Average Error: 16.8 → 0.4
Time: 7.5s
Precision: binary64
Cost: 7616
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\left(\ell + \ell\right) + \ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\left(\ell + \ell\right) + \ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)
(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))
↓
(FPCore (J l K U)
 :precision binary64
 (+
  U
  (* (cos (/ K 2.0)) (* J (+ (+ l l) (* l (* 0.3333333333333333 (* l l))))))))
double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos(K / 2.0)) + U;
}
↓
double code(double J, double l, double K, double U) {
	return U + (cos(K / 2.0) * (J * ((l + l) + (l * (0.3333333333333333 * (l * l))))));
}

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

Alternatives

Alternative 1
Error0.6
Cost7104
\[U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell + \ell\right)\right)\]
Alternative 2
Error8.8
Cost448
\[U + 2 \cdot \left(J \cdot \ell\right)\]
Alternative 3
Error61.7
Cost64
\[-1\]
Alternative 4
Error61.7
Cost64
\[1\]

Error

Derivation

  1. Initial program 16.8

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

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  3. Simplified0.4

    \[\leadsto \left(J \cdot \color{blue}{\left(\left(\ell + \ell\right) + 0.3333333333333333 \cdot {\ell}^{3}\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
  4. Using strategy rm

  5. Applied unpow3_binary64_8260.4

    \[\leadsto \left(J \cdot \left(\left(\ell + \ell\right) + 0.3333333333333333 \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot \ell\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  6. Applied associate-*r*_binary64_7000.4

    \[\leadsto \left(J \cdot \left(\left(\ell + \ell\right) + \color{blue}{\left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right) \cdot \ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  7. Simplified0.4

    \[\leadsto \color{blue}{U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\left(\ell + \ell\right) + \ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\]

  8. Final simplification0.4

    \[\leadsto U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\left(\ell + \ell\right) + \ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2021044 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))