Average Error: 29.0 → 0.6
Time: 19.5s
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 r88861 = x;
        double r88862 = exp(r88861);
        double r88863 = 2.0;
        double r88864 = r88862 - r88863;
        double r88865 = -r88861;
        double r88866 = exp(r88865);
        double r88867 = r88864 + r88866;
        return r88867;
}
↓
double f(double x) {
        double r88868 = x;
        double r88869 = 0.002777777777777778;
        double r88870 = 6.0;
        double r88871 = pow(r88868, r88870);
        double r88872 = 0.08333333333333333;
        double r88873 = 4.0;
        double r88874 = pow(r88868, r88873);
        double r88875 = r88872 * r88874;
        double r88876 = fma(r88869, r88871, r88875);
        double r88877 = fma(r88868, r88868, r88876);
        return r88877;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.0

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

  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 2019326 +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))))