Average Error: 29.9 → 0.6
Time: 21.8s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\]
\left(e^{x} - 2\right) + e^{-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) + \left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)
double f(double x) {
        double r5247660 = x;
        double r5247661 = exp(r5247660);
        double r5247662 = 2.0;
        double r5247663 = r5247661 - r5247662;
        double r5247664 = -r5247660;
        double r5247665 = exp(r5247664);
        double r5247666 = r5247663 + r5247665;
        return r5247666;
}
↓
double f(double x) {
        double r5247667 = x;
        double r5247668 = r5247667 * r5247667;
        double r5247669 = r5247667 * r5247668;
        double r5247670 = 0.002777777777777778;
        double r5247671 = r5247669 * r5247670;
        double r5247672 = r5247671 * r5247669;
        double r5247673 = 0.08333333333333333;
        double r5247674 = r5247668 * r5247668;
        double r5247675 = r5247673 * r5247674;
        double r5247676 = r5247668 + r5247675;
        double r5247677 = r5247672 + r5247676;
        return r5247677;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.9

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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