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

Percentage Accurate: 88.9% → 98.5%

Time: 11.3s

Alternatives: 10

Speedup: 1.3×

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: 88.9% 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: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (/ (/ 1.0 (fma (* x_m z) z x_m)) y_m))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((1.0 / fma((x_m * z), z, x_m)) / y_m));
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(Float64(1.0 / fma(Float64(x_m * z), z, x_m)) / y_m)))
end

Wolfram

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

Derivation

1. Initial program 94.0%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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 94.3

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

4. Applied egg-rr 94.3%

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. *-lowering-*.f64 95.6

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

6. Applied egg-rr 95.6%

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

Alternative 2: 92.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{x\_m \cdot \mathsf{fma}\left(y\_m, z \cdot z, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x\_m \cdot \left(z \cdot z\right)\right)}\\ \end{array}\right) \end{array} \]

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 4e+25)
     (/ 1.0 (* x_m (fma y_m (* z z) y_m)))
     (/ 1.0 (* y_m (* x_m (* z z))))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 4e+25) {
		tmp = 1.0 / (x_m * fma(y_m, (z * z), y_m));
	} else {
		tmp = 1.0 / (y_m * (x_m * (z * z)));
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+25)
		tmp = Float64(1.0 / Float64(x_m * fma(y_m, Float64(z * z), y_m)));
	else
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * Float64(z * z))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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 should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 4e+25], N[(1.0 / N[(x$95$m * N[(y$95$m * N[(z * z), $MachinePrecision] + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.00000000000000036e25

   1. Initial program 99.7%

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

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. *-lowering-*.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   if 4.00000000000000036e25 < (*.f64 z z)

   1. Initial program 87.6%

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

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. *-lowering-*.f64 87.6

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

   4. Applied egg-rr 87.6%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

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

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

      13. *-lowering-*.f64 88.3

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

   6. Applied egg-rr 88.3%

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

   7. Taylor expanded in z around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 88.3

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

   9. Simplified 88.3%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(y, z \cdot z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{y\_m \cdot \left(-x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x\_m \cdot \left(z \cdot z\right)\right)}\\ \end{array}\right) \end{array} \]

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 5e-6)
     (/ (fma z z -1.0) (* y_m (- x_m)))
     (/ 1.0 (* y_m (* x_m (* z z))))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-6) {
		tmp = fma(z, z, -1.0) / (y_m * -x_m);
	} else {
		tmp = 1.0 / (y_m * (x_m * (z * z)));
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-6)
		tmp = Float64(fma(z, z, -1.0) / Float64(y_m * Float64(-x_m)));
	else
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * Float64(z * z))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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 should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e-6], N[(N[(z * z + -1.0), $MachinePrecision] / N[(y$95$m * (-x$95$m)), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000041e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

         \[\leadsto -1 \cdot \frac{{z}^{2}}{x \cdot y} + \color{blue}{\frac{1}{x \cdot y} \cdot 1} \]

      2. cancel-sign-sub N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

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

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

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

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

      16. *-lowering-*.f64 99.5

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

   5. Simplified 99.5%

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

   if 5.00000000000000041e-6 < (*.f64 z z)

   1. Initial program 88.2%

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

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. *-lowering-*.f64 88.2

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

   4. Applied egg-rr 88.2%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

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

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

      13. *-lowering-*.f64 88.8

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

   6. Applied egg-rr 88.8%

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

   7. Taylor expanded in z around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 88.0

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

   9. Simplified 88.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{y \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{y\_m \cdot \left(-x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(y\_m \cdot \left(z \cdot z\right)\right)}\\ \end{array}\right) \end{array} \]

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 5e-6)
     (/ (fma z z -1.0) (* y_m (- x_m)))
     (/ 1.0 (* x_m (* y_m (* z z))))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-6) {
		tmp = fma(z, z, -1.0) / (y_m * -x_m);
	} else {
		tmp = 1.0 / (x_m * (y_m * (z * z)));
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-6)
		tmp = Float64(fma(z, z, -1.0) / Float64(y_m * Float64(-x_m)));
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(y_m * Float64(z * z))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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 should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e-6], N[(N[(z * z + -1.0), $MachinePrecision] / N[(y$95$m * (-x$95$m)), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(y$95$m * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000041e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

         \[\leadsto -1 \cdot \frac{{z}^{2}}{x \cdot y} + \color{blue}{\frac{1}{x \cdot y} \cdot 1} \]

