Average Error: 29.7 → 0.6
Time: 5.1s
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 r135234 = x;
        double r135235 = exp(r135234);
        double r135236 = 2.0;
        double r135237 = r135235 - r135236;
        double r135238 = -r135234;
        double r135239 = exp(r135238);
        double r135240 = r135237 + r135239;
        return r135240;
}
↓
double f(double x) {
        double r135241 = x;
        double r135242 = 2.0;
        double r135243 = pow(r135241, r135242);
        double r135244 = 0.002777777777777778;
        double r135245 = 6.0;
        double r135246 = pow(r135241, r135245);
        double r135247 = r135244 * r135246;
        double r135248 = 0.08333333333333333;
        double r135249 = 4.0;
        double r135250 = pow(r135241, r135249);
        double r135251 = r135248 * r135250;
        double r135252 = r135247 + r135251;
        double r135253 = r135243 + r135252;
        return r135253;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.7

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

  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 2020056 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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