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

Percentage Accurate: 99.9% → 99.9%

Time: 8.8s

Alternatives: 11

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, 0.5× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (fma (log y) x (- (- z) y)))

C

double code(double x, double y, double z) {
	return fma(log(y), x, (-z - y));
}

Julia

function code(x, y, z)
	return fma(log(y), x, Float64(Float64(-z) - y))
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   2. associate--l+ 99.9%

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

   3. *-commutative 99.9%

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

   4. fma-def 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(\log y, x, \left(-z\right) - y\right) \]

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(-z\right) - x \cdot \log \left(\frac{1}{y}\right)\right) - y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (- (- z) (* x (log (/ 1.0 y)))) y))

C

double code(double x, double y, double z) {
	return (-z - (x * log((1.0 / y)))) - 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 - (x * log((1.0d0 / y)))) - y
end function

Java

public static double code(double x, double y, double z) {
	return (-z - (x * Math.log((1.0 / y)))) - y;
}

Python

def code(x, y, z):
	return (-z - (x * math.log((1.0 / y)))) - y

Julia

function code(x, y, z)
	return Float64(Float64(Float64(-z) - Float64(x * log(Float64(1.0 / y)))) - y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (-z - (x * log((1.0 / y)))) - y;
end

Wolfram

code[x_, y_, z_] := N[(N[((-z) - N[(x * N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around inf 99.9%

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

3. Final simplification 99.9%

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

Alternative 3: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{+154} \lor \neg \left(x \leq 5.2 \cdot 10^{+121}\right):\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9.8e+154) (not (<= x 5.2e+121)))
   (- (* (log y) x) y)
   (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.8e+154) || !(x <= 5.2e+121)) {
		tmp = (log(y) * x) - 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 <= (-9.8d+154)) .or. (.not. (x <= 5.2d+121))) then
        tmp = (log(y) * x) - 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 <= -9.8e+154) || !(x <= 5.2e+121)) {
		tmp = (Math.log(y) * x) - y;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -9.8e+154) or not (x <= 5.2e+121):
		tmp = (math.log(y) * x) - y
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9.8e+154) || !(x <= 5.2e+121))
		tmp = Float64(Float64(log(y) * x) - y);
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -9.8e+154) || ~((x <= 5.2e+121)))
		tmp = (log(y) * x) - y;
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9.8e+154], N[Not[LessEqual[x, 5.2e+121]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.8000000000000003e154 or 5.1999999999999998e121 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 90.7%

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

   if -9.8000000000000003e154 < x < 5.1999999999999998e121

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 87.0%

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

      1. neg-mul-1 87.0%

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

   4. Simplified 87.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{+154} \lor \neg \left(x \leq 5.2 \cdot 10^{+121}\right):\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

Alternative 4: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+206} \lor \neg \left(x \leq 2.2 \cdot 10^{+123}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.15e+206) (not (<= x 2.2e+123))) (* (log y) x) (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.15e+206) || !(x <= 2.2e+123)) {
		tmp = log(y) * x;
	} 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 <= (-1.15d+206)) .or. (.not. (x <= 2.2d+123))) then
        tmp = log(y) * x
    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 <= -1.15e+206) || !(x <= 2.2e+123)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.15e+206) or not (x <= 2.2e+123):
		tmp = math.log(y) * x
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.15e+206) || !(x <= 2.2e+123))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.15e+206) || ~((x <= 2.2e+123)))
		tmp = log(y) * x;
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.15e+206], N[Not[LessEqual[x, 2.2e+123]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15000000000000008e206 or 2.19999999999999992e123 < x

   1. Initial program 99.7%

      \[\left(x \cdot \log y - z\right) - y \]
   2. 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. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around inf 78.0%

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

   if -1.15000000000000008e206 < x < 2.19999999999999992e123

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 86.0%

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

      1. neg-mul-1 86.0%

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

   4. Simplified 86.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+206} \lor \neg \left(x \leq 2.2 \cdot 10^{+123}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

