Average Error: 30.1 → 0.7
Time: 16.5s
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 r49584 = x;
        double r49585 = exp(r49584);
        double r49586 = 2.0;
        double r49587 = r49585 - r49586;
        double r49588 = -r49584;
        double r49589 = exp(r49588);
        double r49590 = r49587 + r49589;
        return r49590;
}
↓
double f(double x) {
        double r49591 = x;
        double r49592 = 2.0;
        double r49593 = pow(r49591, r49592);
        double r49594 = 0.002777777777777778;
        double r49595 = 6.0;
        double r49596 = pow(r49591, r49595);
        double r49597 = r49594 * r49596;
        double r49598 = 0.08333333333333333;
        double r49599 = 4.0;
        double r49600 = pow(r49591, r49599);
        double r49601 = r49598 * r49600;
        double r49602 = r49597 + r49601;
        double r49603 = r49593 + r49602;
        return r49603;
}

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

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

  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 2019303 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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