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

Percentage Accurate: 99.9% → 99.9%

Time: 9.7s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Log[y], $MachinePrecision] * x + N[((-y) - 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. sub-neg 99.9%

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

   2. associate-+l- 99.9%

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

   3. *-commutative 99.9%

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

   4. add-sqr-sqrt 0.0%

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

   5. sqrt-unprod 62.1%

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

   6. sqr-neg 62.1%

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

   7. sqrt-unprod 69.4%

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

   8. add-sqr-sqrt 69.4%

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

   9. fma-neg 69.5%

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

   10. add-cube-cbrt 68.8%

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

   11. fma-neg 68.8%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 73.7%

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

   14. sqr-neg 73.7%

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

   15. sqrt-unprod 99.0%

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

   16. add-sqr-sqrt 99.2%

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

   17. fma-define 99.2%

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

   18. add-cube-cbrt 99.9%

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

   19. +-commutative 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l- 99.9%

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

   2. +-commutative 99.9%

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

   3. associate--r+ 99.9%

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

   4. fma-neg 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot x\\ t_1 := t\_0 - z\\ t_2 := \left(-y\right) - z\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -64000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+24}:\\ \;\;\;\;t\_0 - y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log y) x)) (t_1 (- t_0 z)) (t_2 (- (- y) z)))
   (if (<= z -1.15e+116)
     t_1
     (if (<= z -64000000000000.0)
       t_2
       (if (<= z -1.35e-40) t_1 (if (<= z 1.65e+24) (- t_0 y) t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = log(y) * x;
	double t_1 = t_0 - z;
	double t_2 = -y - z;
	double tmp;
	if (z <= -1.15e+116) {
		tmp = t_1;
	} else if (z <= -64000000000000.0) {
		tmp = t_2;
	} else if (z <= -1.35e-40) {
		tmp = t_1;
	} else if (z <= 1.65e+24) {
		tmp = t_0 - y;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = log(y) * x
    t_1 = t_0 - z
    t_2 = -y - z
    if (z <= (-1.15d+116)) then
        tmp = t_1
    else if (z <= (-64000000000000.0d0)) then
        tmp = t_2
    else if (z <= (-1.35d-40)) then
        tmp = t_1
    else if (z <= 1.65d+24) then
        tmp = t_0 - y
    else
        tmp = t_2
    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 t_2 = -y - z;
	double tmp;
	if (z <= -1.15e+116) {
		tmp = t_1;
	} else if (z <= -64000000000000.0) {
		tmp = t_2;
	} else if (z <= -1.35e-40) {
		tmp = t_1;
	} else if (z <= 1.65e+24) {
		tmp = t_0 - y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log(y) * x
	t_1 = t_0 - z
	t_2 = -y - z
	tmp = 0
	if z <= -1.15e+116:
		tmp = t_1
	elif z <= -64000000000000.0:
		tmp = t_2
	elif z <= -1.35e-40:
		tmp = t_1
	elif z <= 1.65e+24:
		tmp = t_0 - y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(y) * x)
	t_1 = Float64(t_0 - z)
	t_2 = Float64(Float64(-y) - z)
	tmp = 0.0
	if (z <= -1.15e+116)
		tmp = t_1;
	elseif (z <= -64000000000000.0)
		tmp = t_2;
	elseif (z <= -1.35e-40)
		tmp = t_1;
	elseif (z <= 1.65e+24)
		tmp = Float64(t_0 - y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log(y) * x;
	t_1 = t_0 - z;
	t_2 = -y - z;
	tmp = 0.0;
	if (z <= -1.15e+116)
		tmp = t_1;
	elseif (z <= -64000000000000.0)
		tmp = t_2;
	elseif (z <= -1.35e-40)
		tmp = t_1;
	elseif (z <= 1.65e+24)
		tmp = t_0 - y;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[((-y) - z), $MachinePrecision]}, If[LessEqual[z, -1.15e+116], t$95$1, If[LessEqual[z, -64000000000000.0], t$95$2, If[LessEqual[z, -1.35e-40], t$95$1, If[LessEqual[z, 1.65e+24], N[(t$95$0 - y), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.14999999999999997e116 or -6.4e13 < z < -1.35e-40

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 92.7%

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

   if -1.14999999999999997e116 < z < -6.4e13 or 1.6499999999999999e24 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 92.6%

