Average Error: 0.0 → 0.0
Time: 1.0m
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 r847566 = 1.0;
        double r847567 = x;
        double r847568 = r847567 * r847567;
        double r847569 = r847566 - r847568;
        double r847570 = -r847569;
        double r847571 = exp(r847570);
        return r847571;
}
↓
double f(double x) {
        double r847572 = x;
        double r847573 = -1.0;
        double r847574 = fma(r847572, r847572, r847573);
        double r847575 = exp(r847574);
        return r847575;
}

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 2019149 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))