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

Average Error: 0.0 → 0.0

Time: 2.4s

Precision: binary64

Cost: 448

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 * (y + 1.0d0))
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 * (y + 1.0));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	8.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	8.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-35}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \]

Alternative 3
Error	15.9
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-35}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 4
Error	18.0
Cost	196

\[\begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-35}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 5
Error	36.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))