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

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

Alternatives: 7

Speedup: 1.0×

• 58.7% 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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 2: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+39} \lor \neg \left(x \leq 2.65 \cdot 10^{-5}\right):\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.45e+39) (not (<= x 2.65e-5)))
   (exp (- x z))
   (exp (- (* y (log y)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.45e+39) || !(x <= 2.65e-5)) {
		tmp = exp((x - z));
	} else {
		tmp = exp(((y * log(y)) - z));
	}
	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 <= (-2.45d+39)) .or. (.not. (x <= 2.65d-5))) then
        tmp = exp((x - z))
    else
        tmp = exp(((y * log(y)) - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.45e+39) || !(x <= 2.65e-5)) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.exp(((y * Math.log(y)) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.45e+39) or not (x <= 2.65e-5):
		tmp = math.exp((x - z))
	else:
		tmp = math.exp(((y * math.log(y)) - z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.45e+39) || !(x <= 2.65e-5))
		tmp = exp(Float64(x - z));
	else
		tmp = exp(Float64(Float64(y * log(y)) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.45e+39) || ~((x <= 2.65e-5)))
		tmp = exp((x - z));
	else
		tmp = exp(((y * log(y)) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.45e+39], N[Not[LessEqual[x, 2.65e-5]], $MachinePrecision]], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.44999999999999994e39 or 2.65e-5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 94.8%

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

   if -2.44999999999999994e39 < x < 2.65e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+39} \lor \neg \left(x \leq 2.65 \cdot 10^{-5}\right):\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+20} \lor \neg \left(x \leq 1.3 \cdot 10^{-5}\right):\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{{y}^{y}}{e^{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.08e+20) (not (<= x 1.3e-5)))
   (exp (- x z))
   (/ (pow y y) (exp z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.08e+20) || !(x <= 1.3e-5)) {
		tmp = exp((x - z));
	} else {
		tmp = pow(y, y) / exp(z);
	}
	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 <= (-1.08d+20)) .or. (.not. (x <= 1.3d-5))) then
        tmp = exp((x - z))
    else
        tmp = (y ** y) / exp(z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.08e+20) || !(x <= 1.3e-5)) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y) / Math.exp(z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.08e+20) or not (x <= 1.3e-5):
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y) / math.exp(z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.08e+20) || !(x <= 1.3e-5))
		tmp = exp(Float64(x - z));
	else
		tmp = Float64((y ^ y) / exp(z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.08e+20) || ~((x <= 1.3e-5)))
		tmp = exp((x - z));
	else
		tmp = (y ^ y) / exp(z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.08e+20], N[Not[LessEqual[x, 1.3e-5]], $MachinePrecision]], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[(N[Power[y, y], $MachinePrecision] / N[Exp[z], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.08e20 or 1.29999999999999992e-5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 94.1%

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

   if -1.08e20 < x < 1.29999999999999992e-5

   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 86.9%

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

      4. *-commutative 86.9%

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

      5. exp-to-pow 86.9%

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

   3. Simplified 86.9%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x - z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 87.7%

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

      1. exp-neg 87.7%

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

      2. associate-*l/ 87.7%

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

      3. *-lft-identity 87.7%

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

   7. Simplified 87.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+20} \lor \neg \left(x \leq 1.3 \cdot 10^{-5}\right):\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{{y}^{y}}{e^{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 180000 \lor \neg \left(y \leq 2.6 \cdot 10^{+67}\right) \land y \leq 1.6 \cdot 10^{+188}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y 180000.0) (and (not (<= y 2.6e+67)) (<= y 1.6e+188)))
   (exp (- x z))
   (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= 180000.0) || (!(y <= 2.6e+67) && (y <= 1.6e+188))) {
		tmp = exp((x - z));
	} else {
		tmp = pow(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 ((y <= 180000.0d0) .or. (.not. (y <= 2.6d+67)) .and. (y <= 1.6d+188)) then
        tmp = exp((x - z))
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= 180000.0) || (!(y <= 2.6e+67) && (y <= 1.6e+188))) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= 180000.0) or (not (y <= 2.6e+67) and (y <= 1.6e+188)):
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= 180000.0) || (!(y <= 2.6e+67) && (y <= 1.6e+188)))
		tmp = exp(Float64(x - z));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= 180000.0) || (~((y <= 2.6e+67)) && (y <= 1.6e+188)))
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, 180000.0], And[N[Not[LessEqual[y, 2.6e+67]], $MachinePrecision], LessEqual[y, 1.6e+188]]], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8e5 or 2.6e67 < y < 1.59999999999999985e188

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 92.8%

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

   if 1.8e5 < y < 2.6e67 or 1.59999999999999985e188 < y

   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 69.3%

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

      4. *-commutative 69.3%

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

      5. exp-to-pow 69.3%

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

   3. Simplified 69.3%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x - z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 74.7%

