Average Error: 0.0 → 0.0
Time: 6.4s
Precision: 64
\[x \cdot y + \left(1 - x\right) \cdot z\]
↓
\[\mathsf{fma}\left(1 - x, z, x \cdot y\right)\]
x \cdot y + \left(1 - x\right) \cdot z
↓
\mathsf{fma}\left(1 - x, z, x \cdot y\right)
double f(double x, double y, double z) {
        double r211241 = x;
        double r211242 = y;
        double r211243 = r211241 * r211242;
        double r211244 = 1.0;
        double r211245 = r211244 - r211241;
        double r211246 = z;
        double r211247 = r211245 * r211246;
        double r211248 = r211243 + r211247;
        return r211248;
}
↓
double f(double x, double y, double z) {
        double r211249 = 1.0;
        double r211250 = x;
        double r211251 = r211249 - r211250;
        double r211252 = z;
        double r211253 = y;
        double r211254 = r211250 * r211253;
        double r211255 = fma(r211251, r211252, r211254);
        return r211255;
}

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. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019306 +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)))