Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A

Average Error: 0.1 → 0.1

Time: 6.8s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 - (x * y))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - (x * (x * y))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 - (x * y));
}

↓

public static double code(double x, double y) {
	return x - (x * (x * y));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ x \cdot \left(1 - x \cdot y\right) \]
   distribute-lft-out-- [<=] 0.1	\[ \color{blue}{x \cdot 1 - x \cdot \left(x \cdot y\right)} \]
   *-rgt-identity [=>] 0.1	\[ \color{blue}{x} - x \cdot \left(x \cdot y\right) \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	15.0
Cost	649

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+140} \lor \neg \left(y \leq 3.4 \cdot 10^{+115}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.1
Cost	448

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

Alternative 3
Error	22.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023039 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))