Average Error: 29.6 → 0.7
Time: 10.3s
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 r70236 = x;
        double r70237 = exp(r70236);
        double r70238 = 2.0;
        double r70239 = r70237 - r70238;
        double r70240 = -r70236;
        double r70241 = exp(r70240);
        double r70242 = r70239 + r70241;
        return r70242;
}
↓
double f(double x) {
        double r70243 = x;
        double r70244 = 2.0;
        double r70245 = pow(r70243, r70244);
        double r70246 = 0.002777777777777778;
        double r70247 = 6.0;
        double r70248 = pow(r70243, r70247);
        double r70249 = r70246 * r70248;
        double r70250 = 0.08333333333333333;
        double r70251 = 4.0;
        double r70252 = pow(r70243, r70251);
        double r70253 = r70250 * r70252;
        double r70254 = r70249 + r70253;
        double r70255 = r70245 + r70254;
        return r70255;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.6

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

  2. Taylor expanded around 0 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2020043 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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