Average Error: 29.6 → 0.6
Time: 18.2s
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 r53031 = x;
        double r53032 = exp(r53031);
        double r53033 = 2.0;
        double r53034 = r53032 - r53033;
        double r53035 = -r53031;
        double r53036 = exp(r53035);
        double r53037 = r53034 + r53036;
        return r53037;
}
↓
double f(double x) {
        double r53038 = x;
        double r53039 = 2.0;
        double r53040 = pow(r53038, r53039);
        double r53041 = 0.002777777777777778;
        double r53042 = 6.0;
        double r53043 = pow(r53038, r53042);
        double r53044 = r53041 * r53043;
        double r53045 = 0.08333333333333333;
        double r53046 = 4.0;
        double r53047 = pow(r53038, r53046);
        double r53048 = r53045 * r53047;
        double r53049 = r53044 + r53048;
        double r53050 = r53040 + r53049;
        return r53050;
}

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))))