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

Percentage Accurate: 99.9% → 99.9%

Time: 8.6s

Alternatives: 9

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: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return fma(x, log(y), (-z - y));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l- 99.8%

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

   2. fma-neg 99.8%

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

   3. +-commutative 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+14} \lor \neg \left(z \leq 2.65 \cdot 10^{-20}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.4e+14) (not (<= z 2.65e-20)))
   (- (- z) y)
   (- (* x (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.4e+14) || !(z <= 2.65e-20)) {
		tmp = -z - y;
	} else {
		tmp = (x * log(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 ((z <= (-3.4d+14)) .or. (.not. (z <= 2.65d-20))) then
        tmp = -z - y
    else
        tmp = (x * log(y)) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.4e+14) || !(z <= 2.65e-20)) {
		tmp = -z - y;
	} else {
		tmp = (x * Math.log(y)) - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.4e+14) or not (z <= 2.65e-20):
		tmp = -z - y
	else:
		tmp = (x * math.log(y)) - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.4e+14) || !(z <= 2.65e-20))
		tmp = Float64(Float64(-z) - y);
	else
		tmp = Float64(Float64(x * log(y)) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.4e+14) || ~((z <= 2.65e-20)))
		tmp = -z - y;
	else
		tmp = (x * log(y)) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.4e+14], N[Not[LessEqual[z, 2.65e-20]], $MachinePrecision]], N[((-z) - y), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.4e14 or 2.6500000000000001e-20 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 78.3%

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

      1. neg-mul-1 78.3%

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

   4. Simplified 78.3%

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

   if -3.4e14 < z < 2.6500000000000001e-20

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 91.0%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+14} \lor \neg \left(z \leq 2.65 \cdot 10^{-20}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \]

Alternative 3: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log y\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-7} \lor \neg \left(z \leq 0.0023\right):\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;t_0 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log y))))
   (if (or (<= z -1.15e-7) (not (<= z 0.0023))) (- t_0 z) (- t_0 y))))

C

double code(double x, double y, double z) {
	double t_0 = x * log(y);
	double tmp;
	if ((z <= -1.15e-7) || !(z <= 0.0023)) {
		tmp = t_0 - z;
	} 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 ((z <= (-1.15d-7)) .or. (.not. (z <= 0.0023d0))) then
        tmp = t_0 - z
    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 ((z <= -1.15e-7) || !(z <= 0.0023)) {
		tmp = t_0 - z;
	} else {
		tmp = t_0 - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log(y)
	tmp = 0
	if (z <= -1.15e-7) or not (z <= 0.0023):
		tmp = t_0 - z
	else:
		tmp = t_0 - y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(y))
	tmp = 0.0
	if ((z <= -1.15e-7) || !(z <= 0.0023))
		tmp = Float64(t_0 - z);
	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 ((z <= -1.15e-7) || ~((z <= 0.0023)))
		tmp = t_0 - z;
	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[Or[LessEqual[z, -1.15e-7], N[Not[LessEqual[z, 0.0023]], $MachinePrecision]], N[(t$95$0 - z), $MachinePrecision], N[(t$95$0 - y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.14999999999999997e-7 or 0.0023 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 99.9%

      \[\leadsto \color{blue}{\left(-1 \cdot y + -1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right)\right) - z} \]
   3. Step-by-step derivation

      1. distribute-lft-out 99.9%

         \[\leadsto \color{blue}{-1 \cdot \left(y + x \cdot \log \left(\frac{1}{y}\right)\right)} - z \]

      2. mul-1-neg 99.9%

         \[\leadsto \color{blue}{\left(-\left(y + x \cdot \log \left(\frac{1}{y}\right)\right)\right)} - z \]

      3. *-commutative 99.9%

         \[\leadsto \left(-\left(y + \color{blue}{\log \left(\frac{1}{y}\right) \cdot x}\right)\right) - z \]

      4. log-rec 99.9%

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

      5. cancel-sign-sub-inv 99.9%

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

      6. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in y around 0 89.1%

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

   if -1.14999999999999997e-7 < z < 0.0023

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 92.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-7} \lor \neg \left(z \leq 0.0023\right):\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \]

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+173} \lor \neg \left(x \leq 6.6 \cdot 10^{+97}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.2e+173) (not (<= x 6.6e+97))) (* x (log y)) (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.2e+173) || !(x <= 6.6e+97)) {
		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 <= (-3.2d+173)) .or. (.not. (x <= 6.6d+97))) 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 <= -3.2e+173) || !(x <= 6.6e+97)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.2e+173) or not (x <= 6.6e+97):
		tmp = x * math.log(y)
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.2e+173) || !(x <= 6.6e+97))
		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 <= -3.2e+173) || ~((x <= 6.6e+97)))
		tmp = x * log(y);
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.2e+173], N[Not[LessEqual[x, 6.6e+97]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000003e173 or 6.6000000000000003e97 < x

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 99.6%

      \[\leadsto \color{blue}{\left(-1 \cdot y + -1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right)\right) - z} \]
   3. Step-by-step derivation

      1. distribute-lft-out 99.6%

         \[\leadsto \color{blue}{-1 \cdot \left(y + x \cdot \log \left(\frac{1}{y}\right)\right)} - z \]

      2. mul-1-neg 99.6%

         \[\leadsto \color{blue}{\left(-\left(y + x \cdot \log \left(\frac{1}{y}\right)\right)\right)} - z \]

