Average Error: 29.0 → 0.6
Time: 21.9s
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 r87458 = x;
        double r87459 = exp(r87458);
        double r87460 = 2.0;
        double r87461 = r87459 - r87460;
        double r87462 = -r87458;
        double r87463 = exp(r87462);
        double r87464 = r87461 + r87463;
        return r87464;
}
↓
double f(double x) {
        double r87465 = x;
        double r87466 = 2.0;
        double r87467 = pow(r87465, r87466);
        double r87468 = 0.002777777777777778;
        double r87469 = 6.0;
        double r87470 = pow(r87465, r87469);
        double r87471 = r87468 * r87470;
        double r87472 = 0.08333333333333333;
        double r87473 = 4.0;
        double r87474 = pow(r87465, r87473);
        double r87475 = r87472 * r87474;
        double r87476 = r87471 + r87475;
        double r87477 = r87467 + r87476;
        return r87477;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.0
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.0

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

  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 2019326 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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