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

Percentage Accurate: 88.8% → 98.2%

Time: 10.5s

Alternatives: 9

Speedup: 0.8×

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: 88.8% 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: 98.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Step-by-step derivation

   1. associate-/l/ N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + z \cdot z\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot z\right)}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 89.5%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right) \]

3. Simplified 89.5%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(y \cdot \left(z \cdot z + \color{blue}{1}\right)\right)\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\left(z \cdot z\right) \cdot y + \color{blue}{1 \cdot y}\right)\right)\right) \]

   3. *-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\left(z \cdot z\right) \cdot y + y\right)\right)\right) \]

   4. distribute-rgt-in N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(z \cdot z\right) \cdot y\right) \cdot x + \color{blue}{y \cdot x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(y \cdot \left(z \cdot z\right)\right) \cdot x + y \cdot x\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(y \cdot z\right) \cdot z\right) \cdot x + y \cdot x\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(y \cdot z\right) \cdot \left(z \cdot x\right) + \color{blue}{y} \cdot x\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(y \cdot z\right) \cdot \left(z \cdot x\right) + x \cdot \color{blue}{y}\right)\right) \]

   9. fma-define N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{fma}\left(y \cdot z, \color{blue}{z \cdot x}, x \cdot y\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(y \cdot z\right), \color{blue}{\left(z \cdot x\right)}, \left(x \cdot y\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\color{blue}{z} \cdot x\right), \left(x \cdot y\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{*.f64}\left(z, \color{blue}{x}\right), \left(x \cdot y\right)\right)\right) \]

   13. *-lowering-*.f64 98.0%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{*.f64}\left(z, x\right), \mathsf{*.f64}\left(x, y\right)\right)\right) \]

6. Applied egg-rr 98.0%

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

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\left(y \cdot z\right) \cdot \left(z \cdot x\right)\right), \color{blue}{\left(x \cdot y\right)}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(y \cdot \left(z \cdot \left(z \cdot x\right)\right)\right), \left(\color{blue}{x} \cdot y\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(z \cdot \left(z \cdot x\right)\right)\right), \left(\color{blue}{x} \cdot y\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(z \cdot x\right)\right)\right), \left(x \cdot y\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, x\right)\right)\right), \left(x \cdot y\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, x\right)\right)\right), \left(y \cdot \color{blue}{x}\right)\right)\right) \]

   7. *-lowering-*.f64 96.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, x\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right)\right) \]

8. Applied egg-rr 96.1%

   \[\leadsto \frac{1}{\color{blue}{y \cdot \left(z \cdot \left(z \cdot x\right)\right) + y \cdot x}} \]
9. Add Preprocessing

Alternative 2: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot x\_m\right) \cdot \left(y\_m \cdot z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e+15)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z z))))
     (/ 1.0 (* (* z x_m) (* y_m z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+15) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = 1.0 / ((z * x_m) * (y_m * z));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+15) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z * z)))
    else
        tmp = 1.0d0 / ((z * x_m) * (y_m * z))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+15) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = 1.0 / ((z * x_m) * (y_m * z));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 5e+15:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)))
	else:
		tmp = 1.0 / ((z * x_m) * (y_m * z))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+15)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(1.0 / Float64(Float64(z * x_m) * Float64(y_m * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e+15)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	else
		tmp = 1.0 / ((z * x_m) * (y_m * z));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+15], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z * x$95$m), $MachinePrecision] * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e15

   1. Initial program 99.7%

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

   if 5e15 < (*.f64 z z)

   1. Initial program 80.5%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + z \cdot z\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot z\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right) \]

   3. Simplified 80.6%

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

      1. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}{1}\right)}\right) \]

      3. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{\left(1 + z \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(1 + z \cdot z\right) \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(1 + z \cdot z\right), \color{blue}{\left(x \cdot y\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(z \cdot z\right)\right), \left(\color{blue}{x} \cdot y\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, z\right)\right), \left(x \cdot y\right)\right)\right) \]

      9. *-lowering-*.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   6. Applied egg-rr 79.6%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\color{blue}{\left({z}^{2}\right)}, \mathsf{*.f64}\left(x, y\right)\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(z \cdot z\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right) \]

      2. *-lowering-*.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right) \]