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

      1. *-commutative 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in x around 0 92.1%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 180000 \lor \neg \left(y \leq 2.6 \cdot 10^{+67}\right) \land y \leq 1.6 \cdot 10^{+188}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-z}\\ \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-297}:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+47}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- z))))
   (if (<= z -29000000.0)
     t_0
     (if (<= z 1.7e-297) (pow y y) (if (<= z 1.1e+47) (exp x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = exp(-z);
	double tmp;
	if (z <= -29000000.0) {
		tmp = t_0;
	} else if (z <= 1.7e-297) {
		tmp = pow(y, y);
	} else if (z <= 1.1e+47) {
		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 <= (-29000000.0d0)) then
        tmp = t_0
    else if (z <= 1.7d-297) then
        tmp = y ** y
    else if (z <= 1.1d+47) 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 <= -29000000.0) {
		tmp = t_0;
	} else if (z <= 1.7e-297) {
		tmp = Math.pow(y, y);
	} else if (z <= 1.1e+47) {
		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 <= -29000000.0:
		tmp = t_0
	elif z <= 1.7e-297:
		tmp = math.pow(y, y)
	elif z <= 1.1e+47:
		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 <= -29000000.0)
		tmp = t_0;
	elseif (z <= 1.7e-297)
		tmp = y ^ y;
	elseif (z <= 1.1e+47)
		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 <= -29000000.0)
		tmp = t_0;
	elseif (z <= 1.7e-297)
		tmp = y ^ y;
	elseif (z <= 1.1e+47)
		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, -29000000.0], t$95$0, If[LessEqual[z, 1.7e-297], N[Power[y, y], $MachinePrecision], If[LessEqual[z, 1.1e+47], N[Exp[x], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.9e7 or 1.1e47 < z

   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 79.7%

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

      4. *-commutative 79.7%

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

      5. exp-to-pow 79.7%

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

   3. Simplified 79.7%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x - z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 75.5%

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

      1. exp-neg 75.5%

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

      2. associate-*l/ 75.5%

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

      3. *-lft-identity 75.5%

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

   7. Simplified 75.5%

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

   8. Taylor expanded in y around 0 85.0%

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

      1. rec-exp 85.0%

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

   10. Simplified 85.0%

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

   if -2.9e7 < z < 1.69999999999999991e-297

   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 90.9%

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

      4. *-commutative 90.9%

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

      5. exp-to-pow 90.9%

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

   3. Simplified 90.9%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x - z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 90.3%

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

      1. *-commutative 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in x around 0 78.8%

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

   if 1.69999999999999991e-297 < z < 1.1e47

   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 83.3%

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

      4. *-commutative 83.3%

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

      5. exp-to-pow 83.3%

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

   3. Simplified 83.3%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x - z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 85.0%

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

      1. *-commutative 85.0%

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

   7. Simplified 85.0%

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

   8. Taylor expanded in y around 0 68.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;e^{-z}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-297}:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+47}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -700 \lor \neg \left(z \leq 1.95 \cdot 10^{+46}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -700.0) (not (<= z 1.95e+46))) (exp (- z)) (exp x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -700.0) || !(z <= 1.95e+46)) {
		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 <= (-700.0d0)) .or. (.not. (z <= 1.95d+46))) 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 <= -700.0) || !(z <= 1.95e+46)) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -700.0) or not (z <= 1.95e+46):
		tmp = math.exp(-z)
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -700.0) || !(z <= 1.95e+46))
		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 <= -700.0) || ~((z <= 1.95e+46)))
		tmp = exp(-z);
	else
		tmp = exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -700.0], N[Not[LessEqual[z, 1.95e+46]], $MachinePrecision]], N[Exp[(-z)], $MachinePrecision], N[Exp[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -700 or 1.94999999999999997e46 < z

   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 79.8%

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

      4. *-commutative 79.8%

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

      5. exp-to-pow 79.8%

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

   3. Simplified 79.8%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x - z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 75.7%

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

      1. exp-neg 75.7%

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

      2. associate-*l/ 75.7%

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

      3. *-lft-identity 75.7%

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

   7. Simplified 75.7%

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

   8. Taylor expanded in y around 0 85.1%

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

      1. rec-exp 85.1%

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

   10. Simplified 85.1%

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

   if -700 < z < 1.94999999999999997e46

   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 86.9%

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

      4. *-commutative 86.9%

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

      5. exp-to-pow 86.9%

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

   3. Simplified 86.9%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x - z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 87.4%

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

      1. *-commutative 87.4%

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

   7. Simplified 87.4%

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

   8. Taylor expanded in y around 0 66.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -700 \lor \neg \left(z \leq 1.95 \cdot 10^{+46}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.7% 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. 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 83.6%

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

   4. *-commutative 83.6%

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

   5. exp-to-pow 83.6%

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

3. Simplified 83.6%

   \[\leadsto \color{blue}{{y}^{y} \cdot e^{x - z}} \]
4. Add Preprocessing

5. Taylor expanded in z around 0 68.0%

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

   1. *-commutative 68.0%

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

7. Simplified 68.0%

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

8. Taylor expanded in y around 0 50.7%

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

9. Final simplification 50.7%

   \[\leadsto e^{x} \]
10. Add Preprocessing

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 2024031 
(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)))