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

Percentage Accurate: 99.9% → 99.9%

Time: 6.5s

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: 99.9% 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: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+50} \lor \neg \left(x \leq 3.8 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.2e+50) (not (<= x 3.8e-17)))
   (fma (log y) x (- y))
   (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.2e+50) || !(x <= 3.8e-17)) {
		tmp = fma(log(y), x, -y);
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.2e+50) || !(x <= 3.8e-17))
		tmp = fma(log(y), x, Float64(-y));
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.2e+50], N[Not[LessEqual[x, 3.8e-17]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.19999999999999983e50 or 3.8000000000000001e-17 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. *-lft-identity N/A

         \[\leadsto x \cdot \log y - \color{blue}{1 \cdot y} \]

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{x \cdot \log y + -1 \cdot y} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} + -1 \cdot y \]

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if -3.19999999999999983e50 < x < 3.8000000000000001e-17

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+50} \lor \neg \left(x \leq 3.8 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+123} \lor \neg \left(x \leq 7.5 \cdot 10^{+179}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.95e+123) (not (<= x 7.5e+179))) (* (log y) x) (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.95e+123) || !(x <= 7.5e+179)) {
		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 <= (-1.95d+123)) .or. (.not. (x <= 7.5d+179))) 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 <= -1.95e+123) || !(x <= 7.5e+179)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.95e+123) or not (x <= 7.5e+179):
		tmp = math.log(y) * x
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.95e+123) || !(x <= 7.5e+179))
		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 <= -1.95e+123) || ~((x <= 7.5e+179)))
		tmp = log(y) * x;
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.95e+123], N[Not[LessEqual[x, 7.5e+179]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.94999999999999996e123 or 7.50000000000000007e179 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 78.7

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

   8. Applied rewrites 78.7%

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

   if -1.94999999999999996e123 < x < 7.50000000000000007e179

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 83.8

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

   5. Applied rewrites 83.8%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+123} \lor \neg \left(x \leq 7.5 \cdot 10^{+179}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

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 Float64(fma(log(y), x, Float64(-z)) - y)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[Log[y], $MachinePrecision] * x + (-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 x around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 5: 66.2% accurate, 18.7× 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

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 66.8

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

5. Applied rewrites 66.8%

   \[\leadsto \color{blue}{\left(-z\right)} - y \]
6. Add Preprocessing

Alternative 6: 33.8% accurate, 37.3× 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

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

   2. lower-neg.f64 32.2

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

5. Applied rewrites 32.2%

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

Reproduce

herbie shell --seed 2024338 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))