   9. Simplified 79.6%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(z \cdot \color{blue}{\left(z \cdot \left(x \cdot y\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(z \cdot \left(\left(z \cdot x\right) \cdot \color{blue}{y}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(z \cdot \left(y \cdot \color{blue}{\left(z \cdot x\right)}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(z \cdot y\right) \cdot \color{blue}{\left(z \cdot x\right)}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(y \cdot z\right) \cdot \left(\color{blue}{z} \cdot x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(z \cdot x\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(z \cdot y\right), \left(\color{blue}{z} \cdot x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \left(\color{blue}{z} \cdot x\right)\right)\right) \]

       9. *-lowering-*.f64 96.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{*.f64}\left(z, \color{blue}{x}\right)\right)\right) \]

   11. Applied egg-rr 96.7%

       \[\leadsto \frac{1}{\color{blue}{\left(z \cdot y\right) \cdot \left(z \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot x\right) \cdot \left(y \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(y\_m \cdot \left(1 + z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot x\_m\right) \cdot \left(y\_m \cdot z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e+15)
     (/ 1.0 (* x_m (* y_m (+ 1.0 (* z z)))))
     (/ 1.0 (* (* z x_m) (* y_m z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+15) {
		tmp = 1.0 / (x_m * (y_m * (1.0 + (z * z))));
	} else {
		tmp = 1.0 / ((z * x_m) * (y_m * z));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+15) then
        tmp = 1.0d0 / (x_m * (y_m * (1.0d0 + (z * z))))
    else
        tmp = 1.0d0 / ((z * x_m) * (y_m * z))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+15) {
		tmp = 1.0 / (x_m * (y_m * (1.0 + (z * z))));
	} else {
		tmp = 1.0 / ((z * x_m) * (y_m * z));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 5e+15:
		tmp = 1.0 / (x_m * (y_m * (1.0 + (z * z))))
	else:
		tmp = 1.0 / ((z * x_m) * (y_m * z))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+15)
		tmp = Float64(1.0 / Float64(x_m * Float64(y_m * Float64(1.0 + Float64(z * z)))));
	else
		tmp = Float64(1.0 / Float64(Float64(z * x_m) * Float64(y_m * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e+15)
		tmp = 1.0 / (x_m * (y_m * (1.0 + (z * z))));
	else
		tmp = 1.0 / ((z * x_m) * (y_m * z));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+15], N[(1.0 / N[(x$95$m * N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z * x$95$m), $MachinePrecision] * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e15

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + z \cdot z\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot z\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 99.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]
   4. Add Preprocessing

   if 5e15 < (*.f64 z z)

   1. Initial program 80.5%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + z \cdot z\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot z\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right) \]

   3. Simplified 80.6%

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

      1. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}{1}\right)}\right) \]

      3. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{\left(1 + z \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(1 + z \cdot z\right) \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(1 + z \cdot z\right), \color{blue}{\left(x \cdot y\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(z \cdot z\right)\right), \left(\color{blue}{x} \cdot y\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, z\right)\right), \left(x \cdot y\right)\right)\right) \]

      9. *-lowering-*.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   6. Applied egg-rr 79.6%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\color{blue}{\left({z}^{2}\right)}, \mathsf{*.f64}\left(x, y\right)\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(z \cdot z\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right) \]

      2. *-lowering-*.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right) \]

   9. Simplified 79.6%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(z \cdot \color{blue}{\left(z \cdot \left(x \cdot y\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(z \cdot \left(\left(z \cdot x\right) \cdot \color{blue}{y}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(z \cdot \left(y \cdot \color{blue}{\left(z \cdot x\right)}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(z \cdot y\right) \cdot \color{blue}{\left(z \cdot x\right)}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(y \cdot z\right) \cdot \left(\color{blue}{z} \cdot x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(z \cdot x\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(z \cdot y\right), \left(\color{blue}{z} \cdot x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \left(\color{blue}{z} \cdot x\right)\right)\right) \]

       9. *-lowering-*.f64 96.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{*.f64}\left(z, \color{blue}{x}\right)\right)\right) \]

