Average Error: 29.7 → 0.7
Time: 7.2s
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 r142237 = x;
        double r142238 = exp(r142237);
        double r142239 = 2.0;
        double r142240 = r142238 - r142239;
        double r142241 = -r142237;
        double r142242 = exp(r142241);
        double r142243 = r142240 + r142242;
        return r142243;
}
↓
double f(double x) {
        double r142244 = x;
        double r142245 = 0.002777777777777778;
        double r142246 = 6.0;
        double r142247 = pow(r142244, r142246);
        double r142248 = 0.08333333333333333;
        double r142249 = 4.0;
        double r142250 = pow(r142244, r142249);
        double r142251 = r142248 * r142250;
        double r142252 = fma(r142245, r142247, r142251);
        double r142253 = fma(r142244, r142244, r142252);
        return r142253;
}

Error

Bits error versus x

Target

Original29.7
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.7

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

  2. Taylor expanded around 0 0.7

    \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)}\]

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020036 +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))))