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

Percentage Accurate: 100.0% → 100.0%

Time: 6.1s

Alternatives: 8

Speedup: 1.0×

• 58.1% 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: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;t_0 \leq 200:\\ \;\;\;\;{y}^{y} \cdot e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= t_0 200.0) (* (pow y y) (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 <= 200.0) {
		tmp = pow(y, y) * 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 <= 200.0d0) then
        tmp = (y ** y) * 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 <= 200.0) {
		tmp = Math.pow(y, y) * 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 <= 200.0:
		tmp = math.pow(y, y) * 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 <= 200.0)
		tmp = Float64((y ^ y) * 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 <= 200.0)
		tmp = (y ^ y) * 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, 200.0], N[(N[Power[y, y], $MachinePrecision] * N[Exp[N[(x - z), $MachinePrecision]], $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)) < 200

   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 99.9%

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

      4. *-commutative 99.9%

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

      5. exp-to-pow 100.0%

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

   3. Simplified 100.0%

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

   if 200 < (*.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 89.8%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 200:\\ \;\;\;\;{y}^{y} \cdot e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-8}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.8e-8) (exp (- x z)) (exp (- (* y (log y)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.8e-8) {
		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 (y <= 1.8d-8) 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 (y <= 1.8e-8) {
		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 y <= 1.8e-8:
		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 (y <= 1.8e-8)
		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 (y <= 1.8e-8)
		tmp = exp((x - z));
	else
		tmp = exp(((y * log(y)) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.8e-8], 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 y < 1.79999999999999991e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.8%

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

   if 1.79999999999999991e-8 < 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 89.9%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-8}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-124}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-45}:\\ \;\;\;\;e^{-z}\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.6e-124)
   (exp x)
   (if (<= y 8.2e-45) (exp (- z)) (if (<= y 0.0008) (exp x) (pow y y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.6e-124) {
		tmp = exp(x);
	} else if (y <= 8.2e-45) {
		tmp = exp(-z);
	} else if (y <= 0.0008) {
		tmp = exp(x);
	} 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 <= 3.6d-124) then
        tmp = exp(x)
    else if (y <= 8.2d-45) then
        tmp = exp(-z)
    else if (y <= 0.0008d0) then
        tmp = exp(x)
    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 <= 3.6e-124) {
		tmp = Math.exp(x);
	} else if (y <= 8.2e-45) {
		tmp = Math.exp(-z);
	} else if (y <= 0.0008) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.6e-124:
		tmp = math.exp(x)
	elif y <= 8.2e-45:
		tmp = math.exp(-z)
	elif y <= 0.0008:
		tmp = math.exp(x)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.6e-124)
		tmp = exp(x);
	elseif (y <= 8.2e-45)
		tmp = exp(Float64(-z));
	elseif (y <= 0.0008)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.6e-124)
		tmp = exp(x);
	elseif (y <= 8.2e-45)
		tmp = exp(-z);
	elseif (y <= 0.0008)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.6e-124], N[Exp[x], $MachinePrecision], If[LessEqual[y, 8.2e-45], N[Exp[(-z)], $MachinePrecision], If[LessEqual[y, 0.0008], N[Exp[x], $MachinePrecision], N[Power[y, y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.6000000000000001e-124 or 8.1999999999999998e-45 < y < 8.00000000000000038e-4

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. exp-diff 99.9%

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

      5. *-commutative 99.9%

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

      6. exp-to-pow 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 99.7%

      \[\leadsto \color{blue}{\frac{1}{e^{z - x}}} \]

   6. Taylor expanded in z around 0 76.1%

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

      1. exp-neg 76.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{x}}}} \]

      2. remove-double-div 76.1%

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

   8. Simplified 76.1%

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

   if 3.6000000000000001e-124 < y < 8.1999999999999998e-45

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 81.4%

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

   4. Taylor expanded in y around 0 81.4%

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

   if 8.00000000000000038e-4 < 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 89.8%

