Average Error: 29.0 → 0.7
Time: 3.9s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)
double f(double x) {
        double r109720 = x;
        double r109721 = exp(r109720);
        double r109722 = 2.0;
        double r109723 = r109721 - r109722;
        double r109724 = -r109720;
        double r109725 = exp(r109724);
        double r109726 = r109723 + r109725;
        return r109726;
}
↓
double f(double x) {
        double r109727 = x;
        double r109728 = 0.002777777777777778;
        double r109729 = 6.0;
        double r109730 = pow(r109727, r109729);
        double r109731 = 0.08333333333333333;
        double r109732 = 4.0;
        double r109733 = pow(r109727, r109732);
        double r109734 = r109731 * r109733;
        double r109735 = fma(r109728, r109730, r109734);
        double r109736 = fma(r109727, r109727, r109735);
        return r109736;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.0

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

  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. Simplified0.7

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

  4. Final simplification0.7

    \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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