Average Error: 0.0 → 0.0
Time: 16.5s
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 r1310295 = 1.0;
        double r1310296 = x;
        double r1310297 = r1310296 * r1310296;
        double r1310298 = r1310295 - r1310297;
        double r1310299 = -r1310298;
        double r1310300 = exp(r1310299);
        return r1310300;
}
↓
double f(double x) {
        double r1310301 = x;
        double r1310302 = -1.0;
        double r1310303 = fma(r1310301, r1310301, r1310302);
        double r1310304 = exp(r1310303);
        return r1310304;
}

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 2019142 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))