Average Error: 29.2 → 0.6
Time: 10.4s
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 r137268 = x;
        double r137269 = exp(r137268);
        double r137270 = 2.0;
        double r137271 = r137269 - r137270;
        double r137272 = -r137268;
        double r137273 = exp(r137272);
        double r137274 = r137271 + r137273;
        return r137274;
}
↓
double f(double x) {
        double r137275 = x;
        double r137276 = 2.0;
        double r137277 = pow(r137275, r137276);
        double r137278 = 0.002777777777777778;
        double r137279 = 6.0;
        double r137280 = pow(r137275, r137279);
        double r137281 = r137278 * r137280;
        double r137282 = 0.08333333333333333;
        double r137283 = 4.0;
        double r137284 = pow(r137275, r137283);
        double r137285 = r137282 * r137284;
        double r137286 = r137281 + r137285;
        double r137287 = r137277 + r137286;
        return r137287;
}

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 2020046 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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