Average Error: 0.1 → 0.1
Time: 3.7s
Precision: 64
\[x - \left(y \cdot 4\right) \cdot z\]
↓
\[\mathsf{fma}\left(y \cdot 4, -z, x\right)\]
x - \left(y \cdot 4\right) \cdot z
↓
\mathsf{fma}\left(y \cdot 4, -z, x\right)
double f(double x, double y, double z) {
        double r136476 = x;
        double r136477 = y;
        double r136478 = 4.0;
        double r136479 = r136477 * r136478;
        double r136480 = z;
        double r136481 = r136479 * r136480;
        double r136482 = r136476 - r136481;
        return r136482;
}
↓
double f(double x, double y, double z) {
        double r136483 = y;
        double r136484 = 4.0;
        double r136485 = r136483 * r136484;
        double r136486 = z;
        double r136487 = -r136486;
        double r136488 = x;
        double r136489 = fma(r136485, r136487, r136488);
        return r136489;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[x - \left(y \cdot 4\right) \cdot z\]

  2. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{x - 4 \cdot \left(z \cdot y\right)}\]

  3. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot 4, -z, x\right)}\]

  4. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y \cdot 4, -z, x\right)\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (* y 4) z)))