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

Percentage Accurate: 100.0% → 100.0%

Time: 3.4s

Alternatives: 7

Speedup: 1.0×

• 59.1% 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. Final simplification 100.0%

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

Alternative 2: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ t_1 := e^{x - z}\\ \mathbf{if}\;t_0 \leq 400000000000:\\ \;\;\;\;{y}^{y} \cdot t_1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))) (t_1 (exp (- x z))))
   (if (<= t_0 400000000000.0)
     (* (pow y y) t_1)
     (if (<= t_0 2e+84) t_1 (exp (- t_0 z))))))

C

double code(double x, double y, double z) {
	double t_0 = y * log(y);
	double t_1 = exp((x - z));
	double tmp;
	if (t_0 <= 400000000000.0) {
		tmp = pow(y, y) * t_1;
	} else if (t_0 <= 2e+84) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = y * log(y)
    t_1 = exp((x - z))
    if (t_0 <= 400000000000.0d0) then
        tmp = (y ** y) * t_1
    else if (t_0 <= 2d+84) then
        tmp = t_1
    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 t_1 = Math.exp((x - z));
	double tmp;
	if (t_0 <= 400000000000.0) {
		tmp = Math.pow(y, y) * t_1;
	} else if (t_0 <= 2e+84) {
		tmp = t_1;
	} else {
		tmp = Math.exp((t_0 - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.log(y)
	t_1 = math.exp((x - z))
	tmp = 0
	if t_0 <= 400000000000.0:
		tmp = math.pow(y, y) * t_1
	elif t_0 <= 2e+84:
		tmp = t_1
	else:
		tmp = math.exp((t_0 - z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * log(y))
	t_1 = exp(Float64(x - z))
	tmp = 0.0
	if (t_0 <= 400000000000.0)
		tmp = Float64((y ^ y) * t_1);
	elseif (t_0 <= 2e+84)
		tmp = t_1;
	else
		tmp = exp(Float64(t_0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * log(y);
	t_1 = exp((x - z));
	tmp = 0.0;
	if (t_0 <= 400000000000.0)
		tmp = (y ^ y) * t_1;
	elseif (t_0 <= 2e+84)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 400000000000.0], N[(N[Power[y, y], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 2e+84], t$95$1, N[Exp[N[(t$95$0 - z), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (log.f64 y)) < 4e11

   1. Initial program 100.0%

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

      1. exp-diff 87.2%

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

      2. +-commutative 87.2%

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

      3. exp-sum 87.2%

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

      4. associate-*r/ 87.2%

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

      5. *-commutative 87.2%

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

      6. exp-to-pow 87.2%

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

      7. exp-diff 100.0%

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

   3. Simplified 100.0%

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

   if 4e11 < (*.f64 y (log.f64 y)) < 2.00000000000000012e84

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 92.8%

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

   if 2.00000000000000012e84 < (*.f64 y (log.f64 y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 92.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 400000000000:\\ \;\;\;\;{y}^{y} \cdot e^{x - z}\\ \mathbf{elif}\;y \cdot \log y \leq 2 \cdot 10^{+84}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \]

Alternative 3: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -32000000 \lor \neg \left(x \leq 5.2 \cdot 10^{-7}\right):\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -32000000.0) (not (<= x 5.2e-7)))
   (exp (- x z))
   (exp (- (* y (log y)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -32000000.0) || !(x <= 5.2e-7)) {
		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 <= (-32000000.0d0)) .or. (.not. (x <= 5.2d-7))) 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 <= -32000000.0) || !(x <= 5.2e-7)) {
		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 <= -32000000.0) or not (x <= 5.2e-7):
		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 <= -32000000.0) || !(x <= 5.2e-7))
		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 <= -32000000.0) || ~((x <= 5.2e-7)))
		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, -32000000.0], N[Not[LessEqual[x, 5.2e-7]], $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 < -3.2e7 or 5.19999999999999998e-7 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 94.1%

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

   if -3.2e7 < x < 5.19999999999999998e-7

   1. Initial program 100.0%

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

   2. 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 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -32000000 \lor \neg \left(x \leq 5.2 \cdot 10^{-7}\right):\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \]

