Average Error: 29.2 → 0.6
Time: 5.6s
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 r119228 = x;
        double r119229 = exp(r119228);
        double r119230 = 2.0;
        double r119231 = r119229 - r119230;
        double r119232 = -r119228;
        double r119233 = exp(r119232);
        double r119234 = r119231 + r119233;
        return r119234;
}
↓
double f(double x) {
        double r119235 = x;
        double r119236 = 2.0;
        double r119237 = pow(r119235, r119236);
        double r119238 = 0.002777777777777778;
        double r119239 = 6.0;
        double r119240 = pow(r119235, r119239);
        double r119241 = r119238 * r119240;
        double r119242 = 0.08333333333333333;
        double r119243 = 4.0;
        double r119244 = pow(r119235, r119243);
        double r119245 = r119242 * r119244;
        double r119246 = r119241 + r119245;
        double r119247 = r119237 + r119246;
        return r119247;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.2

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

  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 2020059 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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