Numeric.SpecFunctions:logGammaCorrection from math-functions-0.1.5.2

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 4

Speedup: 1.2×

• 26.3% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * 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%

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

Alternative 2: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.002:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 0.002) -1.0 (* (* 2.0 x) x)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.002) {
		tmp = -1.0;
	} else {
		tmp = (2.0 * x) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 0.002d0) then
        tmp = -1.0d0
    else
        tmp = (2.0d0 * x) * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.002) {
		tmp = -1.0;
	} else {
		tmp = (2.0 * x) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.002:
		tmp = -1.0
	else:
		tmp = (2.0 * x) * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.002)
		tmp = -1.0;
	else
		tmp = Float64(Float64(2.0 * x) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.002)
		tmp = -1.0;
	else
		tmp = (2.0 * x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.002], -1.0, N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2e-3

   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 99.2%

      \[\leadsto \color{blue}{-1} \]

   if 2e-3 < (*.f64 x x)

   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 0.8%

      \[\leadsto \color{blue}{-1} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot {x}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 99.5

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

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 50.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.002:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 0.002) -1.0 (+ x x)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.002) {
		tmp = -1.0;
	} else {
		tmp = x + x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 0.002d0) then
        tmp = -1.0d0
    else
        tmp = x + x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.002) {
		tmp = -1.0;
	} else {
		tmp = x + x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.002:
		tmp = -1.0
	else:
		tmp = x + x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.002)
		tmp = -1.0;
	else
		tmp = Float64(x + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.002)
		tmp = -1.0;
	else
		tmp = x + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.002], -1.0, N[(x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2e-3

   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 99.2%

      \[\leadsto \color{blue}{-1} \]

   if 2e-3 < (*.f64 x x)

   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 0.8%

      \[\leadsto \color{blue}{-1} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot {x}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 99.5

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

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot x} \]
   9. Step-by-step derivation

   10. Applied rewrites 3.6%

       \[\leadsto x + \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 49.2% 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 48.8%

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

Reproduce

herbie shell --seed 2024320 
(FPCore (x)
  :name "Numeric.SpecFunctions:logGammaCorrection from math-functions-0.1.5.2"
  :precision binary64
  (- (* (* x x) 2.0) 1.0))