Average Error: 30.1 → 0.6
Time: 19.5s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(x \cdot x + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360}\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(x \cdot x + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360}\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}
double f(double x) {
        double r5142751 = x;
        double r5142752 = exp(r5142751);
        double r5142753 = 2.0;
        double r5142754 = r5142752 - r5142753;
        double r5142755 = -r5142751;
        double r5142756 = exp(r5142755);
        double r5142757 = r5142754 + r5142756;
        return r5142757;
}
↓
double f(double x) {
        double r5142758 = x;
        double r5142759 = r5142758 * r5142758;
        double r5142760 = r5142758 * r5142759;
        double r5142761 = r5142760 * r5142760;
        double r5142762 = 0.002777777777777778;
        double r5142763 = r5142761 * r5142762;
        double r5142764 = r5142759 + r5142763;
        double r5142765 = r5142759 * r5142759;
        double r5142766 = 0.08333333333333333;
        double r5142767 = r5142765 * r5142766;
        double r5142768 = r5142764 + r5142767;
        return r5142768;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.1

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

  2. Simplified30.1

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

  3. Taylor expanded around 0 0.6

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019172 
(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))))