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

Percentage Accurate: 100.0% → 100.0%

Time: 3.7s

Alternatives: 6

Speedup: 1.0×

• 58.2% 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: 94.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{+14}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= t_0 4e+14) (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 <= 4e+14) {
		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 <= 4d+14) 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 <= 4e+14) {
		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 <= 4e+14:
		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 <= 4e+14)
		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 <= 4e+14)
		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, 4e+14], 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)) < 4e14

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 98.2%

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

   if 4e14 < (*.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 93.0%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 4 \cdot 10^{+14}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \]

Alternative 3: 73.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-z}\\ \mathbf{if}\;x \leq -2.85 \cdot 10^{+51}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-82}:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-281}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.82 \cdot 10^{-65}:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;x \leq 780000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- z))))
   (if (<= x -2.85e+51)
     (exp x)
     (if (<= x -3.4e-82)
       (pow y y)
       (if (<= x -1.95e-281)
         t_0
         (if (<= x 1.82e-65) (pow y y) (if (<= x 780000.0) t_0 (exp x))))))))

C

double code(double x, double y, double z) {
	double t_0 = exp(-z);
	double tmp;
	if (x <= -2.85e+51) {
		tmp = exp(x);
	} else if (x <= -3.4e-82) {
		tmp = pow(y, y);
	} else if (x <= -1.95e-281) {
		tmp = t_0;
	} else if (x <= 1.82e-65) {
		tmp = pow(y, y);
	} else if (x <= 780000.0) {
		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 <= (-2.85d+51)) then
        tmp = exp(x)
    else if (x <= (-3.4d-82)) then
        tmp = y ** y
    else if (x <= (-1.95d-281)) then
        tmp = t_0
    else if (x <= 1.82d-65) then
        tmp = y ** y
    else if (x <= 780000.0d0) 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 <= -2.85e+51) {
		tmp = Math.exp(x);
	} else if (x <= -3.4e-82) {
		tmp = Math.pow(y, y);
	} else if (x <= -1.95e-281) {
		tmp = t_0;
	} else if (x <= 1.82e-65) {
		tmp = Math.pow(y, y);
	} else if (x <= 780000.0) {
		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 <= -2.85e+51:
		tmp = math.exp(x)
	elif x <= -3.4e-82:
		tmp = math.pow(y, y)
	elif x <= -1.95e-281:
		tmp = t_0
	elif x <= 1.82e-65:
		tmp = math.pow(y, y)
	elif x <= 780000.0:
		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 <= -2.85e+51)
		tmp = exp(x);
	elseif (x <= -3.4e-82)
		tmp = y ^ y;
	elseif (x <= -1.95e-281)
		tmp = t_0;
	elseif (x <= 1.82e-65)
		tmp = y ^ y;
	elseif (x <= 780000.0)
		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 <= -2.85e+51)
		tmp = exp(x);
	elseif (x <= -3.4e-82)
		tmp = y ^ y;
	elseif (x <= -1.95e-281)
		tmp = t_0;
	elseif (x <= 1.82e-65)
		tmp = y ^ y;
	elseif (x <= 780000.0)
		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, -2.85e+51], N[Exp[x], $MachinePrecision], If[LessEqual[x, -3.4e-82], N[Power[y, y], $MachinePrecision], If[LessEqual[x, -1.95e-281], t$95$0, If[LessEqual[x, 1.82e-65], N[Power[y, y], $MachinePrecision], If[LessEqual[x, 780000.0], t$95$0, N[Exp[x], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8500000000000001e51 or 7.8e5 < 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 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. Taylor expanded in z around 0 67.3%

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

      1. *-commutative 67.3%

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

   6. Simplified 67.3%

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

   7. Taylor expanded in y around 0 85.6%

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

   if -2.8500000000000001e51 < x < -3.39999999999999975e-82 or -1.9500000000000001e-281 < x < 1.82e-65

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

      3. exp-sum 86.6%

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

      4. *-commutative 86.6%

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

      5. exp-to-pow 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in z around 0 74.9%

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

      1. *-commutative 74.9%

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

   6. Simplified 74.9%

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

   7. Taylor expanded in x around 0 75.7%

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

   if -3.39999999999999975e-82 < x < -1.9500000000000001e-281 or 1.82e-65 < x < 7.8e5

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 77.3%

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

      1. neg-mul-1 77.3%

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

   4. Simplified 77.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+51}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-82}:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-281}:\\ \;\;\;\;e^{-z}\\ \mathbf{elif}\;x \leq 1.82 \cdot 10^{-65}:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;x \leq 780000:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \]

Alternative 4: 73.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+61} \lor \neg \left(x \leq 780000\right):\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.15e+61) (not (<= x 780000.0))) (exp x) (exp (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.15e+61) || !(x <= 780000.0)) {
		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 <= (-1.15d+61)) .or. (.not. (x <= 780000.0d0))) 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 <= -1.15e+61) || !(x <= 780000.0)) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.exp(-z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.15e+61) or not (x <= 780000.0):
		tmp = math.exp(x)
	else:
		tmp = math.exp(-z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.15e+61) || !(x <= 780000.0))
		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 <= -1.15e+61) || ~((x <= 780000.0)))
		tmp = exp(x);
	else
		tmp = exp(-z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.15e+61], N[Not[LessEqual[x, 780000.0]], $MachinePrecision]], N[Exp[x], $MachinePrecision], N[Exp[(-z)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e61 or 7.8e5 < 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 71.4%

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

      4. *-commutative 71.4%

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

      5. exp-to-pow 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in z around 0 68.0%

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

      1. *-commutative 68.0%

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

   6. Simplified 68.0%

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

   7. Taylor expanded in y around 0 86.8%

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

   if -1.15e61 < x < 7.8e5

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 65.3%

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

      1. neg-mul-1 65.3%

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

   4. Simplified 65.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+61} \lor \neg \left(x \leq 780000\right):\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \]

Alternative 5: 87.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{+191}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.65e+191) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.65e+191) {
		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 <= 1.65d+191) 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 <= 1.65e+191) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.65e+191:
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.65e+191)
		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 <= 1.65e+191)
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.65e+191], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.6499999999999999e191

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 87.9%

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

   if 1.6499999999999999e191 < 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 67.3%

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

      4. *-commutative 67.3%

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

      5. exp-to-pow 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in z around 0 80.8%

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

      1. *-commutative 80.8%

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

   6. Simplified 80.8%

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

   7. Taylor expanded in x around 0 94.3%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{+191}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]

Alternative 6: 52.2% 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. 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 78.5%

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

   4. *-commutative 78.5%

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

   5. exp-to-pow 78.5%

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

3. Simplified 78.5%

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

4. Taylor expanded in z around 0 66.1%

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

   1. *-commutative 66.1%

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

6. Simplified 66.1%

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

7. Taylor expanded in y around 0 53.2%

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

8. 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 2023336 
(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)))