Average Error: 30.0 → 0.7
Time: 6.1s
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 r125632 = x;
        double r125633 = exp(r125632);
        double r125634 = 2.0;
        double r125635 = r125633 - r125634;
        double r125636 = -r125632;
        double r125637 = exp(r125636);
        double r125638 = r125635 + r125637;
        return r125638;
}
↓
double f(double x) {
        double r125639 = x;
        double r125640 = 2.0;
        double r125641 = pow(r125639, r125640);
        double r125642 = 0.002777777777777778;
        double r125643 = 6.0;
        double r125644 = pow(r125639, r125643);
        double r125645 = r125642 * r125644;
        double r125646 = 0.08333333333333333;
        double r125647 = 4.0;
        double r125648 = pow(r125639, r125647);
        double r125649 = r125646 * r125648;
        double r125650 = r125645 + r125649;
        double r125651 = r125641 + r125650;
        return r125651;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 30.0

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

  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 2020057 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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