Average Error: 29.6 → 0.5
Time: 5.1s
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 r108300 = x;
        double r108301 = exp(r108300);
        double r108302 = 2.0;
        double r108303 = r108301 - r108302;
        double r108304 = -r108300;
        double r108305 = exp(r108304);
        double r108306 = r108303 + r108305;
        return r108306;
}
↓
double f(double x) {
        double r108307 = x;
        double r108308 = 2.0;
        double r108309 = pow(r108307, r108308);
        double r108310 = 0.002777777777777778;
        double r108311 = 6.0;
        double r108312 = pow(r108307, r108311);
        double r108313 = r108310 * r108312;
        double r108314 = 0.08333333333333333;
        double r108315 = 4.0;
        double r108316 = pow(r108307, r108315);
        double r108317 = r108314 * r108316;
        double r108318 = r108313 + r108317;
        double r108319 = r108309 + r108318;
        return r108319;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.6

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

  2. Taylor expanded around 0 0.5

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

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2019352 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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