Average Error: 29.9 → 0.6
Time: 5.5s
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 r97293 = x;
        double r97294 = exp(r97293);
        double r97295 = 2.0;
        double r97296 = r97294 - r97295;
        double r97297 = -r97293;
        double r97298 = exp(r97297);
        double r97299 = r97296 + r97298;
        return r97299;
}
↓
double f(double x) {
        double r97300 = x;
        double r97301 = 0.002777777777777778;
        double r97302 = 6.0;
        double r97303 = pow(r97300, r97302);
        double r97304 = 0.08333333333333333;
        double r97305 = 4.0;
        double r97306 = pow(r97300, r97305);
        double r97307 = r97304 * r97306;
        double r97308 = fma(r97301, r97303, r97307);
        double r97309 = fma(r97300, r97300, r97308);
        return r97309;
}

Error

Bits error versus x

Target

Original29.9
Target0.0
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 2020033 +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))))