Average Error: 29.5 → 0.7
Time: 11.3s
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 r92436 = x;
        double r92437 = exp(r92436);
        double r92438 = 2.0;
        double r92439 = r92437 - r92438;
        double r92440 = -r92436;
        double r92441 = exp(r92440);
        double r92442 = r92439 + r92441;
        return r92442;
}
↓
double f(double x) {
        double r92443 = x;
        double r92444 = 2.0;
        double r92445 = pow(r92443, r92444);
        double r92446 = 0.002777777777777778;
        double r92447 = 6.0;
        double r92448 = pow(r92443, r92447);
        double r92449 = r92446 * r92448;
        double r92450 = 0.08333333333333333;
        double r92451 = 4.0;
        double r92452 = pow(r92443, r92451);
        double r92453 = r92450 * r92452;
        double r92454 = r92449 + r92453;
        double r92455 = r92445 + r92454;
        return r92455;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.5

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

  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 2020047 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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