Average Error: 30.1 → 0.6
Time: 21.4s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(x \cdot x + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(x \cdot x + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}
double f(double x) {
        double r4147605 = x;
        double r4147606 = exp(r4147605);
        double r4147607 = 2.0;
        double r4147608 = r4147606 - r4147607;
        double r4147609 = -r4147605;
        double r4147610 = exp(r4147609);
        double r4147611 = r4147608 + r4147610;
        return r4147611;
}
↓
double f(double x) {
        double r4147612 = x;
        double r4147613 = r4147612 * r4147612;
        double r4147614 = r4147612 * r4147613;
        double r4147615 = 0.002777777777777778;
        double r4147616 = r4147614 * r4147615;
        double r4147617 = r4147616 * r4147614;
        double r4147618 = r4147613 + r4147617;
        double r4147619 = r4147613 * r4147613;
        double r4147620 = 0.08333333333333333;
        double r4147621 = r4147619 * r4147620;
        double r4147622 = r4147618 + r4147621;
        return r4147622;
}

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

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

  5. Final simplification0.6

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

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))))