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

Average Error: 0.0 → 0.0

Time: 23.6s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

   [Start] 0.0	\[ \left(x \cdot y + x\right) + y \]
   rational_best-simplify-3 [=>] 0.0	\[ \color{blue}{y + \left(x \cdot y + x\right)} \]
   rational_best-simplify-47 [=>] 0.0	\[ \color{blue}{x + \left(x \cdot y + y\right)} \]
   rational_best-simplify-3 [=>] 0.0	\[ x + \color{blue}{\left(y + x \cdot y\right)} \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	25.3
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -1.92 \cdot 10^{+174}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{+77}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	15.4
Cost	588

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+174}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+78}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1020000000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Error	0.9
Cost	584

\[\begin{array}{l} t_0 := \left(1 + x\right) \cdot y\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-8}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	12.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(1 + y\right) \cdot x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-150}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) \cdot y\\ \end{array} \]

Alternative 5
Error	12.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(1 + y\right) \cdot x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-150}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \]

Alternative 6
Error	27.8
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-87}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7
Error	35.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023099 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))