Average Error: 30.3 → 0.5
Time: 3.8s
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 r83436 = x;
        double r83437 = exp(r83436);
        double r83438 = 2.0;
        double r83439 = r83437 - r83438;
        double r83440 = -r83436;
        double r83441 = exp(r83440);
        double r83442 = r83439 + r83441;
        return r83442;
}
↓
double f(double x) {
        double r83443 = x;
        double r83444 = 0.002777777777777778;
        double r83445 = 6.0;
        double r83446 = pow(r83443, r83445);
        double r83447 = 0.08333333333333333;
        double r83448 = 4.0;
        double r83449 = pow(r83443, r83448);
        double r83450 = r83447 * r83449;
        double r83451 = fma(r83444, r83446, r83450);
        double r83452 = fma(r83443, r83443, r83451);
        return r83452;
}

Error

Bits error versus x

Target

Original30.3
Target0.0
Herbie0.5
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.3

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)\]

Reproduce

herbie shell --seed 2020018 +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))))