Average Error: 29.0 → 0.7
Time: 4.5s
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 r107574 = x;
        double r107575 = exp(r107574);
        double r107576 = 2.0;
        double r107577 = r107575 - r107576;
        double r107578 = -r107574;
        double r107579 = exp(r107578);
        double r107580 = r107577 + r107579;
        return r107580;
}
↓
double f(double x) {
        double r107581 = x;
        double r107582 = 2.0;
        double r107583 = pow(r107581, r107582);
        double r107584 = 0.002777777777777778;
        double r107585 = 6.0;
        double r107586 = pow(r107581, r107585);
        double r107587 = r107584 * r107586;
        double r107588 = 0.08333333333333333;
        double r107589 = 4.0;
        double r107590 = pow(r107581, r107589);
        double r107591 = r107588 * r107590;
        double r107592 = r107587 + r107591;
        double r107593 = r107583 + r107592;
        return r107593;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.0

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

  2. Taylor expanded around 0 0.7

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

  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 2020060 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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