Average Error: 29.9 → 0.5
Time: 23.3s
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 r122504 = x;
        double r122505 = exp(r122504);
        double r122506 = 2.0;
        double r122507 = r122505 - r122506;
        double r122508 = -r122504;
        double r122509 = exp(r122508);
        double r122510 = r122507 + r122509;
        return r122510;
}
↓
double f(double x) {
        double r122511 = x;
        double r122512 = 0.002777777777777778;
        double r122513 = 6.0;
        double r122514 = pow(r122511, r122513);
        double r122515 = 0.08333333333333333;
        double r122516 = 4.0;
        double r122517 = pow(r122511, r122516);
        double r122518 = r122515 * r122517;
        double r122519 = fma(r122512, r122514, r122518);
        double r122520 = fma(r122511, r122511, r122519);
        return r122520;
}

Error

Bits error versus x

Target

Original29.9
Target0.0
Herbie0.5
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.9

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

  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 2019199 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))