Average Error: 0.0 → 0.0
Time: 10.7s
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 r41237 = 1.0;
        double r41238 = x;
        double r41239 = r41238 * r41238;
        double r41240 = r41237 - r41239;
        double r41241 = -r41240;
        double r41242 = exp(r41241);
        return r41242;
}
↓
double f(double x) {
        double r41243 = x;
        double r41244 = 1.0;
        double r41245 = -r41244;
        double r41246 = fma(r41243, r41243, r41245);
        double r41247 = exp(r41246);
        return r41247;
}

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 2019179 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1.0 (* x x)))))