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

Percentage Accurate: 99.8% → 99.8%

Time: 9.7s

Alternatives: 7

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 99.9%

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

Alternative 2: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{+74}:\\ \;\;\;\;t\_0 - z\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+110}:\\ \;\;\;\;\left(0 - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_0 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log y))))
   (if (<= x -1.1e+74)
     (- t_0 z)
     (if (<= x 1.25e+110) (- (- 0.0 z) y) (- t_0 y)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log(y);
	double tmp;
	if (x <= -1.1e+74) {
		tmp = t_0 - z;
	} else if (x <= 1.25e+110) {
		tmp = (0.0 - z) - y;
	} else {
		tmp = t_0 - 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) :: t_0
    real(8) :: tmp
    t_0 = x * log(y)
    if (x <= (-1.1d+74)) then
        tmp = t_0 - z
    else if (x <= 1.25d+110) then
        tmp = (0.0d0 - z) - y
    else
        tmp = t_0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log(y);
	double tmp;
	if (x <= -1.1e+74) {
		tmp = t_0 - z;
	} else if (x <= 1.25e+110) {
		tmp = (0.0 - z) - y;
	} else {
		tmp = t_0 - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log(y)
	tmp = 0
	if x <= -1.1e+74:
		tmp = t_0 - z
	elif x <= 1.25e+110:
		tmp = (0.0 - z) - y
	else:
		tmp = t_0 - y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -1.1e+74)
		tmp = Float64(t_0 - z);
	elseif (x <= 1.25e+110)
		tmp = Float64(Float64(0.0 - z) - y);
	else
		tmp = Float64(t_0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log(y);
	tmp = 0.0;
	if (x <= -1.1e+74)
		tmp = t_0 - z;
	elseif (x <= 1.25e+110)
		tmp = (0.0 - z) - y;
	else
		tmp = t_0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.1e+74], N[(t$95$0 - z), $MachinePrecision], If[LessEqual[x, 1.25e+110], N[(N[(0.0 - z), $MachinePrecision] - y), $MachinePrecision], N[(t$95$0 - y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1000000000000001e74

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \log y\right), \color{blue}{z}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), z\right) \]

      3. log-lowering-log.f64 86.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), z\right) \]

   5. Simplified 86.7%

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

   if -1.1000000000000001e74 < x < 1.24999999999999995e110

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(y + z\right)\right) \]

      2. distribute-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

      4. sub-neg N/A

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

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), \color{blue}{y}\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

      7. --lowering--.f64 90.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

   5. Simplified 90.6%

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

   if 1.24999999999999995e110 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \log y\right), \color{blue}{y}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), y\right) \]

      3. log-lowering-log.f64 89.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), y\right) \]

   5. Simplified 89.7%

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

Alternative 3: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log y - y\\ \mathbf{if}\;x \leq -2.55 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+109}:\\ \;\;\;\;\left(0 - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x (log y)) y)))
   (if (<= x -2.55e+17) t_0 (if (<= x 8.5e+109) (- (- 0.0 z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * log(y)) - y;
	double tmp;
	if (x <= -2.55e+17) {
		tmp = t_0;
	} else if (x <= 8.5e+109) {
		tmp = (0.0 - z) - y;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (x * log(y)) - y
    if (x <= (-2.55d+17)) then
        tmp = t_0
    else if (x <= 8.5d+109) then
        tmp = (0.0d0 - z) - y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * Math.log(y)) - y;
	double tmp;
	if (x <= -2.55e+17) {
		tmp = t_0;
	} else if (x <= 8.5e+109) {
		tmp = (0.0 - z) - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * math.log(y)) - y
	tmp = 0
	if x <= -2.55e+17:
		tmp = t_0
	elif x <= 8.5e+109:
		tmp = (0.0 - z) - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if (x <= -2.55e+17)
		tmp = t_0;
	elseif (x <= 8.5e+109)
		tmp = Float64(Float64(0.0 - z) - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * log(y)) - y;
	tmp = 0.0;
	if (x <= -2.55e+17)
		tmp = t_0;
	elseif (x <= 8.5e+109)
		tmp = (0.0 - z) - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[x, -2.55e+17], t$95$0, If[LessEqual[x, 8.5e+109], N[(N[(0.0 - z), $MachinePrecision] - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.55e17 or 8.5000000000000004e109 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \log y\right), \color{blue}{y}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), y\right) \]