      2. cancel-sign-sub N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

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

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

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

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

      16. *-lowering-*.f64 99.5

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

   5. Simplified 99.5%

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

   if 5.00000000000000041e-6 < (*.f64 z z)

   1. Initial program 88.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 87.3

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

   5. Simplified 87.3%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{y \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{y\_m \cdot \left(-x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot \left(y\_m \cdot y\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 5e-6)
     (/ (fma z z -1.0) (* y_m (- x_m)))
     (/ y_m (* x_m (* y_m y_m)))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-6) {
		tmp = fma(z, z, -1.0) / (y_m * -x_m);
	} else {
		tmp = y_m / (x_m * (y_m * y_m));
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-6)
		tmp = Float64(fma(z, z, -1.0) / Float64(y_m * Float64(-x_m)));
	else
		tmp = Float64(y_m / Float64(x_m * Float64(y_m * y_m)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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 should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e-6], N[(N[(z * z + -1.0), $MachinePrecision] / N[(y$95$m * (-x$95$m)), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(x$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000041e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

         \[\leadsto -1 \cdot \frac{{z}^{2}}{x \cdot y} + \color{blue}{\frac{1}{x \cdot y} \cdot 1} \]

      2. cancel-sign-sub N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

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

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

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

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

      16. *-lowering-*.f64 99.5

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

   5. Simplified 99.5%

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

   if 5.00000000000000041e-6 < (*.f64 z z)

   1. Initial program 88.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 25.3

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

   5. Simplified 25.3%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

      4. /-lowering-/.f64 24.7

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

   7. Applied egg-rr 24.7%

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

   9. Applied egg-rr 36.4%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{y \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(y \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2.35 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot \left(y\_m \cdot y\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 2.35e+48)
     (/ (/ 1.0 x_m) y_m)
     (/ y_m (* x_m (* y_m y_m)))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2.35e+48) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = y_m / (x_m * (y_m * y_m));
	}
	return x_s * (y_s * tmp);
}

Fortran

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 should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    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
    real(8) :: tmp
    if ((z * z) <= 2.35d+48) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = y_m / (x_m * (y_m * y_m))
    end if
    code = x_s * (y_s * tmp)
end function

Java

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;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2.35e+48) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = y_m / (x_m * (y_m * y_m));
	}
	return x_s * (y_s * tmp);
}

Python

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] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 2.35e+48:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = y_m / (x_m * (y_m * y_m))
	return x_s * (y_s * tmp)

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2.35e+48)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(y_m / Float64(x_m * Float64(y_m * y_m)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

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 = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 2.35e+48)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = y_m / (x_m * (y_m * y_m));
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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 should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2.35e+48], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(y$95$m / N[(x$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.35000000000000006e48

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 94.7%

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

   if 2.35000000000000006e48 < (*.f64 z z)

   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 22.3

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

   5. Simplified 22.3%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

      4. /-lowering-/.f64 21.6

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

   7. Applied egg-rr 21.6%

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

   9. Applied egg-rr 34.8%

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

Alternative 7: 69.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\frac{1}{x\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot \left(y\_m \cdot y\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 4e+46) (/ 1.0 (* x_m y_m)) (/ y_m (* x_m (* y_m y_m)))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 4e+46) {
		tmp = 1.0 / (x_m * y_m);
	} else {
		tmp = y_m / (x_m * (y_m * y_m));
	}
	return x_s * (y_s * tmp);
}

Fortran

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 should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    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
    real(8) :: tmp
    if ((z * z) <= 4d+46) then
        tmp = 1.0d0 / (x_m * y_m)
    else
        tmp = y_m / (x_m * (y_m * y_m))
    end if
    code = x_s * (y_s * tmp)
end function

