Average Error: 29.2 → 0.6
Time: 5.0s
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 r104576 = x;
        double r104577 = exp(r104576);
        double r104578 = 2.0;
        double r104579 = r104577 - r104578;
        double r104580 = -r104576;
        double r104581 = exp(r104580);
        double r104582 = r104579 + r104581;
        return r104582;
}
↓
double f(double x) {
        double r104583 = x;
        double r104584 = 0.002777777777777778;
        double r104585 = 6.0;
        double r104586 = pow(r104583, r104585);
        double r104587 = 0.08333333333333333;
        double r104588 = 4.0;
        double r104589 = pow(r104583, r104588);
        double r104590 = r104587 * r104589;
        double r104591 = fma(r104584, r104586, r104590);
        double r104592 = fma(r104583, r104583, r104591);
        return r104592;
}

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 2020059 +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))))