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

Percentage Accurate: 88.9% → 99.1%

Time: 8.0s

Alternatives: 12

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / x) / (y * (1.0 + (z * z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 88.9% 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: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot \mathsf{fma}\left(z, z, 1\right)}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y\_m}}{z \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 2e+306)
     (/ (/ 1.0 (* x_m (fma z z 1.0))) y_m)
     (/ (/ 1.0 (* z y_m)) (* z x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+306) {
		tmp = (1.0 / (x_m * fma(z, z, 1.0))) / y_m;
	} else {
		tmp = (1.0 / (z * y_m)) / (z * x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+306)
		tmp = Float64(Float64(1.0 / Float64(x_m * fma(z, z, 1.0))) / y_m);
	else
		tmp = Float64(Float64(1.0 / Float64(z * y_m)) / Float64(z * x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+306], N[(N[(1.0 / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / N[(z * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000003e306

   1. Initial program 96.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r/ N/A

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

      11. /-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.7

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 96.7

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

   4. Applied egg-rr 96.7%

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

   if 2.00000000000000003e306 < (*.f64 z z)

   1. Initial program 79.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.4

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

   5. Simplified 79.4%

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

      1. metadata-eval N/A

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

      2. remove-double-neg N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. frac-times N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y}}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(z, z, -1\right)}{x\_m \cdot y\_m}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(\left(z \cdot z\right) \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x\_m \cdot \left(z \cdot y\_m\right)\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 2e-6)
     (- (/ (fma z z -1.0) (* x_m y_m)))
     (if (<= (* z z) 5e+302)
       (/ 1.0 (* y_m (* (* z z) x_m)))
       (/ 1.0 (* z (* x_m (* z y_m)))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e-6) {
		tmp = -(fma(z, z, -1.0) / (x_m * y_m));
	} else if ((z * z) <= 5e+302) {
		tmp = 1.0 / (y_m * ((z * z) * x_m));
	} else {
		tmp = 1.0 / (z * (x_m * (z * y_m)));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-6)
		tmp = Float64(-Float64(fma(z, z, -1.0) / Float64(x_m * y_m)));
	elseif (Float64(z * z) <= 5e+302)
		tmp = Float64(1.0 / Float64(y_m * Float64(Float64(z * z) * x_m)));
	else
		tmp = Float64(1.0 / Float64(z * Float64(x_m * Float64(z * y_m))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e-6], (-N[(N[(z * z + -1.0), $MachinePrecision] / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), If[LessEqual[N[(z * z), $MachinePrecision], 5e+302], N[(1.0 / N[(y$95$m * N[(N[(z * z), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z * N[(x$95$m * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 1.99999999999999991e-6

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

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

      2. cancel-sign-sub N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 99.0

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

   5. Simplified 99.0%

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

   if 1.99999999999999991e-6 < (*.f64 z z) < 5e302

   1. Initial program 88.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 86.6

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

   5. Simplified 86.6%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 86.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 86.7

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

   7. Applied egg-rr 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 88.3

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

   9. Applied egg-rr 88.3%

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

   if 5e302 < (*.f64 z z)

   1. Initial program 79.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.7

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

   5. Simplified 79.7%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 98.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 98.4

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

   7. Applied egg-rr 98.4%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(z, z, -1\right)}{x \cdot y}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{y \cdot \left(\left(z \cdot z\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(z \cdot y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot \left(z \cdot z + 1\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{x\_m \cdot \mathsf{fma}\left(y\_m, z \cdot z, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y\_m \cdot \left(z \cdot x\_m\right)\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m (+ (* z z) 1.0)) 2e+305)
     (/ 1.0 (* x_m (fma y_m (* z z) y_m)))
     (/ 1.0 (* z (* y_m (* z x_m))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * ((z * z) + 1.0)) <= 2e+305) {
		tmp = 1.0 / (x_m * fma(y_m, (z * z), y_m));
	} else {
		tmp = 1.0 / (z * (y_m * (z * x_m)));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * Float64(Float64(z * z) + 1.0)) <= 2e+305)
		tmp = Float64(1.0 / Float64(x_m * fma(y_m, Float64(z * z), y_m)));
	else
		tmp = Float64(1.0 / Float64(z * Float64(y_m * Float64(z * x_m))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * N[(N[(z * z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 2e+305], N[(1.0 / N[(x$95$m * N[(y$95$m * N[(z * z), $MachinePrecision] + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z * N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 1.9999999999999999e305

   1. Initial program 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 95.3

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 95.3

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

   4. Applied egg-rr 95.3%

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

   if 1.9999999999999999e305 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.8

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

   5. Simplified 76.8%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 91.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 91.6

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

   7. Applied egg-rr 91.6%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-*.f64 98.0

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

   9. Applied egg-rr 98.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z \cdot z + 1\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(y, z \cdot z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y\_m}}{z \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 5e+302)
     (/ (/ 1.0 y_m) (* x_m (fma z z 1.0)))
     (/ (/ 1.0 (* z y_m)) (* z x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+302) {
		tmp = (1.0 / y_m) / (x_m * fma(z, z, 1.0));
	} else {
		tmp = (1.0 / (z * y_m)) / (z * x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+302)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(z * y_m)) / Float64(z * x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+302], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e302

