Average Error: 0.2 → 0.2
Time: 1.9s
Precision: binary64
\[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]
↓
\[\mathsf{fma}\left(-0.41379310344827586, y, y \cdot \left(3 \cdot x\right)\right) \]
(FPCore (x y) :precision binary64 (* (* (- x (/ 16.0 116.0)) 3.0) y))
↓
(FPCore (x y) :precision binary64 (fma -0.41379310344827586 y (* y (* 3.0 x))))
double code(double x, double y) {
	return ((x - (16.0 / 116.0)) * 3.0) * y;
}
↓
double code(double x, double y) {
	return fma(-0.41379310344827586, y, (y * (3.0 * x)));
}

function code(x, y)
	return Float64(Float64(Float64(x - Float64(16.0 / 116.0)) * 3.0) * y)
end
↓
function code(x, y)
	return fma(-0.41379310344827586, y, Float64(y * Float64(3.0 * x)))
end

code[x_, y_] := N[(N[(N[(x - N[(16.0 / 116.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] * y), $MachinePrecision]
↓
code[x_, y_] := N[(-0.41379310344827586 * y + N[(y * N[(3.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y
↓
\mathsf{fma}\left(-0.41379310344827586, y, y \cdot \left(3 \cdot x\right)\right)

Error

Bits error versus x

Bits error versus y

Target

Original0.2
Target0.2
Herbie0.2
\[y \cdot \left(x \cdot 3 - 0.41379310344827586\right) \]

Derivation

  1. Initial program 0.2

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

  2. Taylor expanded in y around inf 0.3

    \[\leadsto \color{blue}{3 \cdot \left(y \cdot \left(x - 0.13793103448275862\right)\right)} \]

  3. Applied egg-rr0.3

    \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) + -0.13793103448275862 \cdot \left(3 \cdot y\right)} \]

  4. Applied egg-rr0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.41379310344827586, y, y \cdot \left(3 \cdot x\right)\right)} \]

  5. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(-0.41379310344827586, y, y \cdot \left(3 \cdot x\right)\right) \]

Reproduce

herbie shell --seed 2022160 
(FPCore (x y)
  :name "Data.Colour.CIE:cieLAB from colour-2.3.3, A"
  :precision binary64

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

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