Average Error: 29.6 → 0.6
Time: 9.4s
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 r78460 = x;
        double r78461 = exp(r78460);
        double r78462 = 2.0;
        double r78463 = r78461 - r78462;
        double r78464 = -r78460;
        double r78465 = exp(r78464);
        double r78466 = r78463 + r78465;
        return r78466;
}
↓
double f(double x) {
        double r78467 = x;
        double r78468 = 2.0;
        double r78469 = pow(r78467, r78468);
        double r78470 = 0.002777777777777778;
        double r78471 = 6.0;
        double r78472 = pow(r78467, r78471);
        double r78473 = r78470 * r78472;
        double r78474 = 0.08333333333333333;
        double r78475 = 4.0;
        double r78476 = pow(r78467, r78475);
        double r78477 = r78474 * r78476;
        double r78478 = r78473 + r78477;
        double r78479 = r78469 + r78478;
        return r78479;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.6

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

  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 2019304 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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