Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.8%

Time: 8.9s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \log y - z\right) - y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (- (* x (log y)) z) y))

C

double code(double x, double y, double z) {
	return ((x * log(y)) - z) - 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 = ((x * log(y)) - z) - y
end function

Java

public static double code(double x, double y, double z) {
	return ((x * Math.log(y)) - z) - y;
}

Python

def code(x, y, z):
	return ((x * math.log(y)) - z) - y

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * log(y)) - z) - y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * log(y)) - z) - y;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \log y - z\right) - y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (- (* x (log y)) z) y))

C

double code(double x, double y, double z) {
	return ((x * log(y)) - z) - 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 = ((x * log(y)) - z) - y
end function

Java

public static double code(double x, double y, double z) {
	return ((x * Math.log(y)) - z) - y;
}

Python

def code(x, y, z):
	return ((x * math.log(y)) - z) - y

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * log(y)) - z) - y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * log(y)) - z) - y;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt[3]{y}\right)\\ \left(x \cdot \left(t_0 + 2 \cdot t_0\right) - z\right) - y \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (cbrt y)))) (- (- (* x (+ t_0 (* 2.0 t_0))) z) y)))

C

double code(double x, double y, double z) {
	double t_0 = log(cbrt(y));
	return ((x * (t_0 + (2.0 * t_0))) - z) - y;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log(Math.cbrt(y));
	return ((x * (t_0 + (2.0 * t_0))) - z) - y;
}

Julia

function code(x, y, z)
	t_0 = log(cbrt(y))
	return Float64(Float64(Float64(x * Float64(t_0 + Float64(2.0 * t_0))) - z) - y)
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[Power[y, 1/3], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(x * N[(t$95$0 + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Initial program 99.9%

   \[\left(x \cdot \log y - z\right) - y \]
2. Step-by-step derivation

   1. add-log-exp 50.2%

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

   2. *-commutative 50.2%

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

   3. exp-to-pow 50.2%

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

3. Applied egg-rr 50.2%

   \[\leadsto \left(\color{blue}{\log \left({y}^{x}\right)} - z\right) - y \]
4. Step-by-step derivation

   1. add-cube-cbrt 50.2%

      \[\leadsto \left(\log \left({\color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)}}^{x}\right) - z\right) - y \]

   2. unpow-prod-down 50.2%

      \[\leadsto \left(\log \color{blue}{\left({\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{x} \cdot {\left(\sqrt[3]{y}\right)}^{x}\right)} - z\right) - y \]

   3. log-prod 50.2%

      \[\leadsto \left(\color{blue}{\left(\log \left({\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{x}\right) + \log \left({\left(\sqrt[3]{y}\right)}^{x}\right)\right)} - z\right) - y \]

   4. pow2 50.2%

      \[\leadsto \left(\left(\log \left({\color{blue}{\left({\left(\sqrt[3]{y}\right)}^{2}\right)}}^{x}\right) + \log \left({\left(\sqrt[3]{y}\right)}^{x}\right)\right) - z\right) - y \]

5. Applied egg-rr 50.2%

   \[\leadsto \left(\color{blue}{\left(\log \left({\left({\left(\sqrt[3]{y}\right)}^{2}\right)}^{x}\right) + \log \left({\left(\sqrt[3]{y}\right)}^{x}\right)\right)} - z\right) - y \]
6. Step-by-step derivation

   1. log-pow 50.6%

      \[\leadsto \left(\left(\color{blue}{x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{2}\right)} + \log \left({\left(\sqrt[3]{y}\right)}^{x}\right)\right) - z\right) - y \]

   2. log-pow 99.8%

      \[\leadsto \left(\left(x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{2}\right) + \color{blue}{x \cdot \log \left(\sqrt[3]{y}\right)}\right) - z\right) - y \]

   3. distribute-lft-out 99.9%

      \[\leadsto \left(\color{blue}{x \cdot \left(\log \left({\left(\sqrt[3]{y}\right)}^{2}\right) + \log \left(\sqrt[3]{y}\right)\right)} - z\right) - y \]

   4. log-pow 99.9%

      \[\leadsto \left(x \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{y}\right)} + \log \left(\sqrt[3]{y}\right)\right) - z\right) - y \]

7. Simplified 99.9%

   \[\leadsto \left(\color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} - z\right) - y \]

