Average Error: 0.0 → 0.0
Time: 5.5s
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 r727999 = 1.0;
        double r728000 = x;
        double r728001 = r728000 * r728000;
        double r728002 = r727999 - r728001;
        double r728003 = -r728002;
        double r728004 = exp(r728003);
        return r728004;
}
↓
double f(double x) {
        double r728005 = x;
        double r728006 = -1.0;
        double r728007 = fma(r728005, r728005, r728006);
        double r728008 = exp(r728007);
        return r728008;
}

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 2019165 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))