Average Error: 29.3 → 0.7
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 r135515 = x;
        double r135516 = exp(r135515);
        double r135517 = 2.0;
        double r135518 = r135516 - r135517;
        double r135519 = -r135515;
        double r135520 = exp(r135519);
        double r135521 = r135518 + r135520;
        return r135521;
}
↓
double f(double x) {
        double r135522 = x;
        double r135523 = 0.002777777777777778;
        double r135524 = 6.0;
        double r135525 = pow(r135522, r135524);
        double r135526 = 0.08333333333333333;
        double r135527 = 4.0;
        double r135528 = pow(r135522, r135527);
        double r135529 = r135526 * r135528;
        double r135530 = fma(r135523, r135525, r135529);
        double r135531 = fma(r135522, r135522, r135530);
        return r135531;
}

Error

Bits error versus x

Target

Original29.3
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.3

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

  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 2020062 +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))))