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

Percentage Accurate: 88.9% → 98.5%

Time: 8.2s

Alternatives: 11

Speedup: 0.7×

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, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m, x\_m, y\_m \cdot \left(\left(z \cdot z\right) \cdot x\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(z \cdot x\_m\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e+80)
     (/ 1.0 (fma y_m x_m (* y_m (* (* z z) x_m))))
     (/ (/ 1.0 (* y_m (* z x_m))) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+80) {
		tmp = 1.0 / fma(y_m, x_m, (y_m * ((z * z) * x_m)));
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+80)
		tmp = Float64(1.0 / fma(y_m, x_m, Float64(y_m * Float64(Float64(z * z) * x_m))));
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(z * x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+80], N[(1.0 / N[(y$95$m * x$95$m + N[(y$95$m * N[(N[(z * z), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999961e80

   1. Initial program 98.5%

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

      1. div-inv N/A

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

      2. associate-/l/ N/A

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

      3. associate-/r/ N/A

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

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

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

      5. associate-/l/ N/A

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

      6. associate-/r/ N/A

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

      7. /-rgt-identity N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-lowering-*.f64 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-lft-in N/A

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

      4. fma-define N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 99.0%

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

   6. Applied egg-rr 99.0%

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

   if 4.99999999999999961e80 < (*.f64 z z)

   1. Initial program 74.3%

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

   3. Taylor expanded in z around inf

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

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

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 76.1%

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

   5. Simplified 76.1%

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

      1. associate-/l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 94.0%

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

   7. Applied egg-rr 94.0%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, x, y \cdot \left(\left(z \cdot z\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 - z \cdot z}{y\_m \cdot x\_m}\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{\left(z \cdot z\right) \cdot x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot x\_m}}{z \cdot y\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e-5)
     (/ (- 1.0 (* z z)) (* y_m x_m))
     (if (<= (* z z) 2e+307)
       (/ (/ 1.0 (* (* z z) x_m)) y_m)
       (/ (/ 1.0 (* z x_m)) (* z y_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	} else if ((z * z) <= 2e+307) {
		tmp = (1.0 / ((z * z) * x_m)) / y_m;
	} else {
		tmp = (1.0 / (z * x_m)) / (z * y_m);
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d-5) then
        tmp = (1.0d0 - (z * z)) / (y_m * x_m)
    else if ((z * z) <= 2d+307) then
        tmp = (1.0d0 / ((z * z) * x_m)) / y_m
    else
        tmp = (1.0d0 / (z * x_m)) / (z * y_m)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	} else if ((z * z) <= 2e+307) {
		tmp = (1.0 / ((z * z) * x_m)) / y_m;
	} else {
		tmp = (1.0 / (z * x_m)) / (z * y_m);
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 5e-5:
		tmp = (1.0 - (z * z)) / (y_m * x_m)
	elif (z * z) <= 2e+307:
		tmp = (1.0 / ((z * z) * x_m)) / y_m
	else:
		tmp = (1.0 / (z * x_m)) / (z * y_m)
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-5)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y_m * x_m));
	elseif (Float64(z * z) <= 2e+307)
		tmp = Float64(Float64(1.0 / Float64(Float64(z * z) * x_m)) / y_m);
	else
		tmp = Float64(Float64(1.0 / Float64(z * x_m)) / Float64(z * y_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e-5)
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	elseif ((z * z) <= 2e+307)
		tmp = (1.0 / ((z * z) * x_m)) / y_m;
	else
		tmp = (1.0 / (z * x_m)) / (z * y_m);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e-5], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+307], N[(N[(1.0 / N[(N[(z * z), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.7%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 98.7%

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

   7. Simplified 98.7%

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

      1. associate-/l/ N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

      7. associate-/r/ N/A

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

      8. /-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.3%

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

   9. Applied egg-rr 99.3%

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

   if 5.00000000000000024e-5 < (*.f64 z z) < 1.99999999999999997e307

   1. Initial program 83.8%

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

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

      2. associate-/l/ N/A

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

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

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 87.0%

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

   5. Simplified 87.0%

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

   if 1.99999999999999997e307 < (*.f64 z z)

