Average Error: 17.0 → 0.6
Time: 20.0s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \left(2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \ell\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
U + \left(2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \ell\right)
double f(double J, double l, double K, double U) {
        double r1582540 = J;
        double r1582541 = l;
        double r1582542 = exp(r1582541);
        double r1582543 = -r1582541;
        double r1582544 = exp(r1582543);
        double r1582545 = r1582542 - r1582544;
        double r1582546 = r1582540 * r1582545;
        double r1582547 = K;
        double r1582548 = 2.0;
        double r1582549 = r1582547 / r1582548;
        double r1582550 = cos(r1582549);
        double r1582551 = r1582546 * r1582550;
        double r1582552 = U;
        double r1582553 = r1582551 + r1582552;
        return r1582553;
}
↓
double f(double J, double l, double K, double U) {
        double r1582554 = U;
        double r1582555 = 2.0;
        double r1582556 = J;
        double r1582557 = r1582555 * r1582556;
        double r1582558 = K;
        double r1582559 = r1582558 / r1582555;
        double r1582560 = cos(r1582559);
        double r1582561 = l;
        double r1582562 = r1582560 * r1582561;
        double r1582563 = r1582557 * r1582562;
        double r1582564 = r1582554 + r1582563;
        return r1582564;
}

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 17.0

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

  2. Simplified17.0

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

  3. Taylor expanded around 0 0.6

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

  5. Applied add-sqr-sqrt0.9

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

  6. Applied associate-*l*0.8

    \[\leadsto U + \cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\sqrt{2} \cdot \left(J \cdot \ell\right)\right)\right)}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity0.8

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

  9. Applied associate-*l*0.8

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

  10. Simplified0.6

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

  11. Final simplification0.6

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

Reproduce

herbie shell --seed 2019156 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))