Average Error: 30.1 → 0.6
Time: 10.1s
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 r113520 = x;
        double r113521 = exp(r113520);
        double r113522 = 2.0;
        double r113523 = r113521 - r113522;
        double r113524 = -r113520;
        double r113525 = exp(r113524);
        double r113526 = r113523 + r113525;
        return r113526;
}
↓
double f(double x) {
        double r113527 = x;
        double r113528 = 2.0;
        double r113529 = pow(r113527, r113528);
        double r113530 = 0.002777777777777778;
        double r113531 = 6.0;
        double r113532 = pow(r113527, r113531);
        double r113533 = r113530 * r113532;
        double r113534 = 0.08333333333333333;
        double r113535 = 4.0;
        double r113536 = pow(r113527, r113535);
        double r113537 = r113534 * r113536;
        double r113538 = r113533 + r113537;
        double r113539 = r113529 + r113538;
        return r113539;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.1

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

  2. Taylor expanded around 0 0.6

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

  3. Final simplification0.6

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

Reproduce

herbie shell --seed 2020042 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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