Average Error: 29.8 → 0.8
Time: 5.3s
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 r110862 = x;
        double r110863 = exp(r110862);
        double r110864 = 2.0;
        double r110865 = r110863 - r110864;
        double r110866 = -r110862;
        double r110867 = exp(r110866);
        double r110868 = r110865 + r110867;
        return r110868;
}
↓
double f(double x) {
        double r110869 = x;
        double r110870 = 2.0;
        double r110871 = pow(r110869, r110870);
        double r110872 = 0.002777777777777778;
        double r110873 = 6.0;
        double r110874 = pow(r110869, r110873);
        double r110875 = r110872 * r110874;
        double r110876 = 0.08333333333333333;
        double r110877 = 4.0;
        double r110878 = pow(r110869, r110877);
        double r110879 = r110876 * r110878;
        double r110880 = r110875 + r110879;
        double r110881 = r110871 + r110880;
        return r110881;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.1
Herbie0.8
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.8

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

  2. Taylor expanded around 0 0.8

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

  3. Final simplification0.8

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

Reproduce

herbie shell --seed 2020049 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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