Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (- (* x (log y)) z) y))

C

double code(double x, double y, double z) {
	return ((x * log(y)) - z) - y;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x * log(y)) - z) - y
end function

Java

public static double code(double x, double y, double z) {
	return ((x * Math.log(y)) - z) - y;
}

Python

def code(x, y, z):
	return ((x * math.log(y)) - z) - y

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (- (* x (log y)) z) y))

C

double code(double x, double y, double z) {
	return ((x * log(y)) - z) - y;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x * log(y)) - z) - y
end function

Java

public static double code(double x, double y, double z) {
	return ((x * Math.log(y)) - z) - y;
}

Python

def code(x, y, z):
	return ((x * math.log(y)) - z) - y

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (- (* x (log y)) y) z))

C

double code(double x, double y, double z) {
	return ((x * log(y)) - y) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x * log(y)) - y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x * Math.log(y)) - y) - z;
}

Python

def code(x, y, z):
	return ((x * math.log(y)) - y) - z

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around inf 85.9%

   \[\leadsto \color{blue}{z \cdot \left(\frac{x \cdot \log y}{z} - 1\right)} - y \]
4. Step-by-step derivation

   1. *-commutative 85.9%

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

   2. associate-/l* 85.9%

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

   3. fmm-def 85.9%

      \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(\log y, \frac{x}{z}, -1\right)} - y \]

   4. metadata-eval 85.9%

      \[\leadsto z \cdot \mathsf{fma}\left(\log y, \frac{x}{z}, \color{blue}{-1}\right) - y \]

5. Simplified 85.9%

   \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(\log y, \frac{x}{z}, -1\right)} - y \]

6. Taylor expanded in z around 0 99.9%

   \[\leadsto \color{blue}{\left(-1 \cdot z + x \cdot \log y\right) - y} \]
7. Step-by-step derivation

   1. neg-mul-1 99.9%

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

   2. associate--l+ 99.9%

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

   3. +-commutative 99.9%

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

   4. sub-neg 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

   \[\leadsto \left(x \cdot \log y - y\right) - z \]
10. Add Preprocessing

Alternative 2: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log y\\ \mathbf{if}\;y \leq 2.15 \cdot 10^{-15}:\\ \;\;\;\;t\_0 - z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+57} \lor \neg \left(y \leq 1.3 \cdot 10^{+147}\right):\\ \;\;\;\;t\_0 - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log y))))
   (if (<= y 2.15e-15)
     (- t_0 z)
     (if (or (<= y 2.5e+57) (not (<= y 1.3e+147))) (- t_0 y) (- (- z) y)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log(y);
	double tmp;
	if (y <= 2.15e-15) {
		tmp = t_0 - z;
	} else if ((y <= 2.5e+57) || !(y <= 1.3e+147)) {
		tmp = t_0 - y;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * log(y)
    if (y <= 2.15d-15) then
        tmp = t_0 - z
    else if ((y <= 2.5d+57) .or. (.not. (y <= 1.3d+147))) then
        tmp = t_0 - y
    else
        tmp = -z - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log(y);
	double tmp;
	if (y <= 2.15e-15) {
		tmp = t_0 - z;
	} else if ((y <= 2.5e+57) || !(y <= 1.3e+147)) {
		tmp = t_0 - y;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log(y)
	tmp = 0
	if y <= 2.15e-15:
		tmp = t_0 - z
	elif (y <= 2.5e+57) or not (y <= 1.3e+147):
		tmp = t_0 - y
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(y))
	tmp = 0.0
	if (y <= 2.15e-15)
		tmp = Float64(t_0 - z);
	elseif ((y <= 2.5e+57) || !(y <= 1.3e+147))
		tmp = Float64(t_0 - y);
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log(y);
	tmp = 0.0;
	if (y <= 2.15e-15)
		tmp = t_0 - z;
	elseif ((y <= 2.5e+57) || ~((y <= 1.3e+147)))
		tmp = t_0 - y;
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.15e-15], N[(t$95$0 - z), $MachinePrecision], If[Or[LessEqual[y, 2.5e+57], N[Not[LessEqual[y, 1.3e+147]], $MachinePrecision]], N[(t$95$0 - y), $MachinePrecision], N[((-z) - y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.1499999999999998e-15

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 94.3%

      \[\leadsto \color{blue}{x \cdot \log y - z} \]

   if 2.1499999999999998e-15 < y < 2.49999999999999986e57 or 1.2999999999999999e147 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 89.6%

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

   if 2.49999999999999986e57 < y < 1.2999999999999999e147

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 88.1%

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

      1. neg-mul-1 30.9%

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

   5. Simplified 88.1%

      \[\leadsto \color{blue}{\left(-z\right)} - y \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+57} \lor \neg \left(y \leq 1.3 \cdot 10^{+147}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+103} \lor \neg \left(x \leq 3.1 \cdot 10^{+31}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.65e+103) (not (<= x 3.1e+31)))
   (- (* x (log y)) y)
   (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.65e+103) || !(x <= 3.1e+31)) {
		tmp = (x * log(y)) - y;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.65d+103)) .or. (.not. (x <= 3.1d+31))) then
        tmp = (x * log(y)) - y
    else
        tmp = -z - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.65e+103) || !(x <= 3.1e+31)) {
		tmp = (x * Math.log(y)) - y;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.65e+103) or not (x <= 3.1e+31):
		tmp = (x * math.log(y)) - y
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.65e+103) || !(x <= 3.1e+31))
		tmp = Float64(Float64(x * log(y)) - y);
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.65e+103) || ~((x <= 3.1e+31)))
		tmp = (x * log(y)) - y;
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.65e+103], N[Not[LessEqual[x, 3.1e+31]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65000000000000004e103 or 3.1000000000000002e31 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 86.9%

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

   if -1.65000000000000004e103 < x < 3.1000000000000002e31

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 90.4%

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

      1. neg-mul-1 39.4%

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

   5. Simplified 90.4%

      \[\leadsto \color{blue}{\left(-z\right)} - y \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+103} \lor \neg \left(x \leq 3.1 \cdot 10^{+31}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.2% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-15}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 1.9e-15) (- z) (- y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.9e-15) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.9d-15) then
        tmp = -z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.9e-15) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.9e-15:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.9e-15)
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.9e-15)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.9e-15], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 1.9000000000000001e-15

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 94.3%

      \[\leadsto \color{blue}{x \cdot \log y - z} \]

   4. Taylor expanded in x around 0 46.2%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   5. Step-by-step derivation

      1. neg-mul-1 46.2%

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

   6. Simplified 46.2%

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

   if 1.9000000000000001e-15 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 64.1%

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

      1. neg-mul-1 64.1%

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

   5. Simplified 64.1%

      \[\leadsto \color{blue}{-y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-15}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.5% accurate, 26.8× speedup

Math

\[\begin{array}{l} \\ \left(-z\right) - y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (- z) y))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return -z - y

Julia

function code(x, y, z)
	return Float64(Float64(-z) - y)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[((-z) - y), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 67.4%

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

   1. neg-mul-1 29.6%

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

5. Simplified 67.4%

   \[\leadsto \color{blue}{\left(-z\right)} - y \]

6. Final simplification 67.4%

   \[\leadsto \left(-z\right) - y \]
7. Add Preprocessing

Alternative 6: 33.3% accurate, 53.5× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- y))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return -y

Julia

function code(x, y, z)
	return Float64(-y)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := (-y)

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf 39.2%

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

   1. neg-mul-1 39.2%

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

5. Simplified 39.2%

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

6. Final simplification 39.2%

   \[\leadsto -y \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))