Average Error: 29.2 → 0.6
Time: 5.7s
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 r82549 = x;
        double r82550 = exp(r82549);
        double r82551 = 2.0;
        double r82552 = r82550 - r82551;
        double r82553 = -r82549;
        double r82554 = exp(r82553);
        double r82555 = r82552 + r82554;
        return r82555;
}
↓
double f(double x) {
        double r82556 = x;
        double r82557 = 2.0;
        double r82558 = pow(r82556, r82557);
        double r82559 = 0.002777777777777778;
        double r82560 = 6.0;
        double r82561 = pow(r82556, r82560);
        double r82562 = r82559 * r82561;
        double r82563 = 0.08333333333333333;
        double r82564 = 4.0;
        double r82565 = pow(r82556, r82564);
        double r82566 = r82563 * r82565;
        double r82567 = r82562 + r82566;
        double r82568 = r82558 + r82567;
        return r82568;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.2

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

  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 2020039 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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