Average Error: 0.0 → 0.0
Time: 1.8s
Precision: 64
\[x \cdot y + \left(1 - x\right) \cdot z\]
↓
\[\mathsf{fma}\left(x, y, 1 \cdot \left(z - x \cdot z\right)\right)\]
x \cdot y + \left(1 - x\right) \cdot z
↓
\mathsf{fma}\left(x, y, 1 \cdot \left(z - x \cdot z\right)\right)
double f(double x, double y, double z) {
        double r254538 = x;
        double r254539 = y;
        double r254540 = r254538 * r254539;
        double r254541 = 1.0;
        double r254542 = r254541 - r254538;
        double r254543 = z;
        double r254544 = r254542 * r254543;
        double r254545 = r254540 + r254544;
        return r254545;
}
↓
double f(double x, double y, double z) {
        double r254546 = x;
        double r254547 = y;
        double r254548 = 1.0;
        double r254549 = z;
        double r254550 = r254546 * r254549;
        double r254551 = r254549 - r254550;
        double r254552 = r254548 * r254551;
        double r254553 = fma(r254546, r254547, r254552);
        return r254553;
}

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

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

  4. Applied flip3--12.0

    \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{{1}^{3} - {x}^{3}}{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)}} \cdot z\right)\]

  5. Applied associate-*l/14.1

    \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{\left({1}^{3} - {x}^{3}\right) \cdot z}{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)}}\right)\]

  6. Taylor expanded around 0 0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020083 +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)))