Average Error: 0.0 → 0.0
Time: 13.7s
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 r1202102 = 1.0;
        double r1202103 = x;
        double r1202104 = r1202103 * r1202103;
        double r1202105 = r1202102 - r1202104;
        double r1202106 = -r1202105;
        double r1202107 = exp(r1202106);
        return r1202107;
}
↓
double f(double x) {
        double r1202108 = x;
        double r1202109 = -1.0;
        double r1202110 = fma(r1202108, r1202108, r1202109);
        double r1202111 = exp(r1202110);
        return r1202111;
}

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 2019146 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))