Average Error: 29.0 → 0.6
Time: 19.9s
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 r90778 = x;
        double r90779 = exp(r90778);
        double r90780 = 2.0;
        double r90781 = r90779 - r90780;
        double r90782 = -r90778;
        double r90783 = exp(r90782);
        double r90784 = r90781 + r90783;
        return r90784;
}
↓
double f(double x) {
        double r90785 = x;
        double r90786 = 0.002777777777777778;
        double r90787 = 6.0;
        double r90788 = pow(r90785, r90787);
        double r90789 = 0.08333333333333333;
        double r90790 = 4.0;
        double r90791 = pow(r90785, r90790);
        double r90792 = r90789 * r90791;
        double r90793 = fma(r90786, r90788, r90792);
        double r90794 = fma(r90785, r90785, r90793);
        return r90794;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.0

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

  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 2019326 +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))))