   11. Applied egg-rr 96.7%

       \[\leadsto \frac{1}{\color{blue}{\left(z \cdot y\right) \cdot \left(z \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot x\right) \cdot \left(y \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 3.15 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{\frac{-1}{-1 - z \cdot z}}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot x\_m\right) \cdot \left(y\_m \cdot z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 3.15e+127)
     (/ (/ (/ -1.0 (- -1.0 (* z z))) x_m) y_m)
     (/ 1.0 (* (* z x_m) (* y_m z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 3.15e+127) {
		tmp = ((-1.0 / (-1.0 - (z * z))) / x_m) / y_m;
	} else {
		tmp = 1.0 / ((z * x_m) * (y_m * z));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 3.15d+127) then
        tmp = (((-1.0d0) / ((-1.0d0) - (z * z))) / x_m) / y_m
    else
        tmp = 1.0d0 / ((z * x_m) * (y_m * z))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 3.15e+127) {
		tmp = ((-1.0 / (-1.0 - (z * z))) / x_m) / y_m;
	} else {
		tmp = 1.0 / ((z * x_m) * (y_m * z));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 3.15e+127:
		tmp = ((-1.0 / (-1.0 - (z * z))) / x_m) / y_m
	else:
		tmp = 1.0 / ((z * x_m) * (y_m * z))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 3.15e+127)
		tmp = Float64(Float64(Float64(-1.0 / Float64(-1.0 - Float64(z * z))) / x_m) / y_m);
	else
		tmp = Float64(1.0 / Float64(Float64(z * x_m) * Float64(y_m * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 3.15e+127)
		tmp = ((-1.0 / (-1.0 - (z * z))) / x_m) / y_m;
	else
		tmp = 1.0 / ((z * x_m) * (y_m * z));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 3.15e+127], N[(N[(N[(-1.0 / N[(-1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(N[(z * x$95$m), $MachinePrecision] * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.15000000000000024e127

   1. Initial program 92.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + z \cdot z\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot z\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 92.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right) \]

   3. Simplified 92.5%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(y \cdot \left(z \cdot z + \color{blue}{1}\right)\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\left(z \cdot z\right) \cdot y + \color{blue}{1 \cdot y}\right)\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\left(z \cdot z\right) \cdot y + y\right)\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(z \cdot z\right) \cdot y\right) \cdot x + \color{blue}{y \cdot x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(y \cdot \left(z \cdot z\right)\right) \cdot x + y \cdot x\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(y \cdot z\right) \cdot z\right) \cdot x + y \cdot x\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(y \cdot z\right) \cdot \left(z \cdot x\right) + \color{blue}{y} \cdot x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(y \cdot z\right) \cdot \left(z \cdot x\right) + x \cdot \color{blue}{y}\right)\right) \]

      9. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{fma}\left(y \cdot z, \color{blue}{z \cdot x}, x \cdot y\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(y \cdot z\right), \color{blue}{\left(z \cdot x\right)}, \left(x \cdot y\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\color{blue}{z} \cdot x\right), \left(x \cdot y\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{*.f64}\left(z, \color{blue}{x}\right), \left(x \cdot y\right)\right)\right) \]

      13. *-lowering-*.f64 98.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{*.f64}\left(z, x\right), \mathsf{*.f64}\left(x, y\right)\right)\right) \]

   6. Applied egg-rr 98.1%

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

      1. inv-pow N/A

         \[\leadsto {\left(\left(y \cdot z\right) \cdot \left(z \cdot x\right) + x \cdot y\right)}^{\color{blue}{-1}} \]

      2. +-commutative N/A

         \[\leadsto {\left(x \cdot y + \left(y \cdot z\right) \cdot \left(z \cdot x\right)\right)}^{-1} \]

      3. *-commutative N/A

         \[\leadsto {\left(y \cdot x + \left(y \cdot z\right) \cdot \left(z \cdot x\right)\right)}^{-1} \]

      4. associate-*l* N/A

         \[\leadsto {\left(y \cdot x + y \cdot \left(z \cdot \left(z \cdot x\right)\right)\right)}^{-1} \]

      5. distribute-lft-out N/A

         \[\leadsto {\left(y \cdot \left(x + z \cdot \left(z \cdot x\right)\right)\right)}^{-1} \]

      6. unpow-prod-down N/A

         \[\leadsto {y}^{-1} \cdot \color{blue}{{\left(x + z \cdot \left(z \cdot x\right)\right)}^{-1}} \]

      7. *-lft-identity N/A

         \[\leadsto {y}^{-1} \cdot {\left(1 \cdot x + z \cdot \left(z \cdot x\right)\right)}^{-1} \]

      8. associate-*r* N/A

         \[\leadsto {y}^{-1} \cdot {\left(1 \cdot x + \left(z \cdot z\right) \cdot x\right)}^{-1} \]

      9. distribute-rgt-in N/A

         \[\leadsto {y}^{-1} \cdot {\left(x \cdot \left(1 + z \cdot z\right)\right)}^{-1} \]

