Average Error: 17.6 → 0.4
Time: 24.2s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(\left(\frac{1}{3} \cdot {\ell}^{3}\right) \cdot J + J \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right), \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(\left(\frac{1}{3} \cdot {\ell}^{3}\right) \cdot J + J \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right), \cos \left(\frac{K}{2}\right), U\right)
double f(double J, double l, double K, double U) {
        double r85605 = J;
        double r85606 = l;
        double r85607 = exp(r85606);
        double r85608 = -r85606;
        double r85609 = exp(r85608);
        double r85610 = r85607 - r85609;
        double r85611 = r85605 * r85610;
        double r85612 = K;
        double r85613 = 2.0;
        double r85614 = r85612 / r85613;
        double r85615 = cos(r85614);
        double r85616 = r85611 * r85615;
        double r85617 = U;
        double r85618 = r85616 + r85617;
        return r85618;
}
↓
double f(double J, double l, double K, double U) {
        double r85619 = 0.3333333333333333;
        double r85620 = l;
        double r85621 = 3.0;
        double r85622 = pow(r85620, r85621);
        double r85623 = r85619 * r85622;
        double r85624 = J;
        double r85625 = r85623 * r85624;
        double r85626 = 0.016666666666666666;
        double r85627 = 5.0;
        double r85628 = pow(r85620, r85627);
        double r85629 = 2.0;
        double r85630 = r85629 * r85620;
        double r85631 = fma(r85626, r85628, r85630);
        double r85632 = r85624 * r85631;
        double r85633 = r85625 + r85632;
        double r85634 = K;
        double r85635 = 2.0;
        double r85636 = r85634 / r85635;
        double r85637 = cos(r85636);
        double r85638 = U;
        double r85639 = fma(r85633, r85637, r85638);
        return r85639;
}

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. Simplified17.6

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  6. Applied fma-udef0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019235 +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))