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

      1. neg-mul-1 92.6%

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

   5. Simplified 92.6%

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

   if -1.35e-40 < z < 1.6499999999999999e24

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 95.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+116}:\\ \;\;\;\;\log y \cdot x - z\\ \mathbf{elif}\;z \leq -64000000000000:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-40}:\\ \;\;\;\;\log y \cdot x - z\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+24}:\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.15 \cdot 10^{+126} \lor \neg \left(x \leq -1.1 \cdot 10^{+95}\right) \land \left(x \leq -3.5 \cdot 10^{+41} \lor \neg \left(x \leq 7.2 \cdot 10^{+125}\right)\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.15e+126)
         (and (not (<= x -1.1e+95))
              (or (<= x -3.5e+41) (not (<= x 7.2e+125)))))
   (* (log y) x)
   (- (- y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.15e+126) || (!(x <= -1.1e+95) && ((x <= -3.5e+41) || !(x <= 7.2e+125)))) {
		tmp = log(y) * x;
	} else {
		tmp = -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 ((x <= (-4.15d+126)) .or. (.not. (x <= (-1.1d+95))) .and. (x <= (-3.5d+41)) .or. (.not. (x <= 7.2d+125))) then
        tmp = log(y) * x
    else
        tmp = -y - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.15e+126) || (!(x <= -1.1e+95) && ((x <= -3.5e+41) || !(x <= 7.2e+125)))) {
		tmp = Math.log(y) * x;
	} else {
		tmp = -y - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.15e+126) or (not (x <= -1.1e+95) and ((x <= -3.5e+41) or not (x <= 7.2e+125))):
		tmp = math.log(y) * x
	else:
		tmp = -y - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.15e+126) || (!(x <= -1.1e+95) && ((x <= -3.5e+41) || !(x <= 7.2e+125))))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(-y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4.15e+126) || (~((x <= -1.1e+95)) && ((x <= -3.5e+41) || ~((x <= 7.2e+125)))))
		tmp = log(y) * x;
	else
		tmp = -y - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.15e+126], And[N[Not[LessEqual[x, -1.1e+95]], $MachinePrecision], Or[LessEqual[x, -3.5e+41], N[Not[LessEqual[x, 7.2e+125]], $MachinePrecision]]]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[((-y) - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.14999999999999995e126 or -1.0999999999999999e95 < x < -3.4999999999999999e41 or 7.2000000000000007e125 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-+l- 99.8%

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

      3. *-commutative 99.8%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 68.9%

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

      6. sqr-neg 68.9%

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

      7. sqrt-unprod 83.5%

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

      8. add-sqr-sqrt 83.5%

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

      9. fma-neg 83.5%

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

      10. add-cube-cbrt 83.3%

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

      11. fma-neg 83.3%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 73.2%

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

      14. sqr-neg 73.2%

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

      15. sqrt-unprod 99.5%

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

      16. add-sqr-sqrt 99.6%

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

      17. fma-define 99.6%

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

      18. add-cube-cbrt 99.8%

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

      19. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 74.0%

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

   if -4.14999999999999995e126 < x < -1.0999999999999999e95 or -3.4999999999999999e41 < x < 7.2000000000000007e125

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 86.4%

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

      1. neg-mul-1 86.4%

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

   5. Simplified 86.4%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.15 \cdot 10^{+126} \lor \neg \left(x \leq -1.1 \cdot 10^{+95}\right) \land \left(x \leq -3.5 \cdot 10^{+41} \lor \neg \left(x \leq 7.2 \cdot 10^{+125}\right)\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-52} \lor \neg \left(z \leq 1.4 \cdot 10^{+24}\right):\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.25e-52) (not (<= z 1.4e+24)))
   (- (- y) z)
   (- (* (log y) x) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e-52) || !(z <= 1.4e+24)) {
		tmp = -y - z;
	} else {
		tmp = (log(y) * x) - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.25d-52)) .or. (.not. (z <= 1.4d+24))) then
        tmp = -y - z
    else
        tmp = (log(y) * x) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e-52) || !(z <= 1.4e+24)) {
		tmp = -y - z;
	} else {
		tmp = (Math.log(y) * x) - y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.25e-52) or not (z <= 1.4e+24):
		tmp = -y - z
	else:
		tmp = (math.log(y) * x) - y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.25e-52) || !(z <= 1.4e+24))
		tmp = Float64(Float64(-y) - z);
	else
		tmp = Float64(Float64(log(y) * x) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.25e-52) || ~((z <= 1.4e+24)))
		tmp = -y - z;
	else
		tmp = (log(y) * x) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.25e-52], N[Not[LessEqual[z, 1.4e+24]], $MachinePrecision]], N[((-y) - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25e-52 or 1.4000000000000001e24 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 84.4%

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

      1. neg-mul-1 84.4%

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

   5. Simplified 84.4%

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

   if -1.25e-52 < z < 1.4000000000000001e24

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 95.6%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-52} \lor \neg \left(z \leq 1.4 \cdot 10^{+24}\right):\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - y\\ \end{array} \]
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 99.9%

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

3. Final simplification 99.9%

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

Alternative 7: 52.1% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 11200000:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 11200000.0) (- z) (- y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 11200000.0) {
		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 <= 11200000.0d0) 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 <= 11200000.0) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 11200000.0:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 11200000.0)
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 11200000.0)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 11200000.0], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 1.12e7

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-+l- 99.8%

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

      3. *-commutative 99.8%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 92.5%

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

      6. sqr-neg 92.5%

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

      7. sqrt-unprod 92.5%

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

      8. add-sqr-sqrt 92.5%

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

      9. fma-neg 92.5%

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

      10. add-cube-cbrt 91.5%

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

      11. fma-neg 91.5%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 96.9%

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

      14. sqr-neg 96.9%

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

      15. sqrt-unprod 98.8%

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

      16. add-sqr-sqrt 98.9%

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

      17. fma-define 98.9%

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

      18. add-cube-cbrt 99.9%

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

      19. +-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf 47.8%

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

      1. neg-mul-1 47.8%

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

   7. Simplified 47.8%

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

   if 1.12e7 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 55.2%

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

      1. mul-1-neg 55.2%

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

   5. Simplified 55.2%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 11200000:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.4% accurate, 26.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[((-y) - 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 61.9%

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

   1. neg-mul-1 61.9%

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

5. Simplified 61.9%

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

6. Final simplification 61.9%

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

Alternative 9: 33.9% accurate, 53.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf 30.3%

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

   1. mul-1-neg 30.3%

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

5. Simplified 30.3%

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

6. Final simplification 30.3%

   \[\leadsto -y \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))