Average Error: 28.9 → 0.6
Time: 6.9s
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 r122354 = x;
        double r122355 = exp(r122354);
        double r122356 = 2.0;
        double r122357 = r122355 - r122356;
        double r122358 = -r122354;
        double r122359 = exp(r122358);
        double r122360 = r122357 + r122359;
        return r122360;
}
↓
double f(double x) {
        double r122361 = x;
        double r122362 = 0.002777777777777778;
        double r122363 = 6.0;
        double r122364 = pow(r122361, r122363);
        double r122365 = 0.08333333333333333;
        double r122366 = 4.0;
        double r122367 = pow(r122361, r122366);
        double r122368 = r122365 * r122367;
        double r122369 = fma(r122362, r122364, r122368);
        double r122370 = fma(r122361, r122361, r122369);
        return r122370;
}

Error

Bits error versus x

Target

Original28.9
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 28.9

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

  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 2020064 +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))))