Average Error: 29.8 → 0.7
Time: 17.0s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + \left(x \cdot x + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + \left(x \cdot x + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)
double f(double x) {
        double r4666830 = x;
        double r4666831 = exp(r4666830);
        double r4666832 = 2.0;
        double r4666833 = r4666831 - r4666832;
        double r4666834 = -r4666830;
        double r4666835 = exp(r4666834);
        double r4666836 = r4666833 + r4666835;
        return r4666836;
}
↓
double f(double x) {
        double r4666837 = x;
        double r4666838 = r4666837 * r4666837;
        double r4666839 = r4666838 * r4666838;
        double r4666840 = 0.08333333333333333;
        double r4666841 = r4666839 * r4666840;
        double r4666842 = r4666837 * r4666838;
        double r4666843 = 0.002777777777777778;
        double r4666844 = r4666842 * r4666843;
        double r4666845 = r4666844 * r4666842;
        double r4666846 = r4666838 + r4666845;
        double r4666847 = r4666841 + r4666846;
        return r4666847;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.8

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

  2. Simplified29.8

    \[\leadsto \color{blue}{\left(e^{x} - 2\right) - \frac{-1}{e^{x}}}\]

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

    \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + \left(x \cdot x + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\]

Reproduce

herbie shell --seed 2019169 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))