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

Percentage Accurate: 99.9% → 99.9%

Time: 8.0s

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, 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 2: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+175} \lor \neg \left(x \leq 2.86 \cdot 10^{+91}\right):\\ \;\;\;\;y + x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.2e+175) (not (<= x 2.86e+91)))
   (+ y (* x (log y)))
   (- (+ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.2e+175) || !(x <= 2.86e+91)) {
		tmp = y + (x * log(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 <= (-4.2d+175)) .or. (.not. (x <= 2.86d+91))) then
        tmp = y + (x * log(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 <= -4.2e+175) || !(x <= 2.86e+91)) {
		tmp = y + (x * Math.log(y));
	} else {
		tmp = -(y + z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.2e+175) or not (x <= 2.86e+91):
		tmp = y + (x * math.log(y))
	else:
		tmp = -(y + z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.2e+175) || !(x <= 2.86e+91))
		tmp = Float64(y + Float64(x * log(y)));
	else
		tmp = Float64(-Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4.2e+175) || ~((x <= 2.86e+91)))
		tmp = y + (x * log(y));
	else
		tmp = -(y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.2e+175], N[Not[LessEqual[x, 2.86e+91]], $MachinePrecision]], N[(y + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(y + z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -4.1999999999999998e175 or 2.8600000000000001e91 < x

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. +-commutative 99.7%

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

      3. sub-neg 99.7%

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

      4. add-sqr-sqrt 41.6%

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

      5. associate-*r* 41.6%

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

      6. fma-def 41.6%

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

      7. add-sqr-sqrt 5.3%

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

      8. sqrt-unprod 17.9%

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

      9. sqr-neg 17.9%

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

      10. sqrt-unprod 22.6%

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

      11. add-sqr-sqrt 25.3%

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

   3. Applied egg-rr 25.3%

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

   4. Taylor expanded in z around 0 80.6%

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

   if -4.1999999999999998e175 < x < 2.8600000000000001e91

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 83.1%

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

      1. neg-mul-1 83.1%

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

   4. Simplified 83.1%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+175} \lor \neg \left(x \leq 2.86 \cdot 10^{+91}\right):\\ \;\;\;\;y + x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\left(y + z\right)\\ \end{array} \]

Alternative 3: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2400 \lor \neg \left(z \leq 1.7 \cdot 10^{+82}\right):\\ \;\;\;\;-\left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2400.0) (not (<= z 1.7e+82))) (- (+ y z)) (- (* x (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2400.0) || !(z <= 1.7e+82)) {
		tmp = -(y + z);
	} else {
		tmp = (x * log(y)) - 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 ((z <= (-2400.0d0)) .or. (.not. (z <= 1.7d+82))) then
        tmp = -(y + z)
    else
        tmp = (x * log(y)) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2400.0) || !(z <= 1.7e+82)) {
		tmp = -(y + z);
	} else {
		tmp = (x * Math.log(y)) - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2400.0) or not (z <= 1.7e+82):
		tmp = -(y + z)
	else:
		tmp = (x * math.log(y)) - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2400.0) || !(z <= 1.7e+82))
		tmp = Float64(-Float64(y + z));
	else
		tmp = Float64(Float64(x * log(y)) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2400.0) || ~((z <= 1.7e+82)))
		tmp = -(y + z);
	else
		tmp = (x * log(y)) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2400.0], N[Not[LessEqual[z, 1.7e+82]], $MachinePrecision]], (-N[(y + z), $MachinePrecision]), N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2400 or 1.69999999999999997e82 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 83.1%

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

      1. neg-mul-1 83.1%

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

   4. Simplified 83.1%

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

   if -2400 < z < 1.69999999999999997e82

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 91.9%

      \[\leadsto \color{blue}{\log y \cdot x - y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2400 \lor \neg \left(z \leq 1.7 \cdot 10^{+82}\right):\\ \;\;\;\;-\left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \]

Alternative 4: 66.3% accurate, 26.8× speedup

Math

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

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 99.8%

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

2. Taylor expanded in x around 0 65.0%

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

   1. neg-mul-1 65.0%

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

4. Simplified 65.0%

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

5. Final simplification 65.0%

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

Alternative 5: 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 99.8%

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

2. Taylor expanded in y around inf 32.7%

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

   1. neg-mul-1 32.7%

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

4. Simplified 32.7%

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

5. Final simplification 32.7%

   \[\leadsto -y \]

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-sqr-sqrt 46.0%

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

   5. associate-*r* 46.0%

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

   6. fma-def 46.0%

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

   7. add-sqr-sqrt 7.4%

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

   8. sqrt-unprod 6.9%

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

   9. sqr-neg 6.9%

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

   10. sqrt-unprod 7.1%

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

   11. add-sqr-sqrt 7.9%

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

3. Applied egg-rr 7.9%

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

4. Taylor expanded in y around inf 2.3%

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

5. Final simplification 2.3%

   \[\leadsto y \]

Alternative 7: 2.3% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 z)

C

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

Java

public static double code(double x, double y, double z) {
	return z;
}

Python

def code(x, y, z):
	return z

Julia

function code(x, y, z)
	return z
end

MATLAB

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

Wolfram

code[x_, y_, z_] := z

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-sqr-sqrt 46.0%

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

   5. associate-*r* 46.0%

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

   6. fma-def 46.0%

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

   7. add-sqr-sqrt 7.4%

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

   8. sqrt-unprod 6.9%

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

   9. sqr-neg 6.9%

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

   10. sqrt-unprod 7.1%

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

   11. add-sqr-sqrt 7.9%

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

3. Applied egg-rr 7.9%

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

4. Taylor expanded in z around inf 2.4%

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

5. Final simplification 2.4%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023268 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))