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

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

Alternatives: 8

Speedup: 1.0×

• 51.6% 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: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+36}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+55}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.12e+36)
   (exp x)
   (if (<= x 2.7e+55) (exp (- (* y (log y)) z)) (* (pow y y) (exp x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.12e+36) {
		tmp = exp(x);
	} else if (x <= 2.7e+55) {
		tmp = exp(((y * log(y)) - 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 <= (-1.12d+36)) then
        tmp = exp(x)
    else if (x <= 2.7d+55) then
        tmp = exp(((y * log(y)) - 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 <= -1.12e+36) {
		tmp = Math.exp(x);
	} else if (x <= 2.7e+55) {
		tmp = Math.exp(((y * Math.log(y)) - z));
	} else {
		tmp = Math.pow(y, y) * Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.12e+36:
		tmp = math.exp(x)
	elif x <= 2.7e+55:
		tmp = math.exp(((y * math.log(y)) - z))
	else:
		tmp = math.pow(y, y) * math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.12e+36)
		tmp = exp(x);
	elseif (x <= 2.7e+55)
		tmp = exp(Float64(Float64(y * log(y)) - 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 <= -1.12e+36)
		tmp = exp(x);
	elseif (x <= 2.7e+55)
		tmp = exp(((y * log(y)) - z));
	else
		tmp = (y ^ y) * exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.12e+36], N[Exp[x], $MachinePrecision], If[LessEqual[x, 2.7e+55], N[Exp[N[(N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision], N[(N[Power[y, y], $MachinePrecision] * N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.11999999999999999e36

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 77.8%

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

   if -1.11999999999999999e36 < x < 2.69999999999999977e55

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.1%

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

   if 2.69999999999999977e55 < 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 94.6%

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

      4. *-commutative 94.6%

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

      5. exp-to-pow 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in z around 0 94.7%

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

      1. *-commutative 94.7%

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

   6. Simplified 94.7%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+36}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+55}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \]

Alternative 3: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -650000000 \lor \neg \left(z \leq 2.15 \cdot 10^{+130}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -650000000.0) (not (<= z 2.15e+130)))
   (exp (- z))
   (* (pow y y) (exp x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -650000000.0) || !(z <= 2.15e+130)) {
		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 <= (-650000000.0d0)) .or. (.not. (z <= 2.15d+130))) 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 <= -650000000.0) || !(z <= 2.15e+130)) {
		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 <= -650000000.0) or not (z <= 2.15e+130):
		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 <= -650000000.0) || !(z <= 2.15e+130))
		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 <= -650000000.0) || ~((z <= 2.15e+130)))
		tmp = exp(-z);
	else
		tmp = (y ^ y) * exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -650000000.0], N[Not[LessEqual[z, 2.15e+130]], $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 < -6.5e8 or 2.14999999999999992e130 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 85.6%

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

      1. neg-mul-1 85.6%

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

   4. Simplified 85.6%

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

   if -6.5e8 < z < 2.14999999999999992e130

   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 80.0%

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

      4. *-commutative 80.0%

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

      5. exp-to-pow 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in z around 0 80.0%

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

      1. *-commutative 80.0%

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

   6. Simplified 80.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -650000000 \lor \neg \left(z \leq 2.15 \cdot 10^{+130}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \]

Alternative 4: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+35}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-215}:\\ \;\;\;\;\frac{{y}^{y}}{e^{z}}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.45e+35)
   (exp x)
   (if (<= x 2.15e-215) (/ (pow y y) (exp z)) (* (pow y y) (exp x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.45e+35) {
		tmp = exp(x);
	} else if (x <= 2.15e-215) {
		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 <= (-2.45d+35)) then
        tmp = exp(x)
    else if (x <= 2.15d-215) 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 <= -2.45e+35) {
		tmp = Math.exp(x);
	} else if (x <= 2.15e-215) {
		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 <= -2.45e+35:
		tmp = math.exp(x)
	elif x <= 2.15e-215:
		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 <= -2.45e+35)
		tmp = exp(x);
	elseif (x <= 2.15e-215)
		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 <= -2.45e+35)
		tmp = exp(x);
	elseif (x <= 2.15e-215)
		tmp = (y ^ y) / exp(z);
	else
		tmp = (y ^ y) * exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.45e+35], N[Exp[x], $MachinePrecision], If[LessEqual[x, 2.15e-215], 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 < -2.45000000000000013e35

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 77.8%

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

   if -2.45000000000000013e35 < x < 2.15000000000000012e-215

   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 89.2%

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

      4. *-commutative 89.2%

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

      5. exp-to-pow 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in x around 0 93.4%

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

      1. *-commutative 93.4%

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

      2. exp-to-pow 93.4%

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

      3. *-commutative 93.4%

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

      4. exp-sum 99.8%

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

      5. sub-neg 99.8%

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

      6. exp-diff 93.4%

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

      7. *-commutative 93.4%

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

      8. exp-to-pow 93.4%

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

   6. Simplified 93.4%

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

   if 2.15000000000000012e-215 < 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 87.1%

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

      4. *-commutative 87.1%

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

      5. exp-to-pow 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in z around 0 87.4%

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

      1. *-commutative 87.4%

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

   6. Simplified 87.4%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+35}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-215}:\\ \;\;\;\;\frac{{y}^{y}}{e^{z}}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \]

