Average Error: 29.9 → 0.6
Time: 39.5s
Precision: 64
\[\left(e^{x} - 2.0\right) + e^{-x}\]
↓
\[\left(x \cdot x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{360} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot \left(x \cdot x\right)\]
\left(e^{x} - 2.0\right) + e^{-x}
↓
\left(x \cdot x + \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)\right) + \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot \left(x \cdot x\right)
double f(double x) {
        double r5472975 = x;
        double r5472976 = exp(r5472975);
        double r5472977 = 2.0;
        double r5472978 = r5472976 - r5472977;
        double r5472979 = -r5472975;
        double r5472980 = exp(r5472979);
        double r5472981 = r5472978 + r5472980;
        return r5472981;
}
↓
double f(double x) {
        double r5472982 = x;
        double r5472983 = r5472982 * r5472982;
        double r5472984 = r5472982 * r5472983;
        double r5472985 = 0.002777777777777778;
        double r5472986 = r5472985 * r5472984;
        double r5472987 = r5472984 * r5472986;
        double r5472988 = r5472983 + r5472987;
        double r5472989 = 0.08333333333333333;
        double r5472990 = r5472983 * r5472989;
        double r5472991 = r5472990 * r5472983;
        double r5472992 = r5472988 + r5472991;
        return r5472992;
}

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
\[4.0 \cdot {\left(\sinh \left(\frac{x}{2.0}\right)\right)}^{2.0}\]

Derivation

  1. Initial program 29.9

    \[\left(e^{x} - 2.0\right) + e^{-x}\]

  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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019165 
(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))))