Average Error: 29.2 → 0.6
Time: 5.7s
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 r108710 = x;
        double r108711 = exp(r108710);
        double r108712 = 2.0;
        double r108713 = r108711 - r108712;
        double r108714 = -r108710;
        double r108715 = exp(r108714);
        double r108716 = r108713 + r108715;
        return r108716;
}
↓
double f(double x) {
        double r108717 = x;
        double r108718 = 2.0;
        double r108719 = pow(r108717, r108718);
        double r108720 = 0.002777777777777778;
        double r108721 = 6.0;
        double r108722 = pow(r108717, r108721);
        double r108723 = r108720 * r108722;
        double r108724 = 0.08333333333333333;
        double r108725 = 4.0;
        double r108726 = pow(r108717, r108725);
        double r108727 = r108724 * r108726;
        double r108728 = r108723 + r108727;
        double r108729 = r108719 + r108728;
        return r108729;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.2

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

  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 2020046 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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