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

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 6

Speedup: 1.0×

• 51.7% 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. Final simplification 100.0%

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

Alternative 2: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+17}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= t_0 2e+17) (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 <= 2e+17) {
		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 <= 2d+17) 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 <= 2e+17) {
		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 <= 2e+17:
		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 <= 2e+17)
		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 <= 2e+17)
		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, 2e+17], 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)) < 2e17

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.8%

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

   if 2e17 < (*.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 91.8%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 2 \cdot 10^{+17}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \]

Alternative 3: 73.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-z}\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{-19}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-240}:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- z))))
   (if (<= x -2.1e-19)
     (exp x)
     (if (<= x -1.65e-134)
       t_0
       (if (<= x 3.1e-240) (pow y y) (if (<= x 9e+28) t_0 (exp x)))))))

C

double code(double x, double y, double z) {
	double t_0 = exp(-z);
	double tmp;
	if (x <= -2.1e-19) {
		tmp = exp(x);
	} else if (x <= -1.65e-134) {
		tmp = t_0;
	} else if (x <= 3.1e-240) {
		tmp = pow(y, y);
	} else if (x <= 9e+28) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = exp(-z)
    if (x <= (-2.1d-19)) then
        tmp = exp(x)
    else if (x <= (-1.65d-134)) then
        tmp = t_0
    else if (x <= 3.1d-240) then
        tmp = y ** y
    else if (x <= 9d+28) then
        tmp = t_0
    else
        tmp = exp(x)
    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 (x <= -2.1e-19) {
		tmp = Math.exp(x);
	} else if (x <= -1.65e-134) {
		tmp = t_0;
	} else if (x <= 3.1e-240) {
		tmp = Math.pow(y, y);
	} else if (x <= 9e+28) {
		tmp = t_0;
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.exp(-z)
	tmp = 0
	if x <= -2.1e-19:
		tmp = math.exp(x)
	elif x <= -1.65e-134:
		tmp = t_0
	elif x <= 3.1e-240:
		tmp = math.pow(y, y)
	elif x <= 9e+28:
		tmp = t_0
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x, y, z)
	t_0 = exp(Float64(-z))
	tmp = 0.0
	if (x <= -2.1e-19)
		tmp = exp(x);
	elseif (x <= -1.65e-134)
		tmp = t_0;
	elseif (x <= 3.1e-240)
		tmp = y ^ y;
	elseif (x <= 9e+28)
		tmp = t_0;
	else
		tmp = exp(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = exp(-z);
	tmp = 0.0;
	if (x <= -2.1e-19)
		tmp = exp(x);
	elseif (x <= -1.65e-134)
		tmp = t_0;
	elseif (x <= 3.1e-240)
		tmp = y ^ y;
	elseif (x <= 9e+28)
		tmp = t_0;
	else
		tmp = exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Exp[(-z)], $MachinePrecision]}, If[LessEqual[x, -2.1e-19], N[Exp[x], $MachinePrecision], If[LessEqual[x, -1.65e-134], t$95$0, If[LessEqual[x, 3.1e-240], N[Power[y, y], $MachinePrecision], If[LessEqual[x, 9e+28], t$95$0, N[Exp[x], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0999999999999999e-19 or 8.9999999999999994e28 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 91.2%

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

      1. log-pow 81.6%

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

      2. exp-sum 72.4%

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

      3. rem-exp-log 72.4%

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

   4. Simplified 72.4%

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

   5. Taylor expanded in y around 0 84.0%

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

   if -2.0999999999999999e-19 < x < -1.6500000000000001e-134 or 3.10000000000000017e-240 < x < 8.9999999999999994e28

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.0%

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

      1. log-pow 84.8%

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

      2. div-exp 77.8%

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

      3. rem-exp-log 77.8%

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

   4. Simplified 77.8%

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

   5. Taylor expanded in y around 0 77.3%

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

      1. rec-exp 77.3%

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

   7. Simplified 77.3%

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

   if -1.6500000000000001e-134 < x < 3.10000000000000017e-240

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 71.3%

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

      1. log-pow 71.3%

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

      2. exp-sum 71.3%

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

      3. rem-exp-log 71.3%

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

   4. Simplified 71.3%

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

   5. Taylor expanded in x around 0 71.3%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-19}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-134}:\\ \;\;\;\;e^{-z}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-240}:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+28}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \]

