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

Percentage Accurate: 99.9% → 99.8%

Time: 9.5s

Alternatives: 3

Speedup: 1.0×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(1 - x \cdot y\right) \end{array} \]

FPCore

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

C

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 * (1.0d0 - (x * y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - x \cdot y\right) \end{array} \]

FPCore

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

C

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 * (1.0d0 - (x * y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{-1}{\mathsf{fma}\left(x, y, -1\right)}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x (/ -1.0 (fma x y -1.0))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \left(1 - x \cdot y\right) \]
2. Add Preprocessing

4. Applied egg-rr 69.3%

   \[\leadsto x \cdot \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot \left(y \cdot y\right), -1\right)}{\mathsf{fma}\left(x, y, 1\right)}} \]
5. Step-by-step derivation

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. /-lowering-/.f64 N/A

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

   4. clear-num N/A

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

   5. neg-sub0 N/A

      \[\leadsto \frac{x}{\frac{1}{\frac{\color{blue}{0 - \left(x \cdot \left(x \cdot \left(y \cdot y\right)\right) + -1\right)}}{x \cdot y + 1}}} \]

   6. +-commutative N/A

      \[\leadsto \frac{x}{\frac{1}{\frac{0 - \color{blue}{\left(-1 + x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)}}{x \cdot y + 1}}} \]

   7. associate--r+ N/A

      \[\leadsto \frac{x}{\frac{1}{\frac{\color{blue}{\left(0 - -1\right) - x \cdot \left(x \cdot \left(y \cdot y\right)\right)}}{x \cdot y + 1}}} \]

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. associate-*r* N/A

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

   11. swap-sqr N/A

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

   12. +-commutative N/A

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

   13. flip-- N/A

       \[\leadsto \frac{x}{\frac{1}{\color{blue}{1 - x \cdot y}}} \]

   14. metadata-eval N/A

       \[\leadsto \frac{x}{\frac{1}{\color{blue}{1 \cdot 1} - x \cdot y}} \]

   15. *-rgt-identity N/A

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

   16. frac-2neg N/A

       \[\leadsto \frac{x}{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(1 \cdot 1 - \left(x \cdot y\right) \cdot 1\right)\right)}}} \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{x}{\frac{-1}{\mathsf{fma}\left(x, y, -1\right)}}} \]
7. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - x \cdot y\right) \end{array} \]

FPCore

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

C

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 * (1.0d0 - (x * y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \left(1 - x \cdot y\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 3: 51.3% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

   \[x \cdot \left(1 - x \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 52.4%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))