Average Error: 29.5 → 0.7
Time: 4.3s
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 r106184 = x;
        double r106185 = exp(r106184);
        double r106186 = 2.0;
        double r106187 = r106185 - r106186;
        double r106188 = -r106184;
        double r106189 = exp(r106188);
        double r106190 = r106187 + r106189;
        return r106190;
}
↓
double f(double x) {
        double r106191 = x;
        double r106192 = 2.0;
        double r106193 = pow(r106191, r106192);
        double r106194 = 0.002777777777777778;
        double r106195 = 6.0;
        double r106196 = pow(r106191, r106195);
        double r106197 = r106194 * r106196;
        double r106198 = 0.08333333333333333;
        double r106199 = 4.0;
        double r106200 = pow(r106191, r106199);
        double r106201 = r106198 * r106200;
        double r106202 = r106197 + r106201;
        double r106203 = r106193 + r106202;
        return r106203;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.5

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

  2. 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 2020047 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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