      10. unpow-prod-down N/A

          \[\leadsto {\left(y \cdot \left(x \cdot \left(1 + z \cdot z\right)\right)\right)}^{\color{blue}{-1}} \]

      11. *-commutative N/A

          \[\leadsto {\left(\left(x \cdot \left(1 + z \cdot z\right)\right) \cdot y\right)}^{-1} \]

      12. unpow-prod-down N/A

          \[\leadsto {\left(x \cdot \left(1 + z \cdot z\right)\right)}^{-1} \cdot \color{blue}{{y}^{-1}} \]

      13. inv-pow N/A

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

      14. inv-pow N/A

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

      15. div-inv N/A

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

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{x \cdot \left(1 + z \cdot z\right)}\right), \color{blue}{y}\right) \]

   8. Applied egg-rr 93.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-1}{-1 - z \cdot z}}{x}}{y}} \]

   if 3.15000000000000024e127 < z

   1. Initial program 70.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + z \cdot z\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot z\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 70.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right) \]

   3. Simplified 70.9%

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

      1. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}{1}\right)}\right) \]

      3. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{\left(1 + z \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(1 + z \cdot z\right) \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(1 + z \cdot z\right), \color{blue}{\left(x \cdot y\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(z \cdot z\right)\right), \left(\color{blue}{x} \cdot y\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, z\right)\right), \left(x \cdot y\right)\right)\right) \]

      9. *-lowering-*.f64 68.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   6. Applied egg-rr 68.1%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\color{blue}{\left({z}^{2}\right)}, \mathsf{*.f64}\left(x, y\right)\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(z \cdot z\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right) \]

      2. *-lowering-*.f64 68.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right) \]

   9. Simplified 68.1%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(z \cdot \color{blue}{\left(z \cdot \left(x \cdot y\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(z \cdot \left(\left(z \cdot x\right) \cdot \color{blue}{y}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(z \cdot \left(y \cdot \color{blue}{\left(z \cdot x\right)}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(z \cdot y\right) \cdot \color{blue}{\left(z \cdot x\right)}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(y \cdot z\right) \cdot \left(\color{blue}{z} \cdot x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(z \cdot x\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(z \cdot y\right), \left(\color{blue}{z} \cdot x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \left(\color{blue}{z} \cdot x\right)\right)\right) \]

       9. *-lowering-*.f64 97.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{*.f64}\left(z, \color{blue}{x}\right)\right)\right) \]

   11. Applied egg-rr 97.4%

       \[\leadsto \frac{1}{\color{blue}{\left(z \cdot y\right) \cdot \left(z \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.15 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{\frac{-1}{-1 - z \cdot z}}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot x\right) \cdot \left(y \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot x\_m\right) \cdot \left(y\_m \cdot z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 2.1e+68)
     (/ (/ 1.0 y_m) (* x_m (+ 1.0 (* z z))))
     (/ 1.0 (* (* z x_m) (* y_m z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 2.1e+68) {
		tmp = (1.0 / y_m) / (x_m * (1.0 + (z * z)));
	} else {
		tmp = 1.0 / ((z * x_m) * (y_m * z));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2.1d+68) then
        tmp = (1.0d0 / y_m) / (x_m * (1.0d0 + (z * z)))
    else
        tmp = 1.0d0 / ((z * x_m) * (y_m * z))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 2.1e+68) {
		tmp = (1.0 / y_m) / (x_m * (1.0 + (z * z)));
	} else {
		tmp = 1.0 / ((z * x_m) * (y_m * z));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 2.1e+68:
		tmp = (1.0 / y_m) / (x_m * (1.0 + (z * z)))
	else:
		tmp = 1.0 / ((z * x_m) * (y_m * z))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 2.1e+68)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(1.0 / Float64(Float64(z * x_m) * Float64(y_m * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 2.1e+68)
		tmp = (1.0 / y_m) / (x_m * (1.0 + (z * z)));
	else
		tmp = 1.0 / ((z * x_m) * (y_m * z));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 2.1e+68], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z * x$95$m), $MachinePrecision] * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.10000000000000001e68

   1. Initial program 93.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + z \cdot z\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot z\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 93.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right) \]

