Average Error: 16.5 → 0.3
Time: 55.7s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[J \cdot \left((\frac{1}{60} \cdot \left({\ell}^{5}\right) + \left(\ell \cdot (\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right))_* \cdot \cos \left(\frac{K}{2}\right)\right) + U\]
double f(double J, double l, double K, double U) {
        double r12849752 = J;
        double r12849753 = l;
        double r12849754 = exp(r12849753);
        double r12849755 = -r12849753;
        double r12849756 = exp(r12849755);
        double r12849757 = r12849754 - r12849756;
        double r12849758 = r12849752 * r12849757;
        double r12849759 = K;
        double r12849760 = 2.0;
        double r12849761 = r12849759 / r12849760;
        double r12849762 = cos(r12849761);
        double r12849763 = r12849758 * r12849762;
        double r12849764 = U;
        double r12849765 = r12849763 + r12849764;
        return r12849765;
}
↓
double f(double J, double l, double K, double U) {
        double r12849766 = J;
        double r12849767 = 0.016666666666666666;
        double r12849768 = l;
        double r12849769 = 5.0;
        double r12849770 = pow(r12849768, r12849769);
        double r12849771 = 0.3333333333333333;
        double r12849772 = r12849768 * r12849768;
        double r12849773 = 2.0;
        double r12849774 = fma(r12849771, r12849772, r12849773);
        double r12849775 = r12849768 * r12849774;
        double r12849776 = fma(r12849767, r12849770, r12849775);
        double r12849777 = K;
        double r12849778 = r12849777 / r12849773;
        double r12849779 = cos(r12849778);
        double r12849780 = r12849776 * r12849779;
        double r12849781 = r12849766 * r12849780;
        double r12849782 = U;
        double r12849783 = r12849781 + r12849782;
        return r12849783;
}


\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
J \cdot \left((\frac{1}{60} \cdot \left({\ell}^{5}\right) + \left(\ell \cdot (\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right))_* \cdot \cos \left(\frac{K}{2}\right)\right) + U

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 16.5

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  5. Applied associate-*l*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))