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

Percentage Accurate: 99.9% → 99.8%

Time: 12.9s

Alternatives: 8

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (- (* (log (/ 1.0 y)) (- x)) z) y))

C

double code(double x, double y, double z) {
	return ((log((1.0 / y)) * -x) - 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 = ((log((1.0d0 / y)) * -x) - z) - y
end function

Java

public static double code(double x, double y, double z) {
	return ((Math.log((1.0 / y)) * -x) - z) - y;
}

Python

def code(x, y, z):
	return ((math.log((1.0 / y)) * -x) - z) - y

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \left(\log \left(\frac{1}{y}\right) \cdot \left(-x\right) - z\right) - y \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+37} \lor \neg \left(z \leq 1.95 \cdot 10^{+90}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.5e+37) (not (<= z 1.95e+90)))
   (- (- z) y)
   (- (* x (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.5e+37) || !(z <= 1.95e+90)) {
		tmp = -z - y;
	} else {
		tmp = (x * log(y)) - 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 ((z <= (-1.5d+37)) .or. (.not. (z <= 1.95d+90))) then
        tmp = -z - y
    else
        tmp = (x * log(y)) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.5e+37) || !(z <= 1.95e+90)) {
		tmp = -z - y;
	} else {
		tmp = (x * Math.log(y)) - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.5e+37) or not (z <= 1.95e+90):
		tmp = -z - y
	else:
		tmp = (x * math.log(y)) - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.5e+37) || !(z <= 1.95e+90))
		tmp = Float64(Float64(-z) - y);
	else
		tmp = Float64(Float64(x * log(y)) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.5e+37) || ~((z <= 1.95e+90)))
		tmp = -z - y;
	else
		tmp = (x * log(y)) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.5e+37], N[Not[LessEqual[z, 1.95e+90]], $MachinePrecision]], N[((-z) - y), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.50000000000000011e37 or 1.9500000000000001e90 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 90.2%

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

      1. neg-mul-1 90.2%

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

   5. Simplified 90.2%

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

   if -1.50000000000000011e37 < z < 1.9500000000000001e90

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 91.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+37} \lor \neg \left(z \leq 1.95 \cdot 10^{+90}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+139} \lor \neg \left(x \leq 1.9 \cdot 10^{+180}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.15e+139) (not (<= x 1.9e+180))) (* x (log y)) (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.15e+139) || !(x <= 1.9e+180)) {
		tmp = x * log(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 <= (-1.15d+139)) .or. (.not. (x <= 1.9d+180))) then
        tmp = x * log(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 <= -1.15e+139) || !(x <= 1.9e+180)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.15e+139) or not (x <= 1.9e+180):
		tmp = x * math.log(y)
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.15e+139) || !(x <= 1.9e+180))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.15e+139) || ~((x <= 1.9e+180)))
		tmp = x * log(y);
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.15e+139], N[Not[LessEqual[x, 1.9e+180]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e139 or 1.9e180 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 99.8%

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

   4. Taylor expanded in x around inf 94.0%

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

   5. Taylor expanded in x around inf 83.2%

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

      1. *-commutative 83.2%

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

      2. associate-*r* 83.2%

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

      3. neg-mul-1 83.2%

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

      4. log-rec 83.2%

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

      5. remove-double-neg 83.2%

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

   7. Simplified 83.2%

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

   if -1.15e139 < x < 1.9e180

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 84.3%

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

      1. neg-mul-1 84.3%

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

   5. Simplified 84.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+139} \lor \neg \left(x \leq 1.9 \cdot 10^{+180}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 5: 51.9% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+70}:\\ \;\;\;\;y - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 2.5e+70) (- y z) (- y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5e+70) {
		tmp = y - 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 <= 2.5d+70) then
        tmp = y - 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 <= 2.5e+70) {
		tmp = y - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5e+70:
		tmp = y - z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5e+70)
		tmp = Float64(y - z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.5e+70)
		tmp = y - z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5e+70], N[(y - z), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 2.5000000000000001e70

   1. Initial program 99.9%

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

      1. flip-- 49.2%

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

      2. div-inv 49.1%

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

   4. Applied egg-rr 38.1%

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

   5. Taylor expanded in x around 0 50.0%

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

   if 2.5000000000000001e70 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 68.5%

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

      1. mul-1-neg 68.5%

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

   5. Simplified 68.5%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+70}:\\ \;\;\;\;y - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.8% 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 68.7%

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

   1. neg-mul-1 68.7%

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

5. Simplified 68.7%

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

6. Final simplification 68.7%

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

Alternative 7: 34.1% 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 33.5%

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

   1. mul-1-neg 33.5%

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

5. Simplified 33.5%

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

6. Final simplification 33.5%

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

Alternative 8: 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 99.9%

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

   1. flip-- 42.6%

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

   2. div-inv 42.5%

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

4. Applied egg-rr 25.9%

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

5. Taylor expanded in y around inf 2.2%

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

6. Final simplification 2.2%

   \[\leadsto y \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))