Average Error: 0.0 → 0.0
Time: 13.6s
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 r26086 = 1.0;
        double r26087 = x;
        double r26088 = r26087 * r26087;
        double r26089 = r26086 - r26088;
        double r26090 = -r26089;
        double r26091 = exp(r26090);
        return r26091;
}
↓
double f(double x) {
        double r26092 = x;
        double r26093 = 1.0;
        double r26094 = -r26093;
        double r26095 = fma(r26092, r26092, r26094);
        double r26096 = exp(r26095);
        return r26096;
}

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 2019306 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1 (* x x)))))