Average Error: 0.1 → 0.0
Time: 9.3s
Precision: 64
\[x - \left(y \cdot 4\right) \cdot z\]
↓
\[\mathsf{fma}\left(\left(-z\right) \cdot y, 4, x\right)\]
x - \left(y \cdot 4\right) \cdot z
↓
\mathsf{fma}\left(\left(-z\right) \cdot y, 4, x\right)
double f(double x, double y, double z) {
        double r137499 = x;
        double r137500 = y;
        double r137501 = 4.0;
        double r137502 = r137500 * r137501;
        double r137503 = z;
        double r137504 = r137502 * r137503;
        double r137505 = r137499 - r137504;
        return r137505;
}
↓
double f(double x, double y, double z) {
        double r137506 = z;
        double r137507 = -r137506;
        double r137508 = y;
        double r137509 = r137507 * r137508;
        double r137510 = 4.0;
        double r137511 = x;
        double r137512 = fma(r137509, r137510, r137511);
        return r137512;
}

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. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (* y 4) z)))