Average Error: 0.0 → 0.0
Time: 7.6s
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 r380456 = 1.0;
        double r380457 = x;
        double r380458 = r380457 * r380457;
        double r380459 = r380456 - r380458;
        double r380460 = -r380459;
        double r380461 = exp(r380460);
        return r380461;
}
↓
double f(double x) {
        double r380462 = x;
        double r380463 = -1.0;
        double r380464 = fma(r380462, r380462, r380463);
        double r380465 = exp(r380464);
        return r380465;
}

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 2019152 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))