Average Error: 29.2 → 0.6
Time: 36.2s
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 r61891 = x;
        double r61892 = exp(r61891);
        double r61893 = 2.0;
        double r61894 = r61892 - r61893;
        double r61895 = -r61891;
        double r61896 = exp(r61895);
        double r61897 = r61894 + r61896;
        return r61897;
}
↓
double f(double x) {
        double r61898 = x;
        double r61899 = 0.002777777777777778;
        double r61900 = 6.0;
        double r61901 = pow(r61898, r61900);
        double r61902 = 0.08333333333333333;
        double r61903 = 4.0;
        double r61904 = pow(r61898, r61903);
        double r61905 = r61902 * r61904;
        double r61906 = fma(r61899, r61901, r61905);
        double r61907 = fma(r61898, r61898, r61906);
        return r61907;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.2

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

  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 2020046 +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))))