Average Error: 0.1 → 0.1
Time: 2.8s
Precision: binary64
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
↓
\[\mathsf{fma}\left(z, z, y \cdot x\right) + 2 \cdot \left(z \cdot z\right) \]
\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z
↓
\mathsf{fma}\left(z, z, y \cdot x\right) + 2 \cdot \left(z \cdot z\right)

(FPCore (x y z)
 :precision binary64
 (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))
↓
(FPCore (x y z) :precision binary64 (+ (fma z z (* y x)) (* 2.0 (* z z))))
double code(double x, double y, double z) {
	return (((x * y) + (z * z)) + (z * z)) + (z * z);
}
↓
double code(double x, double y, double z) {
	return fma(z, z, (y * x)) + (2.0 * (z * z));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.1
\[\left(3 \cdot z\right) \cdot z + y \cdot x \]

Derivation

  1. Initial program 0.1

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

  2. Applied associate-+l+_binary640.1

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

  3. Simplified0.1

    \[\leadsto \left(x \cdot y + z \cdot z\right) + \color{blue}{2 \cdot \left(z \cdot z\right)} \]

  4. Taylor expanded in x around 0 0.1

    \[\leadsto \color{blue}{\left(y \cdot x + {z}^{2}\right)} + 2 \cdot \left(z \cdot z\right) \]

  5. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, y \cdot x\right)} + 2 \cdot \left(z \cdot z\right) \]

  6. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(z, z, y \cdot x\right) + 2 \cdot \left(z \cdot z\right) \]

Reproduce

herbie shell --seed 2022089 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (+ (* (* 3.0 z) z) (* y x))

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