\[x \cdot y - x \]

↓

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

(FPCore (x y) :precision binary64 (- (* x y) x))

↓

(FPCore (x y) :precision binary64 (fma y x (- x)))

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

↓

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

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

↓

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

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

↓

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

x \cdot y - x

↓

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

Derivation

Initial program 0.0
\[x \cdot y - x \]
Taylor expanded in x around 0 0.0
\[\leadsto \color{blue}{\left(y - 1\right) \cdot x} \]
Simplified0
\[\leadsto \color{blue}{\mathsf{fma}\left(y, x, -x\right)} \]
Proof
(fma.f64 y x (neg.f64 x)): 0 points increase in error, 0 points decrease in error
(Rewrite<= fma-neg_binary64 (-.f64 (*.f64 y x) x)): 5 points increase in error, 0 points decrease in error
(-.f64 (*.f64 y x) (Rewrite<= *-lft-identity_binary64 (*.f64 1 x))): 0 points increase in error, 0 points decrease in error
(Rewrite=> distribute-rgt-out--_binary64 (*.f64 x (-.f64 y 1))): 3 points increase in error, 4 points decrease in error
(Rewrite<= *-commutative_binary64 (*.f64 (-.f64 y 1) x)): 0 points increase in error, 0 points decrease in error
Final simplification0
\[\leadsto \mathsf{fma}\left(y, x, -x\right) \]

Alternatives

Alternative 1
Error	1.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.721179670456387:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.002337342508980307:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	0.0
Cost	320

\[x \cdot \left(y + -1\right) \]

Alternative 3
Error	27.4
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2022291 
(FPCore (x y)
  :name "Data.Histogram.Bin.LogBinD:$cbinSizeN from histogram-fill-0.8.4.1"
  :precision binary64
  (- (* x y) x))

Error

Derivation

Alternatives

Error

Reproduce