   1. Initial program 68.2%

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

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

      2. associate-/l/ N/A

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

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

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 68.2%

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

   5. Simplified 68.2%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{z}}{z}}{y} \]

      3. associate-/l/ N/A

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

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

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-lowering-*.f64 96.4%

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

   7. Applied egg-rr 96.4%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{\left(z \cdot z\right) \cdot x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot x}}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \left(z \cdot z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(z \cdot x\_m\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e+80)
     (/ (/ 1.0 y_m) (* x_m (+ (* z z) 1.0)))
     (/ (/ 1.0 (* y_m (* z x_m))) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+80) {
		tmp = (1.0 / y_m) / (x_m * ((z * z) + 1.0));
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+80) then
        tmp = (1.0d0 / y_m) / (x_m * ((z * z) + 1.0d0))
    else
        tmp = (1.0d0 / (y_m * (z * x_m))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+80) {
		tmp = (1.0 / y_m) / (x_m * ((z * z) + 1.0));
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 5e+80:
		tmp = (1.0 / y_m) / (x_m * ((z * z) + 1.0))
	else:
		tmp = (1.0 / (y_m * (z * x_m))) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+80)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * Float64(Float64(z * z) + 1.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(z * x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e+80)
		tmp = (1.0 / y_m) / (x_m * ((z * z) + 1.0));
	else
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+80], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(N[(z * z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999961e80

   1. Initial program 98.5%

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

      1. associate-/l/ N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-lowering-*.f64 98.5%

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

   4. Applied egg-rr 98.5%

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

   if 4.99999999999999961e80 < (*.f64 z z)

   1. Initial program 74.3%

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

   3. Taylor expanded in z around inf

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

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

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 76.1%

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

   5. Simplified 76.1%

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

      1. associate-/l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 94.0%

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

   7. Applied egg-rr 94.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \left(z \cdot z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x\_m \cdot \left(z \cdot z + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(z \cdot x\_m\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e+80)
     (/ 1.0 (* y_m (* x_m (+ (* z z) 1.0))))
     (/ (/ 1.0 (* y_m (* z x_m))) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+80) {
		tmp = 1.0 / (y_m * (x_m * ((z * z) + 1.0)));
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+80) then
        tmp = 1.0d0 / (y_m * (x_m * ((z * z) + 1.0d0)))
    else
        tmp = (1.0d0 / (y_m * (z * x_m))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+80) {
		tmp = 1.0 / (y_m * (x_m * ((z * z) + 1.0)));
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 5e+80:
		tmp = 1.0 / (y_m * (x_m * ((z * z) + 1.0)))
	else:
		tmp = (1.0 / (y_m * (z * x_m))) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+80)
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * Float64(Float64(z * z) + 1.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(z * x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e+80)
		tmp = 1.0 / (y_m * (x_m * ((z * z) + 1.0)));
	else
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+80], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(N[(z * z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999961e80

   1. Initial program 98.5%

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

      1. div-inv N/A

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

      2. associate-/l/ N/A

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

      3. associate-/r/ N/A

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

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

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

      5. associate-/l/ N/A

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

      6. associate-/r/ N/A

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

      7. /-rgt-identity N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-lowering-*.f64 99.0%

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

   4. Applied egg-rr 99.0%

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

   if 4.99999999999999961e80 < (*.f64 z z)

   1. Initial program 74.3%

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

   3. Taylor expanded in z around inf

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

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

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 76.1%

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

   5. Simplified 76.1%

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

      1. associate-/l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 94.0%

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

   7. Applied egg-rr 94.0%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \left(z \cdot z + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\frac{-1}{\left(-1 - z \cdot z\right) \cdot \left(y\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(z \cdot x\_m\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 2e+65)
     (/ -1.0 (* (- -1.0 (* z z)) (* y_m x_m)))
     (/ (/ 1.0 (* y_m (* z x_m))) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+65) {
		tmp = -1.0 / ((-1.0 - (z * z)) * (y_m * x_m));
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+65) then
        tmp = (-1.0d0) / (((-1.0d0) - (z * z)) * (y_m * x_m))
    else
        tmp = (1.0d0 / (y_m * (z * x_m))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+65) {
		tmp = -1.0 / ((-1.0 - (z * z)) * (y_m * x_m));
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 2e+65:
		tmp = -1.0 / ((-1.0 - (z * z)) * (y_m * x_m))
	else:
		tmp = (1.0 / (y_m * (z * x_m))) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+65)
		tmp = Float64(-1.0 / Float64(Float64(-1.0 - Float64(z * z)) * Float64(y_m * x_m)));
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(z * x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 2e+65)
		tmp = -1.0 / ((-1.0 - (z * z)) * (y_m * x_m));
	else
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+65], N[(-1.0 / N[(N[(-1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e65

