Average Error: 0.0 → 0.0
Time: 13.7s
Precision: 64
\[x \cdot y + \left(1 - x\right) \cdot z\]
↓
\[\mathsf{fma}\left(x, y - z, 1 \cdot z\right)\]
x \cdot y + \left(1 - x\right) \cdot z
↓
\mathsf{fma}\left(x, y - z, 1 \cdot z\right)
double f(double x, double y, double z) {
        double r8953842 = x;
        double r8953843 = y;
        double r8953844 = r8953842 * r8953843;
        double r8953845 = 1.0;
        double r8953846 = r8953845 - r8953842;
        double r8953847 = z;
        double r8953848 = r8953846 * r8953847;
        double r8953849 = r8953844 + r8953848;
        return r8953849;
}
↓
double f(double x, double y, double z) {
        double r8953850 = x;
        double r8953851 = y;
        double r8953852 = z;
        double r8953853 = r8953851 - r8953852;
        double r8953854 = 1.0;
        double r8953855 = r8953854 * r8953852;
        double r8953856 = fma(r8953850, r8953853, r8953855);
        return r8953856;
}

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. Using strategy rm

  4. Applied add-sqr-sqrt32.5

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

  5. Taylor expanded around 0 0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +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)))