Average Error: 0.0 → 0.0
Time: 12.7s
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 r1472133 = 1.0;
        double r1472134 = x;
        double r1472135 = r1472134 * r1472134;
        double r1472136 = r1472133 - r1472135;
        double r1472137 = -r1472136;
        double r1472138 = exp(r1472137);
        return r1472138;
}
↓
double f(double x) {
        double r1472139 = x;
        double r1472140 = -1.0;
        double r1472141 = fma(r1472139, r1472139, r1472140);
        double r1472142 = exp(r1472141);
        return r1472142;
}

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 2019163 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))