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

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

Alternatives: 6

Speedup: 1.0×

• 57.1% 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: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;t\_0 \leq 100000000:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= t_0 100000000.0) (exp (- x z)) (exp (- t_0 z)))))

C

double code(double x, double y, double z) {
	double t_0 = y * log(y);
	double tmp;
	if (t_0 <= 100000000.0) {
		tmp = exp((x - z));
	} else {
		tmp = exp((t_0 - 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) :: t_0
    real(8) :: tmp
    t_0 = y * log(y)
    if (t_0 <= 100000000.0d0) then
        tmp = exp((x - z))
    else
        tmp = exp((t_0 - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.log(y);
	double tmp;
	if (t_0 <= 100000000.0) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.exp((t_0 - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.log(y)
	tmp = 0
	if t_0 <= 100000000.0:
		tmp = math.exp((x - z))
	else:
		tmp = math.exp((t_0 - z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * log(y))
	tmp = 0.0
	if (t_0 <= 100000000.0)
		tmp = exp(Float64(x - z));
	else
		tmp = exp(Float64(t_0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * log(y);
	tmp = 0.0;
	if (t_0 <= 100000000.0)
		tmp = exp((x - z));
	else
		tmp = exp((t_0 - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 100000000.0], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 - z), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 1e8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

   if 1e8 < (*.f64 y (log.f64 y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 93.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 100000000:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+103}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq -23500000:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;x \leq 125:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.1e+103)
   (exp x)
   (if (<= x -23500000.0) (pow y y) (if (<= x 125.0) (exp (- z)) (exp x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e+103) {
		tmp = exp(x);
	} else if (x <= -23500000.0) {
		tmp = pow(y, y);
	} else if (x <= 125.0) {
		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 (x <= (-2.1d+103)) then
        tmp = exp(x)
    else if (x <= (-23500000.0d0)) then
        tmp = y ** y
    else if (x <= 125.0d0) 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 (x <= -2.1e+103) {
		tmp = Math.exp(x);
	} else if (x <= -23500000.0) {
		tmp = Math.pow(y, y);
	} else if (x <= 125.0) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.1e+103:
		tmp = math.exp(x)
	elif x <= -23500000.0:
		tmp = math.pow(y, y)
	elif x <= 125.0:
		tmp = math.exp(-z)
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.1e+103)
		tmp = exp(x);
	elseif (x <= -23500000.0)
		tmp = y ^ y;
	elseif (x <= 125.0)
		tmp = exp(Float64(-z));
	else
		tmp = exp(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.1e+103)
		tmp = exp(x);
	elseif (x <= -23500000.0)
		tmp = y ^ y;
	elseif (x <= 125.0)
		tmp = exp(-z);
	else
		tmp = exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.1e+103], N[Exp[x], $MachinePrecision], If[LessEqual[x, -23500000.0], N[Power[y, y], $MachinePrecision], If[LessEqual[x, 125.0], N[Exp[(-z)], $MachinePrecision], N[Exp[x], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1000000000000002e103 or 125 < 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 80.2%

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

      4. *-commutative 80.2%

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

      5. exp-to-pow 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in z around 0 75.6%

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

      1. *-commutative 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in y around 0 85.1%

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

   if -2.1000000000000002e103 < x < -2.35e7

   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 47.1%

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

      4. *-commutative 47.1%

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

      5. exp-to-pow 47.1%

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

   3. Simplified 47.1%

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

   5. Taylor expanded in z around 0 29.4%

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

      1. *-commutative 29.4%

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

   7. Simplified 29.4%

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

   8. Taylor expanded in x around 0 71.5%

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

   if -2.35e7 < x < 125

   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 94.7%

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

      4. *-commutative 94.7%

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

      5. exp-to-pow 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in x around 0 94.2%

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

      1. exp-neg 94.2%

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

      2. associate-*l/ 94.2%

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

      3. *-lft-identity 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in y around 0 81.2%

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

      1. rec-exp 81.3%

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

   10. Simplified 81.3%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+103}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq -23500000:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;x \leq 125:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -460000000 \lor \neg \left(x \leq 135\right):\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -460000000.0) (not (<= x 135.0))) (exp x) (exp (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -460000000.0) || !(x <= 135.0)) {
		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 <= (-460000000.0d0)) .or. (.not. (x <= 135.0d0))) 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 <= -460000000.0) || !(x <= 135.0)) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.exp(-z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -460000000.0) or not (x <= 135.0):
		tmp = math.exp(x)
	else:
		tmp = math.exp(-z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -460000000.0) || !(x <= 135.0))
		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 <= -460000000.0) || ~((x <= 135.0)))
		tmp = exp(x);
	else
		tmp = exp(-z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -460000000.0], N[Not[LessEqual[x, 135.0]], $MachinePrecision]], N[Exp[x], $MachinePrecision], N[Exp[(-z)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.6e8 or 135 < 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 76.2%

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

      4. *-commutative 76.2%

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

      5. exp-to-pow 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in z around 0 69.8%

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

      1. *-commutative 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in y around 0 78.2%

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

   if -4.6e8 < x < 135

   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 94.0%

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

      4. *-commutative 94.0%

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

      5. exp-to-pow 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in x around 0 94.3%

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

      1. exp-neg 94.3%

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

      2. associate-*l/ 94.3%

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

      3. *-lft-identity 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in y around 0 80.7%

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

      1. rec-exp 80.7%

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

   10. Simplified 80.7%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -460000000 \lor \neg \left(x \leq 135\right):\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+79}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 4e+79) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+79) {
		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 <= 4d+79) 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 <= 4e+79) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4e+79:
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4e+79)
		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 <= 4e+79)
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e+79], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999987e79

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 97.2%

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

   if 3.99999999999999987e79 < 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 71.6%

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

      4. *-commutative 71.6%

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

      5. exp-to-pow 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in z around 0 74.2%

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

      1. *-commutative 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in x around 0 90.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+79}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.1% 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 85.5%

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

   4. *-commutative 85.5%

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

   5. exp-to-pow 85.5%

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

3. Simplified 85.5%

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

5. Taylor expanded in z around 0 67.9%

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

   1. *-commutative 67.9%

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

7. Simplified 67.9%

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

8. Taylor expanded in y around 0 54.5%

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

9. Final simplification 54.5%

   \[\leadsto e^{x} \]
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 2024036 
(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)))