Average Error: 17.1 → 0.4
Time: 9.1s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\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
double f(double J, double l, double K, double U) {
        double r144545 = J;
        double r144546 = l;
        double r144547 = exp(r144546);
        double r144548 = -r144546;
        double r144549 = exp(r144548);
        double r144550 = r144547 - r144549;
        double r144551 = r144545 * r144550;
        double r144552 = K;
        double r144553 = 2.0;
        double r144554 = r144552 / r144553;
        double r144555 = cos(r144554);
        double r144556 = r144551 * r144555;
        double r144557 = U;
        double r144558 = r144556 + r144557;
        return r144558;
}
↓
double f(double J, double l, double K, double U) {
        double r144559 = J;
        double r144560 = 0.3333333333333333;
        double r144561 = l;
        double r144562 = 3.0;
        double r144563 = pow(r144561, r144562);
        double r144564 = 0.016666666666666666;
        double r144565 = 5.0;
        double r144566 = pow(r144561, r144565);
        double r144567 = 2.0;
        double r144568 = r144567 * r144561;
        double r144569 = fma(r144564, r144566, r144568);
        double r144570 = fma(r144560, r144563, r144569);
        double r144571 = r144559 * r144570;
        double r144572 = K;
        double r144573 = 2.0;
        double r144574 = r144572 / r144573;
        double r144575 = cos(r144574);
        double r144576 = r144571 * r144575;
        double r144577 = U;
        double r144578 = r144576 + r144577;
        return r144578;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.1

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

  2. Simplified17.1

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

  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

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2020089 +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))