Average Error: 29.6 → 0.6
Time: 18.1s
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 r42763 = x;
        double r42764 = exp(r42763);
        double r42765 = 2.0;
        double r42766 = r42764 - r42765;
        double r42767 = -r42763;
        double r42768 = exp(r42767);
        double r42769 = r42766 + r42768;
        return r42769;
}
↓
double f(double x) {
        double r42770 = x;
        double r42771 = 2.0;
        double r42772 = pow(r42770, r42771);
        double r42773 = 0.002777777777777778;
        double r42774 = 6.0;
        double r42775 = pow(r42770, r42774);
        double r42776 = r42773 * r42775;
        double r42777 = 0.08333333333333333;
        double r42778 = 4.0;
        double r42779 = pow(r42770, r42778);
        double r42780 = r42777 * r42779;
        double r42781 = r42776 + r42780;
        double r42782 = r42772 + r42781;
        return r42782;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.6

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

  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 2019304 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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