Average Error: 0.2 → 0.1
Time: 3.1s
Precision: binary64
\[\left(x \cdot 3\right) \cdot y - z \]
↓
\[\mathsf{fma}\left(x, 3 \cdot y, -z\right) + \mathsf{fma}\left(-z, 1, z\right) \]
\left(x \cdot 3\right) \cdot y - z
↓
\mathsf{fma}\left(x, 3 \cdot y, -z\right) + \mathsf{fma}\left(-z, 1, z\right)

(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))
↓
(FPCore (x y z)
 :precision binary64
 (+ (fma x (* 3.0 y) (- z)) (fma (- z) 1.0 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(x, (3.0 * y), -z) + fma(-z, 1.0, z);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Initial program 0.2

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

  2. Applied associate-*l*_binary640.1

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

  3. Applied *-un-lft-identity_binary640.1

    \[\leadsto x \cdot \left(3 \cdot y\right) - \color{blue}{1 \cdot z} \]

  4. Applied prod-diff_binary640.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3 \cdot y, -z \cdot 1\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right)} \]

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, 3 \cdot y, -z\right) + \mathsf{fma}\left(-z, 1, z\right) \]

Reproduce

herbie shell --seed 2021275 
(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))