Average Error: 0.0 → 0.0
Time: 22.6s
Precision: 64
\[x \cdot y + \left(1.0 - x\right) \cdot z\]
↓
\[\mathsf{fma}\left(y - z, x, z \cdot 1.0\right)\]
x \cdot y + \left(1.0 - x\right) \cdot z
↓
\mathsf{fma}\left(y - z, x, z \cdot 1.0\right)
double f(double x, double y, double z) {
        double r8852542 = x;
        double r8852543 = y;
        double r8852544 = r8852542 * r8852543;
        double r8852545 = 1.0;
        double r8852546 = r8852545 - r8852542;
        double r8852547 = z;
        double r8852548 = r8852546 * r8852547;
        double r8852549 = r8852544 + r8852548;
        return r8852549;
}
↓
double f(double x, double y, double z) {
        double r8852550 = y;
        double r8852551 = z;
        double r8852552 = r8852550 - r8852551;
        double r8852553 = x;
        double r8852554 = 1.0;
        double r8852555 = r8852551 * r8852554;
        double r8852556 = fma(r8852552, r8852553, r8852555);
        return r8852556;
}

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. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  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

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019165 +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)))