Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Percentage Accurate: 89.5% → 98.9%

Time: 12.1s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * 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 = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / x) / (y * (1.0 + (z * z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * 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 = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / x) / (y * (1.0 + (z * z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z \cdot z\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{x \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{\left(z \cdot x\right) \cdot y\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z z)))))
   (*
    y_s
    (if (<= t_0 4e+307) (/ 1.0 (* x t_0)) (/ (/ 1.0 z) (* (* z x) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 4e+307) {
		tmp = 1.0 / (x * t_0);
	} else {
		tmp = (1.0 / z) / ((z * x) * y_m);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 4d+307) then
        tmp = 1.0d0 / (x * t_0)
    else
        tmp = (1.0d0 / z) / ((z * x) * y_m)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 4e+307) {
		tmp = 1.0 / (x * t_0);
	} else {
		tmp = (1.0 / z) / ((z * x) * y_m);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	t_0 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 4e+307:
		tmp = 1.0 / (x * t_0)
	else:
		tmp = (1.0 / z) / ((z * x) * y_m)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 4e+307)
		tmp = Float64(1.0 / Float64(x * t_0));
	else
		tmp = Float64(Float64(1.0 / z) / Float64(Float64(z * x) * y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 4e+307)
		tmp = 1.0 / (x * t_0);
	else
		tmp = (1.0 / z) / ((z * x) * y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 4e+307], N[(1.0 / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(N[(z * x), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 3.99999999999999994e307

   1. Initial program 96.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   if 3.99999999999999994e307 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 78.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 95.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{x}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* z z) 1e-9) (/ (/ 1.0 x) y_m) (/ 1.0 (* y_m (* z (* z x)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1e-9) {
		tmp = (1.0 / x) / y_m;
	} else {
		tmp = 1.0 / (y_m * (z * (z * x)));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 1d-9) then
        tmp = (1.0d0 / x) / y_m
    else
        tmp = 1.0d0 / (y_m * (z * (z * x)))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1e-9) {
		tmp = (1.0 / x) / y_m;
	} else {
		tmp = 1.0 / (y_m * (z * (z * x)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z * z) <= 1e-9:
		tmp = (1.0 / x) / y_m
	else:
		tmp = 1.0 / (y_m * (z * (z * x)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-9)
		tmp = Float64(Float64(1.0 / x) / y_m);
	else
		tmp = Float64(1.0 / Float64(y_m * Float64(z * Float64(z * x))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 1e-9)
		tmp = (1.0 / x) / y_m;
	else
		tmp = 1.0 / (y_m * (z * (z * x)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1e-9], N[(N[(1.0 / x), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(y$95$m * N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000006e-9

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 1.00000000000000006e-9 < (*.f64 z z)

   1. Initial program 86.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1:\\ \;\;\;\;\frac{\frac{1}{x}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x \cdot \left(z \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= (* z z) 1.0) (/ (/ 1.0 x) y_m) (/ 1.0 (* y_m (* x (* z z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1.0) {
		tmp = (1.0 / x) / y_m;
	} else {
		tmp = 1.0 / (y_m * (x * (z * z)));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 1.0d0) then
        tmp = (1.0d0 / x) / y_m
    else
        tmp = 1.0d0 / (y_m * (x * (z * z)))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1.0) {
		tmp = (1.0 / x) / y_m;
	} else {
		tmp = 1.0 / (y_m * (x * (z * z)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z * z) <= 1.0:
		tmp = (1.0 / x) / y_m
	else:
		tmp = 1.0 / (y_m * (x * (z * z)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 1.0)
		tmp = Float64(Float64(1.0 / x) / y_m);
	else
		tmp = Float64(1.0 / Float64(y_m * Float64(x * Float64(z * z))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 1.0)
		tmp = (1.0 / x) / y_m;
	else
		tmp = 1.0 / (y_m * (x * (z * z)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1.0], N[(N[(1.0 / x), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(y$95$m * N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 1 < (*.f64 z z)

