Average Error: 30.2 → 0.6
Time: 4.4s
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 r107683 = x;
        double r107684 = exp(r107683);
        double r107685 = 2.0;
        double r107686 = r107684 - r107685;
        double r107687 = -r107683;
        double r107688 = exp(r107687);
        double r107689 = r107686 + r107688;
        return r107689;
}
↓
double f(double x) {
        double r107690 = x;
        double r107691 = 2.0;
        double r107692 = pow(r107690, r107691);
        double r107693 = 0.002777777777777778;
        double r107694 = 6.0;
        double r107695 = pow(r107690, r107694);
        double r107696 = r107693 * r107695;
        double r107697 = 0.08333333333333333;
        double r107698 = 4.0;
        double r107699 = pow(r107690, r107698);
        double r107700 = r107697 * r107699;
        double r107701 = r107696 + r107700;
        double r107702 = r107692 + r107701;
        return r107702;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.2
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.2

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

  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 2020001 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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