Java

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;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 4e+46) {
		tmp = 1.0 / (x_m * y_m);
	} else {
		tmp = y_m / (x_m * (y_m * y_m));
	}
	return x_s * (y_s * tmp);
}

Python

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] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 4e+46:
		tmp = 1.0 / (x_m * y_m)
	else:
		tmp = y_m / (x_m * (y_m * y_m))
	return x_s * (y_s * tmp)

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+46)
		tmp = Float64(1.0 / Float64(x_m * y_m));
	else
		tmp = Float64(y_m / Float64(x_m * Float64(y_m * y_m)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

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 = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 4e+46)
		tmp = 1.0 / (x_m * y_m);
	else
		tmp = y_m / (x_m * (y_m * y_m));
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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 should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 4e+46], N[(1.0 / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(x$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4e46

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

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

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

      2. *-lowering-*.f64 94.7

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

   5. Simplified 94.7%

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

   if 4e46 < (*.f64 z z)

   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 22.3

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

   5. Simplified 22.3%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

      4. /-lowering-/.f64 21.6

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

   7. Applied egg-rr 21.6%

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

   9. Applied egg-rr 34.8%

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

Alternative 8: 98.0% accurate, 1.3× speedup

Math

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

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (/ 1.0 (* y_m (fma (* x_m z) z x_m))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (1.0 / (y_m * fma((x_m * z), z, x_m))));
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(1.0 / Float64(y_m * fma(Float64(x_m * z), z, x_m)))))
end

Wolfram

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

Derivation

1. Initial program 94.0%

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

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

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

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

   9. *-lowering-*.f64 94.0

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

4. Applied egg-rr 94.0%

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. distribute-lft1-in N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

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

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

   10. distribute-lft-in N/A

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

   11. *-rgt-identity N/A

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

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

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

   13. *-lowering-*.f64 94.4

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

6. Applied egg-rr 94.4%

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

   1. associate-*r* N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 95.5

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

8. Applied egg-rr 95.5%

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

9. Final simplification 95.5%

   \[\leadsto \frac{1}{y \cdot \mathsf{fma}\left(x \cdot z, z, x\right)} \]
10. Add Preprocessing

Alternative 9: 92.0% accurate, 1.3× speedup

Math

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

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (/ 1.0 (* y_m (fma x_m (* z z) x_m))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (1.0 / (y_m * fma(x_m, (z * z), x_m))));
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(1.0 / Float64(y_m * fma(x_m, Float64(z * z), x_m)))))
end

Wolfram

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

Derivation

1. Initial program 94.0%

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

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

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

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

   9. *-lowering-*.f64 94.0

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

4. Applied egg-rr 94.0%

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. distribute-lft1-in N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

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

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

   10. distribute-lft-in N/A

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

   11. *-rgt-identity N/A

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

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

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

   13. *-lowering-*.f64 94.4

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

6. Applied egg-rr 94.4%

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

7. Final simplification 94.4%

   \[\leadsto \frac{1}{y \cdot \mathsf{fma}\left(x, z \cdot z, x\right)} \]
8. Add Preprocessing

Alternative 10: 58.7% accurate, 2.1× speedup

Math

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

FPCore

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 should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (/ 1.0 (* x_m y_m)))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (1.0 / (x_m * y_m)));
}

Fortran

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 should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    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
    code = x_s * (y_s * (1.0d0 / (x_m * y_m)))
end function

Java

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;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (1.0 / (x_m * y_m)));
}

Python

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] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	return x_s * (y_s * (1.0 / (x_m * y_m)))

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(1.0 / Float64(x_m * y_m))))
end

MATLAB

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 = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(x_s, y_s, x_m, y_m, z)
	tmp = x_s * (y_s * (1.0 / (x_m * y_m)));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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 should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := 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 94.0%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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 62.7

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

5. Simplified 62.7%

   \[\leadsto \color{blue}{\frac{1}{x \cdot y}} \]
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 2024204 
(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)))))