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

Percentage Accurate: 99.9% → 99.9%

Time: 6.1s

Alternatives: 5

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 99.9

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-8} \lor \neg \left(z \leq 7 \cdot 10^{-24}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -7e-8) (not (<= z 7e-24))) (- (- z) y) (fma (log y) x (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7e-8) || !(z <= 7e-24)) {
		tmp = -z - y;
	} else {
		tmp = fma(log(y), x, -y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -7e-8) || !(z <= 7e-24))
		tmp = Float64(Float64(-z) - y);
	else
		tmp = fma(log(y), x, Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -7e-8], N[Not[LessEqual[z, 7e-24]], $MachinePrecision]], N[((-z) - y), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.00000000000000048e-8 or 6.9999999999999993e-24 < z

   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 82.3

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

   5. Applied rewrites 82.3%

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

   if -7.00000000000000048e-8 < z < 6.9999999999999993e-24

   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 94.2

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

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, -y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-8} \lor \neg \left(z \leq 7 \cdot 10^{-24}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 65.9% 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 69.4

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

5. Applied rewrites 69.4%

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

Alternative 5: 33.2% 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 34.1

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

5. Applied rewrites 34.1%

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

Reproduce

herbie shell --seed 2024320 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))