Average Error: 29.4 → 0.5
Time: 13.4s
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 r4134616 = x;
        double r4134617 = exp(r4134616);
        double r4134618 = 2.0;
        double r4134619 = r4134617 - r4134618;
        double r4134620 = -r4134616;
        double r4134621 = exp(r4134620);
        double r4134622 = r4134619 + r4134621;
        return r4134622;
}
↓
double f(double x) {
        double r4134623 = x;
        double r4134624 = r4134623 * r4134623;
        double r4134625 = r4134623 * r4134624;
        double r4134626 = r4134625 * r4134625;
        double r4134627 = 0.002777777777777778;
        double r4134628 = r4134626 * r4134627;
        double r4134629 = r4134624 + r4134628;
        double r4134630 = r4134624 * r4134624;
        double r4134631 = 0.08333333333333333;
        double r4134632 = r4134630 * r4134631;
        double r4134633 = r4134629 + r4134632;
        return r4134633;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.0
Herbie0.5
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.4

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

  2. Simplified29.4

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

  3. Taylor expanded around 0 0.5

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

  4. Simplified0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019171 
(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))))