      3. *-commutative 99.6%

         \[\leadsto \left(-\left(y + \color{blue}{\log \left(\frac{1}{y}\right) \cdot x}\right)\right) - z \]

      4. log-rec 99.6%

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

      5. cancel-sign-sub-inv 99.6%

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

      6. *-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in x around inf 72.8%

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

   if -3.2000000000000003e173 < x < 6.6000000000000003e97

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 82.4%

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

      1. neg-mul-1 82.4%

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

   4. Simplified 82.4%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+173} \lor \neg \left(x \leq 6.6 \cdot 10^{+97}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in y around inf 99.8%

   \[\leadsto \color{blue}{\left(-1 \cdot y + -1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right)\right) - z} \]
3. Step-by-step derivation

   1. distribute-lft-out 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(y + x \cdot \log \left(\frac{1}{y}\right)\right)} - z \]

   2. mul-1-neg 99.8%

      \[\leadsto \color{blue}{\left(-\left(y + x \cdot \log \left(\frac{1}{y}\right)\right)\right)} - z \]

   3. *-commutative 99.8%

      \[\leadsto \left(-\left(y + \color{blue}{\log \left(\frac{1}{y}\right) \cdot x}\right)\right) - z \]

   4. log-rec 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. *-commutative 99.8%

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

4. Simplified 99.8%

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

5. Taylor expanded in y around 0 99.8%

   \[\leadsto \color{blue}{\left(-1 \cdot y + x \cdot \log y\right) - z} \]
6. Step-by-step derivation

   1. neg-mul-1 99.8%

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

   2. +-commutative 99.8%

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

   3. sub-neg 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 6: 51.9% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-7}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 0.0069:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.9e-7) (- z) (if (<= z 0.0069) (- y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e-7) {
		tmp = -z;
	} else if (z <= 0.0069) {
		tmp = -y;
	} else {
		tmp = -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 (z <= (-1.9d-7)) then
        tmp = -z
    else if (z <= 0.0069d0) then
        tmp = -y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e-7) {
		tmp = -z;
	} else if (z <= 0.0069) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.9e-7:
		tmp = -z
	elif z <= 0.0069:
		tmp = -y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.9e-7)
		tmp = Float64(-z);
	elseif (z <= 0.0069)
		tmp = Float64(-y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.9e-7)
		tmp = -z;
	elseif (z <= 0.0069)
		tmp = -y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.9e-7], (-z), If[LessEqual[z, 0.0069], (-y), (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.90000000000000007e-7 or 0.0068999999999999999 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 99.9%

      \[\leadsto \color{blue}{\left(-1 \cdot y + -1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right)\right) - z} \]
   3. Step-by-step derivation

      1. distribute-lft-out 99.9%

         \[\leadsto \color{blue}{-1 \cdot \left(y + x \cdot \log \left(\frac{1}{y}\right)\right)} - z \]

      2. mul-1-neg 99.9%

         \[\leadsto \color{blue}{\left(-\left(y + x \cdot \log \left(\frac{1}{y}\right)\right)\right)} - z \]

      3. *-commutative 99.9%

         \[\leadsto \left(-\left(y + \color{blue}{\log \left(\frac{1}{y}\right) \cdot x}\right)\right) - z \]

      4. log-rec 99.9%

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

      5. cancel-sign-sub-inv 99.9%

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

      6. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around inf 64.9%

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

      1. neg-mul-1 64.9%

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

   7. Simplified 64.9%

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

   if -1.90000000000000007e-7 < z < 0.0068999999999999999

   1. Initial program 99.7%

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

   2. Taylor expanded in y around inf 45.6%

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

      1. mul-1-neg 45.6%

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

   4. Simplified 45.6%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-7}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 0.0069:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7: 66.0% 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 99.8%

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

2. Taylor expanded in x around 0 65.1%

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

   1. neg-mul-1 65.1%

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

4. Simplified 65.1%

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

5. Final simplification 65.1%

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

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 99.8%

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

2. Taylor expanded in y around inf 27.4%

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

   1. mul-1-neg 27.4%

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

4. Simplified 27.4%

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

5. Final simplification 27.4%

   \[\leadsto -y \]

Alternative 9: 2.2% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.8%

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

   1. associate--l- 99.8%

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

   2. *-commutative 99.8%

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

   3. add-cube-cbrt 99.2%

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

   4. associate-*r* 99.2%

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

   5. +-commutative 99.2%

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

   6. add-sqr-sqrt 69.8%

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

   7. sqrt-unprod 45.7%

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

   8. sqr-neg 45.7%

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

   9. sqrt-unprod 9.8%

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

   10. add-sqr-sqrt 34.0%

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

   11. fma-neg 34.0%

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

   12. pow2 34.0%

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

   13. add-sqr-sqrt 9.8%

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

   14. sqrt-unprod 45.7%

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

   15. sqr-neg 45.7%

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

   16. sqrt-unprod 69.8%

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

   17. add-sqr-sqrt 99.2%

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

   18. add-sqr-sqrt 28.9%

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

   19. sqrt-unprod 35.0%

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

   20. sqr-neg 35.0%

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

3. Applied egg-rr 34.0%

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

4. Taylor expanded in z around inf 2.3%

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

5. Final simplification 2.3%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023293 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))