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

Percentage Accurate: 88.1% → 98.6%

Time: 12.4s

Alternatives: 15

Speedup: 0.9×

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: 88.1% 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: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot \mathsf{fma}\left(z, z, 1\right)}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{z \cdot \left(z \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

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) 2e+155)
     (/ (/ 1.0 (* x_m (fma z z 1.0))) y_m)
     (/ (/ 1.0 y_m) (* z (* z 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) {
	double tmp;
	if ((z * z) <= 2e+155) {
		tmp = (1.0 / (x_m * fma(z, z, 1.0))) / y_m;
	} else {
		tmp = (1.0 / y_m) / (z * (z * x_m));
	}
	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) <= 2e+155)
		tmp = Float64(Float64(1.0 / Float64(x_m * fma(z, z, 1.0))) / y_m);
	else
		tmp = Float64(Float64(1.0 / y_m) / Float64(z * Float64(z * x_m)));
	end
	return Float64(y_s * Float64(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], 2e+155], N[(N[(1.0 / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(z * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000001e155

   1. Initial program 98.4%

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

      1. associate-/l/ N/A

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

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

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

      3. clear-num N/A

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

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

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

      5. div-inv N/A

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

      6. remove-double-div N/A

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

      7. *-commutative N/A

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

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

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 98.4

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

   4. Applied egg-rr 98.4%

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

   if 2.00000000000000001e155 < (*.f64 z z)

   1. Initial program 69.5%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(y \cdot {z}^{2}\right)}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 69.6

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

   5. Simplified 69.6%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-lowering-*.f64 92.4

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

   7. Applied egg-rr 92.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 88.6

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

   9. Applied egg-rr 88.6%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z \cdot \left(z \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x\_m}}{y\_m \cdot \left(z \cdot z + 1\right)} \leq 0:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot \left(y\_m \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \end{array}\right) \end{array} \]

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 (<= (/ (/ 1.0 x_m) (* y_m (+ (* z z) 1.0))) 0.0)
     (/ y_m (* x_m (* y_m y_m)))
     (/ (/ 1.0 x_m) y_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) {
	double tmp;
	if (((1.0 / x_m) / (y_m * ((z * z) + 1.0))) <= 0.0) {
		tmp = y_m / (x_m * (y_m * y_m));
	} else {
		tmp = (1.0 / x_m) / y_m;
	}
	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 (((1.0d0 / x_m) / (y_m * ((z * z) + 1.0d0))) <= 0.0d0) then
        tmp = y_m / (x_m * (y_m * y_m))
    else
        tmp = (1.0d0 / x_m) / y_m
    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 (((1.0 / x_m) / (y_m * ((z * z) + 1.0))) <= 0.0) {
		tmp = y_m / (x_m * (y_m * y_m));
	} else {
		tmp = (1.0 / x_m) / y_m;
	}
	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 ((1.0 / x_m) / (y_m * ((z * z) + 1.0))) <= 0.0:
		tmp = y_m / (x_m * (y_m * y_m))
	else:
		tmp = (1.0 / x_m) / y_m
	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(Float64(1.0 / x_m) / Float64(y_m * Float64(Float64(z * z) + 1.0))) <= 0.0)
		tmp = Float64(y_m / Float64(x_m * Float64(y_m * y_m)));
	else
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	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 (((1.0 / x_m) / (y_m * ((z * z) + 1.0))) <= 0.0)
		tmp = y_m / (x_m * (y_m * y_m));
	else
		tmp = (1.0 / x_m) / y_m;
	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[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(N[(z * z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(y$95$m / N[(x$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))) < 0.0

   1. Initial program 82.1%

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

      1. frac-2neg N/A

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

      2. neg-sub0 N/A

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

      3. div-sub N/A

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

      4. associate-/l/ N/A

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

      5. distribute-lft-neg-in N/A

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

      6. frac-sub N/A

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

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

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

   4. Applied egg-rr 26.1%

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

   5. Taylor expanded in z around 0

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

   7. Simplified 59.9%

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

   8. Taylor expanded in z around 0

      \[\leadsto \frac{y}{\color{blue}{x \cdot {y}^{2}}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 43.3

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

   10. Simplified 43.3%

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

   if 0.0 < (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))))

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 83.6%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x}}{y \cdot \left(z \cdot z + 1\right)} \leq 0:\\ \;\;\;\;\frac{y}{x \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x\_m}}{y\_m \cdot \left(z \cdot z + 1\right)} \leq 0:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot \left(y\_m \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot y\_m}\\ \end{array}\right) \end{array} \]

