Average Error: 0.1 → 0.1
Time: 4.2s
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 r126085 = x;
        double r126086 = y;
        double r126087 = 4.0;
        double r126088 = r126086 * r126087;
        double r126089 = z;
        double r126090 = r126088 * r126089;
        double r126091 = r126085 - r126090;
        return r126091;
}
↓
double f(double x, double y, double z) {
        double r126092 = y;
        double r126093 = 4.0;
        double r126094 = r126092 * r126093;
        double r126095 = z;
        double r126096 = -r126095;
        double r126097 = x;
        double r126098 = fma(r126094, r126096, r126097);
        return r126098;
}

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 2019198 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A"
  (- x (* (* y 4.0) z)))