Average Error: 0.0 → 0.0
Time: 10.6s
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 r919688 = 1.0;
        double r919689 = x;
        double r919690 = r919689 * r919689;
        double r919691 = r919688 - r919690;
        double r919692 = -r919691;
        double r919693 = exp(r919692);
        return r919693;
}
↓
double f(double x) {
        double r919694 = x;
        double r919695 = -1.0;
        double r919696 = fma(r919694, r919694, r919695);
        double r919697 = exp(r919696);
        return r919697;
}

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 2019129 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))