Average Error: 0.0 → 0.0
Time: 1.6s
Precision: binary64
\[x - \left(y \cdot 4\right) \cdot z \]
↓
\[\mathsf{fma}\left(y, z \cdot -4, x\right) \]
x - \left(y \cdot 4\right) \cdot z
↓
\mathsf{fma}\left(y, z \cdot -4, x\right)

(FPCore (x y z) :precision binary64 (- x (* (* y 4.0) z)))
↓
(FPCore (x y z) :precision binary64 (fma y (* z -4.0) x))
double code(double x, double y, double z) {
	return x - ((y * 4.0) * z);
}
↓
double code(double x, double y, double z) {
	return fma(y, (z * -4.0), x);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Taylor expanded in y around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022125 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (* y 4.0) z)))