Average Error: 29.5 → 0.5
Time: 19.0s
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 r69602 = x;
        double r69603 = exp(r69602);
        double r69604 = 2.0;
        double r69605 = r69603 - r69604;
        double r69606 = -r69602;
        double r69607 = exp(r69606);
        double r69608 = r69605 + r69607;
        return r69608;
}
↓
double f(double x) {
        double r69609 = x;
        double r69610 = 2.0;
        double r69611 = pow(r69609, r69610);
        double r69612 = 0.002777777777777778;
        double r69613 = 6.0;
        double r69614 = pow(r69609, r69613);
        double r69615 = r69612 * r69614;
        double r69616 = 0.08333333333333333;
        double r69617 = 4.0;
        double r69618 = pow(r69609, r69617);
        double r69619 = r69616 * r69618;
        double r69620 = r69615 + r69619;
        double r69621 = r69611 + r69620;
        return r69621;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.5

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

  2. Taylor expanded around 0 0.5

    \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)}\]

  3. Final simplification0.5

    \[\leadsto {x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\]

Reproduce

herbie shell --seed 2019235 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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