   1. Initial program 98.5%

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

      1. div-inv N/A

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

      2. associate-/l/ N/A

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

      3. associate-/r/ N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-/l/ N/A

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

      8. associate-/r/ N/A

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

      9. clear-num N/A

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

      10. associate-/r/ N/A

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

      11. clear-num N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

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

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

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

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

      16. distribute-neg-in N/A

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

      17. metadata-eval N/A

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

      18. unsub-neg N/A

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

      19. --lowering--.f64 N/A

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

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

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

   4. Applied egg-rr 99.1%

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

   if 2e65 < (*.f64 z z)

   1. Initial program 75.3%

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

   3. Taylor expanded in z around inf

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

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

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 76.1%

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

   5. Simplified 76.1%

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

      1. associate-/l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 94.2%

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

   7. Applied egg-rr 94.2%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\frac{-1}{\left(-1 - z \cdot z\right) \cdot \left(y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.7%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 98.7%

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

   7. Simplified 98.7%

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

   if 5.00000000000000024e-5 < (*.f64 z z)

   1. Initial program 76.8%

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

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

      2. associate-/l/ N/A

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

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

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 78.5%

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

   5. Simplified 78.5%

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

      1. associate-/l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 93.6%

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

   7. Applied egg-rr 93.6%

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

4. Final simplification 96.3%

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

Alternative 7: 97.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.7%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 98.7%

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

   7. Simplified 98.7%

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

      1. associate-/l/ N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

      7. associate-/r/ N/A

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

      8. /-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.3%

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

   9. Applied egg-rr 99.3%

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

   if 5.00000000000000024e-5 < (*.f64 z z)

   1. Initial program 76.8%

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

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

      2. associate-/l/ N/A

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

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

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 78.5%

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

   5. Simplified 78.5%

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

      1. associate-/l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 93.6%

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

   7. Applied egg-rr 93.6%

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

4. Final simplification 96.7%

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

Alternative 8: 96.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.7%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 98.7%

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

   7. Simplified 98.7%

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

      1. associate-/l/ N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

      7. associate-/r/ N/A

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

      8. /-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.3%

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

   9. Applied egg-rr 99.3%

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

   if 5.00000000000000024e-5 < (*.f64 z z)

   1. Initial program 76.8%

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

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

      2. associate-/l/ N/A

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

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

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 78.5%

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

   5. Simplified 78.5%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{z}}{z}}{y} \]

      3. associate-/l/ N/A

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

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

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-lowering-*.f64 92.1%

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

   7. Applied egg-rr 92.1%

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

4. Final simplification 96.0%

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

Alternative 9: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.75:\\ \;\;\;\;\frac{1 - z \cdot z}{y\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\left(z \cdot z\right) \cdot y\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 0.75)
     (/ (- 1.0 (* z z)) (* y_m x_m))
     (/ (/ 1.0 x_m) (* (* z z) y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.75) {
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	} else {
		tmp = (1.0 / x_m) / ((z * z) * y_m);
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 0.75d0) then
        tmp = (1.0d0 - (z * z)) / (y_m * x_m)
    else
        tmp = (1.0d0 / x_m) / ((z * z) * y_m)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.75) {
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	} else {
		tmp = (1.0 / x_m) / ((z * z) * y_m);
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 0.75:
		tmp = (1.0 - (z * z)) / (y_m * x_m)
	else:
		tmp = (1.0 / x_m) / ((z * z) * y_m)
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.75)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y_m * x_m));
	else
		tmp = Float64(Float64(1.0 / x_m) / Float64(Float64(z * z) * y_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 0.75)
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	else
		tmp = (1.0 / x_m) / ((z * z) * y_m);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 0.75], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.75

   1. Initial program 99.7%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 98.7%

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

   7. Simplified 98.7%

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

      1. associate-/l/ N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

      7. associate-/r/ N/A

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

      8. /-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.3%

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

   9. Applied egg-rr 99.3%

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

   if 0.75 < (*.f64 z z)

   1. Initial program 76.8%

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

   3. Taylor expanded in z around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 76.2%

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

   5. Simplified 76.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.75:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(z \cdot z\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

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

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

   4. associate-/l/ N/A

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

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

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

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

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

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

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

   8. *-lowering-*.f64 88.8%

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

3. Simplified 88.8%

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

5. Taylor expanded in z around 0

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

   1. /-lowering-/.f64 62.0%

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

7. Simplified 62.0%

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

Alternative 11: 59.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

3. Taylor expanded in z around 0

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

   1. associate-/r* N/A

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

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

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

   3. /-lowering-/.f64 62.4%

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

5. Simplified 62.4%

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

   1. associate-/l/ N/A

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

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

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

   3. *-lowering-*.f64 62.4%

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

7. Applied egg-rr 62.4%

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

Developer Target 1: 93.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024152 
(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)))))