Average Error: 30.2 → 0.6
Time: 4.3s
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 r96398 = x;
        double r96399 = exp(r96398);
        double r96400 = 2.0;
        double r96401 = r96399 - r96400;
        double r96402 = -r96398;
        double r96403 = exp(r96402);
        double r96404 = r96401 + r96403;
        return r96404;
}
↓
double f(double x) {
        double r96405 = x;
        double r96406 = 2.0;
        double r96407 = pow(r96405, r96406);
        double r96408 = 0.002777777777777778;
        double r96409 = 6.0;
        double r96410 = pow(r96405, r96409);
        double r96411 = r96408 * r96410;
        double r96412 = 0.08333333333333333;
        double r96413 = 4.0;
        double r96414 = pow(r96405, r96413);
        double r96415 = r96412 * r96414;
        double r96416 = r96411 + r96415;
        double r96417 = r96407 + r96416;
        return r96417;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 30.2

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

  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 2020001 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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