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

Percentage Accurate: 99.8% → 99.8%

Time: 9.0s

Alternatives: 8

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: 99.8% 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: 99.8% 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. Add Preprocessing

Alternative 2: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+54} \lor \neg \left(z \leq 1.15 \cdot 10^{+32}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.8e+54) (not (<= z 1.15e+32)))
   (- (- z) y)
   (- (* x (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.8e+54) || !(z <= 1.15e+32)) {
		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 <= (-5.8d+54)) .or. (.not. (z <= 1.15d+32))) 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 <= -5.8e+54) || !(z <= 1.15e+32)) {
		tmp = -z - y;
	} else {
		tmp = (x * Math.log(y)) - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5.8e+54) or not (z <= 1.15e+32):
		tmp = -z - y
	else:
		tmp = (x * math.log(y)) - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.8e+54) || !(z <= 1.15e+32))
		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 <= -5.8e+54) || ~((z <= 1.15e+32)))
		tmp = -z - y;
	else
		tmp = (x * log(y)) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.8e+54], N[Not[LessEqual[z, 1.15e+32]], $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 < -5.7999999999999997e54 or 1.15e32 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 86.5%

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

      1. neg-mul-1 86.5%

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

   5. Simplified 86.5%

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

   if -5.7999999999999997e54 < z < 1.15e32

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 90.7%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+54} \lor \neg \left(z \leq 1.15 \cdot 10^{+32}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+197} \lor \neg \left(x \leq 6.8 \cdot 10^{+108}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.9e+197) (not (<= x 6.8e+108))) (* x (log y)) (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.9e+197) || !(x <= 6.8e+108)) {
		tmp = x * log(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 <= (-2.9d+197)) .or. (.not. (x <= 6.8d+108))) then
        tmp = x * log(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 <= -2.9e+197) || !(x <= 6.8e+108)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.9e+197) or not (x <= 6.8e+108):
		tmp = x * math.log(y)
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.9e+197) || !(x <= 6.8e+108))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.9e+197) || ~((x <= 6.8e+108)))
		tmp = x * log(y);
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.9e+197], N[Not[LessEqual[x, 6.8e+108]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.90000000000000002e197 or 6.79999999999999992e108 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate--l+ 99.7%

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

      3. *-commutative 99.7%

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

      4. add-cube-cbrt 98.5%

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

      5. associate-*l* 98.5%

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

      6. fma-define 98.5%

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

      7. pow2 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in x around inf 74.7%

      \[\leadsto \color{blue}{x \cdot \log y} \]
   6. Step-by-step derivation

      1. *-commutative 74.7%

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

   7. Simplified 74.7%

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

   if -2.90000000000000002e197 < x < 6.79999999999999992e108

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 82.4%

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

      1. neg-mul-1 82.4%

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

   5. Simplified 82.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+197} \lor \neg \left(x \leq 6.8 \cdot 10^{+108}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3350000:\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3350000.0) (- (* x (log y)) z) (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3350000.0) {
		tmp = (x * log(y)) - z;
	} 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 (y <= 3350000.0d0) then
        tmp = (x * log(y)) - z
    else
        tmp = -z - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3350000.0) {
		tmp = (x * Math.log(y)) - z;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3350000.0:
		tmp = (x * math.log(y)) - z
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3350000.0)
		tmp = Float64(Float64(x * log(y)) - z);
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3350000.0)
		tmp = (x * log(y)) - z;
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3350000.0], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.35e6

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 93.5%

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

   if 3.35e6 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 82.8%

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

      1. neg-mul-1 82.8%

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

   5. Simplified 82.8%

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

Alternative 5: 52.6% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.3 \cdot 10^{-11}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 5.3e-11) (- z) (- y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.3e-11) {
		tmp = -z;
	} else {
		tmp = -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 (y <= 5.3d-11) then
        tmp = -z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.3e-11) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.3e-11:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.3e-11)
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.3e-11)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.3e-11], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 5.2999999999999998e-11

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate--l+ 99.8%

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

      3. *-commutative 99.8%

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

      4. add-cube-cbrt 99.2%

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

      5. associate-*l* 99.2%

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

      6. fma-define 99.3%

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

      7. pow2 99.3%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in z around inf 53.0%

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

      1. mul-1-neg 53.0%

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

   7. Simplified 53.0%

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

   if 5.2999999999999998e-11 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 62.5%

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

      1. neg-mul-1 62.5%

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

   5. Simplified 62.5%

      \[\leadsto \color{blue}{-y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 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 68.8%

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

   1. neg-mul-1 68.8%

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

5. Simplified 68.8%

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

Alternative 7: 34.3% 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 y around inf 34.6%

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

   1. neg-mul-1 34.6%

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

5. Simplified 34.6%

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

Alternative 8: 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. Step-by-step derivation

   1. sub-neg 99.8%

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

   2. associate--l+ 99.8%

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

   3. *-commutative 99.8%

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

   4. add-cube-cbrt 99.4%

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

   5. associate-*l* 99.4%

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

   6. fma-define 99.4%

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

   7. pow2 99.4%

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

4. Applied egg-rr 99.4%

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

   1. add-sqr-sqrt 32.8%

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

   2. pow2 32.8%

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

   3. fma-undefine 32.8%

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

   4. +-commutative 32.8%

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

   5. add-sqr-sqrt 25.5%

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

   6. sqrt-unprod 20.0%

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

   7. sqr-neg 20.0%

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

   8. sqrt-unprod 7.4%

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

   9. add-sqr-sqrt 14.6%

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

   10. associate-*r* 14.6%

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

   11. unpow2 14.6%

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

   12. add-cube-cbrt 14.7%

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

6. Applied egg-rr 14.7%

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

7. Taylor expanded in z around inf 2.1%

   \[\leadsto \color{blue}{z} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024091 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))