Average Error: 29.2 → 0.6
Time: 5.6s
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 r119676 = x;
        double r119677 = exp(r119676);
        double r119678 = 2.0;
        double r119679 = r119677 - r119678;
        double r119680 = -r119676;
        double r119681 = exp(r119680);
        double r119682 = r119679 + r119681;
        return r119682;
}
↓
double f(double x) {
        double r119683 = x;
        double r119684 = 0.002777777777777778;
        double r119685 = 6.0;
        double r119686 = pow(r119683, r119685);
        double r119687 = 0.08333333333333333;
        double r119688 = 4.0;
        double r119689 = pow(r119683, r119688);
        double r119690 = r119687 * r119689;
        double r119691 = fma(r119684, r119686, r119690);
        double r119692 = fma(r119683, r119683, r119691);
        return r119692;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.2

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

  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 2020039 +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))))