Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B

Average Accuracy: 100.0% → 100.0%

Time: 2.7s

Precision: binary64

Cost: 6720

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

Wolfram

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

↓

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

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

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

   [Start] 100.0	\[ \left(x \cdot y + x\right) + y \]
   +-commutative [=>] 100.0	\[ \color{blue}{\left(x + x \cdot y\right)} + y \]
   associate-+l+ [=>] 100.0	\[ \color{blue}{x + \left(x \cdot y + y\right)} \]
   fma-def [=>] 100.0	\[ x + \color{blue}{\mathsf{fma}\left(x, y, y\right)} \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	86.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-167}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 2
Accuracy	75.8%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3
Accuracy	86.9%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 4
Accuracy	100.0%
Cost	448

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

Alternative 5
Accuracy	72.5%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-74}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6
Accuracy	42.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))