Average Error: 29.5 → 0.6
Time: 19.8s
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 r42087 = x;
        double r42088 = exp(r42087);
        double r42089 = 2.0;
        double r42090 = r42088 - r42089;
        double r42091 = -r42087;
        double r42092 = exp(r42091);
        double r42093 = r42090 + r42092;
        return r42093;
}
↓
double f(double x) {
        double r42094 = x;
        double r42095 = 2.0;
        double r42096 = pow(r42094, r42095);
        double r42097 = 0.002777777777777778;
        double r42098 = 6.0;
        double r42099 = pow(r42094, r42098);
        double r42100 = r42097 * r42099;
        double r42101 = 0.08333333333333333;
        double r42102 = 4.0;
        double r42103 = pow(r42094, r42102);
        double r42104 = r42101 * r42103;
        double r42105 = r42100 + r42104;
        double r42106 = r42096 + r42105;
        return r42106;
}

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))))