Average Error: 30.1 → 0.5
Time: 21.6s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360} + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + x \cdot x\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360} + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + x \cdot x\right)
double f(double x) {
        double r3084245 = x;
        double r3084246 = exp(r3084245);
        double r3084247 = 2.0;
        double r3084248 = r3084246 - r3084247;
        double r3084249 = -r3084245;
        double r3084250 = exp(r3084249);
        double r3084251 = r3084248 + r3084250;
        return r3084251;
}
↓
double f(double x) {
        double r3084252 = x;
        double r3084253 = r3084252 * r3084252;
        double r3084254 = r3084253 * r3084253;
        double r3084255 = r3084253 * r3084254;
        double r3084256 = 0.002777777777777778;
        double r3084257 = r3084255 * r3084256;
        double r3084258 = 0.08333333333333333;
        double r3084259 = r3084254 * r3084258;
        double r3084260 = r3084259 + r3084253;
        double r3084261 = r3084257 + r3084260;
        return r3084261;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.5
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.1

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

  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

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019149 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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