Average Error: 0.0 → 0.0
Time: 4.6s
Precision: 64
\[x \cdot y + \left(1 - x\right) \cdot z\]
↓
\[\mathsf{fma}\left(x, y - z, z \cdot 1\right)\]
x \cdot y + \left(1 - x\right) \cdot z
↓
\mathsf{fma}\left(x, y - z, z \cdot 1\right)
double f(double x, double y, double z) {
        double r236792 = x;
        double r236793 = y;
        double r236794 = r236792 * r236793;
        double r236795 = 1.0;
        double r236796 = r236795 - r236792;
        double r236797 = z;
        double r236798 = r236796 * r236797;
        double r236799 = r236794 + r236798;
        return r236799;
}
↓
double f(double x, double y, double z) {
        double r236800 = x;
        double r236801 = y;
        double r236802 = z;
        double r236803 = r236801 - r236802;
        double r236804 = 1.0;
        double r236805 = r236802 * r236804;
        double r236806 = fma(r236800, r236803, r236805);
        return r236806;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[x \cdot y + \left(1 - x\right) \cdot z\]

  2. Simplified0.0

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

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(1 \cdot z + x \cdot y\right) - x \cdot z}\]

  4. Simplified0.0

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

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, y - z, z \cdot 1\right)\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1 x) z)))