Average Error: 30.1 → 0.6
Time: 5.3s
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 r114379 = x;
        double r114380 = exp(r114379);
        double r114381 = 2.0;
        double r114382 = r114380 - r114381;
        double r114383 = -r114379;
        double r114384 = exp(r114383);
        double r114385 = r114382 + r114384;
        return r114385;
}
↓
double f(double x) {
        double r114386 = x;
        double r114387 = 0.002777777777777778;
        double r114388 = 6.0;
        double r114389 = pow(r114386, r114388);
        double r114390 = 0.08333333333333333;
        double r114391 = 4.0;
        double r114392 = pow(r114386, r114391);
        double r114393 = r114390 * r114392;
        double r114394 = fma(r114387, r114389, r114393);
        double r114395 = fma(r114386, r114386, r114394);
        return r114395;
}

Error

Bits error versus x

Target

Original30.1
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.1

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

  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 2019356 +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))))