Average Error: 29.9 → 0.6
Time: 4.4s
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 r102655 = x;
        double r102656 = exp(r102655);
        double r102657 = 2.0;
        double r102658 = r102656 - r102657;
        double r102659 = -r102655;
        double r102660 = exp(r102659);
        double r102661 = r102658 + r102660;
        return r102661;
}
↓
double f(double x) {
        double r102662 = x;
        double r102663 = 2.0;
        double r102664 = pow(r102662, r102663);
        double r102665 = 0.002777777777777778;
        double r102666 = 6.0;
        double r102667 = pow(r102662, r102666);
        double r102668 = r102665 * r102667;
        double r102669 = 0.08333333333333333;
        double r102670 = 4.0;
        double r102671 = pow(r102662, r102670);
        double r102672 = r102669 * r102671;
        double r102673 = r102668 + r102672;
        double r102674 = r102664 + r102673;
        return r102674;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.9

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

  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 2020021 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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