8. Final simplification 99.9%

   \[\leadsto \left(x \cdot \left(\log \left(\sqrt[3]{y}\right) + 2 \cdot \log \left(\sqrt[3]{y}\right)\right) - z\right) - y \]

Alternative 2: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log y\\ \mathbf{if}\;y \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;t_0 - z\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+127} \lor \neg \left(y \leq 1.7 \cdot 10^{+163}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t_0 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log y))))
   (if (<= y 7.5e+42)
     (- t_0 z)
     (if (or (<= y 4e+127) (not (<= y 1.7e+163))) (- (- z) y) (- t_0 y)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log(y);
	double tmp;
	if (y <= 7.5e+42) {
		tmp = t_0 - z;
	} else if ((y <= 4e+127) || !(y <= 1.7e+163)) {
		tmp = -z - y;
	} else {
		tmp = t_0 - 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) :: t_0
    real(8) :: tmp
    t_0 = x * log(y)
    if (y <= 7.5d+42) then
        tmp = t_0 - z
    else if ((y <= 4d+127) .or. (.not. (y <= 1.7d+163))) then
        tmp = -z - y
    else
        tmp = t_0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log(y);
	double tmp;
	if (y <= 7.5e+42) {
		tmp = t_0 - z;
	} else if ((y <= 4e+127) || !(y <= 1.7e+163)) {
		tmp = -z - y;
	} else {
		tmp = t_0 - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log(y)
	tmp = 0
	if y <= 7.5e+42:
		tmp = t_0 - z
	elif (y <= 4e+127) or not (y <= 1.7e+163):
		tmp = -z - y
	else:
		tmp = t_0 - y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(y))
	tmp = 0.0
	if (y <= 7.5e+42)
		tmp = Float64(t_0 - z);
	elseif ((y <= 4e+127) || !(y <= 1.7e+163))
		tmp = Float64(Float64(-z) - y);
	else
		tmp = Float64(t_0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log(y);
	tmp = 0.0;
	if (y <= 7.5e+42)
		tmp = t_0 - z;
	elseif ((y <= 4e+127) || ~((y <= 1.7e+163)))
		tmp = -z - y;
	else
		tmp = t_0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.5e+42], N[(t$95$0 - z), $MachinePrecision], If[Or[LessEqual[y, 4e+127], N[Not[LessEqual[y, 1.7e+163]], $MachinePrecision]], N[((-z) - y), $MachinePrecision], N[(t$95$0 - y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.50000000000000041e42

   1. Initial program 99.8%

      \[\left(x \cdot \log y - z\right) - y \]
   2. Step-by-step derivation

      1. add-log-exp 46.6%

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

      2. *-commutative 46.6%

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

      3. exp-to-pow 46.6%

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

   3. Applied egg-rr 46.6%

      \[\leadsto \left(\color{blue}{\log \left({y}^{x}\right)} - z\right) - y \]
   4. Step-by-step derivation

      1. add-cube-cbrt 46.6%

         \[\leadsto \left(\log \left({\color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)}}^{x}\right) - z\right) - y \]

      2. unpow-prod-down 46.6%

         \[\leadsto \left(\log \color{blue}{\left({\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{x} \cdot {\left(\sqrt[3]{y}\right)}^{x}\right)} - z\right) - y \]

      3. log-prod 46.6%

         \[\leadsto \left(\color{blue}{\left(\log \left({\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{x}\right) + \log \left({\left(\sqrt[3]{y}\right)}^{x}\right)\right)} - z\right) - y \]

      4. pow2 46.6%

         \[\leadsto \left(\left(\log \left({\color{blue}{\left({\left(\sqrt[3]{y}\right)}^{2}\right)}}^{x}\right) + \log \left({\left(\sqrt[3]{y}\right)}^{x}\right)\right) - z\right) - y \]

   5. Applied egg-rr 46.6%

      \[\leadsto \left(\color{blue}{\left(\log \left({\left({\left(\sqrt[3]{y}\right)}^{2}\right)}^{x}\right) + \log \left({\left(\sqrt[3]{y}\right)}^{x}\right)\right)} - z\right) - y \]
   6. Step-by-step derivation

      1. log-pow 47.4%

         \[\leadsto \left(\left(\color{blue}{x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{2}\right)} + \log \left({\left(\sqrt[3]{y}\right)}^{x}\right)\right) - z\right) - y \]

