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

Percentage Accurate: 99.9% → 99.9%

Time: 10.3s

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

\[\begin{array}{l} \\ \mathsf{fma}\left(\log y, x, \left(0 - y\right) - z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (log y) x (- (- 0.0 y) z)))

C

double code(double x, double y, double z) {
	return fma(log(y), x, ((0.0 - y) - z));
}

Julia

function code(x, y, z)
	return fma(log(y), x, Float64(Float64(0.0 - y) - z))
end

Wolfram

code[x_, y_, z_] := N[(N[Log[y], $MachinePrecision] * x + N[(N[(0.0 - y), $MachinePrecision] - z), $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. associate--l- N/A

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

   2. *-commutative N/A

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

   3. fmm-def N/A

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

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

      \[\leadsto \mathsf{fma.f64}\left(\log y, \color{blue}{x}, \left(\mathsf{neg}\left(\left(z + y\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \left(\mathsf{neg}\left(\left(z + y\right)\right)\right)\right) \]

   6. distribute-neg-in N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \left(\left(\mathsf{neg}\left(z\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \left(\left(\mathsf{neg}\left(z\right)\right) - y\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), y\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{\_.f64}\left(\left(0 - z\right), y\right)\right) \]

   10. --lowering--.f64 99.9%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right)\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(0 - z\right) - y\right)} \]

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(\log y, x, \left(0 - y\right) - z\right) \]
6. Add Preprocessing

Alternative 2: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0 - y\right) - z\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+17}:\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- 0.0 y) z)))
   (if (<= z -4.2e-26) t_0 (if (<= z 4.6e+17) (- (* (log y) x) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (0.0 - y) - z;
	double tmp;
	if (z <= -4.2e-26) {
		tmp = t_0;
	} else if (z <= 4.6e+17) {
		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 = (0.0d0 - y) - z
    if (z <= (-4.2d-26)) then
        tmp = t_0
    else if (z <= 4.6d+17) 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 = (0.0 - y) - z;
	double tmp;
	if (z <= -4.2e-26) {
		tmp = t_0;
	} else if (z <= 4.6e+17) {
		tmp = (Math.log(y) * x) - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (0.0 - y) - z
	tmp = 0
	if z <= -4.2e-26:
		tmp = t_0
	elif z <= 4.6e+17:
		tmp = (math.log(y) * x) - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.0 - y) - z)
	tmp = 0.0
	if (z <= -4.2e-26)
		tmp = t_0;
	elseif (z <= 4.6e+17)
		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 = (0.0 - y) - z;
	tmp = 0.0;
	if (z <= -4.2e-26)
		tmp = t_0;
	elseif (z <= 4.6e+17)
		tmp = (log(y) * x) - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.0 - y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[z, -4.2e-26], t$95$0, If[LessEqual[z, 4.6e+17], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.20000000000000016e-26 or 4.6e17 < z

   1. Initial program 100.0%

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

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

      2. distribute-neg-in N/A

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

      3. +-commutative N/A

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

      4. sub-neg N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), \color{blue}{y}\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

      7. --lowering--.f64 85.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

   5. Simplified 85.2%

      \[\leadsto \color{blue}{\left(0 - z\right) - y} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), y\right) \]

      2. neg-lowering-neg.f64 85.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(z\right), y\right) \]

   7. Applied egg-rr 85.2%

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

   if -4.20000000000000016e-26 < z < 4.6e17

   1. Initial program 99.9%

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \log y\right), \color{blue}{y}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), y\right) \]

      3. log-lowering-log.f64 94.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), y\right) \]

   5. Simplified 94.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-26}:\\ \;\;\;\;\left(0 - y\right) - z\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+17}:\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;\left(0 - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot x\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+130}:\\ \;\;\;\;\left(0 - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log y) x)))
   (if (<= x -8.5e+40) t_0 (if (<= x 1.06e+130) (- (- 0.0 y) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = log(y) * x;
	double tmp;
	if (x <= -8.5e+40) {
		tmp = t_0;
	} else if (x <= 1.06e+130) {
		tmp = (0.0 - y) - z;
	} 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 <= (-8.5d+40)) then
        tmp = t_0
    else if (x <= 1.06d+130) then
        tmp = (0.0d0 - y) - z
    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 <= -8.5e+40) {
		tmp = t_0;
	} else if (x <= 1.06e+130) {
		tmp = (0.0 - y) - z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log(y) * x
	tmp = 0
	if x <= -8.5e+40:
		tmp = t_0
	elif x <= 1.06e+130:
		tmp = (0.0 - y) - z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -8.5e+40)
		tmp = t_0;
	elseif (x <= 1.06e+130)
		tmp = Float64(Float64(0.0 - y) - z);
	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 <= -8.5e+40)
		tmp = t_0;
	elseif (x <= 1.06e+130)
		tmp = (0.0 - y) - z;
	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, -8.5e+40], t$95$0, If[LessEqual[x, 1.06e+130], N[(N[(0.0 - y), $MachinePrecision] - z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.49999999999999996e40 or 1.06e130 < x

   1. Initial program 99.8%

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\log y}\right) \]

      2. log-lowering-log.f64 72.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right) \]

   5. Simplified 72.1%

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

   if -8.49999999999999996e40 < x < 1.06e130

   1. Initial program 100.0%

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

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

      2. distribute-neg-in N/A

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

      3. +-commutative N/A

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

      4. sub-neg N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), \color{blue}{y}\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

      7. --lowering--.f64 86.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{\left(0 - z\right) - y} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), y\right) \]

