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

Average Accuracy: 99.8% → 99.9%

Time: 7.1s

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 99.8%

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

2. Simplified 87.0%

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

   [Start] 99.8	\[ x \cdot \left(1 - x \cdot y\right) \]
   sub-neg [=>] 99.8	\[ x \cdot \color{blue}{\left(1 + \left(-x \cdot y\right)\right)} \]
   +-commutative [=>] 99.8	\[ x \cdot \color{blue}{\left(\left(-x \cdot y\right) + 1\right)} \]
   distribute-rgt-in [=>] 99.9	\[ \color{blue}{\left(-x \cdot y\right) \cdot x + 1 \cdot x} \]
   *-lft-identity [=>] 99.9	\[ \left(-x \cdot y\right) \cdot x + \color{blue}{x} \]
   *-commutative [=>] 99.9	\[ \left(-\color{blue}{y \cdot x}\right) \cdot x + x \]
   distribute-lft-neg-in [=>] 99.9	\[ \color{blue}{\left(\left(-y\right) \cdot x\right)} \cdot x + x \]
   associate-*l* [=>] 87.0	\[ \color{blue}{\left(-y\right) \cdot \left(x \cdot x\right)} + x \]
   fma-def [=>] 87.0	\[ \color{blue}{\mathsf{fma}\left(-y, x \cdot x, x\right)} \]

3. Taylor expanded in y around 0 87.0%

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

4. Simplified 99.9%

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

   [Start] 87.0	\[ -1 \cdot \left(y \cdot {x}^{2}\right) + x \]
   +-commutative [=>] 87.0	\[ \color{blue}{x + -1 \cdot \left(y \cdot {x}^{2}\right)} \]
   mul-1-neg [=>] 87.0	\[ x + \color{blue}{\left(-y \cdot {x}^{2}\right)} \]
   unpow2 [=>] 87.0	\[ x + \left(-y \cdot \color{blue}{\left(x \cdot x\right)}\right) \]
   sub-neg [<=] 87.0	\[ \color{blue}{x - y \cdot \left(x \cdot x\right)} \]
   *-commutative [=>] 87.0	\[ x - \color{blue}{\left(x \cdot x\right) \cdot y} \]
   associate-*l* [=>] 99.9	\[ x - \color{blue}{x \cdot \left(x \cdot y\right)} \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	77.1%
Cost	649

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+65} \lor \neg \left(y \leq 8.5 \cdot 10^{+88}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	448

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

Alternative 3
Accuracy	66.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))