Average Error: 0.0 → 0.0
Time: 17.7s
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 r1779761 = 1.0;
        double r1779762 = x;
        double r1779763 = r1779762 * r1779762;
        double r1779764 = r1779761 - r1779763;
        double r1779765 = -r1779764;
        double r1779766 = exp(r1779765);
        return r1779766;
}
↓
double f(double x) {
        double r1779767 = x;
        double r1779768 = -1.0;
        double r1779769 = fma(r1779767, r1779767, r1779768);
        double r1779770 = exp(r1779769);
        return r1779770;
}

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 2019107 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))