Average Error: 29.6 → 0.6
Time: 15.0s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + x \cdot x\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(\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) + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + x \cdot x
double f(double x) {
        double r1850895 = x;
        double r1850896 = exp(r1850895);
        double r1850897 = 2.0;
        double r1850898 = r1850896 - r1850897;
        double r1850899 = -r1850895;
        double r1850900 = exp(r1850899);
        double r1850901 = r1850898 + r1850900;
        return r1850901;
}
↓
double f(double x) {
        double r1850902 = x;
        double r1850903 = r1850902 * r1850902;
        double r1850904 = r1850902 * r1850903;
        double r1850905 = 0.002777777777777778;
        double r1850906 = r1850904 * r1850905;
        double r1850907 = r1850906 * r1850904;
        double r1850908 = 0.08333333333333333;
        double r1850909 = r1850903 * r1850903;
        double r1850910 = r1850908 * r1850909;
        double r1850911 = r1850907 + r1850910;
        double r1850912 = r1850911 + r1850903;
        return r1850912;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.6

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

  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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019156 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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