Average Error: 29.6 → 0.6
Time: 14.8s
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 r95352 = x;
        double r95353 = exp(r95352);
        double r95354 = 2.0;
        double r95355 = r95353 - r95354;
        double r95356 = -r95352;
        double r95357 = exp(r95356);
        double r95358 = r95355 + r95357;
        return r95358;
}
↓
double f(double x) {
        double r95359 = x;
        double r95360 = 0.002777777777777778;
        double r95361 = 6.0;
        double r95362 = pow(r95359, r95361);
        double r95363 = 0.08333333333333333;
        double r95364 = 4.0;
        double r95365 = pow(r95359, r95364);
        double r95366 = r95363 * r95365;
        double r95367 = fma(r95360, r95362, r95366);
        double r95368 = fma(r95359, r95359, r95367);
        return r95368;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.6

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

  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 2019325 +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))))