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

Percentage Accurate: 99.9% → 99.9%

Time: 12.6s

Alternatives: 11

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg N/A

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

   2. associate--l+ N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. log-lowering-log.f64 N/A

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

   6. --lowering--.f64 N/A

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

   7. neg-lowering-neg.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot x - z\\ \mathbf{if}\;z \leq -9.6 \cdot 10^{+54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (log y) x) z)))
   (if (<= z -9.6e+54) t_0 (if (<= z 1.95e+39) (fma (log y) x (- y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (log(y) * x) - z;
	double tmp;
	if (z <= -9.6e+54) {
		tmp = t_0;
	} else if (z <= 1.95e+39) {
		tmp = fma(log(y), x, -y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(log(y) * x) - z)
	tmp = 0.0
	if (z <= -9.6e+54)
		tmp = t_0;
	elseif (z <= 1.95e+39)
		tmp = fma(log(y), x, Float64(-y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[z, -9.6e+54], t$95$0, If[LessEqual[z, 1.95e+39], N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.59999999999999993e54 or 1.95e39 < z

   1. Initial program 99.9%

      \[\left(x \cdot \log y - z\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 86.2

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

   5. Simplified 86.2%

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

   if -9.59999999999999993e54 < z < 1.95e39

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. neg-lowering-neg.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 93.6

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

   7. Simplified 93.6%

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-y}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+54}:\\ \;\;\;\;\log y \cdot x - z\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot x\\ t_1 := t\_0 - z\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+39}:\\ \;\;\;\;t\_0 - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log y) x)) (t_1 (- t_0 z)))
   (if (<= z -6.8e+58) t_1 (if (<= z 1.65e+39) (- t_0 y) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = log(y) * x;
	double t_1 = t_0 - z;
	double tmp;
	if (z <= -6.8e+58) {
		tmp = t_1;
	} else if (z <= 1.65e+39) {
		tmp = t_0 - y;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = log(y) * x
    t_1 = t_0 - z
    if (z <= (-6.8d+58)) then
        tmp = t_1
    else if (z <= 1.65d+39) then
        tmp = t_0 - y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log(y) * x;
	double t_1 = t_0 - z;
	double tmp;
	if (z <= -6.8e+58) {
		tmp = t_1;
	} else if (z <= 1.65e+39) {
		tmp = t_0 - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log(y) * x
	t_1 = t_0 - z
	tmp = 0
	if z <= -6.8e+58:
		tmp = t_1
	elif z <= 1.65e+39:
		tmp = t_0 - y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(y) * x)
	t_1 = Float64(t_0 - z)
	tmp = 0.0
	if (z <= -6.8e+58)
		tmp = t_1;
	elseif (z <= 1.65e+39)
		tmp = Float64(t_0 - y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log(y) * x;
	t_1 = t_0 - z;
	tmp = 0.0;
	if (z <= -6.8e+58)
		tmp = t_1;
	elseif (z <= 1.65e+39)
		tmp = t_0 - y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - z), $MachinePrecision]}, If[LessEqual[z, -6.8e+58], t$95$1, If[LessEqual[z, 1.65e+39], N[(t$95$0 - y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.8000000000000001e58 or 1.6500000000000001e39 < z

   1. Initial program 99.9%

      \[\left(x \cdot \log y - z\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 86.2

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

   5. Simplified 86.2%

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

   if -6.8000000000000001e58 < z < 1.6500000000000001e39

   1. Initial program 99.8%

      \[\left(x \cdot \log y - z\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. --lowering--.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 93.6

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

   5. Simplified 93.6%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+58}:\\ \;\;\;\;\log y \cdot x - z\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+39}:\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) - y\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{+80}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+80}:\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- z) y)))
   (if (<= z -8.2e+80) t_0 (if (<= z 3.7e+80) (- (* (log y) x) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -z - y;
	double tmp;
	if (z <= -8.2e+80) {
		tmp = t_0;
	} else if (z <= 3.7e+80) {
		tmp = (log(y) * x) - y;
	} else {
		tmp = t_0;
	}
	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 = -z - y
    if (z <= (-8.2d+80)) then
        tmp = t_0
    else if (z <= 3.7d+80) then
        tmp = (log(y) * x) - y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -z - y;
	double tmp;
	if (z <= -8.2e+80) {
		tmp = t_0;
	} else if (z <= 3.7e+80) {
		tmp = (Math.log(y) * x) - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z - y
	tmp = 0
	if z <= -8.2e+80:
		tmp = t_0
	elif z <= 3.7e+80:
		tmp = (math.log(y) * x) - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) - y)
	tmp = 0.0
	if (z <= -8.2e+80)
		tmp = t_0;
	elseif (z <= 3.7e+80)
		tmp = Float64(Float64(log(y) * x) - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z - y;
	tmp = 0.0;
	if (z <= -8.2e+80)
		tmp = t_0;
	elseif (z <= 3.7e+80)
		tmp = (log(y) * x) - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) - y), $MachinePrecision]}, If[LessEqual[z, -8.2e+80], t$95$0, If[LessEqual[z, 3.7e+80], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.20000000000000003e80 or 3.69999999999999996e80 < z

