Average Error: 17.1 → 0.3
Time: 30.9s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right), \frac{1}{60} \cdot \left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\ell \cdot \ell\right)\right)\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right), \frac{1}{60} \cdot \left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\ell \cdot \ell\right)\right)\right) + U
double f(double J, double l, double K, double U) {
        double r2227551 = J;
        double r2227552 = l;
        double r2227553 = exp(r2227552);
        double r2227554 = -r2227552;
        double r2227555 = exp(r2227554);
        double r2227556 = r2227553 - r2227555;
        double r2227557 = r2227551 * r2227556;
        double r2227558 = K;
        double r2227559 = 2.0;
        double r2227560 = r2227558 / r2227559;
        double r2227561 = cos(r2227560);
        double r2227562 = r2227557 * r2227561;
        double r2227563 = U;
        double r2227564 = r2227562 + r2227563;
        return r2227564;
}
↓
double f(double J, double l, double K, double U) {
        double r2227565 = J;
        double r2227566 = K;
        double r2227567 = 2.0;
        double r2227568 = r2227566 / r2227567;
        double r2227569 = cos(r2227568);
        double r2227570 = r2227565 * r2227569;
        double r2227571 = l;
        double r2227572 = r2227571 * r2227571;
        double r2227573 = 0.3333333333333333;
        double r2227574 = fma(r2227572, r2227573, r2227567);
        double r2227575 = 0.016666666666666666;
        double r2227576 = r2227571 * r2227572;
        double r2227577 = r2227576 * r2227572;
        double r2227578 = r2227575 * r2227577;
        double r2227579 = fma(r2227571, r2227574, r2227578);
        double r2227580 = r2227570 * r2227579;
        double r2227581 = U;
        double r2227582 = r2227580 + r2227581;
        return r2227582;
}

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. 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

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

  5. Applied pow10.3

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

  6. Applied pow10.3

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

  7. Applied pow10.3

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

  8. Applied pow-prod-down0.3

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

  9. Applied pow-prod-down0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))