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

Percentage Accurate: 88.4% → 99.6%

Time: 8.1s

Alternatives: 12

Speedup: 1.1×

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: 88.4% 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: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.5 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{\mathsf{fma}\left(x\_m \cdot z\_m, z\_m, x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z\_m}}{\left(x\_m \cdot z\_m\right) \cdot y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z_m 3.5e+191)
     (/ (/ 1.0 y_m) (fma (* x_m z_m) z_m x_m))
     (/ (/ 1.0 z_m) (* (* x_m z_m) y_m))))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 3.5e+191) {
		tmp = (1.0 / y_m) / fma((x_m * z_m), z_m, x_m);
	} else {
		tmp = (1.0 / z_m) / ((x_m * z_m) * y_m);
	}
	return x_s * (y_s * tmp);
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 3.5e+191)
		tmp = Float64(Float64(1.0 / y_m) / fma(Float64(x_m * z_m), z_m, x_m));
	else
		tmp = Float64(Float64(1.0 / z_m) / Float64(Float64(x_m * z_m) * y_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

z_m = N[Abs[z], $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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z$95$m, 3.5e+191], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(N[(x$95$m * z$95$m), $MachinePrecision] * z$95$m + x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z$95$m), $MachinePrecision] / N[(N[(x$95$m * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.4999999999999997e191

   1. Initial program 91.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. /-rgt-identity N/A

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

      11. lower-*.f64 90.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 90.8

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

   4. Applied rewrites 90.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 94.2

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

   6. Applied rewrites 94.2%

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

   if 3.4999999999999997e191 < z

   1. Initial program 71.1%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 99.8%

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

Alternative 2: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((((z_m * z_m) + 1.0) * y_m) <= 1e+307) {
		tmp = (1.0 / x_m) / fma((y_m * z_m), z_m, y_m);
	} else {
		tmp = (1.0 / ((x_m * z_m) * y_m)) / z_m;
	}
	return x_s * (y_s * tmp);
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(z_m * z_m) + 1.0) * y_m) <= 1e+307)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(y_m * z_m), z_m, y_m));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(x_m * z_m) * y_m)) / z_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 9.99999999999999986e306

   1. Initial program 92.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 97.5

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

   4. Applied rewrites 97.5%

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

   if 9.99999999999999986e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 72.5%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 97.6%

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

   9. Applied rewrites 92.8%

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

   11. Applied rewrites 94.4%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z + 1\right) \cdot y \leq 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(x \cdot z\right) \cdot y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((((z_m * z_m) + 1.0) * y_m) <= 1e+307) {
		tmp = 1.0 / (fma((y_m * z_m), z_m, y_m) * x_m);
	} else {
		tmp = (1.0 / ((x_m * z_m) * y_m)) / z_m;
	}
	return x_s * (y_s * tmp);
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(z_m * z_m) + 1.0) * y_m) <= 1e+307)
		tmp = Float64(1.0 / Float64(fma(Float64(y_m * z_m), z_m, y_m) * x_m));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(x_m * z_m) * y_m)) / z_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 9.99999999999999986e306

   1. Initial program 92.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 92.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 92.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 92.0

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

   4. Applied rewrites 92.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 96.6

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 96.6

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

   6. Applied rewrites 96.6%

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

   if 9.99999999999999986e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 72.5%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 97.6%

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

   9. Applied rewrites 92.8%

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

   11. Applied rewrites 94.4%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z + 1\right) \cdot y \leq 10^{+307}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot z, z, y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(x \cdot z\right) \cdot y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((((z_m * z_m) + 1.0) * y_m) <= 1e+307) {
		tmp = 1.0 / (fma((y_m * z_m), z_m, y_m) * x_m);
	} else {
		tmp = (1.0 / z_m) / ((x_m * z_m) * y_m);
	}
	return x_s * (y_s * tmp);
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(z_m * z_m) + 1.0) * y_m) <= 1e+307)
		tmp = Float64(1.0 / Float64(fma(Float64(y_m * z_m), z_m, y_m) * x_m));
	else
		tmp = Float64(Float64(1.0 / z_m) / Float64(Float64(x_m * z_m) * y_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 9.99999999999999986e306

   1. Initial program 92.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 92.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 92.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 92.0

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

   4. Applied rewrites 92.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 96.6

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 96.6

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

   6. Applied rewrites 96.6%

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

   if 9.99999999999999986e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 72.5%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 94.5%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z + 1\right) \cdot y \leq 10^{+307}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot z, z, y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{\left(x \cdot z\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((((z_m * z_m) + 1.0) * y_m) <= 1e+307) {
		tmp = 1.0 / (fma((y_m * z_m), z_m, y_m) * x_m);
	} else {
		tmp = 1.0 / (((x_m * z_m) * y_m) * z_m);
	}
	return x_s * (y_s * tmp);
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(z_m * z_m) + 1.0) * y_m) <= 1e+307)
		tmp = Float64(1.0 / Float64(fma(Float64(y_m * z_m), z_m, y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x_m * z_m) * y_m) * z_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 9.99999999999999986e306

