Average Error: 29.1 → 0.6
Time: 5.1s
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 r99701 = x;
        double r99702 = exp(r99701);
        double r99703 = 2.0;
        double r99704 = r99702 - r99703;
        double r99705 = -r99701;
        double r99706 = exp(r99705);
        double r99707 = r99704 + r99706;
        return r99707;
}
↓
double f(double x) {
        double r99708 = x;
        double r99709 = 0.002777777777777778;
        double r99710 = 6.0;
        double r99711 = pow(r99708, r99710);
        double r99712 = 0.08333333333333333;
        double r99713 = 4.0;
        double r99714 = pow(r99708, r99713);
        double r99715 = r99712 * r99714;
        double r99716 = fma(r99709, r99711, r99715);
        double r99717 = fma(r99708, r99708, r99716);
        return r99717;
}

Error

Bits error versus x

Target

Original29.1
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.1

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

  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 2020083 +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))))