Average Error: 29.0 → 0.6
Time: 20.9s
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 r46180 = x;
        double r46181 = exp(r46180);
        double r46182 = 2.0;
        double r46183 = r46181 - r46182;
        double r46184 = -r46180;
        double r46185 = exp(r46184);
        double r46186 = r46183 + r46185;
        return r46186;
}
↓
double f(double x) {
        double r46187 = x;
        double r46188 = 2.0;
        double r46189 = pow(r46187, r46188);
        double r46190 = 0.002777777777777778;
        double r46191 = 6.0;
        double r46192 = pow(r46187, r46191);
        double r46193 = r46190 * r46192;
        double r46194 = 0.08333333333333333;
        double r46195 = 4.0;
        double r46196 = pow(r46187, r46195);
        double r46197 = r46194 * r46196;
        double r46198 = r46193 + r46197;
        double r46199 = r46189 + r46198;
        return r46199;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.0

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

  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 2019326 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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