   1. Initial program 92.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 92.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 92.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 92.0

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

   4. Applied rewrites 92.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 96.6

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 96.6

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

   6. Applied rewrites 96.6%

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

   if 9.99999999999999986e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 72.5%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 93.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z + 1\right) \cdot y \leq 10^{+307}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot z, z, y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot z\right) \cdot y\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((((z_m * z_m) + 1.0) * y_m) <= 1e+307) {
		tmp = 1.0 / ((fma(z_m, z_m, 1.0) * y_m) * x_m);
	} else {
		tmp = 1.0 / (((x_m * z_m) * y_m) * z_m);
	}
	return x_s * (y_s * tmp);
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(z_m * z_m) + 1.0) * y_m) <= 1e+307)
		tmp = Float64(1.0 / Float64(Float64(fma(z_m, z_m, 1.0) * y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x_m * z_m) * y_m) * z_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 9.99999999999999986e306

   1. Initial program 92.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 92.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 92.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 92.0

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

   4. Applied rewrites 92.0%

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

   if 9.99999999999999986e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 72.5%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 93.9%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z + 1\right) \cdot y \leq 10^{+307}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot z\right) \cdot y\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 5 \cdot 10^{+186}:\\ \;\;\;\;\frac{--1}{\left(x\_m \cdot y\_m\right) \cdot \mathsf{fma}\left(z\_m, z\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y\_m \cdot z\_m\right) \cdot \left(x\_m \cdot z\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z_m z_m) 5e+186)
     (/ (- -1.0) (* (* x_m y_m) (fma z_m z_m 1.0)))
     (/ 1.0 (* (* y_m z_m) (* x_m z_m)))))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 5e+186) {
		tmp = -(-1.0) / ((x_m * y_m) * fma(z_m, z_m, 1.0));
	} else {
		tmp = 1.0 / ((y_m * z_m) * (x_m * z_m));
	}
	return x_s * (y_s * tmp);
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 5e+186)
		tmp = Float64(Float64(-(-1.0)) / Float64(Float64(x_m * y_m) * fma(z_m, z_m, 1.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(y_m * z_m) * Float64(x_m * z_m)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

z_m = N[Abs[z], $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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5e+186], N[((--1.0) / N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y$95$m * z$95$m), $MachinePrecision] * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999954e186

   1. Initial program 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

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

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

      17. *-rgt-identity N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 96.7

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lower-fma.f64 96.7

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

   4. Applied rewrites 96.7%

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

   if 4.99999999999999954e186 < (*.f64 z z)

   1. Initial program 75.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 98.8%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+186}:\\ \;\;\;\;\frac{--1}{\left(x \cdot y\right) \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot z\right) \cdot \left(x \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.15 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot z\_m}}{y\_m \cdot z\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z_m 1.15e+126)
     (/ (/ 1.0 y_m) (* (fma z_m z_m 1.0) x_m))
     (/ (/ 1.0 (* x_m z_m)) (* y_m z_m))))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.15e+126) {
		tmp = (1.0 / y_m) / (fma(z_m, z_m, 1.0) * x_m);
	} else {
		tmp = (1.0 / (x_m * z_m)) / (y_m * z_m);
	}
	return x_s * (y_s * tmp);
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 1.15e+126)
		tmp = Float64(Float64(1.0 / y_m) / Float64(fma(z_m, z_m, 1.0) * x_m));
	else
		tmp = Float64(Float64(1.0 / Float64(x_m * z_m)) / Float64(y_m * z_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

z_m = N[Abs[z], $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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z$95$m, 1.15e+126], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.15e126

   1. Initial program 93.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. /-rgt-identity N/A

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

      11. lower-*.f64 92.3

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 92.3

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

   4. Applied rewrites 92.3%

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

   if 1.15e126 < z

   1. Initial program 70.8%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 70.8

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

   5. Applied rewrites 70.8%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{1}{y}}{\mathsf{fma}\left(z, z, 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot z}}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.7

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

   5. Applied rewrites 65.7%

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

   7. Applied rewrites 66.3%

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

   if 1 < z

   1. Initial program 79.0%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   7. Applied rewrites 79.6%

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

   9. Applied rewrites 86.0%

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

4. Final simplification 73.2%

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

Alternative 10: 96.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.7

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

   5. Applied rewrites 65.7%

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

   7. Applied rewrites 66.3%

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

   if 1 < z

   1. Initial program 79.0%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   7. Applied rewrites 94.5%

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

4. Final simplification 76.1%

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

Alternative 11: 57.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 49.2

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

5. Applied rewrites 49.2%

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

7. Applied rewrites 49.6%

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

Alternative 12: 58.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 49.2

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

5. Applied rewrites 49.2%

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

6. Final simplification 49.2%

   \[\leadsto \frac{1}{x \cdot y} \]
7. 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 2024235 
(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)))))