Average Error: 0.0 → 0.0
Time: 2.7s
Precision: binary64
Cost: 6720
\[\left(x \cdot y + x\right) + y \]
↓
\[y + \mathsf{fma}\left(x, y, x\right) \]
(FPCore (x y) :precision binary64 (+ (+ (* x y) x) y))
↓
(FPCore (x y) :precision binary64 (+ y (fma x y x)))
double code(double x, double y) {
	return ((x * y) + x) + y;
}
↓
double code(double x, double y) {
	return y + fma(x, y, x);
}

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

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

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

Error

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Alternatives

Alternative 1
Error25.5
Cost984
\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+75}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.06 \cdot 10^{+49}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-57}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-81}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-160}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
Alternative 2
Error20.7
Cost720
\[\begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-81}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-160}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
Alternative 3
Error20.3
Cost716
\[\begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-55}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-80}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-160}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]
Alternative 4
Error20.3
Cost716
\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-81}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-160}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \]
Alternative 5
Error29.1
Cost460
\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-57}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-81}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-160}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
Alternative 6
Error0.0
Cost448
\[y + \left(x + y \cdot x\right) \]
Alternative 7
Error35.9
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022340 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))