Average Error: 29.3 → 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 r104676 = x;
        double r104677 = exp(r104676);
        double r104678 = 2.0;
        double r104679 = r104677 - r104678;
        double r104680 = -r104676;
        double r104681 = exp(r104680);
        double r104682 = r104679 + r104681;
        return r104682;
}
↓
double f(double x) {
        double r104683 = x;
        double r104684 = 2.0;
        double r104685 = pow(r104683, r104684);
        double r104686 = 0.002777777777777778;
        double r104687 = 6.0;
        double r104688 = pow(r104683, r104687);
        double r104689 = r104686 * r104688;
        double r104690 = 0.08333333333333333;
        double r104691 = 4.0;
        double r104692 = pow(r104683, r104691);
        double r104693 = r104690 * r104692;
        double r104694 = r104689 + r104693;
        double r104695 = r104685 + r104694;
        return r104695;
}

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 2020062 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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