Average Error: 0.1 → 0.1
Time: 2.7s
Precision: binary64
\[\left(x \cdot 3\right) \cdot y - z \]
↓
\[\mathsf{fma}\left(3, y \cdot x, -z\right) \]
\left(x \cdot 3\right) \cdot y - z
↓
\mathsf{fma}\left(3, y \cdot x, -z\right)

(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))
↓
(FPCore (x y z) :precision binary64 (fma 3.0 (* y x) (- z)))
double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}
↓
double code(double x, double y, double z) {
	return fma(3.0, (y * x), -z);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.2
Herbie0.1
\[x \cdot \left(3 \cdot y\right) - z \]

Derivation

  1. Initial program 0.1

    \[\left(x \cdot 3\right) \cdot y - z \]

  2. Taylor expanded in x around 0 0.1

    \[\leadsto \color{blue}{3 \cdot \left(y \cdot x\right)} - z \]

  3. Applied fma-neg_binary640.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(3, y \cdot x, -z\right)} \]

  4. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(3, y \cdot x, -z\right) \]

Reproduce

herbie shell --seed 2022067 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

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

  (- (* (* x 3.0) y) z))