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

Percentage Accurate: 99.9% → 99.9%

Time: 7.1s

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

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

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]

Derivation

1. Initial program 99.8%

   \[\left(x \cdot \log y - z\right) - y \]

2. Final simplification 99.8%

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

Alternative 2: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+185} \lor \neg \left(x \leq -2.8 \cdot 10^{+110}\right) \land \left(x \leq -5.5 \cdot 10^{+53} \lor \neg \left(x \leq 1.7 \cdot 10^{+68}\right)\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.7e+185)
         (and (not (<= x -2.8e+110))
              (or (<= x -5.5e+53) (not (<= x 1.7e+68)))))
   (* x (log y))
   (- (- y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.7e+185) || (!(x <= -2.8e+110) && ((x <= -5.5e+53) || !(x <= 1.7e+68)))) {
		tmp = x * log(y);
	} else {
		tmp = -y - z;
	}
	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 <= (-4.7d+185)) .or. (.not. (x <= (-2.8d+110))) .and. (x <= (-5.5d+53)) .or. (.not. (x <= 1.7d+68))) then
        tmp = x * log(y)
    else
        tmp = -y - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.7e+185) || (!(x <= -2.8e+110) && ((x <= -5.5e+53) || !(x <= 1.7e+68)))) {
		tmp = x * Math.log(y);
	} else {
		tmp = -y - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.7e+185) or (not (x <= -2.8e+110) and ((x <= -5.5e+53) or not (x <= 1.7e+68))):
		tmp = x * math.log(y)
	else:
		tmp = -y - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.7e+185) || (!(x <= -2.8e+110) && ((x <= -5.5e+53) || !(x <= 1.7e+68))))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4.7e+185) || (~((x <= -2.8e+110)) && ((x <= -5.5e+53) || ~((x <= 1.7e+68)))))
		tmp = x * log(y);
	else
		tmp = -y - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.7e+185], And[N[Not[LessEqual[x, -2.8e+110]], $MachinePrecision], Or[LessEqual[x, -5.5e+53], N[Not[LessEqual[x, 1.7e+68]], $MachinePrecision]]]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-y) - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.69999999999999972e185 or -2.79999999999999987e110 < x < -5.49999999999999975e53 or 1.70000000000000008e68 < x

   1. Initial program 99.7%

      \[\left(x \cdot \log y - z\right) - y \]

   2. Taylor expanded in z around 0 87.7%

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

   3. Taylor expanded in x around inf 77.5%

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

   if -4.69999999999999972e185 < x < -2.79999999999999987e110 or -5.49999999999999975e53 < x < 1.70000000000000008e68

   1. Initial program 99.9%

      \[\left(x \cdot \log y - z\right) - y \]

   2. Taylor expanded in x around 0 87.3%

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

      1. neg-mul-1 87.3%

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

   4. Simplified 87.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+185} \lor \neg \left(x \leq -2.8 \cdot 10^{+110}\right) \land \left(x \leq -5.5 \cdot 10^{+53} \lor \neg \left(x \leq 1.7 \cdot 10^{+68}\right)\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]

Alternative 3: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+48} \lor \neg \left(x \leq 1.92 \cdot 10^{+68}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.05e+48) (not (<= x 1.92e+68)))
   (- (* x (log y)) y)
   (- (- y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.05e+48) || !(x <= 1.92e+68)) {
		tmp = (x * log(y)) - y;
	} else {
		tmp = -y - z;
	}
	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 <= (-1.05d+48)) .or. (.not. (x <= 1.92d+68))) then
        tmp = (x * log(y)) - y
    else
        tmp = -y - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.05e+48) || !(x <= 1.92e+68)) {
		tmp = (x * Math.log(y)) - y;
	} else {
		tmp = -y - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.05e+48) or not (x <= 1.92e+68):
		tmp = (x * math.log(y)) - y
	else:
		tmp = -y - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.05e+48) || !(x <= 1.92e+68))
		tmp = Float64(Float64(x * log(y)) - y);
	else
		tmp = Float64(Float64(-y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.05e+48) || ~((x <= 1.92e+68)))
		tmp = (x * log(y)) - y;
	else
		tmp = -y - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.05e+48], N[Not[LessEqual[x, 1.92e+68]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], N[((-y) - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0499999999999999e48 or 1.92000000000000001e68 < x

   1. Initial program 99.7%

      \[\left(x \cdot \log y - z\right) - y \]

   2. Taylor expanded in z around 0 84.4%

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

   if -1.0499999999999999e48 < x < 1.92000000000000001e68

   1. Initial program 100.0%

      \[\left(x \cdot \log y - z\right) - y \]

   2. Taylor expanded in x around 0 88.0%

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

      1. neg-mul-1 88.0%

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

   4. Simplified 88.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+48} \lor \neg \left(x \leq 1.92 \cdot 10^{+68}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]

Alternative 4: 66.0% accurate, 26.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return -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 = -y - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(x \cdot \log y - z\right) - y \]

2. Taylor expanded in x around 0 64.9%

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

   1. neg-mul-1 64.9%

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

4. Simplified 64.9%

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

5. Final simplification 64.9%

   \[\leadsto \left(-y\right) - z \]

Alternative 5: 33.5% 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 99.8%

   \[\left(x \cdot \log y - z\right) - y \]

2. Taylor expanded in y around inf 30.5%

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

   1. neg-mul-1 30.5%

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

4. Simplified 30.5%

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

5. Final simplification 30.5%

   \[\leadsto -y \]

Alternative 6: 2.3% accurate, 107.0× 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 y
end

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 99.8%

   \[\left(x \cdot \log y - z\right) - y \]
2. Step-by-step derivation

   1. associate--l- 99.8%

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

   2. +-commutative 99.8%

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

   3. sub-neg 99.8%

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

   4. *-commutative 99.8%

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

   5. add-sqr-sqrt 50.3%

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

   6. associate-*r* 50.3%

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

   7. fma-def 50.3%

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

   8. add-sqr-sqrt 11.2%

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

   9. sqrt-unprod 14.7%

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

   10. sqr-neg 14.7%

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

   11. sqrt-unprod 11.8%

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

   12. add-sqr-sqrt 13.6%

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

3. Applied egg-rr 13.6%

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

4. Taylor expanded in y around inf 2.6%

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

5. Final simplification 2.6%

   \[\leadsto y \]

Reproduce

herbie shell --seed 2023290 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))