Average Error: 29.9 → 0.5
Time: 20.7s
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 r94899 = x;
        double r94900 = exp(r94899);
        double r94901 = 2.0;
        double r94902 = r94900 - r94901;
        double r94903 = -r94899;
        double r94904 = exp(r94903);
        double r94905 = r94902 + r94904;
        return r94905;
}
↓
double f(double x) {
        double r94906 = x;
        double r94907 = 0.002777777777777778;
        double r94908 = 6.0;
        double r94909 = pow(r94906, r94908);
        double r94910 = 0.08333333333333333;
        double r94911 = 4.0;
        double r94912 = pow(r94906, r94911);
        double r94913 = r94910 * r94912;
        double r94914 = fma(r94907, r94909, r94913);
        double r94915 = fma(r94906, r94906, r94914);
        return r94915;
}

Error

Bits error versus x

Target

Original29.9
Target0.0
Herbie0.5
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.9

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

  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. Simplified0.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)}\]

  4. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)\]

Reproduce

herbie shell --seed 2019212 +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))))