Average Error: 0.0 → 0.0
Time: 3.2s
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 r175161 = 1.0;
        double r175162 = x;
        double r175163 = r175162 * r175162;
        double r175164 = r175161 - r175163;
        double r175165 = -r175164;
        double r175166 = exp(r175165);
        return r175166;
}
↓
double f(double x) {
        double r175167 = x;
        double r175168 = -1.0;
        double r175169 = fma(r175167, r175167, r175168);
        double r175170 = exp(r175169);
        return r175170;
}

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 2019154 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))