Average Error: 17.6 → 0.7
Time: 27.2s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right) + \left(U + \ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right) + \left(U + \ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)
double f(double J, double l, double K, double U) {
        double r6968393 = J;
        double r6968394 = l;
        double r6968395 = exp(r6968394);
        double r6968396 = -r6968394;
        double r6968397 = exp(r6968396);
        double r6968398 = r6968395 - r6968397;
        double r6968399 = r6968393 * r6968398;
        double r6968400 = K;
        double r6968401 = 2.0;
        double r6968402 = r6968400 / r6968401;
        double r6968403 = cos(r6968402);
        double r6968404 = r6968399 * r6968403;
        double r6968405 = U;
        double r6968406 = r6968404 + r6968405;
        return r6968406;
}
↓
double f(double J, double l, double K, double U) {
        double r6968407 = l;
        double r6968408 = 0.5;
        double r6968409 = K;
        double r6968410 = r6968408 * r6968409;
        double r6968411 = cos(r6968410);
        double r6968412 = J;
        double r6968413 = r6968411 * r6968412;
        double r6968414 = r6968407 * r6968413;
        double r6968415 = U;
        double r6968416 = r6968415 + r6968414;
        double r6968417 = r6968414 + r6968416;
        return r6968417;
}

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 17.6

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

  2. Simplified17.6

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

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Taylor expanded around inf 0.7

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

  6. Simplified0.7

    \[\leadsto U + \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell + \ell\right)}\]
  7. Using strategy rm

  8. Applied distribute-rgt-in0.7

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

  9. Applied associate-+r+0.7

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

  10. Final simplification0.7

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

Reproduce

herbie shell --seed 2019171 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))