Average Error: 29.7 → 0.6
Time: 4.4s
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 r93713 = x;
        double r93714 = exp(r93713);
        double r93715 = 2.0;
        double r93716 = r93714 - r93715;
        double r93717 = -r93713;
        double r93718 = exp(r93717);
        double r93719 = r93716 + r93718;
        return r93719;
}
↓
double f(double x) {
        double r93720 = x;
        double r93721 = 0.002777777777777778;
        double r93722 = 6.0;
        double r93723 = pow(r93720, r93722);
        double r93724 = 0.08333333333333333;
        double r93725 = 4.0;
        double r93726 = pow(r93720, r93725);
        double r93727 = r93724 * r93726;
        double r93728 = fma(r93721, r93723, r93727);
        double r93729 = fma(r93720, r93720, r93728);
        return r93729;
}

Error

Bits error versus x

Target

Original29.7
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.7

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

  2. Taylor expanded around 0 0.6

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

  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 2019344 +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))))