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

Percentage Accurate: 99.8% → 99.9%

Time: 11.5s

Alternatives: 7

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: 99.8% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l- 99.8%

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

   2. *-commutative 99.8%

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

   3. fma-neg 99.9%

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

   4. +-commutative 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(\log y, x, \left(-z\right) - y\right) \]
6. Add Preprocessing

Alternative 2: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+28} \lor \neg \left(z \leq 3.7 \cdot 10^{-22}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.5e+28) (not (<= z 3.7e-22)))
   (- (- z) y)
   (- (* (log y) x) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.5e+28) || !(z <= 3.7e-22)) {
		tmp = -z - y;
	} else {
		tmp = (log(y) * x) - 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 <= (-2.5d+28)) .or. (.not. (z <= 3.7d-22))) then
        tmp = -z - y
    else
        tmp = (log(y) * x) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.5e+28) || !(z <= 3.7e-22)) {
		tmp = -z - y;
	} else {
		tmp = (Math.log(y) * x) - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.5e+28) or not (z <= 3.7e-22):
		tmp = -z - y
	else:
		tmp = (math.log(y) * x) - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.5e+28) || !(z <= 3.7e-22))
		tmp = Float64(Float64(-z) - y);
	else
		tmp = Float64(Float64(log(y) * x) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.5e+28) || ~((z <= 3.7e-22)))
		tmp = -z - y;
	else
		tmp = (log(y) * x) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.5e+28], N[Not[LessEqual[z, 3.7e-22]], $MachinePrecision]], N[((-z) - y), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.49999999999999979e28 or 3.7e-22 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 84.0%

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

      1. neg-mul-1 84.0%

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

   5. Simplified 84.0%

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

   if -2.49999999999999979e28 < z < 3.7e-22

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 92.7%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+28} \lor \neg \left(z \leq 3.7 \cdot 10^{-22}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+49} \lor \neg \left(x \leq 2.7 \cdot 10^{+92}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.7e+49) (not (<= x 2.7e+92))) (* (log y) x) (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.7e+49) || !(x <= 2.7e+92)) {
		tmp = log(y) * x;
	} 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 <= (-2.7d+49)) .or. (.not. (x <= 2.7d+92))) then
        tmp = log(y) * x
    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 <= -2.7e+49) || !(x <= 2.7e+92)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.7e+49) or not (x <= 2.7e+92):
		tmp = math.log(y) * x
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.7e+49) || !(x <= 2.7e+92))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.7e+49) || ~((x <= 2.7e+92)))
		tmp = log(y) * x;
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.7e+49], N[Not[LessEqual[x, 2.7e+92]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000001e49 or 2.6999999999999999e92 < x

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. *-commutative 99.6%

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

      3. fma-neg 99.7%

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

      4. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf 75.0%

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

   if -2.7000000000000001e49 < x < 2.6999999999999999e92

   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 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+49} \lor \neg \left(x \leq 2.7 \cdot 10^{+92}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 5: 51.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+27} \lor \neg \left(z \leq 1.45 \cdot 10^{-26}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.4e+27) (not (<= z 1.45e-26))) (- z) (- y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.4e+27) || !(z <= 1.45e-26)) {
		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 ((z <= (-6.4d+27)) .or. (.not. (z <= 1.45d-26))) 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 ((z <= -6.4e+27) || !(z <= 1.45e-26)) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.4e+27) or not (z <= 1.45e-26):
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.4e+27) || !(z <= 1.45e-26))
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.4e+27) || ~((z <= 1.45e-26)))
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.4e+27], N[Not[LessEqual[z, 1.45e-26]], $MachinePrecision]], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if z < -6.4000000000000003e27 or 1.4499999999999999e-26 < z

   1. Initial program 99.9%

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

      1. associate--l- 99.9%

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

      2. *-commutative 99.9%

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

      3. fma-neg 99.9%

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

      4. +-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf 62.6%

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

      1. neg-mul-1 62.6%

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

   7. Simplified 62.6%

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

   if -6.4000000000000003e27 < z < 1.4499999999999999e-26

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 41.8%

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

      1. neg-mul-1 41.8%

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

   5. Simplified 41.8%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+27} \lor \neg \left(z \leq 1.45 \cdot 10^{-26}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.3% 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 99.8%

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

3. Taylor expanded in x around 0 63.5%

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

   1. neg-mul-1 63.5%

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

5. Simplified 63.5%

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

6. Final simplification 63.5%

   \[\leadsto \left(-z\right) - y \]
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 99.8%

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

3. Taylor expanded in y around inf 32.3%

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

   1. neg-mul-1 32.3%

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

5. Simplified 32.3%

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

6. Final simplification 32.3%

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

Reproduce

herbie shell --seed 2024034 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))