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

   4. Taylor expanded in z around 0 83.6%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-124}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-45}:\\ \;\;\;\;e^{-z}\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+29} \lor \neg \left(z \leq 1.22 \cdot 10^{+98}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.8e+29) (not (<= z 1.22e+98))) (exp (- z)) (exp x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.8e+29) || !(z <= 1.22e+98)) {
		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 <= (-4.8d+29)) .or. (.not. (z <= 1.22d+98))) 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 <= -4.8e+29) || !(z <= 1.22e+98)) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.8e+29) or not (z <= 1.22e+98):
		tmp = math.exp(-z)
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.8e+29) || !(z <= 1.22e+98))
		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 <= -4.8e+29) || ~((z <= 1.22e+98)))
		tmp = exp(-z);
	else
		tmp = exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.8e+29], N[Not[LessEqual[z, 1.22e+98]], $MachinePrecision]], N[Exp[(-z)], $MachinePrecision], N[Exp[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8000000000000002e29 or 1.22e98 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 94.2%

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

   4. Taylor expanded in y around 0 83.4%

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

   if -4.8000000000000002e29 < z < 1.22e98

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. exp-diff 83.1%

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

      5. *-commutative 83.1%

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

      6. exp-to-pow 83.2%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in y around 0 69.2%

      \[\leadsto \color{blue}{\frac{1}{e^{z - x}}} \]

   6. Taylor expanded in z around 0 65.6%

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

      1. exp-neg 65.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{x}}}} \]

      2. remove-double-div 65.6%

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

   8. Simplified 65.6%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+29} \lor \neg \left(z \leq 1.22 \cdot 10^{+98}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0185:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 0.0185) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0185) {
		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 <= 0.0185d0) 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 <= 0.0185) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.0185:
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.0185)
		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 <= 0.0185)
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.0185], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0184999999999999991

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.8%

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

   if 0.0184999999999999991 < 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 89.8%

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

   4. Taylor expanded in z around 0 83.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0185:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.4% 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. associate--l+ 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-+l- 100.0%

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

   4. exp-diff 81.6%

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

   5. *-commutative 81.6%

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

   6. exp-to-pow 81.6%

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

3. Simplified 81.6%

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

5. Taylor expanded in y around 0 76.7%

   \[\leadsto \color{blue}{\frac{1}{e^{z - x}}} \]

6. Taylor expanded in z around 0 53.0%

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

   1. exp-neg 53.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{x}}}} \]

   2. remove-double-div 53.0%

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

8. Simplified 53.0%

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

9. Final simplification 53.0%

   \[\leadsto e^{x} \]
10. Add Preprocessing

Alternative 8: 14.3% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ x + 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x 1.0))

C

double code(double x, double y, double z) {
	return x + 1.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return x + 1.0;
}

Python

def code(x, y, z):
	return x + 1.0

Julia

function code(x, y, z)
	return Float64(x + 1.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + 1.0;
end

Wolfram

code[x_, y_, z_] := N[(x + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around 0 83.3%

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

   1. exp-sum 71.1%

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

   2. *-commutative 71.1%

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

   3. exp-to-pow 71.2%

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

   4. *-commutative 71.2%

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

5. Simplified 71.2%

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

6. Taylor expanded in x around 0 35.1%

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

   1. distribute-lft1-in 46.6%

      \[\leadsto \color{blue}{\left(x + 1\right) \cdot {y}^{y}} \]

   2. *-commutative 46.6%

      \[\leadsto \color{blue}{{y}^{y} \cdot \left(x + 1\right)} \]

   3. +-commutative 46.6%

      \[\leadsto {y}^{y} \cdot \color{blue}{\left(1 + x\right)} \]

8. Simplified 46.6%

   \[\leadsto \color{blue}{{y}^{y} \cdot \left(1 + x\right)} \]

9. Taylor expanded in y around 0 13.6%

   \[\leadsto \color{blue}{1 + x} \]

10. Final simplification 13.6%

    \[\leadsto x + 1 \]
11. 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 2024020 
(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)))