double r129105 = pow(r129103, r129104);
        double r129106 = 0.016666666666666666;
        double r129107 = 5.0;
        double r129108 = pow(r129103, r129107);
        double r129109 = 2.0;
        double r129110 = r129109 * r129103;
        double r129111 = fma(r129106, r129108, r129110);
        double r129112 = fma(r129102, r129105, r129111);
        double r129113 = r129101 * r129112;
        double r129114 = K;
        double r129115 = 2.0;
        double r129116 = r129114 / r129115;
        double r129117 = cos(r129116);
        double r129118 = U;
        double r129119 = fma(r129113, r129117, r129118);
        return r129119;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.6

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

  2. Simplified17.6

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

  3. Taylor expanded around 0 0.5

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

  4. Simplified0.5

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

  6. Applied fma-udef0.5

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

  8. Applied fma-def0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2020047 +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))