Average Error: 29.9 → 0.6
Time: 19.1s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)
double f(double x) {
        double r77973 = x;
        double r77974 = exp(r77973);
        double r77975 = 2.0;
        double r77976 = r77974 - r77975;
        double r77977 = -r77973;
        double r77978 = exp(r77977);
        double r77979 = r77976 + r77978;
        return r77979;
}
↓
double f(double x) {
        double r77980 = x;
        double r77981 = 2.0;
        double r77982 = pow(r77980, r77981);
        double r77983 = 0.002777777777777778;
        double r77984 = 6.0;
        double r77985 = pow(r77980, r77984);
        double r77986 = r77983 * r77985;
        double r77987 = 0.08333333333333333;
        double r77988 = 4.0;
        double r77989 = pow(r77980, r77988);
        double r77990 = r77987 * r77989;
        double r77991 = r77986 + r77990;
        double r77992 = r77982 + r77991;
        return r77992;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.9

    \[\left(e^{x} - 2\right) + e^{-x}\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)}\]

  3. Final simplification0.6

    \[\leadsto {x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\]

Reproduce

herbie shell --seed 2019212 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

  :herbie-target
  (* 4 (pow (sinh (/ x 2)) 2))

  (+ (- (exp x) 2) (exp (- x))))