   1. Initial program 99.9%

      \[\left(x \cdot \log y - z\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(y + z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(z + y\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right) + \left(\mathsf{neg}\left(y\right)\right)} \]

      4. sub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. neg-lowering-neg.f64 83.8

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

   5. Simplified 83.8%

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

   if -8.20000000000000003e80 < z < 3.69999999999999996e80

   1. Initial program 99.8%

      \[\left(x \cdot \log y - z\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. --lowering--.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 91.6

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

   5. Simplified 91.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+80}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+80}:\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot x\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+185}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+201}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log y) x)))
   (if (<= x -2.6e+185) t_0 (if (<= x 7e+201) (- (- z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = log(y) * x;
	double tmp;
	if (x <= -2.6e+185) {
		tmp = t_0;
	} else if (x <= 7e+201) {
		tmp = -z - y;
	} else {
		tmp = t_0;
	}
	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 = log(y) * x
    if (x <= (-2.6d+185)) then
        tmp = t_0
    else if (x <= 7d+201) then
        tmp = -z - y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log(y) * x;
	double tmp;
	if (x <= -2.6e+185) {
		tmp = t_0;
	} else if (x <= 7e+201) {
		tmp = -z - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log(y) * x
	tmp = 0
	if x <= -2.6e+185:
		tmp = t_0
	elif x <= 7e+201:
		tmp = -z - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -2.6e+185)
		tmp = t_0;
	elseif (x <= 7e+201)
		tmp = Float64(Float64(-z) - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log(y) * x;
	tmp = 0.0;
	if (x <= -2.6e+185)
		tmp = t_0;
	elseif (x <= 7e+201)
		tmp = -z - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.6e+185], t$95$0, If[LessEqual[x, 7e+201], N[((-z) - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000001e185 or 7.0000000000000004e201 < x

   1. Initial program 99.6%

      \[\left(x \cdot \log y - z\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. log-lowering-log.f64 82.3

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

   5. Simplified 82.3%

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

   if -2.60000000000000001e185 < x < 7.0000000000000004e201

   1. Initial program 99.9%

      \[\left(x \cdot \log y - z\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(y + z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(z + y\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right) + \left(\mathsf{neg}\left(y\right)\right)} \]

      4. sub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. neg-lowering-neg.f64 81.1

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

   5. Simplified 81.1%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+185}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+201}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot \log y - z\right) - y \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto \left(\log y \cdot x - z\right) - y \]
4. Add Preprocessing

Alternative 7: 53.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+55}:\\ \;\;\;\;y - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 1.7e+55) (- y z) (- y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.7e+55) {
		tmp = y - 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 <= 1.7d+55) then
        tmp = y - 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 <= 1.7e+55) {
		tmp = y - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.7e+55:
		tmp = y - z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.7e+55)
		tmp = Float64(y - z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.7e+55)
		tmp = y - z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.7e+55], N[(y - z), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 1.6999999999999999e55

   1. Initial program 99.8%

      \[\left(x \cdot \log y - z\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(y + z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(z + y\right)}\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right) + \left(\mathsf{neg}\left(y\right)\right)} \]

      4. sub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. neg-lowering-neg.f64 57.7

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

   5. Simplified 57.7%

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

   7. Applied egg-rr 49.3%

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

   if 1.6999999999999999e55 < y

   1. Initial program 99.9%

      \[\left(x \cdot \log y - z\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

      2. neg-lowering-neg.f64 64.4

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

   5. Simplified 64.4%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+55}:\\ \;\;\;\;y - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.2% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+56}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 1.1e+56) (- z) (- y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.1e+56) {
		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 <= 1.1d+56) 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 <= 1.1e+56) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.1e+56:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.1e+56)
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.1e+56)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.1e+56], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 1.10000000000000008e56

   1. Initial program 99.8%

      \[\left(x \cdot \log y - z\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-lowering-neg.f64 49.2

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

   5. Simplified 49.2%

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

   if 1.10000000000000008e56 < y

   1. Initial program 99.9%

      \[\left(x \cdot \log y - z\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

      2. neg-lowering-neg.f64 64.4

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

   5. Simplified 64.4%

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

Alternative 9: 66.5% accurate, 18.7× 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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(y + z\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(z + y\right)}\right) \]

   3. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right) + \left(\mathsf{neg}\left(y\right)\right)} \]

   4. sub-neg N/A

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

   5. --lowering--.f64 N/A

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

   6. neg-lowering-neg.f64 66.6

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

5. Simplified 66.6%

   \[\leadsto \color{blue}{\left(-z\right) - y} \]
6. Add Preprocessing

Alternative 10: 34.6% accurate, 37.3× 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. Add Preprocessing

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

   2. neg-lowering-neg.f64 33.6

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

5. Simplified 33.6%

   \[\leadsto \color{blue}{-y} \]
6. Add Preprocessing

Alternative 11: 2.3% accurate, 112.0× 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 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. Add Preprocessing

