Average Error: 29.7 → 0.7
Time: 5.1s
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 r106900 = x;
        double r106901 = exp(r106900);
        double r106902 = 2.0;
        double r106903 = r106901 - r106902;
        double r106904 = -r106900;
        double r106905 = exp(r106904);
        double r106906 = r106903 + r106905;
        return r106906;
}
↓
double f(double x) {
        double r106907 = x;
        double r106908 = 2.0;
        double r106909 = pow(r106907, r106908);
        double r106910 = 0.002777777777777778;
        double r106911 = 6.0;
        double r106912 = pow(r106907, r106911);
        double r106913 = r106910 * r106912;
        double r106914 = 0.08333333333333333;
        double r106915 = 4.0;
        double r106916 = pow(r106907, r106915);
        double r106917 = r106914 * r106916;
        double r106918 = r106913 + r106917;
        double r106919 = r106909 + r106918;
        return r106919;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.7

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

  2. Taylor expanded around 0 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2020027 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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