Average Error: 30.2 → 0.7
Time: 5.2s
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 r101033 = x;
        double r101034 = exp(r101033);
        double r101035 = 2.0;
        double r101036 = r101034 - r101035;
        double r101037 = -r101033;
        double r101038 = exp(r101037);
        double r101039 = r101036 + r101038;
        return r101039;
}
↓
double f(double x) {
        double r101040 = x;
        double r101041 = 2.0;
        double r101042 = pow(r101040, r101041);
        double r101043 = 0.002777777777777778;
        double r101044 = 6.0;
        double r101045 = pow(r101040, r101044);
        double r101046 = r101043 * r101045;
        double r101047 = 0.08333333333333333;
        double r101048 = 4.0;
        double r101049 = pow(r101040, r101048);
        double r101050 = r101047 * r101049;
        double r101051 = r101046 + r101050;
        double r101052 = r101042 + r101051;
        return r101052;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 30.2

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

  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 2019353 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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