Average Error: 30.1 → 0.7
Time: 20.9s
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 r2913081 = x;
        double r2913082 = exp(r2913081);
        double r2913083 = 2.0;
        double r2913084 = r2913082 - r2913083;
        double r2913085 = -r2913081;
        double r2913086 = exp(r2913085);
        double r2913087 = r2913084 + r2913086;
        return r2913087;
}
↓
double f(double x) {
        double r2913088 = x;
        double r2913089 = r2913088 * r2913088;
        double r2913090 = r2913089 * r2913089;
        double r2913091 = r2913089 * r2913090;
        double r2913092 = 0.002777777777777778;
        double r2913093 = r2913091 * r2913092;
        double r2913094 = 0.08333333333333333;
        double r2913095 = r2913090 * r2913094;
        double r2913096 = r2913095 + r2913089;
        double r2913097 = r2913093 + r2913096;
        return r2913097;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.1

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019144 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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