      3. log-lowering-log.f64 83.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), y\right) \]

   5. Simplified 83.9%

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

   if -2.55e17 < x < 8.5000000000000004e109

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(y + z\right)\right) \]

      2. distribute-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

      4. sub-neg N/A

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

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), \color{blue}{y}\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

      7. --lowering--.f64 92.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

   5. Simplified 92.4%

      \[\leadsto \color{blue}{\left(0 - z\right) - y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+128}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+115}:\\ \;\;\;\;\left(0 - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log y))))
   (if (<= x -1.3e+128) t_0 (if (<= x 3.7e+115) (- (- 0.0 z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * log(y);
	double tmp;
	if (x <= -1.3e+128) {
		tmp = t_0;
	} else if (x <= 3.7e+115) {
		tmp = (0.0 - z) - y;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = x * log(y)
    if (x <= (-1.3d+128)) then
        tmp = t_0
    else if (x <= 3.7d+115) then
        tmp = (0.0d0 - z) - y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log(y);
	double tmp;
	if (x <= -1.3e+128) {
		tmp = t_0;
	} else if (x <= 3.7e+115) {
		tmp = (0.0 - z) - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log(y)
	tmp = 0
	if x <= -1.3e+128:
		tmp = t_0
	elif x <= 3.7e+115:
		tmp = (0.0 - z) - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -1.3e+128)
		tmp = t_0;
	elseif (x <= 3.7e+115)
		tmp = Float64(Float64(0.0 - z) - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log(y);
	tmp = 0.0;
	if (x <= -1.3e+128)
		tmp = t_0;
	elseif (x <= 3.7e+115)
		tmp = (0.0 - z) - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e+128], t$95$0, If[LessEqual[x, 3.7e+115], N[(N[(0.0 - z), $MachinePrecision] - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3e128 or 3.70000000000000006e115 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\log y}\right) \]

      2. log-lowering-log.f64 76.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right) \]

   5. Simplified 76.1%

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

   if -1.3e128 < x < 3.70000000000000006e115

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(y + z\right)\right) \]

      2. distribute-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

      4. sub-neg N/A

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

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), \color{blue}{y}\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

      7. --lowering--.f64 88.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

   5. Simplified 88.5%

      \[\leadsto \color{blue}{\left(0 - z\right) - y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 51.5% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+103}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;0 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 6.4e+103) (- 0.0 z) (- 0.0 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.4e+103) {
		tmp = 0.0 - z;
	} else {
		tmp = 0.0 - 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 <= 6.4d+103) then
        tmp = 0.0d0 - z
    else
        tmp = 0.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.4e+103) {
		tmp = 0.0 - z;
	} else {
		tmp = 0.0 - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.4e+103:
		tmp = 0.0 - z
	else:
		tmp = 0.0 - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.4e+103)
		tmp = Float64(0.0 - z);
	else
		tmp = Float64(0.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.4e+103)
		tmp = 0.0 - z;
	else
		tmp = 0.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.4e+103], N[(0.0 - z), $MachinePrecision], N[(0.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.39999999999999985e103

   1. Initial program 99.8%

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

      1. associate--l- N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot \log y + \left(z + y\right)}{\left(x \cdot \log y\right) \cdot \left(x \cdot \log y\right) - \left(z + y\right) \cdot \left(z + y\right)}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot \log y + \left(z + y\right)\right), \color{blue}{\left(\left(x \cdot \log y\right) \cdot \left(x \cdot \log y\right) - \left(z + y\right) \cdot \left(z + y\right)\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \log y\right), \left(z + y\right)\right), \left(\color{blue}{\left(x \cdot \log y\right) \cdot \left(x \cdot \log y\right)} - \left(z + y\right) \cdot \left(z + y\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \log y\right), \left(z + y\right)\right), \left(\color{blue}{\left(x \cdot \log y\right)} \cdot \left(x \cdot \log y\right) - \left(z + y\right) \cdot \left(z + y\right)\right)\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \left(z + y\right)\right), \left(\left(x \cdot \color{blue}{\log y}\right) \cdot \left(x \cdot \log y\right) - \left(z + y\right) \cdot \left(z + y\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \left(y + z\right)\right), \left(\left(x \cdot \log y\right) \cdot \color{blue}{\left(x \cdot \log y\right)} - \left(z + y\right) \cdot \left(z + y\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{+.f64}\left(y, z\right)\right), \left(\left(x \cdot \log y\right) \cdot \color{blue}{\left(x \cdot \log y\right)} - \left(z + y\right) \cdot \left(z + y\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{+.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(\left(\left(x \cdot \log y\right) \cdot \left(x \cdot \log y\right)\right), \color{blue}{\left(\left(z + y\right) \cdot \left(z + y\right)\right)}\right)\right)\right) \]

