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

Average Error: 0.0 → 0.0

Time: 1.9s

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 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{y + \mathsf{fma}\left(x, y, x\right)} \]
   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))): 1 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 simplification 0.0

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

Alternatives

Alternative 1
Error	9.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+16}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \]

Alternative 2
Error	27.5
Cost	456

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

Alternative 3
Error	0.0
Cost	448

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

Alternative 4
Error	0.0
Cost	448

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

Alternative 5
Error	14.1
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+18}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 6
Error	36.0
Cost	64

\[y \]

Reproduce

herbie shell --seed 2022330 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))