Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

Alternatives: 9

Speedup: 1.0×

• 51.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return exp(((x + (y * log(y))) - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = exp(((x + (y * log(y))) - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return exp(((x + (y * log(y))) - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = exp(((x + (y * log(y))) - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return exp(((x + (y * log(y))) - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = exp(((x + (y * log(y))) - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{\left(x + y \cdot \log y\right) - z} \]

2. Final simplification 100.0%

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

Alternative 2: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+56} \lor \neg \left(z \leq 3.7 \cdot 10^{+77}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x + y \cdot \log y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.6e+56) (not (<= z 3.7e+77)))
   (exp (- z))
   (exp (+ x (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.6e+56) || !(z <= 3.7e+77)) {
		tmp = exp(-z);
	} else {
		tmp = exp((x + (y * log(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 <= (-4.6d+56)) .or. (.not. (z <= 3.7d+77))) then
        tmp = exp(-z)
    else
        tmp = exp((x + (y * log(y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.6e+56) || !(z <= 3.7e+77)) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.exp((x + (y * Math.log(y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.6e+56) or not (z <= 3.7e+77):
		tmp = math.exp(-z)
	else:
		tmp = math.exp((x + (y * math.log(y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.6e+56) || !(z <= 3.7e+77))
		tmp = exp(Float64(-z));
	else
		tmp = exp(Float64(x + Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4.6e+56) || ~((z <= 3.7e+77)))
		tmp = exp(-z);
	else
		tmp = exp((x + (y * log(y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.6e+56], N[Not[LessEqual[z, 3.7e+77]], $MachinePrecision]], N[Exp[(-z)], $MachinePrecision], N[Exp[N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.60000000000000029e56 or 3.69999999999999995e77 < z

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]

   2. Taylor expanded in z around inf 92.4%

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

      1. neg-mul-1 92.4%

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

   4. Simplified 92.4%

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

   if -4.60000000000000029e56 < z < 3.69999999999999995e77

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]

   2. Taylor expanded in z around 0 94.8%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+56} \lor \neg \left(z \leq 3.7 \cdot 10^{+77}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x + y \cdot \log y}\\ \end{array} \]

Alternative 3: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+47} \lor \neg \left(z \leq 3.1 \cdot 10^{+75}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.4e+47) (not (<= z 3.1e+75)))
   (exp (- z))
   (* (pow y y) (exp x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.4e+47) || !(z <= 3.1e+75)) {
		tmp = exp(-z);
	} else {
		tmp = pow(y, y) * exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-3.4d+47)) .or. (.not. (z <= 3.1d+75))) then
        tmp = exp(-z)
    else
        tmp = (y ** y) * exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.4e+47) || !(z <= 3.1e+75)) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.pow(y, y) * Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.4e+47) or not (z <= 3.1e+75):
		tmp = math.exp(-z)
	else:
		tmp = math.pow(y, y) * math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.4e+47) || !(z <= 3.1e+75))
		tmp = exp(Float64(-z));
	else
		tmp = Float64((y ^ y) * exp(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.4e+47) || ~((z <= 3.1e+75)))
		tmp = exp(-z);
	else
		tmp = (y ^ y) * exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.4e+47], N[Not[LessEqual[z, 3.1e+75]], $MachinePrecision]], N[Exp[(-z)], $MachinePrecision], N[(N[Power[y, y], $MachinePrecision] * N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3999999999999998e47 or 3.1000000000000001e75 < z

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]

   2. Taylor expanded in z around inf 91.6%

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

      1. neg-mul-1 91.6%

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

   4. Simplified 91.6%

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

   if -3.3999999999999998e47 < z < 3.1000000000000001e75

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

      3. exp-sum 84.1%

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

      4. *-commutative 84.1%

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

      5. exp-to-pow 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in z around 0 80.8%

      \[\leadsto \color{blue}{e^{x} \cdot {y}^{y}} \]
   5. Step-by-step derivation

      1. *-commutative 80.8%

         \[\leadsto \color{blue}{{y}^{y} \cdot e^{x}} \]

