Average Error: 29.0 → 0.7
Time: 4.6s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)
double f(double x) {
        double r106247 = x;
        double r106248 = exp(r106247);
        double r106249 = 2.0;
        double r106250 = r106248 - r106249;
        double r106251 = -r106247;
        double r106252 = exp(r106251);
        double r106253 = r106250 + r106252;
        return r106253;
}
↓
double f(double x) {
        double r106254 = x;
        double r106255 = 2.0;
        double r106256 = pow(r106254, r106255);
        double r106257 = 0.002777777777777778;
        double r106258 = 6.0;
        double r106259 = pow(r106254, r106258);
        double r106260 = r106257 * r106259;
        double r106261 = 0.08333333333333333;
        double r106262 = 4.0;
        double r106263 = pow(r106254, r106262);
        double r106264 = r106261 * r106263;
        double r106265 = r106260 + r106264;
        double r106266 = r106256 + r106265;
        return r106266;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.0

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

  2. Taylor expanded around 0 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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