Average Error: 30.1 → 0.6
Time: 4.8s
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 r114052 = x;
        double r114053 = exp(r114052);
        double r114054 = 2.0;
        double r114055 = r114053 - r114054;
        double r114056 = -r114052;
        double r114057 = exp(r114056);
        double r114058 = r114055 + r114057;
        return r114058;
}
↓
double f(double x) {
        double r114059 = x;
        double r114060 = 0.002777777777777778;
        double r114061 = 6.0;
        double r114062 = pow(r114059, r114061);
        double r114063 = 0.08333333333333333;
        double r114064 = 4.0;
        double r114065 = pow(r114059, r114064);
        double r114066 = r114063 * r114065;
        double r114067 = fma(r114060, r114062, r114066);
        double r114068 = fma(r114059, r114059, r114067);
        return r114068;
}

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 2020065 +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))))