Average Error: 29.5 → 0.6
Time: 12.1s
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 r115453 = x;
        double r115454 = exp(r115453);
        double r115455 = 2.0;
        double r115456 = r115454 - r115455;
        double r115457 = -r115453;
        double r115458 = exp(r115457);
        double r115459 = r115456 + r115458;
        return r115459;
}
↓
double f(double x) {
        double r115460 = x;
        double r115461 = 2.0;
        double r115462 = pow(r115460, r115461);
        double r115463 = 0.002777777777777778;
        double r115464 = 6.0;
        double r115465 = pow(r115460, r115464);
        double r115466 = r115463 * r115465;
        double r115467 = 0.08333333333333333;
        double r115468 = 4.0;
        double r115469 = pow(r115460, r115468);
        double r115470 = r115467 * r115469;
        double r115471 = r115466 + r115470;
        double r115472 = r115462 + r115471;
        return r115472;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.5

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

  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 2019351 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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