      12. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{+.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(\left({\left(x \cdot \log y\right)}^{2}\right), \left(\color{blue}{\left(z + y\right)} \cdot \left(z + y\right)\right)\right)\right)\right) \]

      13. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{+.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(x \cdot \log y\right), 2\right), \left(\color{blue}{\left(z + y\right)} \cdot \left(z + y\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{+.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, \log y\right), 2\right), \left(\left(\color{blue}{z} + y\right) \cdot \left(z + y\right)\right)\right)\right)\right) \]

      15. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{+.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), 2\right), \left(\left(z + y\right) \cdot \left(z + y\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{+.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), 2\right), \mathsf{*.f64}\left(\left(z + y\right), \color{blue}{\left(z + y\right)}\right)\right)\right)\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{+.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), 2\right), \mathsf{*.f64}\left(\left(y + z\right), \left(\color{blue}{z} + y\right)\right)\right)\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{+.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, z\right), \left(\color{blue}{z} + y\right)\right)\right)\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{+.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, z\right), \left(y + \color{blue}{z}\right)\right)\right)\right)\right) \]

      20. +-lowering-+.f64 54.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{+.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, z\right), \mathsf{+.f64}\left(y, \color{blue}{z}\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 54.1%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{z}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 46.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{z}\right)\right) \]

   7. Simplified 46.8%

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

      1. associate-/r/ N/A

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

      2. metadata-eval N/A

         \[\leadsto -1 \cdot z \]

      3. neg-mul-1 N/A

         \[\leadsto \mathsf{neg}\left(z\right) \]

      4. neg-lowering-neg.f64 46.8%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   9. Applied egg-rr 46.8%

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

   if 6.39999999999999985e103 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(y\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{y} \]

      3. --lowering--.f64 67.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{y}\right) \]

   5. Simplified 67.5%

      \[\leadsto \color{blue}{0 - y} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(y\right) \]

      2. neg-lowering-neg.f64 67.5%

         \[\leadsto \mathsf{neg.f64}\left(y\right) \]

   7. Applied egg-rr 67.5%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+103}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;0 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.7% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \left(0 - z\right) - y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (- 0.0 z) y))

C

double code(double x, double y, double z) {
	return (0.0 - 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 = (0.0d0 - z) - y
end function

Java

public static double code(double x, double y, double z) {
	return (0.0 - z) - y;
}

Python

def code(x, y, z):
	return (0.0 - z) - y

Julia

function code(x, y, z)
	return Float64(Float64(0.0 - z) - y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (0.0 - z) - y;
end

Wolfram

code[x_, y_, z_] := N[(N[(0.0 - z), $MachinePrecision] - y), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\left(y + z\right)\right) \]

   2. distribute-neg-in N/A

      \[\leadsto \left(\mathsf{neg}\left(y\right)\right) + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

   4. sub-neg N/A

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

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), \color{blue}{y}\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

   7. --lowering--.f64 67.4%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

5. Simplified 67.4%

   \[\leadsto \color{blue}{\left(0 - z\right) - y} \]
6. Add Preprocessing

Alternative 7: 33.4% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ 0 - y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- 0.0 y))

C

double code(double x, double y, double z) {
	return 0.0 - 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 = 0.0d0 - y
end function

Java

public static double code(double x, double y, double z) {
	return 0.0 - y;
}

Python

def code(x, y, z):
	return 0.0 - y

Julia

function code(x, y, z)
	return Float64(0.0 - y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0 - y;
end

Wolfram

code[x_, y_, z_] := N[(0.0 - y), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(y\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{y} \]

   3. --lowering--.f64 33.5%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{y}\right) \]

5. Simplified 33.5%

   \[\leadsto \color{blue}{0 - y} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(y\right) \]

   2. neg-lowering-neg.f64 33.5%

      \[\leadsto \mathsf{neg.f64}\left(y\right) \]

7. Applied egg-rr 33.5%

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

8. Final simplification 33.5%

   \[\leadsto 0 - y \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024160 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))