Average Error: 0.0 → 0.0
Time: 2.2s
Precision: binary64
Cost: 6720
\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]
\[\left(x \cdot y + x\right) + y \]
↓
\[x + \mathsf{fma}\left(x, y, y\right) \]
(FPCore (x y) :precision binary64 (+ (+ (* x y) x) y))
↓
(FPCore (x y) :precision binary64 (+ x (fma x y y)))
double code(double x, double y) {
	return ((x * y) + x) + y;
}
↓
double code(double x, double y) {
	return x + fma(x, y, y);
}

function code(x, y)
	return Float64(Float64(Float64(x * y) + x) + y)
end
↓
function code(x, y)
	return Float64(x + fma(x, y, y))
end

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

\left(x \cdot y + x\right) + y
↓
x + \mathsf{fma}\left(x, y, y\right)

Error

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{x + \mathsf{fma}\left(x, y, y\right)} \]
    Proof
    (+.f64 x (fma.f64 x y y)): 0 points increase in error, 0 points decrease in error
    (+.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) y))): 3 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 x (*.f64 x y)) y)): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x y) x)) y): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.0

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

Alternatives

Alternative 1
Error9.0
Cost584
\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-37}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]
Alternative 2
Error15.6
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
Alternative 3
Error8.3
Cost452
\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-97}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]
Alternative 4
Error0.0
Cost448
\[y + \left(x + x \cdot y\right) \]
Alternative 5
Error17.6
Cost196
\[\begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
Alternative 6
Error36.1
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022338 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))