Average Error: 29.3 → 0.7
Time: 4.5s
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 r79479 = x;
        double r79480 = exp(r79479);
        double r79481 = 2.0;
        double r79482 = r79480 - r79481;
        double r79483 = -r79479;
        double r79484 = exp(r79483);
        double r79485 = r79482 + r79484;
        return r79485;
}
↓
double f(double x) {
        double r79486 = x;
        double r79487 = 2.0;
        double r79488 = pow(r79486, r79487);
        double r79489 = 0.002777777777777778;
        double r79490 = 6.0;
        double r79491 = pow(r79486, r79490);
        double r79492 = r79489 * r79491;
        double r79493 = 0.08333333333333333;
        double r79494 = 4.0;
        double r79495 = pow(r79486, r79494);
        double r79496 = r79493 * r79495;
        double r79497 = r79492 + r79496;
        double r79498 = r79488 + r79497;
        return r79498;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.3

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

  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 2020062 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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