Average Error: 0.0 → 0.0
Time: 20.9s
Precision: 64
\[x \cdot y + \left(1 - x\right) \cdot z\]
↓
\[\mathsf{fma}\left(y - z, x, z \cdot 1\right)\]
x \cdot y + \left(1 - x\right) \cdot z
↓
\mathsf{fma}\left(y - z, x, z \cdot 1\right)
double f(double x, double y, double z) {
        double r7315055 = x;
        double r7315056 = y;
        double r7315057 = r7315055 * r7315056;
        double r7315058 = 1.0;
        double r7315059 = r7315058 - r7315055;
        double r7315060 = z;
        double r7315061 = r7315059 * r7315060;
        double r7315062 = r7315057 + r7315061;
        return r7315062;
}
↓
double f(double x, double y, double z) {
        double r7315063 = y;
        double r7315064 = z;
        double r7315065 = r7315063 - r7315064;
        double r7315066 = x;
        double r7315067 = 1.0;
        double r7315068 = r7315064 * r7315067;
        double r7315069 = fma(r7315065, r7315066, r7315068);
        return r7315069;
}

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

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

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Taylor expanded around 0 0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

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)))