   6. Simplified 80.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+47} \lor \neg \left(z \leq 3.1 \cdot 10^{+75}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \]

Alternative 4: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.3 \cdot 10^{+25}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 370:\\ \;\;\;\;\frac{{y}^{y}}{e^{z}}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.3e+25)
   (exp x)
   (if (<= x 370.0) (/ (pow y y) (exp z)) (* (pow y y) (exp x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.3e+25) {
		tmp = exp(x);
	} else if (x <= 370.0) {
		tmp = pow(y, y) / exp(z);
	} else {
		tmp = pow(y, y) * exp(x);
	}
	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 <= (-6.3d+25)) then
        tmp = exp(x)
    else if (x <= 370.0d0) then
        tmp = (y ** y) / exp(z)
    else
        tmp = (y ** y) * exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.3e+25) {
		tmp = Math.exp(x);
	} else if (x <= 370.0) {
		tmp = Math.pow(y, y) / Math.exp(z);
	} else {
		tmp = Math.pow(y, y) * Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.3e+25:
		tmp = math.exp(x)
	elif x <= 370.0:
		tmp = math.pow(y, y) / math.exp(z)
	else:
		tmp = math.pow(y, y) * math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.3e+25)
		tmp = exp(x);
	elseif (x <= 370.0)
		tmp = Float64((y ^ y) / exp(z));
	else
		tmp = Float64((y ^ y) * exp(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6.3e+25)
		tmp = exp(x);
	elseif (x <= 370.0)
		tmp = (y ^ y) / exp(z);
	else
		tmp = (y ^ y) * exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.3e+25], N[Exp[x], $MachinePrecision], If[LessEqual[x, 370.0], N[(N[Power[y, y], $MachinePrecision] / N[Exp[z], $MachinePrecision]), $MachinePrecision], N[(N[Power[y, y], $MachinePrecision] * N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.29999999999999973e25

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]

   2. Taylor expanded in x around inf 72.1%

      \[\leadsto e^{\color{blue}{x}} \]

   if -6.29999999999999973e25 < x < 370

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

      3. exp-sum 88.3%

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

      4. *-commutative 88.3%

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

      5. exp-to-pow 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in x around 0 88.5%

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

      1. rec-exp 88.5%

         \[\leadsto \color{blue}{\frac{1}{e^{z}}} \cdot {y}^{y} \]

      2. associate-*l/ 88.5%

         \[\leadsto \color{blue}{\frac{1 \cdot {y}^{y}}{e^{z}}} \]

      3. *-lft-identity 88.5%

         \[\leadsto \frac{\color{blue}{{y}^{y}}}{e^{z}} \]

   6. Simplified 88.5%

      \[\leadsto \color{blue}{\frac{{y}^{y}}{e^{z}}} \]

   if 370 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

      3. exp-sum 97.0%

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

      4. *-commutative 97.0%

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

      5. exp-to-pow 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in z around 0 88.2%

      \[\leadsto \color{blue}{e^{x} \cdot {y}^{y}} \]
   5. Step-by-step derivation

      1. *-commutative 88.2%

         \[\leadsto \color{blue}{{y}^{y} \cdot e^{x}} \]

   6. Simplified 88.2%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.3 \cdot 10^{+25}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 370:\\ \;\;\;\;\frac{{y}^{y}}{e^{z}}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \]

