Average Error: 29.7 → 0.6
Time: 5.0s
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 r133297 = x;
        double r133298 = exp(r133297);
        double r133299 = 2.0;
        double r133300 = r133298 - r133299;
        double r133301 = -r133297;
        double r133302 = exp(r133301);
        double r133303 = r133300 + r133302;
        return r133303;
}
↓
double f(double x) {
        double r133304 = x;
        double r133305 = 2.0;
        double r133306 = pow(r133304, r133305);
        double r133307 = 0.002777777777777778;
        double r133308 = 6.0;
        double r133309 = pow(r133304, r133308);
        double r133310 = r133307 * r133309;
        double r133311 = 0.08333333333333333;
        double r133312 = 4.0;
        double r133313 = pow(r133304, r133312);
        double r133314 = r133311 * r133313;
        double r133315 = r133310 + r133314;
        double r133316 = r133306 + r133315;
        return r133316;
}

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))))