Average Error: 29.5 → 0.6
Time: 19.4s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(x \cdot x + \left(\frac{1}{360} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(x \cdot x + \left(\frac{1}{360} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}
double f(double x) {
        double r5507025 = x;
        double r5507026 = exp(r5507025);
        double r5507027 = 2.0;
        double r5507028 = r5507026 - r5507027;
        double r5507029 = -r5507025;
        double r5507030 = exp(r5507029);
        double r5507031 = r5507028 + r5507030;
        return r5507031;
}
↓
double f(double x) {
        double r5507032 = x;
        double r5507033 = r5507032 * r5507032;
        double r5507034 = 0.002777777777777778;
        double r5507035 = r5507033 * r5507032;
        double r5507036 = r5507034 * r5507035;
        double r5507037 = r5507036 * r5507035;
        double r5507038 = r5507033 + r5507037;
        double r5507039 = r5507033 * r5507033;
        double r5507040 = 0.08333333333333333;
        double r5507041 = r5507039 * r5507040;
        double r5507042 = r5507038 + r5507041;
        return r5507042;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.5

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

  2. Simplified29.5

    \[\leadsto \color{blue}{\left(e^{x} - 2\right) - \frac{-1}{e^{x}}}\]

  3. Taylor expanded around 0 0.6

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019174 
(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))))