Alternative 5: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-19}:\\ \;\;\;\;{y}^{y} \cdot e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.9e-19) (* (pow y y) (exp (- x z))) (exp (- (* y (log y)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.9e-19) {
		tmp = pow(y, y) * 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 (y <= 2.9d-19) then
        tmp = (y ** y) * 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 (y <= 2.9e-19) {
		tmp = Math.pow(y, y) * Math.exp((x - z));
	} else {
		tmp = Math.exp(((y * Math.log(y)) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.9e-19:
		tmp = math.pow(y, y) * 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 (y <= 2.9e-19)
		tmp = Float64((y ^ y) * 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 (y <= 2.9e-19)
		tmp = (y ^ y) * exp((x - z));
	else
		tmp = exp(((y * log(y)) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.9e-19], N[(N[Power[y, y], $MachinePrecision] * N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9e-19

   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 2.9e-19 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 88.8%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-19}:\\ \;\;\;\;{y}^{y} \cdot e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \]

Alternative 6: 74.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-z}\\ \mathbf{if}\;y \leq 5.6 \cdot 10^{-208}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-32}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;y \leq 126:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- z))))
   (if (<= y 5.6e-208)
     (exp x)
     (if (<= y 6.2e-122)
       t_0
       (if (<= y 1.9e-32) (exp x) (if (<= y 126.0) t_0 (pow y y)))))))

C

double code(double x, double y, double z) {
	double t_0 = exp(-z);
	double tmp;
	if (y <= 5.6e-208) {
		tmp = exp(x);
	} else if (y <= 6.2e-122) {
		tmp = t_0;
	} else if (y <= 1.9e-32) {
		tmp = exp(x);
	} else if (y <= 126.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = exp(-z)
    if (y <= 5.6d-208) then
        tmp = exp(x)
    else if (y <= 6.2d-122) then
        tmp = t_0
    else if (y <= 1.9d-32) then
        tmp = exp(x)
    else if (y <= 126.0d0) then
        tmp = t_0
    else
        tmp = y ** y
    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 (y <= 5.6e-208) {
		tmp = Math.exp(x);
	} else if (y <= 6.2e-122) {
		tmp = t_0;
	} else if (y <= 1.9e-32) {
		tmp = Math.exp(x);
	} else if (y <= 126.0) {
		tmp = t_0;
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.exp(-z)
	tmp = 0
	if y <= 5.6e-208:
		tmp = math.exp(x)
	elif y <= 6.2e-122:
		tmp = t_0
	elif y <= 1.9e-32:
		tmp = math.exp(x)
	elif y <= 126.0:
		tmp = t_0
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	t_0 = exp(Float64(-z))
	tmp = 0.0
	if (y <= 5.6e-208)
		tmp = exp(x);
	elseif (y <= 6.2e-122)
		tmp = t_0;
	elseif (y <= 1.9e-32)
		tmp = exp(x);
	elseif (y <= 126.0)
		tmp = t_0;
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = exp(-z);
	tmp = 0.0;
	if (y <= 5.6e-208)
		tmp = exp(x);
	elseif (y <= 6.2e-122)
		tmp = t_0;
	elseif (y <= 1.9e-32)
		tmp = exp(x);
	elseif (y <= 126.0)
		tmp = t_0;
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Exp[(-z)], $MachinePrecision]}, If[LessEqual[y, 5.6e-208], N[Exp[x], $MachinePrecision], If[LessEqual[y, 6.2e-122], t$95$0, If[LessEqual[y, 1.9e-32], N[Exp[x], $MachinePrecision], If[LessEqual[y, 126.0], t$95$0, N[Power[y, y], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.60000000000000003e-208 or 6.1999999999999997e-122 < y < 1.90000000000000004e-32

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 87.2%

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

   if 5.60000000000000003e-208 < y < 6.1999999999999997e-122 or 1.90000000000000004e-32 < y < 126

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 88.2%

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

      1. neg-mul-1 88.2%

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

   4. Simplified 88.2%

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

   if 126 < 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 65.5%

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

      4. *-commutative 65.5%

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

      5. exp-to-pow 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in z around 0 67.7%

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

      1. *-commutative 67.7%

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

   6. Simplified 67.7%

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

   7. Taylor expanded in x around 0 82.0%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-208}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-122}:\\ \;\;\;\;e^{-z}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-32}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;y \leq 126:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]

Alternative 7: 72.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+35} \lor \neg \left(x \leq 5.9 \cdot 10^{+16}\right):\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.6e+35) (not (<= x 5.9e+16))) (exp x) (exp (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.6e+35) || !(x <= 5.9e+16)) {
		tmp = exp(x);
	} else {
		tmp = 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 <= (-4.6d+35)) .or. (.not. (x <= 5.9d+16))) then
        tmp = exp(x)
    else
        tmp = exp(-z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.6e+35) || !(x <= 5.9e+16)) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.exp(-z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.6e+35) or not (x <= 5.9e+16):
		tmp = math.exp(x)
	else:
		tmp = math.exp(-z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.6e+35) || !(x <= 5.9e+16))
		tmp = exp(x);
	else
		tmp = exp(Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4.6e+35) || ~((x <= 5.9e+16)))
		tmp = exp(x);
	else
		tmp = exp(-z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.6e+35], N[Not[LessEqual[x, 5.9e+16]], $MachinePrecision]], N[Exp[x], $MachinePrecision], N[Exp[(-z)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5999999999999996e35 or 5.9e16 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 84.1%

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

   if -4.5999999999999996e35 < x < 5.9e16

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 62.7%

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

      1. neg-mul-1 62.7%

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

   4. Simplified 62.7%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+35} \lor \neg \left(x \leq 5.9 \cdot 10^{+16}\right):\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \]

Alternative 8: 52.3% 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 49.6%

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

3. Final simplification 49.6%

   \[\leadsto e^{x} \]

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 2023318 
(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)))