Average Error: 29.7 → 0.6
Time: 6.3s
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 r133034 = x;
        double r133035 = exp(r133034);
        double r133036 = 2.0;
        double r133037 = r133035 - r133036;
        double r133038 = -r133034;
        double r133039 = exp(r133038);
        double r133040 = r133037 + r133039;
        return r133040;
}
↓
double f(double x) {
        double r133041 = x;
        double r133042 = 2.0;
        double r133043 = pow(r133041, r133042);
        double r133044 = 0.002777777777777778;
        double r133045 = 6.0;
        double r133046 = pow(r133041, r133045);
        double r133047 = r133044 * r133046;
        double r133048 = 0.08333333333333333;
        double r133049 = 4.0;
        double r133050 = pow(r133041, r133049);
        double r133051 = r133048 * r133050;
        double r133052 = r133047 + r133051;
        double r133053 = r133043 + r133052;
        return r133053;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.7

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

  2. Taylor expanded around 0 0.6

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

  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 2019344 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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