Alternative 4: 88.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+47} \lor \neg \left(y \leq 4.9 \cdot 10^{+70}\right) \land y \leq 1.05 \cdot 10^{+175}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y 9.5e+47) (and (not (<= y 4.9e+70)) (<= y 1.05e+175)))
   (exp (- x z))
   (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= 9.5e+47) || (!(y <= 4.9e+70) && (y <= 1.05e+175))) {
		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 <= 9.5d+47) .or. (.not. (y <= 4.9d+70)) .and. (y <= 1.05d+175)) 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 <= 9.5e+47) || (!(y <= 4.9e+70) && (y <= 1.05e+175))) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= 9.5e+47) or (not (y <= 4.9e+70) and (y <= 1.05e+175)):
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= 9.5e+47) || (!(y <= 4.9e+70) && (y <= 1.05e+175)))
		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 <= 9.5e+47) || (~((y <= 4.9e+70)) && (y <= 1.05e+175)))
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, 9.5e+47], And[N[Not[LessEqual[y, 4.9e+70]], $MachinePrecision], LessEqual[y, 1.05e+175]]], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.50000000000000001e47 or 4.90000000000000028e70 < y < 1.05e175

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 91.3%

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

   if 9.50000000000000001e47 < y < 4.90000000000000028e70 or 1.05e175 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 96.1%

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

      1. log-pow 94.2%

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

      2. exp-sum 82.4%

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

      3. rem-exp-log 82.4%

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

   4. Simplified 82.4%

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

   5. Taylor expanded in x around 0 94.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+47} \lor \neg \left(y \leq 4.9 \cdot 10^{+70}\right) \land y \leq 1.05 \cdot 10^{+175}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]

Alternative 5: 73.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-19}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+29}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.1e-19) (exp x) (if (<= x 2.2e+29) (exp (- z)) (exp x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e-19) {
		tmp = exp(x);
	} else if (x <= 2.2e+29) {
		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-19)) then
        tmp = exp(x)
    else if (x <= 2.2d+29) 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-19) {
		tmp = Math.exp(x);
	} else if (x <= 2.2e+29) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.1e-19:
		tmp = math.exp(x)
	elif x <= 2.2e+29:
		tmp = math.exp(-z)
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.1e-19)
		tmp = exp(x);
	elseif (x <= 2.2e+29)
		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-19)
		tmp = exp(x);
	elseif (x <= 2.2e+29)
		tmp = exp(-z);
	else
		tmp = exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.1e-19], N[Exp[x], $MachinePrecision], If[LessEqual[x, 2.2e+29], N[Exp[(-z)], $MachinePrecision], N[Exp[x], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0999999999999999e-19 or 2.2000000000000001e29 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 91.2%

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

      1. log-pow 81.6%

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

      2. exp-sum 72.4%

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

      3. rem-exp-log 72.4%

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

   4. Simplified 72.4%

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

   5. Taylor expanded in y around 0 84.0%

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

   if -2.0999999999999999e-19 < x < 2.2000000000000001e29

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.7%

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

      1. log-pow 89.1%

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

      2. div-exp 79.9%

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

      3. rem-exp-log 79.9%

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

   4. Simplified 79.9%

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

   5. Taylor expanded in y around 0 69.4%

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

      1. rec-exp 69.4%

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

   7. Simplified 69.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-19}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+29}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \]

Alternative 6: 53.2% 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 z around 0 75.2%

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

   1. log-pow 70.6%

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

   2. exp-sum 66.2%

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

   3. rem-exp-log 66.2%

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

4. Simplified 66.2%

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

5. Taylor expanded in y around 0 53.2%

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

6. Final simplification 53.2%

   \[\leadsto e^{x} \]

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 2023279 
(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)))