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

Average Error: 0.0 → 0.0

Time: 1.6s

Precision: binary64

Cost: 448

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x y) :precision binary64 (+ y (* x (+ 1.0 y))))

C

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

↓

double code(double x, double y) {
	return y + (x * (1.0 + y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * y) + x) + y
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y + (x * (1.0d0 + y))
end function

Java

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

↓

public static double code(double x, double y) {
	return y + (x * (1.0 + y));
}

Python

def code(x, y):
	return ((x * y) + x) + y

↓

def code(x, y):
	return y + (x * (1.0 + y))

Julia

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

↓

function code(x, y)
	return Float64(y + Float64(x * Float64(1.0 + y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * y) + x) + y;
end

↓

function tmp = code(x, y)
	tmp = y + (x * (1.0 + y));
end

Wolfram

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

↓

code[x_, y_] := N[(y + N[(x * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

   \[\leadsto \color{blue}{x \cdot \left(1 + y\right)} + y \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	8.3
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2717529438719227 \cdot 10^{+20}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 4.1938992463191097 \cdot 10^{+37}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+51}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 2
Error	1.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2717529438719227 \cdot 10^{+20}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.8267484706792414 \cdot 10^{-24}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \]

Alternative 3
Error	9.6
Cost	192

\[x + y \]

Alternative 4
Error	36.6
Cost	64

\[y \]

Reproduce

herbie shell --seed 2022228 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))