Average Error: 0.1 → 0.1
Time: 1.8s
Precision: binary64
\[x \cdot x - \left(y \cdot 4\right) \cdot z \]
↓
\[\mathsf{fma}\left(x, x, z \cdot \left(y \cdot -4\right)\right) + \mathsf{fma}\left(-z, y \cdot 4, z \cdot \left(y \cdot 4\right)\right) \]
x \cdot x - \left(y \cdot 4\right) \cdot z
↓
\mathsf{fma}\left(x, x, z \cdot \left(y \cdot -4\right)\right) + \mathsf{fma}\left(-z, y \cdot 4, z \cdot \left(y \cdot 4\right)\right)

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  2. Applied prod-diff_binary640.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2022067 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))