Average Error: 30.2 → 0.6
Time: 3.7s
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 r88247 = x;
        double r88248 = exp(r88247);
        double r88249 = 2.0;
        double r88250 = r88248 - r88249;
        double r88251 = -r88247;
        double r88252 = exp(r88251);
        double r88253 = r88250 + r88252;
        return r88253;
}
↓
double f(double x) {
        double r88254 = x;
        double r88255 = 0.002777777777777778;
        double r88256 = 6.0;
        double r88257 = pow(r88254, r88256);
        double r88258 = 0.08333333333333333;
        double r88259 = 4.0;
        double r88260 = pow(r88254, r88259);
        double r88261 = r88258 * r88260;
        double r88262 = fma(r88255, r88257, r88261);
        double r88263 = fma(r88254, r88254, r88262);
        return r88263;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 30.2

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

  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 2020020 +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))))