   3. Simplified 93.1%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l* N/A

         \[\leadsto \frac{1}{y \cdot \color{blue}{\frac{1 + z \cdot z}{\frac{1}{x}}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{y}}{\color{blue}{\frac{1 + z \cdot z}{\frac{1}{x}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{y}\right), \color{blue}{\left(\frac{1 + z \cdot z}{\frac{1}{x}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, y\right), \left(\frac{\color{blue}{1 + z \cdot z}}{\frac{1}{x}}\right)\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, y\right), \left(\left(1 + z \cdot z\right) \cdot \color{blue}{\frac{1}{\frac{1}{x}}}\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, y\right), \left(\left(1 + z \cdot z\right) \cdot \frac{x}{\color{blue}{1}}\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, y\right), \left(\left(1 + z \cdot z\right) \cdot x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, y\right), \left(x \cdot \color{blue}{\left(1 + z \cdot z\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(x, \color{blue}{\left(1 + z \cdot z\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot z\right)}\right)\right)\right) \]

      13. *-lowering-*.f64 92.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right) \]

   6. Applied egg-rr 92.8%

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

   if 2.10000000000000001e68 < z

   1. Initial program 72.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + z \cdot z\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot z\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 72.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right) \]

   3. Simplified 72.5%

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

      1. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}{1}\right)}\right) \]

      3. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{\left(1 + z \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(1 + z \cdot z\right) \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(1 + z \cdot z\right), \color{blue}{\left(x \cdot y\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(z \cdot z\right)\right), \left(\color{blue}{x} \cdot y\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, z\right)\right), \left(x \cdot y\right)\right)\right) \]

      9. *-lowering-*.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   6. Applied egg-rr 72.4%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\color{blue}{\left({z}^{2}\right)}, \mathsf{*.f64}\left(x, y\right)\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(z \cdot z\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right) \]

      2. *-lowering-*.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right) \]

   9. Simplified 72.4%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(z \cdot \color{blue}{\left(z \cdot \left(x \cdot y\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(z \cdot \left(\left(z \cdot x\right) \cdot \color{blue}{y}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(z \cdot \left(y \cdot \color{blue}{\left(z \cdot x\right)}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(z \cdot y\right) \cdot \color{blue}{\left(z \cdot x\right)}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(y \cdot z\right) \cdot \left(\color{blue}{z} \cdot x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(z \cdot x\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(z \cdot y\right), \left(\color{blue}{z} \cdot x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \left(\color{blue}{z} \cdot x\right)\right)\right) \]

       9. *-lowering-*.f64 97.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{*.f64}\left(z, \color{blue}{x}\right)\right)\right) \]

   11. Applied egg-rr 97.9%

       \[\leadsto \frac{1}{\color{blue}{\left(z \cdot y\right) \cdot \left(z \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot x\right) \cdot \left(y \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot x\_m\right) \cdot \left(y\_m \cdot z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e-24)
     (/ (/ 1.0 x_m) y_m)
     (/ 1.0 (* (* z x_m) (* y_m z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-24) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / ((z * x_m) * (y_m * z));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d-24) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = 1.0d0 / ((z * x_m) * (y_m * z))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-24) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / ((z * x_m) * (y_m * z));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 5e-24:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = 1.0 / ((z * x_m) * (y_m * z))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-24)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(1.0 / Float64(Float64(z * x_m) * Float64(y_m * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e-24)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = 1.0 / ((z * x_m) * (y_m * z));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e-24], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(N[(z * x$95$m), $MachinePrecision] * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999998e-24

   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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \color{blue}{y}\right) \]
   4. Step-by-step derivation

   5. Simplified 99.7%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y}} \]

   if 4.9999999999999998e-24 < (*.f64 z z)

   1. Initial program 81.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + z \cdot z\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot z\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 81.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right) \]

   3. Simplified 81.3%

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

      1. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}{1}\right)}\right) \]

      3. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{\left(1 + z \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(1 + z \cdot z\right) \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(1 + z \cdot z\right), \color{blue}{\left(x \cdot y\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(z \cdot z\right)\right), \left(\color{blue}{x} \cdot y\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, z\right)\right), \left(x \cdot y\right)\right)\right) \]

      9. *-lowering-*.f64 80.4%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   6. Applied egg-rr 80.4%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\color{blue}{\left({z}^{2}\right)}, \mathsf{*.f64}\left(x, y\right)\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(z \cdot z\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right) \]

      2. *-lowering-*.f64 79.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right) \]

