Average Error: 29.5 → 0.6
Time: 6.6s
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 r152414 = x;
        double r152415 = exp(r152414);
        double r152416 = 2.0;
        double r152417 = r152415 - r152416;
        double r152418 = -r152414;
        double r152419 = exp(r152418);
        double r152420 = r152417 + r152419;
        return r152420;
}
↓
double f(double x) {
        double r152421 = x;
        double r152422 = 2.0;
        double r152423 = pow(r152421, r152422);
        double r152424 = 0.002777777777777778;
        double r152425 = 6.0;
        double r152426 = pow(r152421, r152425);
        double r152427 = r152424 * r152426;
        double r152428 = 0.08333333333333333;
        double r152429 = 4.0;
        double r152430 = pow(r152421, r152429);
        double r152431 = r152428 * r152430;
        double r152432 = r152427 + r152431;
        double r152433 = r152423 + r152432;
        return r152433;
}

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 2019322 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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