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

Percentage Accurate: 99.9% → 99.9%

Time: 7.8s

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: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+25} \lor \neg \left(z \leq 5.1 \cdot 10^{+118}\right):\\ \;\;\;\;-\left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.9e+25) (not (<= z 5.1e+118)))
   (- (+ y z))
   (- (* x (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.9e+25) || !(z <= 5.1e+118)) {
		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 <= (-3.9d+25)) .or. (.not. (z <= 5.1d+118))) 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 <= -3.9e+25) || !(z <= 5.1e+118)) {
		tmp = -(y + z);
	} else {
		tmp = (x * Math.log(y)) - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.9e+25) or not (z <= 5.1e+118):
		tmp = -(y + z)
	else:
		tmp = (x * math.log(y)) - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.9e+25) || !(z <= 5.1e+118))
		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 <= -3.9e+25) || ~((z <= 5.1e+118)))
		tmp = -(y + z);
	else
		tmp = (x * log(y)) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.9e+25], N[Not[LessEqual[z, 5.1e+118]], $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 < -3.9000000000000002e25 or 5.10000000000000002e118 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 86.1%

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

      1. neg-mul-1 86.1%

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

   4. Simplified 86.1%

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

   if -3.9000000000000002e25 < z < 5.10000000000000002e118

   1. Initial program 99.8%

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

      1. add-cube-cbrt 98.9%

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

      2. pow3 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in z around 0 89.6%

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

      1. pow-base-1 89.6%

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

      2. *-lft-identity 89.6%

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

   6. Simplified 89.6%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+25} \lor \neg \left(z \leq 5.1 \cdot 10^{+118}\right):\\ \;\;\;\;-\left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \]

Alternative 3: 67.4% 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 99.9%

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

2. Taylor expanded in x around 0 68.1%

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

   1. neg-mul-1 68.1%

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

4. Simplified 68.1%

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

5. Final simplification 68.1%

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

Alternative 4: 33.3% 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. flip-- 53.6%

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

   2. clear-num 53.6%

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

   3. fma-def 53.6%

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

   4. pow2 53.6%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \log y, z\right)}{\color{blue}{{\left(x \cdot \log y\right)}^{2}} - z \cdot z}} - y \]

3. Applied egg-rr 53.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(x, \log y, z\right)}{{\left(x \cdot \log y\right)}^{2} - z \cdot z}}} - y \]

4. Taylor expanded in x around 0 68.0%

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

   1. add-sqr-sqrt 36.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{-1}{z}}} \cdot \sqrt{\frac{1}{\frac{-1}{z}}}} - y \]

   2. sqrt-unprod 37.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{-1}{z}} \cdot \frac{1}{\frac{-1}{z}}}} - y \]

   3. associate-/r/ 37.9%

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

   4. metadata-eval 37.9%

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

   5. associate-/r/ 37.9%

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

   6. metadata-eval 37.9%

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

   7. swap-sqr 37.9%

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

   8. metadata-eval 37.9%

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

   9. *-un-lft-identity 37.9%

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

   10. sqrt-unprod 14.8%

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

   11. add-sqr-sqrt 34.4%

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

   12. add-log-exp 24.8%

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

   13. *-un-lft-identity 24.8%

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

   14. log-prod 24.8%

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

   15. metadata-eval 24.8%

       \[\leadsto \left(\color{blue}{0} + \log \left(e^{z}\right)\right) - y \]

   16. add-log-exp 34.4%

       \[\leadsto \left(0 + \color{blue}{z}\right) - y \]

6. Applied egg-rr 34.4%

   \[\leadsto \color{blue}{\left(0 + z\right)} - y \]
7. Step-by-step derivation

   1. +-lft-identity 34.4%

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

8. Simplified 34.4%

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

9. Final simplification 34.4%

   \[\leadsto z - y \]

Reproduce

herbie shell --seed 2023185 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))