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

Percentage Accurate: 99.9% → 99.9%

Time: 8.9s

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

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

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. Add Preprocessing

3. Final simplification 99.8%

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

Alternative 2: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+14} \lor \neg \left(x \leq 6.8 \cdot 10^{+80}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9.2e+14) (not (<= x 6.8e+80)))
   (- (* x (log y)) y)
   (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.2e+14) || !(x <= 6.8e+80)) {
		tmp = (x * log(y)) - y;
	} 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 <= (-9.2d+14)) .or. (.not. (x <= 6.8d+80))) then
        tmp = (x * log(y)) - y
    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 <= -9.2e+14) || !(x <= 6.8e+80)) {
		tmp = (x * Math.log(y)) - y;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -9.2e+14) or not (x <= 6.8e+80):
		tmp = (x * math.log(y)) - y
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9.2e+14) || !(x <= 6.8e+80))
		tmp = Float64(Float64(x * log(y)) - y);
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -9.2e+14) || ~((x <= 6.8e+80)))
		tmp = (x * log(y)) - y;
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9.2e+14], N[Not[LessEqual[x, 6.8e+80]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.2e14 or 6.79999999999999984e80 < x

   1. Initial program 99.7%

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

      1. add-cube-cbrt 98.7%

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

      2. associate-*l* 98.8%

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

      3. fma-neg 98.8%

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

      4. pow2 98.8%

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in x around inf 85.1%

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

   if -9.2e14 < x < 6.79999999999999984e80

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 89.5%

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

      1. neg-mul-1 89.5%

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

   5. Simplified 89.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+14} \lor \neg \left(x \leq 6.8 \cdot 10^{+80}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.6% 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 99.8%

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

3. Taylor expanded in x around 0 63.7%

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

   1. neg-mul-1 63.7%

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

5. Simplified 63.7%

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

6. Final simplification 63.7%

   \[\leadsto \left(-z\right) - y \]
7. Add Preprocessing

Alternative 4: 33.6% 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. Add Preprocessing

3. Taylor expanded in z around inf 90.7%

   \[\leadsto \color{blue}{z \cdot \left(\frac{x \cdot \log y}{z} - 1\right)} - y \]
4. Step-by-step derivation

   1. sub-neg 90.7%

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

   2. metadata-eval 90.7%

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

   3. distribute-rgt-in 90.7%

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

   4. *-commutative 90.7%

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

   5. associate-/l* 90.7%

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

   6. neg-mul-1 90.7%

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

   7. add-sqr-sqrt 43.1%

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

   8. sqrt-unprod 62.2%

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

   9. sqr-neg 62.2%

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

   10. sqrt-unprod 32.1%

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

   11. add-sqr-sqrt 60.1%

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

5. Applied egg-rr 60.1%

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

6. Taylor expanded in y around inf 34.4%

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

   1. neg-mul-1 34.4%

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

8. Simplified 34.4%

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

9. Final simplification 34.4%

   \[\leadsto -y \]
10. Add Preprocessing

Alternative 5: 2.2% 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. Add Preprocessing

3. Taylor expanded in z around inf 90.7%

   \[\leadsto \color{blue}{z \cdot \left(\frac{x \cdot \log y}{z} - 1\right)} - y \]
4. Step-by-step derivation

   1. sub-neg 90.7%

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

   2. metadata-eval 90.7%

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

   3. distribute-rgt-in 90.7%

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

   4. *-commutative 90.7%

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

   5. associate-/l* 90.7%

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

   6. neg-mul-1 90.7%

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

   7. add-sqr-sqrt 43.1%

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

   8. sqrt-unprod 62.2%

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

   9. sqr-neg 62.2%

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

   10. sqrt-unprod 32.1%

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

   11. add-sqr-sqrt 60.1%

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

5. Applied egg-rr 60.1%

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

6. Taylor expanded in y around inf 59.8%

   \[\leadsto \color{blue}{y \cdot \left(\left(-1 \cdot \frac{x \cdot \log \left(\frac{1}{y}\right)}{y} + \frac{z}{y}\right) - 1\right)} \]
7. Step-by-step derivation

   1. sub-neg 59.8%

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

   2. metadata-eval 59.8%

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

   3. +-commutative 59.8%

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

   4. mul-1-neg 59.8%

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

   5. log-rec 59.8%

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

   6. mul-1-neg 59.8%

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

   7. associate-/l* 59.8%

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

   8. distribute-rgt-neg-in 59.8%

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

   9. mul-1-neg 59.8%

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

   10. distribute-frac-neg 59.8%

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

   11. remove-double-neg 59.8%

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

8. Simplified 59.8%

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

9. Taylor expanded in z around inf 2.3%

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

10. Final simplification 2.3%

    \[\leadsto z \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024054 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))