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

Average Error: 0.0 → 0.0

Time: 3.2s

Precision: binary64

Cost: 448

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x, double y) {
	return x + ((1.0 + 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 + ((1.0d0 + 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 + ((1.0 + x) * y);
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

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

2. Taylor expanded in y around 0 0.0

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	16.1
Cost	856

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-194}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-161}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-109}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-98}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-40}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 2
Error	18.1
Cost	724

\[\begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-194}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-161}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-109}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-97}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 3
Error	0.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1600:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-271}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \]

Alternative 4
Error	1.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1600:\\ \;\;\;\;x \cdot \left(1 + y\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+20}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5
Error	1.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1600:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6
Error	7.6
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+20}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7
Error	36.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023076 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))