Average Error: 0.0 → 0.0
Time: 13.7s
Precision: 64
\[e^{-\left(1 - x \cdot x\right)}\]
↓
\[e^{\mathsf{fma}\left(x, x, -1\right)}\]
e^{-\left(1 - x \cdot x\right)}
↓
e^{\mathsf{fma}\left(x, x, -1\right)}
double f(double x) {
        double r21426 = 1.0;
        double r21427 = x;
        double r21428 = r21427 * r21427;
        double r21429 = r21426 - r21428;
        double r21430 = -r21429;
        double r21431 = exp(r21430);
        return r21431;
}
↓
double f(double x) {
        double r21432 = x;
        double r21433 = 1.0;
        double r21434 = -r21433;
        double r21435 = fma(r21432, r21432, r21434);
        double r21436 = exp(r21435);
        return r21436;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[e^{-\left(1 - x \cdot x\right)}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{e^{\mathsf{fma}\left(x, x, -1\right)}}\]

  3. Final simplification0.0

    \[\leadsto e^{\mathsf{fma}\left(x, x, -1\right)}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1 (* x x)))))