Average Error: 17.8 → 0.7
Time: 54.1s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[2 \cdot \left(J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
2 \cdot \left(J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) + U
double f(double J, double l, double K, double U) {
        double r112626 = J;
        double r112627 = l;
        double r112628 = exp(r112627);
        double r112629 = -r112627;
        double r112630 = exp(r112629);
        double r112631 = r112628 - r112630;
        double r112632 = r112626 * r112631;
        double r112633 = K;
        double r112634 = 2.0;
        double r112635 = r112633 / r112634;
        double r112636 = cos(r112635);
        double r112637 = r112632 * r112636;
        double r112638 = U;
        double r112639 = r112637 + r112638;
        return r112639;
}
↓
double f(double J, double l, double K, double U) {
        double r112640 = 2.0;
        double r112641 = J;
        double r112642 = 0.5;
        double r112643 = K;
        double r112644 = r112642 * r112643;
        double r112645 = cos(r112644);
        double r112646 = l;
        double r112647 = r112645 * r112646;
        double r112648 = r112641 * r112647;
        double r112649 = r112640 * r112648;
        double r112650 = U;
        double r112651 = r112649 + r112650;
        return r112651;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 17.8

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

  2. Taylor expanded around 0 0.7

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

  3. Taylor expanded around inf 0.7

    \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right)} + U\]

  4. Final simplification0.7

    \[\leadsto 2 \cdot \left(J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) + U\]

Reproduce

herbie shell --seed 2019322 +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))