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

Percentage Accurate: 99.9% → 99.9%

Time: 10.9s

Alternatives: 10

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 99.8%

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

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sub-neg N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-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: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) - y\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+180}:\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- z) y)))
   (if (<= z -2.1e+113) t_0 (if (<= z 2.9e+180) (- (* (log y) x) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -z - y;
	double tmp;
	if (z <= -2.1e+113) {
		tmp = t_0;
	} else if (z <= 2.9e+180) {
		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 <= (-2.1d+113)) then
        tmp = t_0
    else if (z <= 2.9d+180) 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 <= -2.1e+113) {
		tmp = t_0;
	} else if (z <= 2.9e+180) {
		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 <= -2.1e+113:
		tmp = t_0
	elif z <= 2.9e+180:
		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 <= -2.1e+113)
		tmp = t_0;
	elseif (z <= 2.9e+180)
		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 <= -2.1e+113)
		tmp = t_0;
	elseif (z <= 2.9e+180)
		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, -2.1e+113], t$95$0, If[LessEqual[z, 2.9e+180], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0999999999999999e113 or 2.90000000000000007e180 < 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. lower--.f64 N/A

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

      6. lower-neg.f64 85.0

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

   5. Simplified 85.0%

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

   if -2.0999999999999999e113 < z < 2.90000000000000007e180

   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. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 87.1

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

   5. Simplified 87.1%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+113}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+180}:\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot x\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+106}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log y) x)))
   (if (<= x -3.8e+98) t_0 (if (<= x 6.8e+106) (- (- z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = log(y) * x;
	double tmp;
	if (x <= -3.8e+98) {
		tmp = t_0;
	} else if (x <= 6.8e+106) {
		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 <= (-3.8d+98)) then
        tmp = t_0
    else if (x <= 6.8d+106) 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 <= -3.8e+98) {
		tmp = t_0;
	} else if (x <= 6.8e+106) {
		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 <= -3.8e+98:
		tmp = t_0
	elif x <= 6.8e+106:
		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 <= -3.8e+98)
		tmp = t_0;
	elseif (x <= 6.8e+106)
		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 <= -3.8e+98)
		tmp = t_0;
	elseif (x <= 6.8e+106)
		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, -3.8e+98], t$95$0, If[LessEqual[x, 6.8e+106], N[((-z) - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999999e98 or 6.79999999999999989e106 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 76.8

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

   5. Simplified 76.8%

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

   if -3.7999999999999999e98 < x < 6.79999999999999989e106

   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. lower--.f64 N/A

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

      6. lower-neg.f64 82.6

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

   5. Simplified 82.6%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+98}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+106}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.75e+87) (fma (log y) x (- z)) (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.75e+87) {
		tmp = fma(log(y), x, -z);
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.75e+87)
		tmp = fma(log(y), x, Float64(-z));
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.75e+87], N[(N[Log[y], $MachinePrecision] * x + (-z)), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.74999999999999993e87

   1. Initial program 99.8%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sub-neg N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-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 inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.0

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

   7. Simplified 90.0%

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

   if 1.74999999999999993e87 < y

   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. lower--.f64 N/A

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

      6. lower-neg.f64 87.3

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

   5. Simplified 87.3%

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

Alternative 5: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{+87}:\\ \;\;\;\;\log y \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.75e+87) (- (* (log y) x) z) (- (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.75e+87) {
		tmp = (log(y) * x) - z;
	} 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 (y <= 1.75d+87) then
        tmp = (log(y) * x) - z
    else
        tmp = -z - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.75e+87) {
		tmp = (Math.log(y) * x) - z;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.75e+87:
		tmp = (math.log(y) * x) - z
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.75e+87)
		tmp = Float64(Float64(log(y) * x) - z);
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.75e+87)
		tmp = (log(y) * x) - z;
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.75e+87], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.74999999999999993e87

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 90.0

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

   5. Simplified 90.0%

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

   if 1.74999999999999993e87 < y

   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. lower--.f64 N/A

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

      6. lower-neg.f64 87.3

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

   5. Simplified 87.3%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{+87}:\\ \;\;\;\;\log y \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]
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 99.8%

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

3. Final simplification 99.8%

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

Alternative 7: 52.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+94}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 122000000000:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.3e+94) (- z) (if (<= z 122000000000.0) (- y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.3e+94) {
		tmp = -z;
	} else if (z <= 122000000000.0) {
		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 <= (-1.3d+94)) then
        tmp = -z
    else if (z <= 122000000000.0d0) 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 <= -1.3e+94) {
		tmp = -z;
	} else if (z <= 122000000000.0) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.3e+94:
		tmp = -z
	elif z <= 122000000000.0:
		tmp = -y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.3e+94)
		tmp = Float64(-z);
	elseif (z <= 122000000000.0)
		tmp = Float64(-y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.3e+94)
		tmp = -z;
	elseif (z <= 122000000000.0)
		tmp = -y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.3e+94], (-z), If[LessEqual[z, 122000000000.0], (-y), (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e94 or 1.22e11 < z

