Average Error: 30.1 → 0.7
Time: 6.4s
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 r103591 = x;
        double r103592 = exp(r103591);
        double r103593 = 2.0;
        double r103594 = r103592 - r103593;
        double r103595 = -r103591;
        double r103596 = exp(r103595);
        double r103597 = r103594 + r103596;
        return r103597;
}
↓
double f(double x) {
        double r103598 = x;
        double r103599 = 2.0;
        double r103600 = pow(r103598, r103599);
        double r103601 = 0.002777777777777778;
        double r103602 = 6.0;
        double r103603 = pow(r103598, r103602);
        double r103604 = r103601 * r103603;
        double r103605 = 0.08333333333333333;
        double r103606 = 4.0;
        double r103607 = pow(r103598, r103606);
        double r103608 = r103605 * r103607;
        double r103609 = r103604 + r103608;
        double r103610 = r103600 + r103609;
        return r103610;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.1

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

  2. Taylor expanded around 0 0.7

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

  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 2020034 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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