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 (<= (/ (/ 1.0 x_m) (* y_m (+ (* z z) 1.0))) 0.0)
     (/ y_m (* x_m (* y_m y_m)))
     (/ 1.0 (* x_m y_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) {
	double tmp;
	if (((1.0 / x_m) / (y_m * ((z * z) + 1.0))) <= 0.0) {
		tmp = y_m / (x_m * (y_m * y_m));
	} else {
		tmp = 1.0 / (x_m * y_m);
	}
	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 (((1.0d0 / x_m) / (y_m * ((z * z) + 1.0d0))) <= 0.0d0) then
        tmp = y_m / (x_m * (y_m * y_m))
    else
        tmp = 1.0d0 / (x_m * y_m)
    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 (((1.0 / x_m) / (y_m * ((z * z) + 1.0))) <= 0.0) {
		tmp = y_m / (x_m * (y_m * y_m));
	} else {
		tmp = 1.0 / (x_m * y_m);
	}
	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 ((1.0 / x_m) / (y_m * ((z * z) + 1.0))) <= 0.0:
		tmp = y_m / (x_m * (y_m * y_m))
	else:
		tmp = 1.0 / (x_m * y_m)
	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(Float64(1.0 / x_m) / Float64(y_m * Float64(Float64(z * z) + 1.0))) <= 0.0)
		tmp = Float64(y_m / Float64(x_m * Float64(y_m * y_m)));
	else
		tmp = Float64(1.0 / Float64(x_m * y_m));
	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 (((1.0 / x_m) / (y_m * ((z * z) + 1.0))) <= 0.0)
		tmp = y_m / (x_m * (y_m * y_m));
	else
		tmp = 1.0 / (x_m * y_m);
	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[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(N[(z * z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(y$95$m / N[(x$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))) < 0.0

   1. Initial program 82.1%

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

      1. frac-2neg N/A

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

      2. neg-sub0 N/A

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

      3. div-sub N/A

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

      4. associate-/l/ N/A

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

      5. distribute-lft-neg-in N/A

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

      6. frac-sub N/A

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

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

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

   4. Applied egg-rr 26.1%

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

   5. Taylor expanded in z around 0

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

   7. Simplified 59.9%

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

   8. Taylor expanded in z around 0

      \[\leadsto \frac{y}{\color{blue}{x \cdot {y}^{2}}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 43.3

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

   10. Simplified 43.3%

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

   if 0.0 < (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))))

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

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

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

      2. *-lowering-*.f64 83.3

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

   5. Simplified 83.3%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x}}{y \cdot \left(z \cdot z + 1\right)} \leq 0:\\ \;\;\;\;\frac{y}{x \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot \left(z \cdot z + 1\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(\mathsf{fma}\left(z, z, 1\right) \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot \left(z \cdot x\_m\right)}}{y\_m}\\ \end{array}\right) \end{array} \]

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 (<= (* y_m (+ (* z z) 1.0)) 5e+305)
     (/ 1.0 (* x_m (* (fma z z 1.0) y_m)))
     (/ (/ 1.0 (* z (* z x_m))) y_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) {
	double tmp;
	if ((y_m * ((z * z) + 1.0)) <= 5e+305) {
		tmp = 1.0 / (x_m * (fma(z, z, 1.0) * y_m));
	} else {
		tmp = (1.0 / (z * (z * x_m))) / y_m;
	}
	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(y_m * Float64(Float64(z * z) + 1.0)) <= 5e+305)
		tmp = Float64(1.0 / Float64(x_m * Float64(fma(z, z, 1.0) * y_m)));
	else
		tmp = Float64(Float64(1.0 / Float64(z * Float64(z * x_m))) / y_m);
	end
	return Float64(y_s * Float64(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[(y$95$m * N[(N[(z * z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 5e+305], N[(1.0 / N[(x$95$m * N[(N[(z * z + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 5.00000000000000009e305

   1. Initial program 94.4%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 94.3

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

   4. Applied egg-rr 94.3%

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

   if 5.00000000000000009e305 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 49.8%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(y \cdot {z}^{2}\right)}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 49.8

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

   5. Simplified 49.8%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-lowering-*.f64 86.7

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

   7. Applied egg-rr 86.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. associate-/l/ N/A

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

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

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

      6. associate-/l/ N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 81.6

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

   9. Applied egg-rr 81.6%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z \cdot z + 1\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{x \cdot \left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot \left(z \cdot x\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot \left(z \cdot z + 1\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(\mathsf{fma}\left(z, z, 1\right) \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{z \cdot \left(z \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