      2. log-pow 99.8%

         \[\leadsto \left(\left(x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{2}\right) + \color{blue}{x \cdot \log \left(\sqrt[3]{y}\right)}\right) - z\right) - y \]

      3. distribute-lft-out 99.8%

         \[\leadsto \left(\color{blue}{x \cdot \left(\log \left({\left(\sqrt[3]{y}\right)}^{2}\right) + \log \left(\sqrt[3]{y}\right)\right)} - z\right) - y \]

      4. log-pow 99.8%

         \[\leadsto \left(x \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{y}\right)} + \log \left(\sqrt[3]{y}\right)\right) - z\right) - y \]

   7. Simplified 99.8%

      \[\leadsto \left(\color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} - z\right) - y \]
   8. Step-by-step derivation

      1. add-log-exp 99.8%

         \[\leadsto \left(x \cdot \left(\color{blue}{\log \left(e^{2 \cdot \log \left(\sqrt[3]{y}\right)}\right)} + \log \left(\sqrt[3]{y}\right)\right) - z\right) - y \]

      2. *-commutative 99.8%

         \[\leadsto \left(x \cdot \left(\log \left(e^{\color{blue}{\log \left(\sqrt[3]{y}\right) \cdot 2}}\right) + \log \left(\sqrt[3]{y}\right)\right) - z\right) - y \]

      3. exp-to-pow 99.8%

         \[\leadsto \left(x \cdot \left(\log \color{blue}{\left({\left(\sqrt[3]{y}\right)}^{2}\right)} + \log \left(\sqrt[3]{y}\right)\right) - z\right) - y \]

      4. pow2 99.8%

         \[\leadsto \left(x \cdot \left(\log \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)} + \log \left(\sqrt[3]{y}\right)\right) - z\right) - y \]

      5. log-prod 99.8%

         \[\leadsto \left(x \cdot \color{blue}{\log \left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} - z\right) - y \]

      6. add-cube-cbrt 99.8%

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

      7. expm1-log1p-u 14.8%

         \[\leadsto \left(x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log y\right)\right)} - z\right) - y \]

   9. Applied egg-rr 14.8%

      \[\leadsto \left(x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log y\right)\right)} - z\right) - y \]

   10. Taylor expanded in y around 0 90.0%

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

   if 7.50000000000000041e42 < y < 3.99999999999999982e127 or 1.7000000000000001e163 < y

   1. Initial program 100.0%

      \[\left(x \cdot \log y - z\right) - y \]

   2. Taylor expanded in x around 0 89.3%

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

      1. neg-mul-1 89.3%

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

   4. Simplified 89.3%

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

   if 3.99999999999999982e127 < y < 1.7000000000000001e163

   1. Initial program 99.7%

      \[\left(x \cdot \log y - z\right) - y \]

   2. Taylor expanded in z around 0 90.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+127} \lor \neg \left(y \leq 1.7 \cdot 10^{+163}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \]

Alternative 3: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+30} \lor \neg \left(x \leq 3 \cdot 10^{+173}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.9e+30) (not (<= x 3e+173))) (- (* x (log y)) y) (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.9e+30) || !(x <= 3e+173)) {
		tmp = (x * log(y)) - y;
	} else {
		tmp = -z - 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 ((x <= (-3.9d+30)) .or. (.not. (x <= 3d+173))) then
        tmp = (x * log(y)) - y
    else
        tmp = -z - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.9e+30) || !(x <= 3e+173)) {
		tmp = (x * Math.log(y)) - y;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.9e+30) or not (x <= 3e+173):
		tmp = (x * math.log(y)) - y
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.9e+30) || !(x <= 3e+173))
		tmp = Float64(Float64(x * log(y)) - y);
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.9e+30) || ~((x <= 3e+173)))
		tmp = (x * log(y)) - y;
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.9e+30], N[Not[LessEqual[x, 3e+173]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.90000000000000011e30 or 2.9999999999999998e173 < x

   1. Initial program 99.7%

      \[\left(x \cdot \log y - z\right) - y \]

   2. Taylor expanded in z around 0 86.2%

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

   if -3.90000000000000011e30 < x < 2.9999999999999998e173

   1. Initial program 99.9%

      \[\left(x \cdot \log y - z\right) - y \]