Alternative 5: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0003:\\ \;\;\;\;{y}^{y} \cdot e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{x + y \cdot \log y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.0003) (* (pow y y) (exp (- x z))) (exp (+ x (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0003) {
		tmp = pow(y, y) * exp((x - z));
	} else {
		tmp = exp((x + (y * log(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 <= 0.0003d0) then
        tmp = (y ** y) * exp((x - z))
    else
        tmp = exp((x + (y * log(y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0003) {
		tmp = Math.pow(y, y) * Math.exp((x - z));
	} else {
		tmp = Math.exp((x + (y * Math.log(y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.0003:
		tmp = math.pow(y, y) * math.exp((x - z))
	else:
		tmp = math.exp((x + (y * math.log(y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.0003)
		tmp = Float64((y ^ y) * exp(Float64(x - z)));
	else
		tmp = exp(Float64(x + Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.0003)
		tmp = (y ^ y) * exp((x - z));
	else
		tmp = exp((x + (y * log(y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.0003], N[(N[Power[y, y], $MachinePrecision] * N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.99999999999999974e-4

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

      3. exp-sum 100.0%

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

      4. *-commutative 100.0%

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

      5. exp-to-pow 100.0%

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

   3. Simplified 100.0%

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

   if 2.99999999999999974e-4 < y

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]

   2. Taylor expanded in z around 0 88.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0003:\\ \;\;\;\;{y}^{y} \cdot e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{x + y \cdot \log y}\\ \end{array} \]

Alternative 6: 73.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-z}\\ \mathbf{if}\;z \leq -4200000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-282}:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+68}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- z))))
   (if (<= z -4200000000.0)
     t_0
     (if (<= z 3.1e-282) (pow y y) (if (<= z 1.6e+68) (exp x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = exp(-z);
	double tmp;
	if (z <= -4200000000.0) {
		tmp = t_0;
	} else if (z <= 3.1e-282) {
		tmp = pow(y, y);
	} else if (z <= 1.6e+68) {
		tmp = exp(x);
	} 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 = exp(-z)
    if (z <= (-4200000000.0d0)) then
        tmp = t_0
    else if (z <= 3.1d-282) then
        tmp = y ** y
    else if (z <= 1.6d+68) then
        tmp = exp(x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.exp(-z);
	double tmp;
	if (z <= -4200000000.0) {
		tmp = t_0;
	} else if (z <= 3.1e-282) {
		tmp = Math.pow(y, y);
	} else if (z <= 1.6e+68) {
		tmp = Math.exp(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.exp(-z)
	tmp = 0
	if z <= -4200000000.0:
		tmp = t_0
	elif z <= 3.1e-282:
		tmp = math.pow(y, y)
	elif z <= 1.6e+68:
		tmp = math.exp(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = exp(Float64(-z))
	tmp = 0.0
	if (z <= -4200000000.0)
		tmp = t_0;
	elseif (z <= 3.1e-282)
		tmp = y ^ y;
	elseif (z <= 1.6e+68)
		tmp = exp(x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = exp(-z);
	tmp = 0.0;
	if (z <= -4200000000.0)
		tmp = t_0;
	elseif (z <= 3.1e-282)
		tmp = y ^ y;
	elseif (z <= 1.6e+68)
		tmp = exp(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Exp[(-z)], $MachinePrecision]}, If[LessEqual[z, -4200000000.0], t$95$0, If[LessEqual[z, 3.1e-282], N[Power[y, y], $MachinePrecision], If[LessEqual[z, 1.6e+68], N[Exp[x], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.2e9 or 1.59999999999999997e68 < z

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]

   2. Taylor expanded in z around inf 89.8%

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

      1. neg-mul-1 89.8%

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

   4. Simplified 89.8%

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

   if -4.2e9 < z < 3.10000000000000013e-282

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

      3. exp-sum 82.3%

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

      4. *-commutative 82.3%

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

      5. exp-to-pow 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in z around 0 81.6%

      \[\leadsto \color{blue}{e^{x} \cdot {y}^{y}} \]
   5. Step-by-step derivation

      1. *-commutative 81.6%

         \[\leadsto \color{blue}{{y}^{y} \cdot e^{x}} \]

   6. Simplified 81.6%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x}} \]

   7. Taylor expanded in x around 0 70.7%

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

   if 3.10000000000000013e-282 < z < 1.59999999999999997e68

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]

