Data.Colour.CIE:cieLAB from colour-2.3.3, A

Average Error: 0.2 → 0.2

Time: 14.3s

Precision: 64

Math

\[\left(\left(x - \frac{16.0}{116.0}\right) \cdot 3.0\right) \cdot y\]

↓

\[\mathsf{fma}\left(3.0, x, -0.41379310344827586\right) \cdot y\]

C

double f(double x, double y) {
        double r39450211 = x;
        double r39450212 = 16.0;
        double r39450213 = 116.0;
        double r39450214 = r39450212 / r39450213;
        double r39450215 = r39450211 - r39450214;
        double r39450216 = 3.0;
        double r39450217 = r39450215 * r39450216;
        double r39450218 = y;
        double r39450219 = r39450217 * r39450218;
        return r39450219;
}

↓

double f(double x, double y) {
        double r39450220 = 3.0;
        double r39450221 = x;
        double r39450222 = 0.41379310344827586;
        double r39450223 = -r39450222;
        double r39450224 = fma(r39450220, r39450221, r39450223);
        double r39450225 = y;
        double r39450226 = r39450224 * r39450225;
        return r39450226;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.2
Target	0.2
Herbie	0.2

\[y \cdot \left(x \cdot 3.0 - 0.41379310344827586\right)\]

Derivation

1. Initial program 0.2

   \[\left(\left(x - \frac{16.0}{116.0}\right) \cdot 3.0\right) \cdot y\]

2. Taylor expanded around 0 0.2

   \[\leadsto \color{blue}{3.0 \cdot \left(x \cdot y\right) - 0.41379310344827586 \cdot y}\]

3. Simplified 0.2

   \[\leadsto \color{blue}{y \cdot \left(3.0 \cdot x - 0.41379310344827586\right)}\]
4. Using strategy rm

5. Applied fma-neg 0.2

   \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(3.0, x, -0.41379310344827586\right)}\]

6. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(3.0, x, -0.41379310344827586\right) \cdot y\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y)
  :name "Data.Colour.CIE:cieLAB from colour-2.3.3, A"

  :herbie-target
  (* y (- (* x 3.0) 0.41379310344827586))

  (* (* (- x (/ 16.0 116.0)) 3.0) y))