   2. Taylor expanded in x around 0 87.5%

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

      1. neg-mul-1 87.5%

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

   4. Simplified 87.5%

      \[\leadsto \color{blue}{\left(-z\right)} - y \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+30} \lor \neg \left(x \leq 3 \cdot 10^{+173}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

Alternative 4: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+203} \lor \neg \left(x \leq 1.4 \cdot 10^{+179}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.1e+203) (not (<= x 1.4e+179))) (* x (log y)) (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.1e+203) || !(x <= 1.4e+179)) {
		tmp = x * log(y);
	} else {
		tmp = -z - 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 ((x <= (-3.1d+203)) .or. (.not. (x <= 1.4d+179))) then
        tmp = x * log(y)
    else
        tmp = -z - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.1e+203) || !(x <= 1.4e+179)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.1e+203) or not (x <= 1.4e+179):
		tmp = x * math.log(y)
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.1e+203) || !(x <= 1.4e+179))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.1e+203) || ~((x <= 1.4e+179)))
		tmp = x * log(y);
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.1e+203], N[Not[LessEqual[x, 1.4e+179]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e203 or 1.4e179 < x

   1. Initial program 99.6%

      \[\left(x \cdot \log y - z\right) - y \]

   2. Taylor expanded in z around 0 93.9%

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

      1. fma-neg 93.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, -y\right)} \]

   4. Simplified 93.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, -y\right)} \]

   5. Taylor expanded in x around inf 83.0%

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

   if -3.1e203 < x < 1.4e179

   1. Initial program 99.9%

      \[\left(x \cdot \log y - z\right) - y \]

   2. Taylor expanded in x around 0 83.9%

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

      1. neg-mul-1 83.9%

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

   4. Simplified 83.9%

      \[\leadsto \color{blue}{\left(-z\right)} - y \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+203} \lor \neg \left(x \leq 1.4 \cdot 10^{+179}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

Alternative 5: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \log y - z\right) - y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (- (* x (log y)) z) y))

C

double code(double x, double y, double z) {
	return ((x * log(y)) - z) - 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 = ((x * log(y)) - z) - y
end function

Java

public static double code(double x, double y, double z) {
	return ((x * Math.log(y)) - z) - y;
}

Python

def code(x, y, z):
	return ((x * math.log(y)) - z) - y

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * log(y)) - z) - y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * log(y)) - z) - y;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x \cdot \log y - z\right) - y \]

2. Final simplification 99.9%

   \[\leadsto \left(x \cdot \log y - z\right) - y \]

Alternative 6: 53.0% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{+69}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 5.6e+69) (- z) (- y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.6e+69) {
		tmp = -z;
	} else {
		tmp = -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 <= 5.6d+69) then
        tmp = -z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.6e+69) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.6e+69:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.6e+69)
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.6e+69)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.6e+69], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 5.59999999999999964e69

   1. Initial program 99.8%

      \[\left(x \cdot \log y - z\right) - y \]
   2. Step-by-step derivation

      1. add-log-exp 49.3%

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

      2. *-commutative 49.3%

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

      3. exp-to-pow 49.3%

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

   3. Applied egg-rr 49.3%

      \[\leadsto \left(\color{blue}{\log \left({y}^{x}\right)} - z\right) - y \]
   4. Step-by-step derivation

      1. add-cube-cbrt 49.3%

         \[\leadsto \left(\log \left({\color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)}}^{x}\right) - z\right) - y \]

      2. unpow-prod-down 49.3%

         \[\leadsto \left(\log \color{blue}{\left({\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{x} \cdot {\left(\sqrt[3]{y}\right)}^{x}\right)} - z\right) - y \]

      3. log-prod 49.3%

         \[\leadsto \left(\color{blue}{\left(\log \left({\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{x}\right) + \log \left({\left(\sqrt[3]{y}\right)}^{x}\right)\right)} - z\right) - y \]

      4. pow2 49.3%

         \[\leadsto \left(\left(\log \left({\color{blue}{\left({\left(\sqrt[3]{y}\right)}^{2}\right)}}^{x}\right) + \log \left({\left(\sqrt[3]{y}\right)}^{x}\right)\right) - z\right) - y \]

