Average Error: 30.2 → 0.7
Time: 5.2s
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 r95179 = x;
        double r95180 = exp(r95179);
        double r95181 = 2.0;
        double r95182 = r95180 - r95181;
        double r95183 = -r95179;
        double r95184 = exp(r95183);
        double r95185 = r95182 + r95184;
        return r95185;
}
↓
double f(double x) {
        double r95186 = x;
        double r95187 = 2.0;
        double r95188 = pow(r95186, r95187);
        double r95189 = 0.002777777777777778;
        double r95190 = 6.0;
        double r95191 = pow(r95186, r95190);
        double r95192 = r95189 * r95191;
        double r95193 = 0.08333333333333333;
        double r95194 = 4.0;
        double r95195 = pow(r95186, r95194);
        double r95196 = r95193 * r95195;
        double r95197 = r95192 + r95196;
        double r95198 = r95188 + r95197;
        return r95198;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 30.2

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

  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 2019353 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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