Alternative 4: 72.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-z}\\ \mathbf{if}\;z \leq -8 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-290}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-242}:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+36}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- z))))
   (if (<= z -8e-15)
     t_0
     (if (<= z 1.1e-290)
       (exp x)
       (if (<= z 5.5e-242) (pow y y) (if (<= z 1.8e+36) (exp x) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = exp(-z);
	double tmp;
	if (z <= -8e-15) {
		tmp = t_0;
	} else if (z <= 1.1e-290) {
		tmp = exp(x);
	} else if (z <= 5.5e-242) {
		tmp = pow(y, y);
	} else if (z <= 1.8e+36) {
		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 <= (-8d-15)) then
        tmp = t_0
    else if (z <= 1.1d-290) then
        tmp = exp(x)
    else if (z <= 5.5d-242) then
        tmp = y ** y
    else if (z <= 1.8d+36) 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 <= -8e-15) {
		tmp = t_0;
	} else if (z <= 1.1e-290) {
		tmp = Math.exp(x);
	} else if (z <= 5.5e-242) {
		tmp = Math.pow(y, y);
	} else if (z <= 1.8e+36) {
		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 <= -8e-15:
		tmp = t_0
	elif z <= 1.1e-290:
		tmp = math.exp(x)
	elif z <= 5.5e-242:
		tmp = math.pow(y, y)
	elif z <= 1.8e+36:
		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 <= -8e-15)
		tmp = t_0;
	elseif (z <= 1.1e-290)
		tmp = exp(x);
	elseif (z <= 5.5e-242)
		tmp = y ^ y;
	elseif (z <= 1.8e+36)
		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 <= -8e-15)
		tmp = t_0;
	elseif (z <= 1.1e-290)
		tmp = exp(x);
	elseif (z <= 5.5e-242)
		tmp = y ^ y;
	elseif (z <= 1.8e+36)
		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, -8e-15], t$95$0, If[LessEqual[z, 1.1e-290], N[Exp[x], $MachinePrecision], If[LessEqual[z, 5.5e-242], N[Power[y, y], $MachinePrecision], If[LessEqual[z, 1.8e+36], N[Exp[x], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.0000000000000006e-15 or 1.7999999999999999e36 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 91.6%

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

   3. Taylor expanded in y around 0 82.3%

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

   if -8.0000000000000006e-15 < z < 1.1e-290 or 5.4999999999999998e-242 < z < 1.7999999999999999e36

   1. Initial program 100.0%

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

      1. exp-diff 96.8%

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

      2. +-commutative 96.8%

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

      3. exp-sum 88.7%

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

      4. associate-*r/ 88.7%

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

      5. *-commutative 88.7%

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

      6. exp-to-pow 88.7%

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

      7. exp-diff 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in z around 0 91.9%

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

      1. *-commutative 91.9%

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

   6. Simplified 91.9%

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

   7. Taylor expanded in y around 0 75.6%

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

   if 1.1e-290 < z < 5.4999999999999998e-242

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 87.1%

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

   3. Taylor expanded in z around 0 87.1%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-15}:\\ \;\;\;\;e^{-z}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-290}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-242}:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+36}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \]

Alternative 5: 72.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-15} \lor \neg \left(z \leq 6.5 \cdot 10^{+39}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8e-15) (not (<= z 6.5e+39))) (exp (- z)) (exp x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8e-15) || !(z <= 6.5e+39)) {
		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 <= (-8d-15)) .or. (.not. (z <= 6.5d+39))) 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 <= -8e-15) || !(z <= 6.5e+39)) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8e-15) or not (z <= 6.5e+39):
		tmp = math.exp(-z)
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8e-15) || !(z <= 6.5e+39))
		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 <= -8e-15) || ~((z <= 6.5e+39)))
		tmp = exp(-z);
	else
		tmp = exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8e-15], N[Not[LessEqual[z, 6.5e+39]], $MachinePrecision]], N[Exp[(-z)], $MachinePrecision], N[Exp[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.0000000000000006e-15 or 6.5000000000000001e39 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 91.6%

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

   3. Taylor expanded in y around 0 82.3%

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

   if -8.0000000000000006e-15 < z < 6.5000000000000001e39

   1. Initial program 100.0%

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

      1. exp-diff 97.1%

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

      2. +-commutative 97.1%

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

      3. exp-sum 88.5%

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

      4. associate-*r/ 88.5%

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

      5. *-commutative 88.5%

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

      6. exp-to-pow 88.5%

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

      7. exp-diff 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in z around 0 91.4%

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

      1. *-commutative 91.4%

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

   6. Simplified 91.4%

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

   7. Taylor expanded in y around 0 71.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-15} \lor \neg \left(z \leq 6.5 \cdot 10^{+39}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \]

Alternative 6: 90.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.3 \cdot 10^{+98}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.3e+98) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.3e+98) {
		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 <= 5.3d+98) 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 <= 5.3e+98) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.3e+98:
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.3e+98)
		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 <= 5.3e+98)
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.3e+98], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.29999999999999997e98

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 95.6%

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

   if 5.29999999999999997e98 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 93.2%

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

   3. Taylor expanded in z around 0 86.3%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.3 \cdot 10^{+98}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]

Alternative 7: 51.9% 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. exp-diff 80.5%

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

   2. +-commutative 80.5%

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

   3. exp-sum 73.8%

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

   4. associate-*r/ 73.8%

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

   5. *-commutative 73.8%

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

   6. exp-to-pow 73.8%

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

   7. exp-diff 84.4%

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

3. Simplified 84.4%

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

4. Taylor expanded in z around 0 69.7%

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

   1. *-commutative 69.7%

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

6. Simplified 69.7%

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

7. Taylor expanded in y around 0 54.8%

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

8. Final simplification 54.8%

   \[\leadsto e^{x} \]

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 2023326 
(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)))