Average Error: 0.0 → 0
Time: 631.0ms
Precision: 64
\[x \cdot x + 1\]
↓
\[\mathsf{fma}\left(x, x, 1\right)\]
x \cdot x + 1
↓
\mathsf{fma}\left(x, x, 1\right)
double f(double x) {
        double r105984 = x;
        double r105985 = r105984 * r105984;
        double r105986 = 1.0;
        double r105987 = r105985 + r105986;
        return r105987;
}
↓
double f(double x) {
        double r105988 = x;
        double r105989 = 1.0;
        double r105990 = fma(r105988, r105988, r105989);
        return r105990;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[x \cdot x + 1\]

  2. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 1\right)}\]

  3. Final simplification0

    \[\leadsto \mathsf{fma}\left(x, x, 1\right)\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, A"
  :precision binary64
  (+ (* x x) 1))