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 (<= (* y_m (+ (* z z) 1.0)) 5e+305)
     (/ 1.0 (* x_m (* (fma z z 1.0) y_m)))
     (/ (/ 1.0 y_m) (* z (* z 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) {
	double tmp;
	if ((y_m * ((z * z) + 1.0)) <= 5e+305) {
		tmp = 1.0 / (x_m * (fma(z, z, 1.0) * y_m));
	} else {
		tmp = (1.0 / y_m) / (z * (z * x_m));
	}
	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(y_m * Float64(Float64(z * z) + 1.0)) <= 5e+305)
		tmp = Float64(1.0 / Float64(x_m * Float64(fma(z, z, 1.0) * y_m)));
	else
		tmp = Float64(Float64(1.0 / y_m) / Float64(z * Float64(z * x_m)));
	end
	return Float64(y_s * Float64(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[(y$95$m * N[(N[(z * z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 5e+305], N[(1.0 / N[(x$95$m * N[(N[(z * z + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(z * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 5.00000000000000009e305

   1. Initial program 94.4%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 94.3

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

   4. Applied egg-rr 94.3%

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

   if 5.00000000000000009e305 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 49.8%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(y \cdot {z}^{2}\right)}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 49.8

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

   5. Simplified 49.8%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-lowering-*.f64 86.7

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

   7. Applied egg-rr 86.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 81.7

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

   9. Applied egg-rr 81.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z \cdot z + 1\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{x \cdot \left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z \cdot \left(z \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{z \cdot \left(z \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

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) 2e+148)
     (/ (/ 1.0 y_m) (* x_m (fma z z 1.0)))
     (/ (/ 1.0 y_m) (* z (* z 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) {
	double tmp;
	if ((z * z) <= 2e+148) {
		tmp = (1.0 / y_m) / (x_m * fma(z, z, 1.0));
	} else {
		tmp = (1.0 / y_m) / (z * (z * x_m));
	}
	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) <= 2e+148)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(1.0 / y_m) / Float64(z * Float64(z * x_m)));
	end
	return Float64(y_s * Float64(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], 2e+148], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(z * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e148

   1. Initial program 99.0%

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

      1. clear-num N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. div-inv N/A

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

      7. remove-double-div N/A

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

      8. *-commutative N/A

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

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

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 98.3

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

   4. Applied egg-rr 98.3%

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

   if 2.0000000000000001e148 < (*.f64 z z)

   1. Initial program 69.4%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(y \cdot {z}^{2}\right)}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 69.5

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

   5. Simplified 69.5%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-lowering-*.f64 91.7

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

   7. Applied egg-rr 91.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 88.9

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

   9. Applied egg-rr 88.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z \cdot \left(z \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(z \cdot y\_m, z, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot \left(z \cdot x\_m\right)}}{y\_m}\\ \end{array}\right) \end{array} \]

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+29)
     (/ (/ 1.0 x_m) (fma (* z y_m) z y_m))
     (/ (/ 1.0 (* z (* z x_m))) y_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) {
	double tmp;
	if ((z * z) <= 5e+29) {
		tmp = (1.0 / x_m) / fma((z * y_m), z, y_m);
	} else {
		tmp = (1.0 / (z * (z * x_m))) / y_m;
	}
	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+29)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(z * y_m), z, y_m));
	else
		tmp = Float64(Float64(1.0 / Float64(z * Float64(z * x_m))) / y_m);
	end
	return Float64(y_s * Float64(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+29], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(z * y$95$m), $MachinePrecision] * z + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e29

   1. Initial program 99.6%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

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

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

      6. *-lowering-*.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   if 5.0000000000000001e29 < (*.f64 z z)

   1. Initial program 73.3%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(y \cdot {z}^{2}\right)}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 73.4

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

   5. Simplified 73.4%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-lowering-*.f64 92.2

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

   7. Applied egg-rr 92.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. associate-/l/ N/A

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

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

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

      6. associate-/l/ N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 89.0

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

   9. Applied egg-rr 89.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(z \cdot y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot \left(z \cdot x\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+76}:\\ \;\;\;\;\frac{0 - -1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(x\_m \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(z \cdot \left(z \cdot x\_m\right)\right)}\\ \end{array}\right) \end{array} \]

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 z) 1e+76)
     (/ (- 0.0 -1.0) (* (fma z z 1.0) (* x_m y_m)))
     (/ 1.0 (* y_m (* z (* z 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) {
	double tmp;
	if ((z * z) <= 1e+76) {
		tmp = (0.0 - -1.0) / (fma(z, z, 1.0) * (x_m * y_m));
	} else {
		tmp = 1.0 / (y_m * (z * (z * x_m)));
	}
	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) <= 1e+76)
		tmp = Float64(Float64(0.0 - -1.0) / Float64(fma(z, z, 1.0) * Float64(x_m * y_m)));
	else
		tmp = Float64(1.0 / Float64(y_m * Float64(z * Float64(z * x_m))));
	end
	return Float64(y_s * Float64(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], 1e+76], N[(N[(0.0 - -1.0), $MachinePrecision] / N[(N[(z * z + 1.0), $MachinePrecision] * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y$95$m * N[(z * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e76