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

   2. neg-lowering-neg.f64 33.6

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

5. Simplified 33.6%

   \[\leadsto \color{blue}{-y} \]
6. Step-by-step derivation

   1. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - y} \]

   2. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

   5. --lowering--.f64 N/A

      \[\leadsto \frac{\color{blue}{0 - {y}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

   6. cube-mult N/A

      \[\leadsto \frac{0 - \color{blue}{y \cdot \left(y \cdot y\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{0 - \color{blue}{y \cdot \left(y \cdot y\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{0 - y \cdot \color{blue}{\left(y \cdot y\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{0 - y \cdot \left(y \cdot y\right)}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)} \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \frac{0 - y \cdot \left(y \cdot y\right)}{\color{blue}{0 + \left(y \cdot y + 0 \cdot y\right)}} \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{0 - y \cdot \left(y \cdot y\right)}{0 + \color{blue}{\mathsf{fma}\left(y, y, 0 \cdot y\right)}} \]

   12. *-lowering-*.f64 8.8

       \[\leadsto \frac{0 - y \cdot \left(y \cdot y\right)}{0 + \mathsf{fma}\left(y, y, \color{blue}{0 \cdot y}\right)} \]

7. Applied egg-rr 8.8%

   \[\leadsto \color{blue}{\frac{0 - y \cdot \left(y \cdot y\right)}{0 + \mathsf{fma}\left(y, y, 0 \cdot y\right)}} \]
8. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{0 + \left(y \cdot y + 0 \cdot y\right)}{0 - y \cdot \left(y \cdot y\right)}}} \]

   2. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{0 + \left(y \cdot y + 0 \cdot y\right)} \cdot \left(0 - y \cdot \left(y \cdot y\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{0 + \left(y \cdot y + 0 \cdot y\right)} \cdot \left(0 - y \cdot \left(y \cdot y\right)\right)} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot \left(0 - y \cdot \left(y \cdot y\right)\right) \]

   5. +-lft-identity N/A

      \[\leadsto \frac{1}{\color{blue}{y \cdot y + 0 \cdot y}} \cdot \left(0 - y \cdot \left(y \cdot y\right)\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \frac{1}{\color{blue}{y \cdot \left(y + 0\right)}} \cdot \left(0 - y \cdot \left(y \cdot y\right)\right) \]

   7. +-rgt-identity N/A

      \[\leadsto \frac{1}{y \cdot \color{blue}{y}} \cdot \left(0 - y \cdot \left(y \cdot y\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{y \cdot y}} \cdot \left(0 - y \cdot \left(y \cdot y\right)\right) \]

   9. sub0-neg N/A

      \[\leadsto \frac{1}{y \cdot y} \cdot \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(y \cdot y\right)\right)\right)} \]

   10. neg-lowering-neg.f64 N/A

       \[\leadsto \frac{1}{y \cdot y} \cdot \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(y \cdot y\right)\right)\right)} \]

   11. +-rgt-identity N/A

       \[\leadsto \frac{1}{y \cdot y} \cdot \left(\mathsf{neg}\left(y \cdot \left(y \cdot \color{blue}{\left(y + 0\right)}\right)\right)\right) \]

   12. distribute-rgt-out N/A

       \[\leadsto \frac{1}{y \cdot y} \cdot \left(\mathsf{neg}\left(y \cdot \color{blue}{\left(y \cdot y + 0 \cdot y\right)}\right)\right) \]

   13. +-lft-identity N/A

       \[\leadsto \frac{1}{y \cdot y} \cdot \left(\mathsf{neg}\left(y \cdot \color{blue}{\left(0 + \left(y \cdot y + 0 \cdot y\right)\right)}\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \frac{1}{y \cdot y} \cdot \left(\mathsf{neg}\left(\color{blue}{y \cdot \left(0 + \left(y \cdot y + 0 \cdot y\right)\right)}\right)\right) \]

   15. +-lft-identity N/A

       \[\leadsto \frac{1}{y \cdot y} \cdot \left(\mathsf{neg}\left(y \cdot \color{blue}{\left(y \cdot y + 0 \cdot y\right)}\right)\right) \]

   16. distribute-rgt-out N/A

       \[\leadsto \frac{1}{y \cdot y} \cdot \left(\mathsf{neg}\left(y \cdot \color{blue}{\left(y \cdot \left(y + 0\right)\right)}\right)\right) \]

   17. +-rgt-identity N/A

       \[\leadsto \frac{1}{y \cdot y} \cdot \left(\mathsf{neg}\left(y \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

   18. *-lowering-*.f64 8.8

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

9. Applied egg-rr 8.8%

   \[\leadsto \color{blue}{\frac{1}{y \cdot y} \cdot \left(-y \cdot \left(y \cdot y\right)\right)} \]
10. Step-by-step derivation

    1. neg-mul-1 N/A

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

11. Applied egg-rr 2.3%

    \[\leadsto \color{blue}{y} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))