Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, B

Average Error: 0.2 → 0.1

Time: 1.8s

Precision: 64

Math

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

↓

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

C

double code(double x, double y, double z) {
	return ((double) (((double) (((double) (x * 3.0)) * y)) - z));
}

↓

double code(double x, double y, double z) {
	return ((double) fma(3.0, ((double) (x * y)), ((double) -(z))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	0.2
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 0.2

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

2. Taylor expanded around 0 0.1

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

4. Applied fma-neg 0.1

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

5. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (* x (* 3 y)) z)

  (- (* (* x 3) y) z))