Alternative 5: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+80}:\\ \;\;\;\;\log y \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.3e+80) (- (* (log y) x) z) (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.3e+80) {
		tmp = (log(y) * x) - 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 <= 1.3d+80) then
        tmp = (log(y) * x) - 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 <= 1.3e+80) {
		tmp = (Math.log(y) * x) - z;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.3e+80:
		tmp = (math.log(y) * x) - z
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.3e+80)
		tmp = Float64(Float64(log(y) * x) - z);
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.3e+80)
		tmp = (log(y) * x) - z;
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.3e+80], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.29999999999999991e80

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. associate--l+ 99.9%

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

      3. *-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 87.6%

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

   if 1.29999999999999991e80 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 88.1%

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

      1. neg-mul-1 88.1%

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

   4. Simplified 88.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+80}:\\ \;\;\;\;\log y \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (- (* (log y) x) z) y))

C

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

Java

public static double code(double x, double y, double z) {
	return ((Math.log(y) * x) - z) - y;
}

Python

def code(x, y, z):
	return ((math.log(y) * x) - z) - y

Julia

function code(x, y, z)
	return Float64(Float64(Float64(log(y) * x) - z) - y)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 7: 51.0% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{-29}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-39}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-11} \lor \neg \left(z \leq 3.4 \cdot 10^{+43}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;z - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.16e-29)
   (- z)
   (if (<= z 1.8e-39)
     (- y)
     (if (or (<= z 2.2e-11) (not (<= z 3.4e+43))) (- z) (- z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.16e-29) {
		tmp = -z;
	} else if (z <= 1.8e-39) {
		tmp = -y;
	} else if ((z <= 2.2e-11) || !(z <= 3.4e+43)) {
		tmp = -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 (z <= (-1.16d-29)) then
        tmp = -z
    else if (z <= 1.8d-39) then
        tmp = -y
    else if ((z <= 2.2d-11) .or. (.not. (z <= 3.4d+43))) then
        tmp = -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 (z <= -1.16e-29) {
		tmp = -z;
	} else if (z <= 1.8e-39) {
		tmp = -y;
	} else if ((z <= 2.2e-11) || !(z <= 3.4e+43)) {
		tmp = -z;
	} else {
		tmp = z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.16e-29:
		tmp = -z
	elif z <= 1.8e-39:
		tmp = -y
	elif (z <= 2.2e-11) or not (z <= 3.4e+43):
		tmp = -z
	else:
		tmp = z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.16e-29)
		tmp = Float64(-z);
	elseif (z <= 1.8e-39)
		tmp = Float64(-y);
	elseif ((z <= 2.2e-11) || !(z <= 3.4e+43))
		tmp = Float64(-z);
	else
		tmp = Float64(z - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.16e-29)
		tmp = -z;
	elseif (z <= 1.8e-39)
		tmp = -y;
	elseif ((z <= 2.2e-11) || ~((z <= 3.4e+43)))
		tmp = -z;
	else
		tmp = z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.16e-29], (-z), If[LessEqual[z, 1.8e-39], (-y), If[Or[LessEqual[z, 2.2e-11], N[Not[LessEqual[z, 3.4e+43]], $MachinePrecision]], (-z), N[(z - y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.15999999999999996e-29 or 1.8e-39 < z < 2.2000000000000002e-11 or 3.40000000000000012e43 < z

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. associate--l+ 99.9%

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

      3. *-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in z around inf 65.9%

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

      1. mul-1-neg 65.9%

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

   6. Simplified 65.9%

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

   if -1.15999999999999996e-29 < z < 1.8e-39

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 56.1%

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

      1. neg-mul-1 56.1%

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

   4. Simplified 56.1%

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

   if 2.2000000000000002e-11 < z < 3.40000000000000012e43

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. associate--l+ 99.9%

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

      3. *-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(-z\right) - y\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.9%

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

      2. +-commutative 99.9%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 90.9%