   9. Simplified 79.2%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(z \cdot \color{blue}{\left(z \cdot \left(x \cdot y\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(z \cdot \left(\left(z \cdot x\right) \cdot \color{blue}{y}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(z \cdot \left(y \cdot \color{blue}{\left(z \cdot x\right)}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(z \cdot y\right) \cdot \color{blue}{\left(z \cdot x\right)}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(y \cdot z\right) \cdot \left(\color{blue}{z} \cdot x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(z \cdot x\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(z \cdot y\right), \left(\color{blue}{z} \cdot x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \left(\color{blue}{z} \cdot x\right)\right)\right) \]

       9. *-lowering-*.f64 95.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{*.f64}\left(z, \color{blue}{x}\right)\right)\right) \]

   11. Applied egg-rr 95.7%

       \[\leadsto \frac{1}{\color{blue}{\left(z \cdot y\right) \cdot \left(z \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot x\right) \cdot \left(y \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 92.7%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \color{blue}{y}\right) \]
   4. Step-by-step derivation

   5. Simplified 66.4%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y}} \]

   if 1 < z

   1. Initial program 80.1%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + z \cdot z\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot z\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 80.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right) \]

   3. Simplified 80.2%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(y \cdot \left(z \cdot z + \color{blue}{1}\right)\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\left(z \cdot z\right) \cdot y + \color{blue}{1 \cdot y}\right)\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\left(z \cdot z\right) \cdot y + y\right)\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(z \cdot z\right) \cdot y\right) \cdot x + \color{blue}{y \cdot x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(y \cdot \left(z \cdot z\right)\right) \cdot x + y \cdot x\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(y \cdot z\right) \cdot z\right) \cdot x + y \cdot x\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(y \cdot z\right) \cdot \left(z \cdot x\right) + \color{blue}{y} \cdot x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(y \cdot z\right) \cdot \left(z \cdot x\right) + x \cdot \color{blue}{y}\right)\right) \]

      9. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{fma}\left(y \cdot z, \color{blue}{z \cdot x}, x \cdot y\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(y \cdot z\right), \color{blue}{\left(z \cdot x\right)}, \left(x \cdot y\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\color{blue}{z} \cdot x\right), \left(x \cdot y\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{*.f64}\left(z, \color{blue}{x}\right), \left(x \cdot y\right)\right)\right) \]

      13. *-lowering-*.f64 96.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{*.f64}\left(z, x\right), \mathsf{*.f64}\left(x, y\right)\right)\right) \]

   6. Applied egg-rr 96.9%

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\left(y \cdot z\right) \cdot \left(z \cdot x\right)\right), \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(y \cdot \left(z \cdot \left(z \cdot x\right)\right)\right), \left(\color{blue}{x} \cdot y\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(z \cdot \left(z \cdot x\right)\right)\right), \left(\color{blue}{x} \cdot y\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(z \cdot x\right)\right)\right), \left(x \cdot y\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, x\right)\right)\right), \left(x \cdot y\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, x\right)\right)\right), \left(y \cdot \color{blue}{x}\right)\right)\right) \]

      7. *-lowering-*.f64 90.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, x\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right)\right) \]

   8. Applied egg-rr 90.9%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(y \cdot {z}^{2}\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{{z}^{2}}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(y \cdot x\right) \cdot {\color{blue}{z}}^{2}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot {z}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot {z}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left({z}^{2}\right)}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{z}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 80.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right) \]

   11. Simplified 80.0%

       \[\leadsto \color{blue}{\frac{1}{y \cdot \left(x \cdot \left(z \cdot z\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 73.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 92.7%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \color{blue}{y}\right) \]
   4. Step-by-step derivation

   5. Simplified 66.4%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y}} \]

   if 1 < z

   1. Initial program 80.1%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + z \cdot z\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot z\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 80.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right) \]

   3. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(y \cdot {z}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot {z}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 79.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right) \]

   7. Simplified 79.9%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(z \cdot z\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 58.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Step-by-step derivation

   1. associate-/l/ N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + z \cdot z\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot z\right)}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 89.5%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right) \]

3. Simplified 89.5%

   \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in z around 0

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

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot y\right)}\right) \]

   2. *-lowering-*.f64 54.8%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

7. Simplified 54.8%

   \[\leadsto \color{blue}{\frac{1}{x \cdot y}} \]

8. Final simplification 54.8%

   \[\leadsto \frac{1}{y \cdot x} \]
9. Add Preprocessing

Developer Target 1: 92.6% 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 2024138 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* y (+ 1 (* z z))) -inf.0) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 868074325056725200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x)))))

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