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

Percentage Accurate: 99.8% → 99.9%

Time: 9.1s

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: 99.8% 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

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

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 99.8%

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

   1. associate--l- 99.8%

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

   2. fma-neg 99.8%

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

   3. +-commutative 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+32} \lor \neg \left(x \leq 2.75 \cdot 10^{+38}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.2e+32) (not (<= x 2.75e+38)))
   (- (* x (log y)) y)
   (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.2e+32) || !(x <= 2.75e+38)) {
		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 <= (-1.2d+32)) .or. (.not. (x <= 2.75d+38))) 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 <= -1.2e+32) || !(x <= 2.75e+38)) {
		tmp = (x * Math.log(y)) - y;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.2e+32) or not (x <= 2.75e+38):
		tmp = (x * math.log(y)) - y
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.2e+32) || !(x <= 2.75e+38))
		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 <= -1.2e+32) || ~((x <= 2.75e+38)))
		tmp = (x * log(y)) - y;
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.2e+32], N[Not[LessEqual[x, 2.75e+38]], $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 < -1.19999999999999996e32 or 2.7500000000000002e38 < x

   1. Initial program 99.6%

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

   2. Taylor expanded in z around 0 84.3%

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

   if -1.19999999999999996e32 < x < 2.7500000000000002e38

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 90.6%

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

      1. neg-mul-1 90.6%

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

   4. Simplified 90.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+32} \lor \neg \left(x \leq 2.75 \cdot 10^{+38}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

Alternative 3: 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]

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 4: 65.5% 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. Taylor expanded in x around 0 63.1%

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

   1. neg-mul-1 63.1%

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

4. Simplified 63.1%

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

5. Final simplification 63.1%

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

Alternative 5: 33.0% 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 99.8%

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

2. Taylor expanded in y around inf 29.6%

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

   1. neg-mul-1 29.6%

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

4. Simplified 29.6%

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

5. Final simplification 29.6%

   \[\leadsto -y \]

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. add-cube-cbrt 99.2%

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

   5. associate-*l* 99.2%

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

   6. fma-def 99.2%

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

   7. pow2 99.2%

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

   8. add-sqr-sqrt 18.9%

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

   9. sqrt-unprod 38.2%

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

   10. sqr-neg 38.2%

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

   11. sqrt-unprod 30.2%

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

   12. add-sqr-sqrt 36.4%

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

3. Applied egg-rr 36.4%

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

4. Taylor expanded in y around inf 2.1%

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

5. Final simplification 2.1%

   \[\leadsto y \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))