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

Percentage Accurate: 99.9% → 99.9%

Time: 8.2s

Alternatives: 4

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 99.9%

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

Alternative 2: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+44} \lor \neg \left(x \leq 2.3 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.2e+44) (not (<= x 2.3e-36)))
   (- (* x (log y)) y)
   (- (- y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.2e+44) || !(x <= 2.3e-36)) {
		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 <= (-3.2d+44)) .or. (.not. (x <= 2.3d-36))) 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 <= -3.2e+44) || !(x <= 2.3e-36)) {
		tmp = (x * Math.log(y)) - y;
	} else {
		tmp = -y - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.2e+44) or not (x <= 2.3e-36):
		tmp = (x * math.log(y)) - y
	else:
		tmp = -y - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.2e+44) || !(x <= 2.3e-36))
		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 <= -3.2e+44) || ~((x <= 2.3e-36)))
		tmp = (x * log(y)) - y;
	else
		tmp = -y - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.2e+44], N[Not[LessEqual[x, 2.3e-36]], $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 < -3.20000000000000004e44 or 2.29999999999999996e-36 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. add-cube-cbrt 98.9%

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}\right) \cdot \sqrt[3]{\log y}\right)} \cdot x - z\right) - y \]

      3. associate-*l* 98.9%

         \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}\right) \cdot \left(\sqrt[3]{\log y} \cdot x\right)} - z\right) - y \]

      4. fma-neg 98.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}, \sqrt[3]{\log y} \cdot x, -z\right)} - y \]

      5. pow2 98.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\log y}\right)}^{2}}, \sqrt[3]{\log y} \cdot x, -z\right) - y \]

   4. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -z\right)} - y \]

   5. Taylor expanded in x around inf 76.2%

      \[\leadsto \color{blue}{x \cdot \log y} - y \]
   6. Step-by-step derivation

      1. *-commutative 76.2%

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

   7. Simplified 76.2%

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

   if -3.20000000000000004e44 < x < 2.29999999999999996e-36

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 93.1%

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

      1. neg-mul-1 93.1%

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

   5. Simplified 93.1%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+44} \lor \neg \left(x \leq 2.3 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.4% 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 99.9%

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

3. Taylor expanded in x around 0 67.6%

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

   1. neg-mul-1 67.6%

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

5. Simplified 67.6%

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

6. Final simplification 67.6%

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

Alternative 4: 33.6% 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 z around inf 92.2%

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

   1. sub-neg 92.2%

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

   2. distribute-rgt-in 92.2%

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

   3. *-commutative 92.2%

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

   4. associate-/l* 92.2%

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

   5. metadata-eval 92.2%

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

   6. neg-mul-1 92.2%

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

   7. add-sqr-sqrt 50.0%

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

   8. sqrt-unprod 61.1%

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

   9. sqr-neg 61.1%

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

   10. sqrt-unprod 27.9%

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

   11. add-sqr-sqrt 55.7%

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

5. Applied egg-rr 55.7%

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

6. Taylor expanded in y around inf 33.0%

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

   1. neg-mul-1 33.0%

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

8. Simplified 33.0%

   \[\leadsto \color{blue}{-y} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))