Average Error: 0.2 → 0.2
Time: 2.4s
Precision: 64
\[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y\]
↓
\[\mathsf{fma}\left(3 \cdot 1, x, -0.413793103448275856\right) \cdot y\]
\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y
↓
\mathsf{fma}\left(3 \cdot 1, x, -0.413793103448275856\right) \cdot y
double code(double x, double y) {
	return ((double) (((double) (((double) (x - ((double) (16.0 / 116.0)))) * 3.0)) * y));
}
↓
double code(double x, double y) {
	return ((double) (((double) fma(((double) (3.0 * 1.0)), x, ((double) -(0.41379310344827586)))) * y));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.2
Target0.2
Herbie0.2
\[y \cdot \left(x \cdot 3 - 0.413793103448275856\right)\]

Derivation

  1. Initial program 0.2

    \[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y\]

  2. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{\left(3 \cdot x - 0.413793103448275856\right)} \cdot y\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.2

    \[\leadsto \left(3 \cdot \color{blue}{\left(1 \cdot x\right)} - 0.413793103448275856\right) \cdot y\]

  5. Applied associate-*r*0.2

    \[\leadsto \left(\color{blue}{\left(3 \cdot 1\right) \cdot x} - 0.413793103448275856\right) \cdot y\]

  6. Applied fma-neg0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(3 \cdot 1, x, -0.413793103448275856\right)} \cdot y\]

  7. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(3 \cdot 1, x, -0.413793103448275856\right) \cdot y\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x y)
  :name "Data.Colour.CIE:cieLAB from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (* y (- (* x 3) 0.41379310344827586))

  (* (* (- x (/ 16 116)) 3) y))