   1. Initial program 96.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

      13. /-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 96.8

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

   4. Applied egg-rr 96.8%

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

   if 5e302 < (*.f64 z z)

   1. Initial program 79.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.7

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

   5. Simplified 79.7%

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

      1. metadata-eval N/A

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

      2. remove-double-neg N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. frac-times N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y}}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{1}{y\_m \cdot \mathsf{fma}\left(x\_m, z \cdot z, x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y\_m}}{z \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 2e+306)
     (/ 1.0 (* y_m (fma x_m (* z z) x_m)))
     (/ (/ 1.0 (* z y_m)) (* z x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+306) {
		tmp = 1.0 / (y_m * fma(x_m, (z * z), x_m));
	} else {
		tmp = (1.0 / (z * y_m)) / (z * x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+306)
		tmp = Float64(1.0 / Float64(y_m * fma(x_m, Float64(z * z), x_m)));
	else
		tmp = Float64(Float64(1.0 / Float64(z * y_m)) / Float64(z * x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+306], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z * z), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000003e306

   1. Initial program 96.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r/ N/A

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

      11. /-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.7

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 96.7

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

   4. Applied egg-rr 96.7%

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

      1. lift-fma.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lower-*.f64 95.5

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

      21. lift-fma.f64 N/A

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

   6. Applied egg-rr 95.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 95.5

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. distribute-rgt-out N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lower-fma.f64 N/A

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

   8. Applied egg-rr 96.1%

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

   if 2.00000000000000003e306 < (*.f64 z z)

   1. Initial program 79.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.4

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

   5. Simplified 79.4%

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

      1. metadata-eval N/A

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

      2. remove-double-neg N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. frac-times N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{1}{y \cdot \mathsf{fma}\left(x, z \cdot z, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y}}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{1}{y\_m \cdot \mathsf{fma}\left(x\_m, z \cdot z, x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot \left(z \cdot y\_m\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 2e+306)
     (/ 1.0 (* y_m (fma x_m (* z z) x_m)))
     (/ (/ 1.0 (* x_m (* z y_m))) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+306) {
		tmp = 1.0 / (y_m * fma(x_m, (z * z), x_m));
	} else {
		tmp = (1.0 / (x_m * (z * y_m))) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+306)
		tmp = Float64(1.0 / Float64(y_m * fma(x_m, Float64(z * z), x_m)));
	else
		tmp = Float64(Float64(1.0 / Float64(x_m * Float64(z * y_m))) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+306], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z * z), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x$95$m * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000003e306

   1. Initial program 96.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r/ N/A

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

      11. /-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.7

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 96.7

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

   4. Applied egg-rr 96.7%

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

      1. lift-fma.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lower-*.f64 95.5

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

      21. lift-fma.f64 N/A

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

   6. Applied egg-rr 95.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 95.5

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. distribute-rgt-out N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lower-fma.f64 N/A

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

   8. Applied egg-rr 96.1%

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

   if 2.00000000000000003e306 < (*.f64 z z)

   1. Initial program 79.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.4

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

   5. Simplified 79.4%

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

      1. metadata-eval N/A

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

      2. remove-double-neg N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. frac-times N/A

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

      6. remove-double-neg N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

      13. lift-/.f64 N/A

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

      14. remove-double-div N/A

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

      15. *-rgt-identity N/A

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

      16. *-rgt-identity N/A

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

      17. *-commutative N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 99.9

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

Alternative 7: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{y\_m \cdot \mathsf{fma}\left(x\_m, z \cdot z, x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x\_m \cdot \left(z \cdot y\_m\right)\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 5e+302)
     (/ 1.0 (* y_m (fma x_m (* z z) x_m)))
     (/ 1.0 (* z (* x_m (* z y_m))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+302) {
		tmp = 1.0 / (y_m * fma(x_m, (z * z), x_m));
	} else {
		tmp = 1.0 / (z * (x_m * (z * y_m)));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+302)
		tmp = Float64(1.0 / Float64(y_m * fma(x_m, Float64(z * z), x_m)));
	else
		tmp = Float64(1.0 / Float64(z * Float64(x_m * Float64(z * y_m))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+302], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z * z), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z * N[(x$95$m * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e302

   1. Initial program 96.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r/ N/A

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

      11. /-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.7

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 96.7

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

   4. Applied egg-rr 96.7%

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

      1. lift-fma.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lower-*.f64 95.5

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

      21. lift-fma.f64 N/A

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

   6. Applied egg-rr 95.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 95.5

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. distribute-rgt-out N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lower-fma.f64 N/A

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

   8. Applied egg-rr 96.0%

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

   if 5e302 < (*.f64 z z)