      2. neg-lowering-neg.f64 86.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(z\right), y\right) \]

   7. Applied egg-rr 86.1%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+40}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+130}:\\ \;\;\;\;\left(0 - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+63}:\\ \;\;\;\;\log y \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;\left(0 - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4e+63) (- (* (log y) x) z) (- (- 0.0 y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e+63) {
		tmp = (log(y) * x) - z;
	} else {
		tmp = (0.0 - y) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.4d+63) then
        tmp = (log(y) * x) - z
    else
        tmp = (0.0d0 - y) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e+63) {
		tmp = (Math.log(y) * x) - z;
	} else {
		tmp = (0.0 - y) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.4e+63:
		tmp = (math.log(y) * x) - z
	else:
		tmp = (0.0 - y) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4e+63)
		tmp = Float64(Float64(log(y) * x) - z);
	else
		tmp = Float64(Float64(0.0 - y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.4e+63)
		tmp = (log(y) * x) - z;
	else
		tmp = (0.0 - y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.4e+63], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision], N[(N[(0.0 - y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4e63

   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

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \log y\right), \color{blue}{z}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), z\right) \]

      3. log-lowering-log.f64 91.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), z\right) \]

   5. Simplified 91.4%

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

   if 2.4e63 < y

   1. Initial program 100.0%

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

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

      2. distribute-neg-in N/A

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

      3. +-commutative N/A

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

      4. sub-neg N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), \color{blue}{y}\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

      7. --lowering--.f64 89.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

   5. Simplified 89.9%

      \[\leadsto \color{blue}{\left(0 - z\right) - y} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), y\right) \]

      2. neg-lowering-neg.f64 89.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(z\right), y\right) \]

   7. Applied egg-rr 89.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+63}:\\ \;\;\;\;\log y \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;\left(0 - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 6: 51.9% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1600000000000:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;0 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1600000000000.0) (- 0.0 z) (- 0.0 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1600000000000.0) {
		tmp = 0.0 - z;
	} else {
		tmp = 0.0 - 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) :: tmp
    if (y <= 1600000000000.0d0) then
        tmp = 0.0d0 - z
    else
        tmp = 0.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1600000000000.0) {
		tmp = 0.0 - z;
	} else {
		tmp = 0.0 - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1600000000000.0:
		tmp = 0.0 - z
	else:
		tmp = 0.0 - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1600000000000.0)
		tmp = Float64(0.0 - z);
	else
		tmp = Float64(0.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1600000000000.0)
		tmp = 0.0 - z;
	else
		tmp = 0.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1600000000000.0], N[(0.0 - z), $MachinePrecision], N[(0.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.6e12

   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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 53.1%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 53.1%

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

   if 1.6e12 < y

   1. Initial program 100.0%

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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 66.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{y}\right) \]

   5. Simplified 66.4%

      \[\leadsto \color{blue}{0 - y} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 66.4%

         \[\leadsto \mathsf{neg.f64}\left(y\right) \]

   7. Applied egg-rr 66.4%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1600000000000:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;0 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.1% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \left(0 - y\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (- 0.0 y) z))

C

double code(double x, double y, double z) {
	return (0.0 - 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 = (0.0d0 - y) - z
end function

Java

public static double code(double x, double y, double z) {
	return (0.0 - y) - z;
}

Python

def code(x, y, z):
	return (0.0 - y) - z

Julia

function code(x, y, z)
	return Float64(Float64(0.0 - y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (0.0 - y) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(0.0 - y), $MachinePrecision] - z), $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

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

   2. distribute-neg-in N/A

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

   3. +-commutative N/A

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

   4. sub-neg N/A

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

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

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), \color{blue}{y}\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

   7. --lowering--.f64 68.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

5. Simplified 68.8%

   \[\leadsto \color{blue}{\left(0 - z\right) - y} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), y\right) \]

   2. neg-lowering-neg.f64 68.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(z\right), y\right) \]

7. Applied egg-rr 68.8%

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

8. Final simplification 68.8%

   \[\leadsto \left(0 - y\right) - z \]
9. Add Preprocessing

Alternative 8: 33.6% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ 0 - y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- 0.0 y))

C

double code(double x, double y, double z) {
	return 0.0 - 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 = 0.0d0 - y
end function

Java

public static double code(double x, double y, double z) {
	return 0.0 - y;
}

Python

def code(x, y, z):
	return 0.0 - y

Julia

function code(x, y, z)
	return Float64(0.0 - y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0 - y;
end

Wolfram

code[x_, y_, z_] := N[(0.0 - y), $MachinePrecision]

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

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 31.6%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{y}\right) \]

5. Simplified 31.6%

   \[\leadsto \color{blue}{0 - y} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 31.6%

      \[\leadsto \mathsf{neg.f64}\left(y\right) \]

7. Applied egg-rr 31.6%

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

8. Final simplification 31.6%

   \[\leadsto 0 - y \]
9. Add Preprocessing

Alternative 9: 2.3% 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 99.9%

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

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

   2. distribute-neg-in N/A

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

   3. +-commutative N/A

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

   4. sub-neg N/A

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

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

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), \color{blue}{y}\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

   7. --lowering--.f64 68.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

5. Simplified 68.8%

   \[\leadsto \color{blue}{\left(0 - z\right) - y} \]

7. Applied egg-rr 7.2%

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

8. Taylor expanded in z around inf

   \[\leadsto \color{blue}{z} \]
9. Step-by-step derivation

10. Simplified 2.2%

    \[\leadsto \color{blue}{z} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))