Average Error: 29.9 → 0.6
Time: 14.3s
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 r102301 = x;
        double r102302 = exp(r102301);
        double r102303 = 2.0;
        double r102304 = r102302 - r102303;
        double r102305 = -r102301;
        double r102306 = exp(r102305);
        double r102307 = r102304 + r102306;
        return r102307;
}
↓
double f(double x) {
        double r102308 = x;
        double r102309 = 0.002777777777777778;
        double r102310 = 6.0;
        double r102311 = pow(r102308, r102310);
        double r102312 = 0.08333333333333333;
        double r102313 = 4.0;
        double r102314 = pow(r102308, r102313);
        double r102315 = r102312 * r102314;
        double r102316 = fma(r102309, r102311, r102315);
        double r102317 = fma(r102308, r102308, r102316);
        return r102317;
}

Error

Bits error versus x

Target

Original29.9
Target0.1
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.9

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

  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 2019306 +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))))