Average Error: 30.1 → 0.6
Time: 4.9s
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 r115130 = x;
        double r115131 = exp(r115130);
        double r115132 = 2.0;
        double r115133 = r115131 - r115132;
        double r115134 = -r115130;
        double r115135 = exp(r115134);
        double r115136 = r115133 + r115135;
        return r115136;
}
↓
double f(double x) {
        double r115137 = x;
        double r115138 = 2.0;
        double r115139 = pow(r115137, r115138);
        double r115140 = 0.002777777777777778;
        double r115141 = 6.0;
        double r115142 = pow(r115137, r115141);
        double r115143 = r115140 * r115142;
        double r115144 = 0.08333333333333333;
        double r115145 = 4.0;
        double r115146 = pow(r115137, r115145);
        double r115147 = r115144 * r115146;
        double r115148 = r115143 + r115147;
        double r115149 = r115139 + r115148;
        return r115149;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.1

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

  2. Taylor expanded around 0 0.6

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

  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 2019356 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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