Average Error: 29.6 → 0.6
Time: 18.1s
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 r54055 = x;
        double r54056 = exp(r54055);
        double r54057 = 2.0;
        double r54058 = r54056 - r54057;
        double r54059 = -r54055;
        double r54060 = exp(r54059);
        double r54061 = r54058 + r54060;
        return r54061;
}
↓
double f(double x) {
        double r54062 = x;
        double r54063 = 2.0;
        double r54064 = pow(r54062, r54063);
        double r54065 = 0.002777777777777778;
        double r54066 = 6.0;
        double r54067 = pow(r54062, r54066);
        double r54068 = r54065 * r54067;
        double r54069 = 0.08333333333333333;
        double r54070 = 4.0;
        double r54071 = pow(r54062, r54070);
        double r54072 = r54069 * r54071;
        double r54073 = r54068 + r54072;
        double r54074 = r54064 + r54073;
        return r54074;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.6

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

  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 2019325 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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