Average Error: 17.6 → 0.4
Time: 27.2s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J, \mathsf{fma}\left(\ell, \left(\ell \cdot \ell\right) \cdot \frac{1}{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, \ell + \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(J, \mathsf{fma}\left(\ell, \left(\ell \cdot \ell\right) \cdot \frac{1}{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, \ell + \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right), U\right)
double f(double J, double l, double K, double U) {
        double r5541369 = J;
        double r5541370 = l;
        double r5541371 = exp(r5541370);
        double r5541372 = -r5541370;
        double r5541373 = exp(r5541372);
        double r5541374 = r5541371 - r5541373;
        double r5541375 = r5541369 * r5541374;
        double r5541376 = K;
        double r5541377 = 2.0;
        double r5541378 = r5541376 / r5541377;
        double r5541379 = cos(r5541378);
        double r5541380 = r5541375 * r5541379;
        double r5541381 = U;
        double r5541382 = r5541380 + r5541381;
        return r5541382;
}
↓
double f(double J, double l, double K, double U) {
        double r5541383 = J;
        double r5541384 = l;
        double r5541385 = r5541384 * r5541384;
        double r5541386 = 0.3333333333333333;
        double r5541387 = r5541385 * r5541386;
        double r5541388 = 0.016666666666666666;
        double r5541389 = 5.0;
        double r5541390 = pow(r5541384, r5541389);
        double r5541391 = r5541384 + r5541384;
        double r5541392 = fma(r5541388, r5541390, r5541391);
        double r5541393 = fma(r5541384, r5541387, r5541392);
        double r5541394 = K;
        double r5541395 = 2.0;
        double r5541396 = r5541394 / r5541395;
        double r5541397 = cos(r5541396);
        double r5541398 = r5541393 * r5541397;
        double r5541399 = U;
        double r5541400 = fma(r5541383, r5541398, r5541399);
        return r5541400;
}

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

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

  3. Simplified0.4

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

  5. Applied associate-*l*0.4

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

  7. Applied fma-def0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))