Average Error: 29.8 → 0.6
Time: 5.3s
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 r102826 = x;
        double r102827 = exp(r102826);
        double r102828 = 2.0;
        double r102829 = r102827 - r102828;
        double r102830 = -r102826;
        double r102831 = exp(r102830);
        double r102832 = r102829 + r102831;
        return r102832;
}
↓
double f(double x) {
        double r102833 = x;
        double r102834 = 0.002777777777777778;
        double r102835 = 6.0;
        double r102836 = pow(r102833, r102835);
        double r102837 = 0.08333333333333333;
        double r102838 = 4.0;
        double r102839 = pow(r102833, r102838);
        double r102840 = r102837 * r102839;
        double r102841 = fma(r102834, r102836, r102840);
        double r102842 = fma(r102833, r102833, r102841);
        return r102842;
}

Error

Bits error versus x

Target

Original29.8
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.8

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

  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. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020002 +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))))