   2. Taylor expanded in x around inf 69.8%

      \[\leadsto e^{\color{blue}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4200000000:\\ \;\;\;\;e^{-z}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-282}:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+68}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \]

Alternative 7: 73.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -450000000000 \lor \neg \left(z \leq 1.8 \cdot 10^{+66}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -450000000000.0) (not (<= z 1.8e+66))) (exp (- z)) (exp x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -450000000000.0) || !(z <= 1.8e+66)) {
		tmp = exp(-z);
	} else {
		tmp = exp(x);
	}
	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 <= (-450000000000.0d0)) .or. (.not. (z <= 1.8d+66))) then
        tmp = exp(-z)
    else
        tmp = exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -450000000000.0) || !(z <= 1.8e+66)) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -450000000000.0) or not (z <= 1.8e+66):
		tmp = math.exp(-z)
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -450000000000.0) || !(z <= 1.8e+66))
		tmp = exp(Float64(-z));
	else
		tmp = exp(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -450000000000.0) || ~((z <= 1.8e+66)))
		tmp = exp(-z);
	else
		tmp = exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -450000000000.0], N[Not[LessEqual[z, 1.8e+66]], $MachinePrecision]], N[Exp[(-z)], $MachinePrecision], N[Exp[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.5e11 or 1.8e66 < z

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]

   2. Taylor expanded in z around inf 90.5%

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

      1. neg-mul-1 90.5%

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

   4. Simplified 90.5%

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

   if -4.5e11 < z < 1.8e66

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]

   2. Taylor expanded in x around inf 65.7%

      \[\leadsto e^{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -450000000000 \lor \neg \left(z \leq 1.8 \cdot 10^{+66}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \]

Alternative 8: 53.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e^{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (exp x))

C

double code(double x, double y, double z) {
	return exp(x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = exp(x)
end function

Java

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

Python

def code(x, y, z):
	return math.exp(x)

Julia

function code(x, y, z)
	return exp(x)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Exp[x], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{\left(x + y \cdot \log y\right) - z} \]

2. Taylor expanded in x around inf 52.9%

   \[\leadsto e^{\color{blue}{x}} \]

3. Final simplification 52.9%

   \[\leadsto e^{x} \]

Alternative 9: 14.1% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ x + 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x 1.0))

C

double code(double x, double y, double z) {
	return x + 1.0;
}

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 + 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return x + 1.0;
}

Python

def code(x, y, z):
	return x + 1.0

Julia

function code(x, y, z)
	return Float64(x + 1.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + 1.0;
end

Wolfram

code[x_, y_, z_] := N[(x + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{\left(x + y \cdot \log y\right) - z} \]
2. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. associate--l+ 100.0%

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

   3. exp-sum 82.8%

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

   4. *-commutative 82.8%

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

   5. exp-to-pow 82.8%

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

3. Simplified 82.8%

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

4. Taylor expanded in z around 0 66.1%

   \[\leadsto \color{blue}{e^{x} \cdot {y}^{y}} \]
5. Step-by-step derivation

   1. *-commutative 66.1%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x}} \]

6. Simplified 66.1%

   \[\leadsto \color{blue}{{y}^{y} \cdot e^{x}} \]

7. Taylor expanded in x around 0 33.6%

   \[\leadsto \color{blue}{x \cdot {y}^{y} + {y}^{y}} \]
8. Step-by-step derivation

   1. distribute-lft1-in 43.1%

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

   2. *-commutative 43.1%

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

9. Simplified 43.1%

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

10. Taylor expanded in y around 0 14.0%

    \[\leadsto \color{blue}{1 + x} \]

11. Final simplification 14.0%

    \[\leadsto x + 1 \]

Developer target: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023293 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (exp (+ (- x z) (* (log y) y)))

  (exp (- (+ x (* y (log y))) z)))