Average Error: 0.1 → 0.0
Time: 10.8s
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 r151343 = x;
        double r151344 = y;
        double r151345 = 4.0;
        double r151346 = r151344 * r151345;
        double r151347 = z;
        double r151348 = r151346 * r151347;
        double r151349 = r151343 - r151348;
        return r151349;
}
↓
double f(double x, double y, double z) {
        double r151350 = z;
        double r151351 = -r151350;
        double r151352 = y;
        double r151353 = r151351 * r151352;
        double r151354 = 4.0;
        double r151355 = x;
        double r151356 = fma(r151353, r151354, r151355);
        return r151356;
}

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)))