   1. Initial program 79.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.7

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

   5. Simplified 79.7%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 98.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 98.4

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

   7. Applied egg-rr 98.4%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{y \cdot \mathsf{fma}\left(x, z \cdot z, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(z \cdot y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(z, z, -1\right)}{x\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(z \cdot \left(z \cdot x\_m\right)\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 2e-6)
     (- (/ (fma z z -1.0) (* x_m y_m)))
     (/ 1.0 (* y_m (* z (* z x_m))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e-6) {
		tmp = -(fma(z, z, -1.0) / (x_m * y_m));
	} else {
		tmp = 1.0 / (y_m * (z * (z * x_m)));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-6)
		tmp = Float64(-Float64(fma(z, z, -1.0) / Float64(x_m * y_m)));
	else
		tmp = Float64(1.0 / Float64(y_m * Float64(z * Float64(z * x_m))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e-6], (-N[(N[(z * z + -1.0), $MachinePrecision] / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), N[(1.0 / N[(y$95$m * N[(z * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999991e-6

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

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

      2. cancel-sign-sub N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 99.0

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

   5. Simplified 99.0%

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

   if 1.99999999999999991e-6 < (*.f64 z z)

   1. Initial program 83.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.8

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

   5. Simplified 82.8%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 93.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 93.2

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

   7. Applied egg-rr 93.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 83.5

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

   9. Applied egg-rr 83.5%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. lower-*.f64 88.7

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

   11. Applied egg-rr 88.7%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(z, z, -1\right)}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(z, z, -1\right)}{x\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y\_m \cdot \left(z \cdot x\_m\right)\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 2e-6)
     (- (/ (fma z z -1.0) (* x_m y_m)))
     (/ 1.0 (* z (* y_m (* z x_m))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e-6) {
		tmp = -(fma(z, z, -1.0) / (x_m * y_m));
	} else {
		tmp = 1.0 / (z * (y_m * (z * x_m)));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-6)
		tmp = Float64(-Float64(fma(z, z, -1.0) / Float64(x_m * y_m)));
	else
		tmp = Float64(1.0 / Float64(z * Float64(y_m * Float64(z * x_m))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e-6], (-N[(N[(z * z + -1.0), $MachinePrecision] / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), N[(1.0 / N[(z * N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999991e-6

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

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

      2. cancel-sign-sub N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 99.0

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

   5. Simplified 99.0%

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

   if 1.99999999999999991e-6 < (*.f64 z z)

   1. Initial program 83.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.8

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

   5. Simplified 82.8%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 93.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 93.2

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

   7. Applied egg-rr 93.2%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-*.f64 93.2

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

   9. Applied egg-rr 93.2%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(z, z, -1\right)}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(z, z, -1\right)}{x\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x\_m \cdot \left(z \cdot y\_m\right)\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 2e-6)
     (- (/ (fma z z -1.0) (* x_m y_m)))
     (/ 1.0 (* z (* x_m (* z y_m))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e-6) {
		tmp = -(fma(z, z, -1.0) / (x_m * y_m));
	} else {
		tmp = 1.0 / (z * (x_m * (z * y_m)));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-6)
		tmp = Float64(-Float64(fma(z, z, -1.0) / Float64(x_m * y_m)));
	else
		tmp = Float64(1.0 / Float64(z * Float64(x_m * Float64(z * y_m))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e-6], (-N[(N[(z * z + -1.0), $MachinePrecision] / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), N[(1.0 / N[(z * N[(x$95$m * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999991e-6

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

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

      2. cancel-sign-sub N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 99.0

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

   5. Simplified 99.0%

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

   if 1.99999999999999991e-6 < (*.f64 z z)

   1. Initial program 83.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.8

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

   5. Simplified 82.8%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 93.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 93.2

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

   7. Applied egg-rr 93.2%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(z, z, -1\right)}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(z \cdot y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(z, z, -1\right)}{x\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(\left(z \cdot z\right) \cdot y\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 2e-6)
     (- (/ (fma z z -1.0) (* x_m y_m)))
     (/ 1.0 (* x_m (* (* z z) y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e-6) {
		tmp = -(fma(z, z, -1.0) / (x_m * y_m));
	} else {
		tmp = 1.0 / (x_m * ((z * z) * y_m));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-6)
		tmp = Float64(-Float64(fma(z, z, -1.0) / Float64(x_m * y_m)));
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(Float64(z * z) * y_m)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e-6], (-N[(N[(z * z + -1.0), $MachinePrecision] / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), N[(1.0 / N[(x$95$m * N[(N[(z * z), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999991e-6

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

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

      2. cancel-sign-sub N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 99.0

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

   5. Simplified 99.0%

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

   if 1.99999999999999991e-6 < (*.f64 z z)

   1. Initial program 83.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.8

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

   5. Simplified 82.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(z, z, -1\right)}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(z \cdot z\right) \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 60.3

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

5. Simplified 60.3%

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

Developer Target 1: 92.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024212 
(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)))))