Average Error: 30.1 → 0.6
Time: 26.8s
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 r88604 = x;
        double r88605 = exp(r88604);
        double r88606 = 2.0;
        double r88607 = r88605 - r88606;
        double r88608 = -r88604;
        double r88609 = exp(r88608);
        double r88610 = r88607 + r88609;
        return r88610;
}
↓
double f(double x) {
        double r88611 = x;
        double r88612 = 2.0;
        double r88613 = pow(r88611, r88612);
        double r88614 = 0.002777777777777778;
        double r88615 = 6.0;
        double r88616 = pow(r88611, r88615);
        double r88617 = r88614 * r88616;
        double r88618 = 0.08333333333333333;
        double r88619 = 4.0;
        double r88620 = pow(r88611, r88619);
        double r88621 = r88618 * r88620;
        double r88622 = r88617 + r88621;
        double r88623 = r88613 + r88622;
        return r88623;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.1

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

  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 2019294 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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