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

Average Error: 0.01% → 0.01%

Time: 1.9s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.01

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

2. Final simplification 0.01

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

Alternatives

Alternative 1
Error	37.2%
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+29}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{+50}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 2
Error	30.59%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-57}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 3
Error	29.98%
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 4
Error	44.17%
Cost	196

\[\begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 5
Error	56.44%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023089 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))