Average Error: 0.0 → 0.0
Time: 11.1s
Precision: 64
\[x \cdot y + \left(1.0 - x\right) \cdot z\]
↓
\[\mathsf{fma}\left(x, y - z, 1.0 \cdot z\right)\]
x \cdot y + \left(1.0 - x\right) \cdot z
↓
\mathsf{fma}\left(x, y - z, 1.0 \cdot z\right)
double f(double x, double y, double z) {
        double r13407238 = x;
        double r13407239 = y;
        double r13407240 = r13407238 * r13407239;
        double r13407241 = 1.0;
        double r13407242 = r13407241 - r13407238;
        double r13407243 = z;
        double r13407244 = r13407242 * r13407243;
        double r13407245 = r13407240 + r13407244;
        return r13407245;
}
↓
double f(double x, double y, double z) {
        double r13407246 = x;
        double r13407247 = y;
        double r13407248 = z;
        double r13407249 = r13407247 - r13407248;
        double r13407250 = 1.0;
        double r13407251 = r13407250 * r13407248;
        double r13407252 = fma(r13407246, r13407249, r13407251);
        return r13407252;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[x \cdot y + \left(1.0 - x\right) \cdot z\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(1.0 - x, z, y \cdot x\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt32.2

    \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(1.0 - x, z, y \cdot x\right)} \cdot \sqrt{\mathsf{fma}\left(1.0 - x, z, y \cdot x\right)}}\]

  5. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(1.0 \cdot z + x \cdot y\right) - x \cdot z}\]

  6. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y - z, z \cdot 1.0\right)}\]

  7. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, y - z, 1.0 \cdot z\right)\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  (+ (* x y) (* (- 1.0 x) z)))