Average Error: 29.3 → 0.6
Time: 13.4s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[x \cdot x + \left({x}^{6} \cdot \frac{1}{360} + \frac{1}{12} \cdot {x}^{4}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
x \cdot x + \left({x}^{6} \cdot \frac{1}{360} + \frac{1}{12} \cdot {x}^{4}\right)
double f(double x) {
        double r62834 = x;
        double r62835 = exp(r62834);
        double r62836 = 2.0;
        double r62837 = r62835 - r62836;
        double r62838 = -r62834;
        double r62839 = exp(r62838);
        double r62840 = r62837 + r62839;
        return r62840;
}
↓
double f(double x) {
        double r62841 = x;
        double r62842 = r62841 * r62841;
        double r62843 = 6.0;
        double r62844 = pow(r62841, r62843);
        double r62845 = 0.002777777777777778;
        double r62846 = r62844 * r62845;
        double r62847 = 0.08333333333333333;
        double r62848 = 4.0;
        double r62849 = pow(r62841, r62848);
        double r62850 = r62847 * r62849;
        double r62851 = r62846 + r62850;
        double r62852 = r62842 + r62851;
        return r62852;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.3

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

  2. Simplified29.3

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

  3. Taylor expanded around 0 0.6

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))