Average Error: 17.6 → 0.4
Time: 8.0s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), \cos \left(\frac{K}{2}\right), U\right)\]
\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 r157091 = J;
        double r157092 = l;
        double r157093 = exp(r157092);
        double r157094 = -r157092;
        double r157095 = exp(r157094);
        double r157096 = r157093 - r157095;
        double r157097 = r157091 * r157096;
        double r157098 = K;
        double r157099 = 2.0;
        double r157100 = r157098 / r157099;
        double r157101 = cos(r157100);
        double r157102 = r157097 * r157101;
        double r157103 = U;
        double r157104 = r157102 + r157103;
        return r157104;
}
↓
double f(double J, double l, double K, double U) {
        double r157105 = J;
        double r157106 = 0.3333333333333333;
        double r157107 = l;
        double r157108 = 3.0;
        double r157109 = pow(r157107, r157108);
        double r157110 = 0.016666666666666666;
        double r157111 = 5.0;
        double r157112 = pow(r157107, r157111);
        double r157113 = 2.0;
        double r157114 = r157113 * r157107;
        double r157115 = fma(r157110, r157112, r157114);
        double r157116 = fma(r157106, r157109, r157115);
        double r157117 = r157105 * r157116;
        double r157118 = K;
        double r157119 = 2.0;
        double r157120 = r157118 / r157119;
        double r157121 = cos(r157120);
        double r157122 = U;
        double r157123 = fma(r157117, r157121, r157122);
        return r157123;
}

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. 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 2019346 +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))