Numeric.SpecFunctions:logGammaCorrection from math-functions-0.1.5.2

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

Alternatives: 3

Speedup: 1.4×

• 24.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot 2 - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* (* x x) 2.0) 1.0))

C

double code(double x) {
	return ((x * x) * 2.0) - 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) * 2.0d0) - 1.0d0
end function

Java

public static double code(double x) {
	return ((x * x) * 2.0) - 1.0;
}

Python

def code(x):
	return ((x * x) * 2.0) - 1.0

Julia

function code(x)
	return Float64(Float64(Float64(x * x) * 2.0) - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) * 2.0) - 1.0;
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot 2 - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* (* x x) 2.0) 1.0))

C

double code(double x) {
	return ((x * x) * 2.0) - 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) * 2.0d0) - 1.0d0
end function

Java

public static double code(double x) {
	return ((x * x) * 2.0) - 1.0;
}

Python

def code(x):
	return ((x * x) * 2.0) - 1.0

Julia

function code(x)
	return Float64(Float64(Float64(x * x) * 2.0) - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) * 2.0) - 1.0;
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fma (+ x x) x -1.0))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot x, x, -1\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot x}, x, -1\right) \]

   2. count-2-rev N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x + x}, x, -1\right) \]

   3. lower-+.f64 100.0

      \[\leadsto \mathsf{fma}\left(\color{blue}{x + x}, x, -1\right) \]

6. Applied rewrites 100.0%

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

Alternative 2: 50.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 2, -1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x 2.0 -1.0))

C

double code(double x) {
	return fma(x, 2.0, -1.0);
}

Julia

function code(x)
	return fma(x, 2.0, -1.0)
end

Wolfram

code[x_] := N[(x * 2.0 + -1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot x, x, -1\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. count-2-rev N/A

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

   8. flip-+ N/A

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

   9. +-inverses N/A

      \[\leadsto x \cdot \frac{\color{blue}{0}}{x - x} + -1 \]

   10. +-inverses N/A

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

   11. associate-*r/ N/A

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

   12. distribute-lft-out-- N/A

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

   13. flip-+ N/A

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

   14. count-2-rev N/A

       \[\leadsto \color{blue}{2 \cdot x} + -1 \]

   15. *-commutative N/A

       \[\leadsto \color{blue}{x \cdot 2} + -1 \]

   16. lower-fma.f64 56.2

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2, -1\right)} \]

6. Applied rewrites 56.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2, -1\right)} \]
7. Add Preprocessing

Alternative 3: 49.5% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Applied rewrites 54.9%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024329 
(FPCore (x)
  :name "Numeric.SpecFunctions:logGammaCorrection from math-functions-0.1.5.2"
  :precision binary64
  (- (* (* x x) 2.0) 1.0))