   1. Initial program 86.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1:\\ \;\;\;\;\frac{\frac{1}{x}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y\_m \cdot \left(z \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= (* z z) 1.0) (/ (/ 1.0 x) y_m) (/ 1.0 (* x (* y_m (* z z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1.0) {
		tmp = (1.0 / x) / y_m;
	} else {
		tmp = 1.0 / (x * (y_m * (z * z)));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 1.0d0) then
        tmp = (1.0d0 / x) / y_m
    else
        tmp = 1.0d0 / (x * (y_m * (z * z)))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1.0) {
		tmp = (1.0 / x) / y_m;
	} else {
		tmp = 1.0 / (x * (y_m * (z * z)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z * z) <= 1.0:
		tmp = (1.0 / x) / y_m
	else:
		tmp = 1.0 / (x * (y_m * (z * z)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 1.0)
		tmp = Float64(Float64(1.0 / x) / y_m);
	else
		tmp = Float64(1.0 / Float64(x * Float64(y_m * Float64(z * z))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 1.0)
		tmp = (1.0 / x) / y_m;
	else
		tmp = 1.0 / (x * (y_m * (z * z)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1.0], N[(N[(1.0 / x), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(x * N[(y$95$m * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 1 < (*.f64 z z)

   1. Initial program 86.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \frac{1}{y\_m \cdot \left(z \cdot \left(z \cdot x\right) + x\right)} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (/ 1.0 (* y_m (+ (* z (* z x)) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (1.0 / (y_m * ((z * (z * x)) + x)));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (1.0d0 / (y_m * ((z * (z * x)) + x)))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (1.0 / (y_m * ((z * (z * x)) + x)));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	return y_s * (1.0 / (y_m * ((z * (z * x)) + x)))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(1.0 / Float64(y_m * Float64(Float64(z * Float64(z * x)) + x))))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (1.0 / (y_m * ((z * (z * x)) + x)));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(1.0 / N[(y$95$m * N[(N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.9%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

8. Applied egg-rr 0

   \[\leadsto expr\]
9. Add Preprocessing

Alternative 6: 58.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \frac{\frac{1}{x}}{y\_m} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z) :precision binary64 (* y_s (/ (/ 1.0 x) y_m)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	return y_s * ((1.0 / x) / y_m);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * ((1.0d0 / x) / y_m)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * ((1.0 / x) / y_m);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	return y_s * ((1.0 / x) / y_m)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(Float64(1.0 / x) / y_m))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * ((1.0 / x) / y_m);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(N[(1.0 / x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.9%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Add Preprocessing

3. Taylor expanded in z around 0 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]
6. Add Preprocessing

Alternative 7: 58.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \frac{1}{x \cdot y\_m} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z) :precision binary64 (* y_s (/ 1.0 (* x y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (1.0 / (x * y_m));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (1.0d0 / (x * y_m))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (1.0 / (x * y_m));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	return y_s * (1.0 / (x * y_m))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(1.0 / Float64(x * y_m)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (1.0 / (x * y_m));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(1.0 / N[(x * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.9%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in z around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]
8. Add Preprocessing

Developer target: 92.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + z \cdot z\\ t_1 := y \cdot t\_0\\ t_2 := \frac{\frac{1}{y}}{t\_0 \cdot x}\\ \mathbf{if}\;t\_1 < -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{t\_0 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z z))) (t_1 (* y t_0)) (t_2 (/ (/ 1.0 y) (* t_0 x))))
   (if (< t_1 (- INFINITY))
     t_2
     (if (< t_1 8.680743250567252e+305) (/ (/ 1.0 x) (* t_0 y)) t_2))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z * z)
	t_1 = y * t_0
	t_2 = (1.0 / y) / (t_0 * x)
	tmp = 0
	if t_1 < -math.inf:
		tmp = t_2
	elif t_1 < 8.680743250567252e+305:
		tmp = (1.0 / x) / (t_0 * y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z * z))
	t_1 = Float64(y * t_0)
	t_2 = Float64(Float64(1.0 / y) / Float64(t_0 * x))
	tmp = 0.0
	if (t_1 < Float64(-Inf))
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = Float64(Float64(1.0 / x) / Float64(t_0 * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z * z);
	t_1 = y * t_0;
	t_2 = (1.0 / y) / (t_0 * x);
	tmp = 0.0;
	if (t_1 < -Inf)
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = (1.0 / x) / (t_0 * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / y), $MachinePrecision] / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$1, (-Infinity)], t$95$2, If[Less[t$95$1, 8.680743250567252e+305], N[(N[(1.0 / x), $MachinePrecision] / N[(t$95$0 * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Reproduce

herbie shell --seed 2024110 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))