Average Error: 29.3 → 0.7
Time: 23.5s
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 r98585 = x;
        double r98586 = exp(r98585);
        double r98587 = 2.0;
        double r98588 = r98586 - r98587;
        double r98589 = -r98585;
        double r98590 = exp(r98589);
        double r98591 = r98588 + r98590;
        return r98591;
}
↓
double f(double x) {
        double r98592 = x;
        double r98593 = 2.0;
        double r98594 = pow(r98592, r98593);
        double r98595 = 0.002777777777777778;
        double r98596 = 6.0;
        double r98597 = pow(r98592, r98596);
        double r98598 = r98595 * r98597;
        double r98599 = 0.08333333333333333;
        double r98600 = 4.0;
        double r98601 = pow(r98592, r98600);
        double r98602 = r98599 * r98601;
        double r98603 = r98598 + r98602;
        double r98604 = r98594 + r98603;
        return r98604;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.3

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

  2. Taylor expanded around 0 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2019297 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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