Average Error: 29.4 → 0.7
Time: 23.3s
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 r4268833 = x;
        double r4268834 = exp(r4268833);
        double r4268835 = 2.0;
        double r4268836 = r4268834 - r4268835;
        double r4268837 = -r4268833;
        double r4268838 = exp(r4268837);
        double r4268839 = r4268836 + r4268838;
        return r4268839;
}
↓
double f(double x) {
        double r4268840 = x;
        double r4268841 = r4268840 * r4268840;
        double r4268842 = r4268840 * r4268841;
        double r4268843 = 0.002777777777777778;
        double r4268844 = r4268842 * r4268843;
        double r4268845 = r4268844 * r4268842;
        double r4268846 = 0.08333333333333333;
        double r4268847 = r4268841 * r4268841;
        double r4268848 = r4268846 * r4268847;
        double r4268849 = r4268841 + r4268848;
        double r4268850 = r4268845 + r4268849;
        return r4268850;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.4

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019162 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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