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

↓

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

(FPCore (x y) :precision binary64 (* (* (- x (/ 16.0 116.0)) 3.0) y))

↓

(FPCore (x y) :precision binary64 (* (fma x 3.0 -0.41379310344827586) y))

double code(double x, double y) {
	return ((x - (16.0 / 116.0)) * 3.0) * y;
}

↓

double code(double x, double y) {
	return fma(x, 3.0, -0.41379310344827586) * y;
}

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

↓

function code(x, y)
	return Float64(fma(x, 3.0, -0.41379310344827586) * y)
end

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

↓

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

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

↓

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

Original	0.2
Target	0.2
Herbie	0.2

Derivation

Initial program 0.2
\[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]
Simplified0.2
\[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, -0.41379310344827586\right) \cdot y} \]
Proof
(*.f64 (fma.f64 x 3 -12/29) y): 0 points increase in error, 0 points decrease in error
(*.f64 (fma.f64 x 3 (Rewrite<= metadata-eval (*.f64 3 -4/29))) y): 0 points increase in error, 0 points decrease in error
(*.f64 (fma.f64 x 3 (*.f64 3 (Rewrite<= metadata-eval (neg.f64 4/29)))) y): 0 points increase in error, 0 points decrease in error
(*.f64 (fma.f64 x 3 (*.f64 3 (neg.f64 (Rewrite<= metadata-eval (/.f64 16 116))))) y): 0 points increase in error, 0 points decrease in error
(*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x 3) (*.f64 3 (neg.f64 (/.f64 16 116))))) y): 0 points increase in error, 0 points decrease in error
(*.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 x)) (*.f64 3 (neg.f64 (/.f64 16 116)))) y): 0 points increase in error, 0 points decrease in error
(*.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 3 (+.f64 x (neg.f64 (/.f64 16 116))))) y): 0 points increase in error, 0 points decrease in error
(*.f64 (*.f64 3 (Rewrite<= sub-neg_binary64 (-.f64 x (/.f64 16 116)))) y): 0 points increase in error, 0 points decrease in error
(*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 x (/.f64 16 116)) 3)) y): 0 points increase in error, 0 points decrease in error
Final simplification0.2
\[\leadsto \mathsf{fma}\left(x, 3, -0.41379310344827586\right) \cdot y \]

Alternatives

Alternative 1
Error	1.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -0.135 \lor \neg \left(x \leq 0.135\right):\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.41379310344827586 \cdot y\\ \end{array} \]

Alternative 2
Error	1.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -0.135 \lor \neg \left(x \leq 0.135\right):\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.41379310344827586 \cdot y\\ \end{array} \]

Alternative 3
Error	1.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -0.135:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 0.135:\\ \;\;\;\;-0.41379310344827586 \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	448

\[3 \cdot \left(y \cdot \left(x + -0.13793103448275862\right)\right) \]

Alternative 5
Error	0.3
Cost	448

\[\left(3 \cdot y\right) \cdot \left(x + -0.13793103448275862\right) \]

Alternative 6
Error	0.2
Cost	448

\[y \cdot \left(-0.41379310344827586 + x \cdot 3\right) \]

Alternative 7
Error	27.5
Cost	192

\[-0.41379310344827586 \cdot y \]

Reproduce

herbie shell --seed 2022340 
(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))

Error

Target

Derivation

Alternatives

Error

Reproduce