Average Error: 30.1 → 0.7
Time: 17.5s
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 r76345 = x;
        double r76346 = exp(r76345);
        double r76347 = 2.0;
        double r76348 = r76346 - r76347;
        double r76349 = -r76345;
        double r76350 = exp(r76349);
        double r76351 = r76348 + r76350;
        return r76351;
}
↓
double f(double x) {
        double r76352 = x;
        double r76353 = 2.0;
        double r76354 = pow(r76352, r76353);
        double r76355 = 0.002777777777777778;
        double r76356 = 6.0;
        double r76357 = pow(r76352, r76356);
        double r76358 = r76355 * r76357;
        double r76359 = 0.08333333333333333;
        double r76360 = 4.0;
        double r76361 = pow(r76352, r76360);
        double r76362 = r76359 * r76361;
        double r76363 = r76358 + r76362;
        double r76364 = r76354 + r76363;
        return r76364;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.1

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

  2. Taylor expanded around 0 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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