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

Average Error: 0.01% → 0.01%

Time: 2.4s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.01

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

2. Simplified 0.01

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

   [Start] 0.01	\[ \left(x \cdot y + x\right) + y \]
   +-commutative [=>] 0.01	\[ \color{blue}{y + \left(x \cdot y + x\right)} \]
   fma-def [=>] 0.01	\[ y + \color{blue}{\mathsf{fma}\left(x, y, x\right)} \]

3. Final simplification 0.01

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

Alternatives

Alternative 1
Error	39.58%
Cost	856

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-216}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-173}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.96 \cdot 10^{-104}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-76}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-64}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 2
Error	46.1%
Cost	724

\[\begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-216}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-171}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-104}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-76}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-64}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 3
Error	21.54%
Cost	589

\[\begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+100}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{+62} \lor \neg \left(x \leq 6200000000\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 4
Error	15.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 5
Error	15.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-11}:\\ \;\;\;\;x + y \cdot x\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 6
Error	14.69%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -580:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq 6200000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7
Error	0.01%
Cost	448

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

Alternative 8
Error	56.84%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023115 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))