   5. Applied egg-rr 49.3%

      \[\leadsto \left(\color{blue}{\left(\log \left({\left({\left(\sqrt[3]{y}\right)}^{2}\right)}^{x}\right) + \log \left({\left(\sqrt[3]{y}\right)}^{x}\right)\right)} - z\right) - y \]
   6. Step-by-step derivation

      1. log-pow 50.0%

         \[\leadsto \left(\left(\color{blue}{x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{2}\right)} + \log \left({\left(\sqrt[3]{y}\right)}^{x}\right)\right) - z\right) - y \]

      2. log-pow 99.8%

         \[\leadsto \left(\left(x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{2}\right) + \color{blue}{x \cdot \log \left(\sqrt[3]{y}\right)}\right) - z\right) - y \]

      3. distribute-lft-out 99.8%

         \[\leadsto \left(\color{blue}{x \cdot \left(\log \left({\left(\sqrt[3]{y}\right)}^{2}\right) + \log \left(\sqrt[3]{y}\right)\right)} - z\right) - y \]

      4. log-pow 99.8%

         \[\leadsto \left(x \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{y}\right)} + \log \left(\sqrt[3]{y}\right)\right) - z\right) - y \]

   7. Simplified 99.8%

      \[\leadsto \left(\color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} - z\right) - y \]
   8. Step-by-step derivation

      1. add-log-exp 99.8%

         \[\leadsto \left(x \cdot \left(\color{blue}{\log \left(e^{2 \cdot \log \left(\sqrt[3]{y}\right)}\right)} + \log \left(\sqrt[3]{y}\right)\right) - z\right) - y \]

      2. *-commutative 99.8%

         \[\leadsto \left(x \cdot \left(\log \left(e^{\color{blue}{\log \left(\sqrt[3]{y}\right) \cdot 2}}\right) + \log \left(\sqrt[3]{y}\right)\right) - z\right) - y \]

      3. exp-to-pow 99.8%

         \[\leadsto \left(x \cdot \left(\log \color{blue}{\left({\left(\sqrt[3]{y}\right)}^{2}\right)} + \log \left(\sqrt[3]{y}\right)\right) - z\right) - y \]

      4. pow2 99.8%

         \[\leadsto \left(x \cdot \left(\log \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)} + \log \left(\sqrt[3]{y}\right)\right) - z\right) - y \]

      5. log-prod 99.8%

         \[\leadsto \left(x \cdot \color{blue}{\log \left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} - z\right) - y \]

      6. add-cube-cbrt 99.8%

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

      7. expm1-log1p-u 23.9%

         \[\leadsto \left(x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log y\right)\right)} - z\right) - y \]

   9. Applied egg-rr 23.9%

      \[\leadsto \left(x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log y\right)\right)} - z\right) - y \]

   10. Taylor expanded in z around inf 48.1%

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

       1. neg-mul-1 48.1%

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

   12. Simplified 48.1%

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

   if 5.59999999999999964e69 < y

   1. Initial program 99.9%

      \[\left(x \cdot \log y - z\right) - y \]

   2. Taylor expanded in y around inf 68.6%

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

      1. neg-mul-1 68.6%

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

   4. Simplified 68.6%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{+69}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 7: 66.9% accurate, 26.8× speedup

Math

\[\begin{array}{l} \\ \left(-z\right) - y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (- z) y))

C

double code(double x, double y, double z) {
	return -z - 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 = -z - y
end function

Java

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

Python

def code(x, y, z):
	return -z - y

Julia

function code(x, y, z)
	return Float64(Float64(-z) - y)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[((-z) - y), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x \cdot \log y - z\right) - y \]

2. Taylor expanded in x around 0 70.0%

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

   1. neg-mul-1 70.0%

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

4. Simplified 70.0%

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

5. Final simplification 70.0%

   \[\leadsto \left(-z\right) - y \]

Alternative 8: 33.7% accurate, 53.5× speedup

Math

\[\begin{array}{l} \\ -y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- y))

C

double code(double x, double y, double z) {
	return -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 = -y
end function

Java

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

Python

def code(x, y, z):
	return -y

Julia

function code(x, y, z)
	return Float64(-y)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := (-y)

Derivation

1. Initial program 99.9%

   \[\left(x \cdot \log y - z\right) - y \]

2. Taylor expanded in y around inf 34.8%

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

   1. neg-mul-1 34.8%

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

4. Simplified 34.8%

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

5. Final simplification 34.8%

   \[\leadsto -y \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))