Average Error: 12.5 → 0.0
Time: 3.0s
Precision: binary64
\[\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z \]
↓
\[\mathsf{fma}\left(y, x, -y \cdot z\right) + \mathsf{fma}\left(-z, y, y \cdot z\right) \]
\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z
↓
\mathsf{fma}\left(y, x, -y \cdot z\right) + \mathsf{fma}\left(-z, y, y \cdot z\right)

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original12.5
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y \]

Derivation

  1. Initial program 12.5

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

  2. Simplified0.0

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

  3. Taylor expanded in x around 0 0.0

    \[\leadsto \color{blue}{y \cdot x - y \cdot z} \]

  4. Applied prod-diff_binary640.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022081 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, D"
  :precision binary64

  :herbie-target
  (* (- x z) y)

  (- (+ (- (* x y) (* y y)) (* y y)) (* y z)))