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

Percentage Accurate: 99.9% → 99.9%

Time: 11.7s

Alternatives: 4

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. Final simplification 99.9%

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

Alternative 2: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{+162} \lor \neg \left(z \leq 2.9 \cdot 10^{+95}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.52e+162) (not (<= z 2.9e+95)))
   (- (- z) y)
   (- (* x (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.52e+162) || !(z <= 2.9e+95)) {
		tmp = -z - y;
	} 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 <= (-1.52d+162)) .or. (.not. (z <= 2.9d+95))) then
        tmp = -z - y
    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 <= -1.52e+162) || !(z <= 2.9e+95)) {
		tmp = -z - y;
	} else {
		tmp = (x * Math.log(y)) - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.52e+162) or not (z <= 2.9e+95):
		tmp = -z - y
	else:
		tmp = (x * math.log(y)) - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.52e+162) || !(z <= 2.9e+95))
		tmp = Float64(Float64(-z) - y);
	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 <= -1.52e+162) || ~((z <= 2.9e+95)))
		tmp = -z - y;
	else
		tmp = (x * log(y)) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.52e+162], N[Not[LessEqual[z, 2.9e+95]], $MachinePrecision]], N[((-z) - y), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5200000000000001e162 or 2.90000000000000013e95 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 91.8%

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

      1. neg-mul-1 91.8%

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

   4. Simplified 91.8%

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

   if -1.5200000000000001e162 < z < 2.90000000000000013e95

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. add-cube-cbrt 99.2%

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

      3. associate-*r* 99.2%

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

      4. fma-neg 99.2%

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

      5. pow2 99.2%

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

   3. Applied egg-rr 99.2%

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

   4. Taylor expanded in z around 0 86.4%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(x \cdot \log y\right)} - y \]
   5. Step-by-step derivation

      1. pow-base-1 86.4%

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

      2. associate-*r* 86.4%

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

      3. *-lft-identity 86.4%

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

   6. Simplified 86.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{+162} \lor \neg \left(z \leq 2.9 \cdot 10^{+95}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \]

Alternative 3: 66.0% 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 66.2%

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

   1. neg-mul-1 66.2%

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

4. Simplified 66.2%

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

5. Final simplification 66.2%

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

Alternative 4: 33.4% accurate, 35.7× speedup

Math

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

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(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. Step-by-step derivation

   1. add-cube-cbrt 98.7%

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

   2. pow3 98.7%

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

3. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \log y - z}\right)}^{3}} - y \]
4. Step-by-step derivation

   1. rem-cube-cbrt 99.9%

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

   2. sub-neg 99.9%

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

   3. *-commutative 99.9%

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

   4. add-cube-cbrt 99.4%

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

   5. unpow2 99.4%

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

   6. associate-*r* 99.4%

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

   7. fma-udef 99.4%

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

   8. *-commutative 99.4%

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

   9. unpow2 99.4%

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

   10. cbrt-unprod 75.8%

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

   11. pow2 75.8%

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

   12. add-sqr-sqrt 37.2%

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

   13. sqrt-unprod 45.7%

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

   14. sqr-neg 45.7%

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

   15. sqrt-unprod 21.2%

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

   16. add-sqr-sqrt 43.9%

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

5. Applied egg-rr 43.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{{x}^{2}} \cdot \log y, \sqrt[3]{x}, z\right)} - y \]
6. Step-by-step derivation

   1. *-commutative 43.9%

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

7. Simplified 43.9%

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

8. Taylor expanded in x around 0 32.7%

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

9. Final simplification 32.7%

   \[\leadsto z - y \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))