Average Error: 29.7 → 0.6
Time: 16.6s
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 r86576 = x;
        double r86577 = exp(r86576);
        double r86578 = 2.0;
        double r86579 = r86577 - r86578;
        double r86580 = -r86576;
        double r86581 = exp(r86580);
        double r86582 = r86579 + r86581;
        return r86582;
}
↓
double f(double x) {
        double r86583 = x;
        double r86584 = 2.0;
        double r86585 = pow(r86583, r86584);
        double r86586 = 0.002777777777777778;
        double r86587 = 6.0;
        double r86588 = pow(r86583, r86587);
        double r86589 = r86586 * r86588;
        double r86590 = 0.08333333333333333;
        double r86591 = 4.0;
        double r86592 = pow(r86583, r86591);
        double r86593 = r86590 * r86592;
        double r86594 = r86589 + r86593;
        double r86595 = r86585 + r86594;
        return r86595;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.7

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

  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 2019347 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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