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

Average Accuracy: 100.0% → 100.0%

Time: 2.3s

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 100.0%

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

2. Taylor expanded in y around 0 100.0%

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

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	76.0%
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+89}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+104}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 2
Accuracy	87.9%
Cost	589

\[\begin{array}{l} \mathbf{if}\;y \leq -95000000 \lor \neg \left(y \leq 5.7 \cdot 10^{+88}\right) \land y \leq 4 \cdot 10^{+104}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 3
Accuracy	98.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -95000000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) \cdot y\\ \end{array} \]

Alternative 4
Accuracy	99.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(1 + y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) \cdot y\\ \end{array} \]

Alternative 5
Accuracy	72.4%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6
Accuracy	43.5%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023135 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))