   1. Initial program 98.9%

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

      1. associate-/l/ N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. distribute-rgt-neg-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. distribute-rgt-neg-out N/A

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

      12. remove-double-div N/A

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

      13. div-inv N/A

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

      14. neg-lowering-neg.f64 N/A

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

      15. div-inv N/A

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

      16. remove-double-div N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   if 1e76 < (*.f64 z z)

   1. Initial program 72.3%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(y \cdot {z}^{2}\right)}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 72.4

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

   5. Simplified 72.4%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-lowering-*.f64 91.7

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

   7. Applied egg-rr 91.7%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 88.6

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

   9. Applied egg-rr 88.6%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+76}:\\ \;\;\;\;\frac{0 - -1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0002:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(\mathsf{fma}\left(z, z, 1\right) \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(z \cdot \left(z \cdot x\_m\right)\right)}\\ \end{array}\right) \end{array} \]

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 z) 0.0002)
     (/ 1.0 (* x_m (* (fma z z 1.0) y_m)))
     (/ 1.0 (* y_m (* z (* z 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) {
	double tmp;
	if ((z * z) <= 0.0002) {
		tmp = 1.0 / (x_m * (fma(z, z, 1.0) * y_m));
	} else {
		tmp = 1.0 / (y_m * (z * (z * x_m)));
	}
	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) <= 0.0002)
		tmp = Float64(1.0 / Float64(x_m * Float64(fma(z, z, 1.0) * y_m)));
	else
		tmp = Float64(1.0 / Float64(y_m * Float64(z * Float64(z * x_m))));
	end
	return Float64(y_s * Float64(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], 0.0002], N[(1.0 / N[(x$95$m * N[(N[(z * z + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y$95$m * N[(z * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e-4

   1. Initial program 99.6%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   if 2.0000000000000001e-4 < (*.f64 z z)

   1. Initial program 74.4%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(y \cdot {z}^{2}\right)}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 74.5

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

   5. Simplified 74.5%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-lowering-*.f64 92.5

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

   7. Applied egg-rr 92.5%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 89.0

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

   9. Applied egg-rr 89.0%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0002:\\ \;\;\;\;\frac{1}{x \cdot \left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0002:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(z \cdot \left(z \cdot x\_m\right)\right)}\\ \end{array}\right) \end{array} \]

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 z) 0.0002)
     (/ (/ 1.0 x_m) y_m)
     (/ 1.0 (* y_m (* z (* z 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) {
	double tmp;
	if ((z * z) <= 0.0002) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (y_m * (z * (z * x_m)));
	}
	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) <= 0.0002d0) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = 1.0d0 / (y_m * (z * (z * x_m)))
    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) <= 0.0002) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (y_m * (z * (z * x_m)));
	}
	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) <= 0.0002:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = 1.0 / (y_m * (z * (z * x_m)))
	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) <= 0.0002)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(1.0 / Float64(y_m * Float64(z * Float64(z * x_m))));
	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) <= 0.0002)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = 1.0 / (y_m * (z * (z * x_m)));
	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], 0.0002], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(y$95$m * N[(z * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e-4

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 98.6%

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

   if 2.0000000000000001e-4 < (*.f64 z z)

   1. Initial program 74.4%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(y \cdot {z}^{2}\right)}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 74.5

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

   5. Simplified 74.5%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-lowering-*.f64 92.5

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

   7. Applied egg-rr 92.5%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 89.0

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

   9. Applied egg-rr 89.0%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0002:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0002:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot x\_m\right) \cdot \left(z \cdot y\_m\right)}\\ \end{array}\right) \end{array} \]

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) 0.0002)
     (/ (/ 1.0 x_m) y_m)
     (/ 1.0 (* (* z x_m) (* z y_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) {
	double tmp;
	if ((z * z) <= 0.0002) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / ((z * x_m) * (z * y_m));
	}
	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) <= 0.0002d0) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = 1.0d0 / ((z * x_m) * (z * y_m))
    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) <= 0.0002) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / ((z * x_m) * (z * y_m));
	}
	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) <= 0.0002:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = 1.0 / ((z * x_m) * (z * y_m))
	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) <= 0.0002)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(1.0 / Float64(Float64(z * x_m) * Float64(z * y_m)));
	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) <= 0.0002)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = 1.0 / ((z * x_m) * (z * y_m));
	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], 0.0002], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(N[(z * x$95$m), $MachinePrecision] * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e-4