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

      5. sqr-neg 90.9%

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

      6. sqrt-unprod 90.9%

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

      7. add-sqr-sqrt 90.9%

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

   5. Applied egg-rr 90.9%

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

      1. +-commutative 90.9%

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

      2. fma-def 90.9%

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

   7. Simplified 90.9%

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

   8. Taylor expanded in x around 0 56.6%

      \[\leadsto \color{blue}{z - y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{-29}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-39}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-11} \lor \neg \left(z \leq 3.4 \cdot 10^{+43}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;z - y\\ \end{array} \]

Alternative 8: 51.0% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-29} \lor \neg \left(z \leq 1.8 \cdot 10^{-39}\right) \land \left(z \leq 3.6 \cdot 10^{-10} \lor \neg \left(z \leq 6.4 \cdot 10^{+44}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.55e-29)
         (and (not (<= z 1.8e-39)) (or (<= z 3.6e-10) (not (<= z 6.4e+44)))))
   (- z)
   (- y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.55e-29) || (!(z <= 1.8e-39) && ((z <= 3.6e-10) || !(z <= 6.4e+44)))) {
		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 ((z <= (-1.55d-29)) .or. (.not. (z <= 1.8d-39)) .and. (z <= 3.6d-10) .or. (.not. (z <= 6.4d+44))) 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 ((z <= -1.55e-29) || (!(z <= 1.8e-39) && ((z <= 3.6e-10) || !(z <= 6.4e+44)))) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.55e-29) or (not (z <= 1.8e-39) and ((z <= 3.6e-10) or not (z <= 6.4e+44))):
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.55e-29) || (!(z <= 1.8e-39) && ((z <= 3.6e-10) || !(z <= 6.4e+44))))
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.55e-29) || (~((z <= 1.8e-39)) && ((z <= 3.6e-10) || ~((z <= 6.4e+44)))))
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.55e-29], And[N[Not[LessEqual[z, 1.8e-39]], $MachinePrecision], Or[LessEqual[z, 3.6e-10], N[Not[LessEqual[z, 6.4e+44]], $MachinePrecision]]]], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if z < -1.55000000000000013e-29 or 1.8e-39 < z < 3.6e-10 or 6.40000000000000009e44 < z

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. associate--l+ 99.9%

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

      3. *-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in z around inf 65.9%

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

      1. mul-1-neg 65.9%

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

   6. Simplified 65.9%

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

   if -1.55000000000000013e-29 < z < 1.8e-39 or 3.6e-10 < z < 6.40000000000000009e44

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 56.1%

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

      1. neg-mul-1 56.1%

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

   4. Simplified 56.1%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-29} \lor \neg \left(z \leq 1.8 \cdot 10^{-39}\right) \land \left(z \leq 3.6 \cdot 10^{-10} \lor \neg \left(z \leq 6.4 \cdot 10^{+44}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 9: 66.1% 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 71.4%

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

   1. neg-mul-1 71.4%

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

4. Simplified 71.4%

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

5. Final simplification 71.4%

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

Alternative 10: 34.1% 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 99.9%

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

2. Taylor expanded in y around inf 32.9%

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

   1. neg-mul-1 32.9%

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

4. Simplified 32.9%

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

5. Final simplification 32.9%

   \[\leadsto -y \]

Alternative 11: 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 99.9%

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

   1. sub-neg 99.9%

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

   2. associate--l+ 99.9%

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

   3. *-commutative 99.9%

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

   4. fma-def 99.9%

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

3. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(-z\right) - y\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.9%

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

   2. +-commutative 99.9%

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

   3. add-sqr-sqrt 55.7%

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

   4. sqrt-unprod 66.3%

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

   5. sqr-neg 66.3%

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

   6. sqrt-unprod 26.2%

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

   7. add-sqr-sqrt 60.0%

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

5. Applied egg-rr 60.0%

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

   1. +-commutative 60.0%

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

   2. fma-def 60.0%

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

7. Simplified 60.0%

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

8. Taylor expanded in z around inf 2.1%

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

9. Final simplification 2.1%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))