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

Percentage Accurate: 99.9% → 99.9%

Time: 3.8s

Precision: binary64

Cost: 448

• 25.4% of points produce a very large (infinite) output. You may want to add a precondition.

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 99.9%

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

2. Taylor expanded in x around 0 94.1%

   \[\leadsto \color{blue}{-1 \cdot \left(y \cdot {x}^{2}\right) + x} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x - x \cdot \left(x \cdot y\right)} \]
   Step-by-step derivation

   [Start] 94.1%	\[ -1 \cdot \left(y \cdot {x}^{2}\right) + x \]
   +-commutative [=>] 94.1%	\[ \color{blue}{x + -1 \cdot \left(y \cdot {x}^{2}\right)} \]
   mul-1-neg [=>] 94.1%	\[ x + \color{blue}{\left(-y \cdot {x}^{2}\right)} \]
   unpow2 [=>] 94.1%	\[ x + \left(-y \cdot \color{blue}{\left(x \cdot x\right)}\right) \]
   sub-neg [<=] 94.1%	\[ \color{blue}{x - y \cdot \left(x \cdot x\right)} \]
   *-commutative [=>] 94.1%	\[ x - \color{blue}{\left(x \cdot x\right) \cdot y} \]
   associate-*r* [<=] 99.9%	\[ x - \color{blue}{x \cdot \left(x \cdot y\right)} \]

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	448

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

Alternative 2
Accuracy	74.8%
Cost	649

\[\begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{-70} \lor \neg \left(y \leq 42000000000\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	448

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

Alternative 4
Accuracy	50.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023245 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))