Average Error: 29.8 → 0.5
Time: 26.6s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(x \cdot x + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\]
\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) + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)
double f(double x) {
        double r4791178 = x;
        double r4791179 = exp(r4791178);
        double r4791180 = 2.0;
        double r4791181 = r4791179 - r4791180;
        double r4791182 = -r4791178;
        double r4791183 = exp(r4791182);
        double r4791184 = r4791181 + r4791183;
        return r4791184;
}
↓
double f(double x) {
        double r4791185 = x;
        double r4791186 = r4791185 * r4791185;
        double r4791187 = r4791185 * r4791186;
        double r4791188 = 0.002777777777777778;
        double r4791189 = r4791187 * r4791188;
        double r4791190 = r4791189 * r4791187;
        double r4791191 = r4791186 + r4791190;
        double r4791192 = 0.08333333333333333;
        double r4791193 = r4791186 * r4791186;
        double r4791194 = r4791192 * r4791193;
        double r4791195 = r4791191 + r4791194;
        return r4791195;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.0
Herbie0.5
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.8

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019192 
(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))))