Average Error: 0.0 → 0.0
Time: 10.6s
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 r123858 = x;
        double r123859 = y;
        double r123860 = r123858 * r123859;
        double r123861 = 1.0;
        double r123862 = r123861 - r123858;
        double r123863 = z;
        double r123864 = r123862 * r123863;
        double r123865 = r123860 + r123864;
        return r123865;
}
↓
double f(double x, double y, double z) {
        double r123866 = 1.0;
        double r123867 = x;
        double r123868 = r123866 - r123867;
        double r123869 = z;
        double r123870 = y;
        double r123871 = r123867 * r123870;
        double r123872 = fma(r123868, r123869, r123871);
        return r123872;
}

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 2019347 +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)))