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

Percentage Accurate: 99.9% → 99.9%

Time: 11.7s

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: 99.9% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l- 99.9%

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

   2. fma-neg 99.9%

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

   3. +-commutative 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+73} \lor \neg \left(x \leq 3.4 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.6e+73) (not (<= x 3.4e-36)))
   (- (* x (log y)) y)
   (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.6e+73) || !(x <= 3.4e-36)) {
		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 <= (-7.6d+73)) .or. (.not. (x <= 3.4d-36))) 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 <= -7.6e+73) || !(x <= 3.4e-36)) {
		tmp = (x * Math.log(y)) - y;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -7.6e+73) or not (x <= 3.4e-36):
		tmp = (x * math.log(y)) - y
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7.6e+73) || !(x <= 3.4e-36))
		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 <= -7.6e+73) || ~((x <= 3.4e-36)))
		tmp = (x * log(y)) - y;
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7.6e+73], N[Not[LessEqual[x, 3.4e-36]], $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 < -7.60000000000000044e73 or 3.4000000000000003e-36 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0 83.9%

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

   if -7.60000000000000044e73 < x < 3.4000000000000003e-36

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 91.1%

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

      1. neg-mul-1 91.1%

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

   4. Simplified 91.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+73} \lor \neg \left(x \leq 3.4 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

Alternative 3: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+140} \lor \neg \left(x \leq 2.7 \cdot 10^{+136}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.5e+140) (not (<= x 2.7e+136))) (* x (log y)) (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.5e+140) || !(x <= 2.7e+136)) {
		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 <= (-3.5d+140)) .or. (.not. (x <= 2.7d+136))) 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 <= -3.5e+140) || !(x <= 2.7e+136)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.5e+140) or not (x <= 2.7e+136):
		tmp = x * math.log(y)
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.5e+140) || !(x <= 2.7e+136))
		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 <= -3.5e+140) || ~((x <= 2.7e+136)))
		tmp = x * log(y);
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.5e+140], N[Not[LessEqual[x, 2.7e+136]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.49999999999999989e140 or 2.7000000000000002e136 < x

   1. Initial program 99.6%

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

   2. Taylor expanded in z around 0 86.1%

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

      1. fma-neg 86.2%

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

   4. Simplified 86.2%

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

   5. Taylor expanded in x around inf 74.5%

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

   if -3.49999999999999989e140 < x < 2.7000000000000002e136

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 85.6%

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

      1. neg-mul-1 85.6%

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

   4. Simplified 85.6%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+140} \lor \neg \left(x \leq 2.7 \cdot 10^{+136}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

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. Final simplification 99.9%

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

Alternative 5: 67.2% 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. Taylor expanded in x around 0 69.3%

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

   1. neg-mul-1 69.3%

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

4. Simplified 69.3%

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

5. Final simplification 69.3%

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

Alternative 6: 34.0% 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. Taylor expanded in y around inf 33.5%

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

   1. neg-mul-1 33.5%

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

4. Simplified 33.5%

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

5. Final simplification 33.5%

   \[\leadsto -y \]

Alternative 7: 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. Step-by-step derivation

   1. associate--l- 99.9%

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

   2. +-commutative 99.9%

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

   3. sub-neg 99.9%

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

   4. *-commutative 99.9%

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

   5. add-cube-cbrt 99.4%

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

   6. associate-*r* 99.4%

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

   7. fma-def 99.4%

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

   8. pow2 99.4%

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

   9. add-sqr-sqrt 28.5%

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

   10. sqrt-unprod 37.0%

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

   11. sqr-neg 37.0%

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

   12. sqrt-unprod 23.1%

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

   13. add-sqr-sqrt 29.5%

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

3. Applied egg-rr 29.5%

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

4. Taylor expanded in y around inf 2.5%

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

5. Final simplification 2.5%

   \[\leadsto y \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))