Average Error: 0.0 → 0.0
Time: 3.0s
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 r488588 = 1.0;
        double r488589 = x;
        double r488590 = r488589 * r488589;
        double r488591 = r488588 - r488590;
        double r488592 = -r488591;
        double r488593 = exp(r488592);
        return r488593;
}
↓
double f(double x) {
        double r488594 = x;
        double r488595 = -1.0;
        double r488596 = fma(r488594, r488594, r488595);
        double r488597 = exp(r488596);
        return r488597;
}

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 2019152 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))