Average Error: 29.3 → 0.7
Time: 5.4s
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 r110711 = x;
        double r110712 = exp(r110711);
        double r110713 = 2.0;
        double r110714 = r110712 - r110713;
        double r110715 = -r110711;
        double r110716 = exp(r110715);
        double r110717 = r110714 + r110716;
        return r110717;
}
↓
double f(double x) {
        double r110718 = x;
        double r110719 = 2.0;
        double r110720 = pow(r110718, r110719);
        double r110721 = 0.002777777777777778;
        double r110722 = 6.0;
        double r110723 = pow(r110718, r110722);
        double r110724 = r110721 * r110723;
        double r110725 = 0.08333333333333333;
        double r110726 = 4.0;
        double r110727 = pow(r110718, r110726);
        double r110728 = r110725 * r110727;
        double r110729 = r110724 + r110728;
        double r110730 = r110720 + r110729;
        return r110730;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
Target0.1
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 2019362 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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