Average Error: 0.0 → 0.0
Time: 13.4s
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 r32686 = 1.0;
        double r32687 = x;
        double r32688 = r32687 * r32687;
        double r32689 = r32686 - r32688;
        double r32690 = -r32689;
        double r32691 = exp(r32690);
        return r32691;
}
↓
double f(double x) {
        double r32692 = x;
        double r32693 = 1.0;
        double r32694 = -r32693;
        double r32695 = fma(r32692, r32692, r32694);
        double r32696 = exp(r32695);
        return r32696;
}

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 2019304 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1 (* x x)))))