Average Error: 29.7 → 0.6
Time: 4.5s
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 r91774 = x;
        double r91775 = exp(r91774);
        double r91776 = 2.0;
        double r91777 = r91775 - r91776;
        double r91778 = -r91774;
        double r91779 = exp(r91778);
        double r91780 = r91777 + r91779;
        return r91780;
}
↓
double f(double x) {
        double r91781 = x;
        double r91782 = 2.0;
        double r91783 = pow(r91781, r91782);
        double r91784 = 0.002777777777777778;
        double r91785 = 6.0;
        double r91786 = pow(r91781, r91785);
        double r91787 = r91784 * r91786;
        double r91788 = 0.08333333333333333;
        double r91789 = 4.0;
        double r91790 = pow(r91781, r91789);
        double r91791 = r91788 * r91790;
        double r91792 = r91787 + r91791;
        double r91793 = r91783 + r91792;
        return r91793;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.7

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

  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 2019347 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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