Average Error: 17.9 → 0.5
Time: 7.4s
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 r148523 = J;
        double r148524 = l;
        double r148525 = exp(r148524);
        double r148526 = -r148524;
        double r148527 = exp(r148526);
        double r148528 = r148525 - r148527;
        double r148529 = r148523 * r148528;
        double r148530 = K;
        double r148531 = 2.0;
        double r148532 = r148530 / r148531;
        double r148533 = cos(r148532);
        double r148534 = r148529 * r148533;
        double r148535 = U;
        double r148536 = r148534 + r148535;
        return r148536;
}
↓
double f(double J, double l, double K, double U) {
        double r148537 = J;
        double r148538 = 0.3333333333333333;
        double r148539 = l;
        double r148540 = 3.0;
        double r148541 = pow(r148539, r148540);
        double r148542 = 0.016666666666666666;
        double r148543 = 5.0;
        double r148544 = pow(r148539, r148543);
        double r148545 = 2.0;
        double r148546 = r148545 * r148539;
        double r148547 = fma(r148542, r148544, r148546);
        double r148548 = fma(r148538, r148541, r148547);
        double r148549 = r148537 * r148548;
        double r148550 = K;
        double r148551 = 2.0;
        double r148552 = r148550 / r148551;
        double r148553 = cos(r148552);
        double r148554 = U;
        double r148555 = fma(r148549, r148553, r148554);
        return r148555;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.9

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  5. Applied fma-def0.5

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

  6. 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 2020001 +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))