Average Error: 0.0 → 0.0
Time: 2.7s
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 r305009 = 1.0;
        double r305010 = x;
        double r305011 = r305010 * r305010;
        double r305012 = r305009 - r305011;
        double r305013 = -r305012;
        double r305014 = exp(r305013);
        return r305014;
}
↓
double f(double x) {
        double r305015 = x;
        double r305016 = -1.0;
        double r305017 = fma(r305015, r305015, r305016);
        double r305018 = exp(r305017);
        return r305018;
}

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 2019156 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))