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 98.6%

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

   if 2.0000000000000001e-4 < (*.f64 z z)

   1. Initial program 74.4%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(y \cdot {z}^{2}\right)}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 74.5

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

   5. Simplified 74.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. /-rgt-identity N/A

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

      4. associate-/r/ N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. div-inv N/A

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

      9. remove-double-div N/A

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

      10. *-lowering-*.f64 94.7

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

   7. Applied egg-rr 94.7%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0002:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot x\right) \cdot \left(z \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0002:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x\_m \cdot \left(z \cdot y\_m\right)\right)}\\ \end{array}\right) \end{array} \]

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 z) 0.0002)
     (/ (/ 1.0 x_m) y_m)
     (/ 1.0 (* z (* x_m (* z y_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) {
	double tmp;
	if ((z * z) <= 0.0002) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (z * (x_m * (z * y_m)));
	}
	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) <= 0.0002d0) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = 1.0d0 / (z * (x_m * (z * y_m)))
    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) <= 0.0002) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (z * (x_m * (z * y_m)));
	}
	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) <= 0.0002:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = 1.0 / (z * (x_m * (z * y_m)))
	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) <= 0.0002)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(1.0 / Float64(z * Float64(x_m * Float64(z * y_m))));
	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) <= 0.0002)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = 1.0 / (z * (x_m * (z * y_m)));
	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], 0.0002], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(z * N[(x$95$m * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e-4

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 98.6%

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

   if 2.0000000000000001e-4 < (*.f64 z z)

   1. Initial program 74.4%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(y \cdot {z}^{2}\right)}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 74.5

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

   5. Simplified 74.5%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-lowering-*.f64 92.5

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

   7. Applied egg-rr 92.5%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0002:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(z \cdot y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.38:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(z \cdot \left(z \cdot y\_m\right)\right)}\\ \end{array}\right) \end{array} \]

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 0.38) (/ (/ 1.0 x_m) y_m) (/ 1.0 (* x_m (* z (* z y_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) {
	double tmp;
	if (z <= 0.38) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (x_m * (z * (z * y_m)));
	}
	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 <= 0.38d0) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = 1.0d0 / (x_m * (z * (z * y_m)))
    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 <= 0.38) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (x_m * (z * (z * y_m)));
	}
	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 <= 0.38:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = 1.0 / (x_m * (z * (z * y_m)))
	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 <= 0.38)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(z * Float64(z * y_m))));
	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 <= 0.38)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = 1.0 / (x_m * (z * (z * y_m)));
	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, 0.38], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(z * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.38

   1. Initial program 92.6%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 72.0%

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

   if 0.38 < z

   1. Initial program 72.0%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(y \cdot {z}^{2}\right)}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 72.1

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

   5. Simplified 72.1%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 85.3

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

   7. Applied egg-rr 85.3%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.38:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.38:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(\left(z \cdot z\right) \cdot y\_m\right)}\\ \end{array}\right) \end{array} \]

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 0.38) (/ (/ 1.0 x_m) y_m) (/ 1.0 (* x_m (* (* z z) y_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) {
	double tmp;
	if (z <= 0.38) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (x_m * ((z * z) * y_m));
	}
	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 <= 0.38d0) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = 1.0d0 / (x_m * ((z * z) * y_m))
    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 <= 0.38) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (x_m * ((z * z) * y_m));
	}
	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 <= 0.38:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = 1.0 / (x_m * ((z * z) * y_m))
	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 <= 0.38)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(Float64(z * z) * y_m)));
	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 <= 0.38)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = 1.0 / (x_m * ((z * z) * y_m));
	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, 0.38], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(N[(z * z), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.38

   1. Initial program 92.6%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 72.0%

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

   if 0.38 < z

   1. Initial program 72.0%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

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

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(y \cdot {z}^{2}\right)}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 72.1

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

   5. Simplified 72.1%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.38:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(z \cdot z\right) \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.4% accurate, 2.1× speedup

Math

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

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 (* x_m y_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 / (x_m * y_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 / (x_m * y_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 / (x_m * y_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 / (x_m * y_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(x_m * y_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 / (x_m * y_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[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in z around 0

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

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

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

   2. *-lowering-*.f64 56.2

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

5. Simplified 56.2%

   \[\leadsto \color{blue}{\frac{1}{x \cdot y}} \]
6. Add Preprocessing

Developer Target 1: 92.4% accurate, 0.5× 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 2024196 
(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)))))