   1. Initial program 99.9%

      \[\left(x \cdot \log y - z\right) - y \]
   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. lower-neg.f64 62.6

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

   5. Simplified 62.6%

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

   if -1.3e94 < z < 1.22e11

   1. Initial program 99.8%

      \[\left(x \cdot \log y - z\right) - y \]
   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. lower-neg.f64 41.5

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

   5. Simplified 41.5%

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

Alternative 8: 66.3% 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 99.8%

   \[\left(x \cdot \log y - z\right) - y \]
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. lower--.f64 N/A

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

   6. lower-neg.f64 61.1

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

5. Simplified 61.1%

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

Alternative 9: 33.4% 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 99.8%

   \[\left(x \cdot \log y - z\right) - y \]
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. lower-neg.f64 29.0

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

5. Simplified 29.0%

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

Alternative 10: 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 99.8%

   \[\left(x \cdot \log y - z\right) - y \]
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. lower-neg.f64 29.0

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

5. Simplified 29.0%

   \[\leadsto \color{blue}{-y} \]
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. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}{{0}^{3} - {y}^{3}}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}{{0}^{3} - {y}^{3}}}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{1}{\frac{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}{\color{blue}{0} - {y}^{3}}} \]

   6. neg-sub0 N/A

      \[\leadsto \frac{1}{\frac{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}{\color{blue}{\mathsf{neg}\left({y}^{3}\right)}}} \]

   7. cube-neg N/A

      \[\leadsto \frac{1}{\frac{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}}} \]

   8. lift-neg.f64 N/A

      \[\leadsto \frac{1}{\frac{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}{{\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}}^{3}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}} \]

   11. +-lft-identity N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot y + 0 \cdot y}}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}} \]

   12. mul0-lft N/A

       \[\leadsto \frac{1}{\frac{y \cdot y + \color{blue}{0}}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}} \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(y, y, 0\right)}}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}} \]

   14. lift-neg.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, y, 0\right)}{{\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}}^{3}}} \]

   15. cube-neg N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, y, 0\right)}{\color{blue}{\mathsf{neg}\left({y}^{3}\right)}}} \]

   16. lower-neg.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, y, 0\right)}{\color{blue}{\mathsf{neg}\left({y}^{3}\right)}}} \]

   17. cube-mult N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 7.1

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

7. Applied egg-rr 7.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, 0\right)}{-y \cdot \left(y \cdot y\right)}}} \]
8. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. clear-num N/A

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

   6. remove-double-div N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. cube-unmult N/A

       \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{y}^{3}}\right)}{\mathsf{fma}\left(y, y, 0\right)} \]

   11. cube-neg N/A

       \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}}{\mathsf{fma}\left(y, y, 0\right)} \]

   12. lift-neg.f64 N/A

       \[\leadsto \frac{{\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}}^{3}}{\mathsf{fma}\left(y, y, 0\right)} \]

   13. sqr-pow N/A

       \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)}}}{\mathsf{fma}\left(y, y, 0\right)} \]

   14. unpow-prod-down N/A

       \[\leadsto \frac{\color{blue}{{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{\mathsf{fma}\left(y, y, 0\right)} \]

   15. lift-neg.f64 N/A

       \[\leadsto \frac{{\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{\mathsf{fma}\left(y, y, 0\right)} \]

   16. lift-neg.f64 N/A

       \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{\mathsf{fma}\left(y, y, 0\right)} \]

   17. sqr-neg N/A

       \[\leadsto \frac{{\color{blue}{\left(y \cdot y\right)}}^{\left(\frac{3}{2}\right)}}{\mathsf{fma}\left(y, y, 0\right)} \]

   18. unpow-prod-down N/A

       \[\leadsto \frac{\color{blue}{{y}^{\left(\frac{3}{2}\right)} \cdot {y}^{\left(\frac{3}{2}\right)}}}{\mathsf{fma}\left(y, y, 0\right)} \]

   19. sqr-pow N/A

       \[\leadsto \frac{\color{blue}{{y}^{3}}}{\mathsf{fma}\left(y, y, 0\right)} \]

   20. lift-fma.f64 N/A

       \[\leadsto \frac{{y}^{3}}{\color{blue}{y \cdot y + 0}} \]

   21. lift-*.f64 N/A

       \[\leadsto \frac{{y}^{3}}{\color{blue}{y \cdot y} + 0} \]

   22. +-rgt-identity N/A

       \[\leadsto \frac{{y}^{3}}{\color{blue}{y \cdot y}} \]

   23. lift-*.f64 N/A

       \[\leadsto \frac{{y}^{3}}{\color{blue}{y \cdot y}} \]

   24. pow2 N/A

       \[\leadsto \frac{{y}^{3}}{\color{blue}{{y}^{2}}} \]

9. Applied egg-rr 2.4%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))