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

Percentage Accurate: 88.0% → 98.5%

Time: 8.9s

Alternatives: 11

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.3× speedup

Math

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

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 (+ 1.0 (* z z))) 5e+302)
     (/ (pow (fma y_m (* z z) y_m) -1.0) x_m)
     (pow (* z (* (* z x_m) y_m)) -1.0)))))

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 * (1.0 + (z * z))) <= 5e+302) {
		tmp = pow(fma(y_m, (z * z), y_m), -1.0) / x_m;
	} else {
		tmp = pow((z * ((z * x_m) * y_m)), -1.0);
	}
	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(1.0 + Float64(z * z))) <= 5e+302)
		tmp = Float64((fma(y_m, Float64(z * z), y_m) ^ -1.0) / x_m);
	else
		tmp = Float64(z * Float64(Float64(z * x_m) * y_m)) ^ -1.0;
	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[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[(N[Power[N[(y$95$m * N[(z * z), $MachinePrecision] + y$95$m), $MachinePrecision], -1.0], $MachinePrecision] / x$95$m), $MachinePrecision], N[Power[N[(z * N[(N[(z * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 5e302

   1. Initial program 91.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. inv-pow N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 91.7

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 91.7

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

   4. Applied rewrites 91.7%

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

   if 5e302 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 73.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 79.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 79.9

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

   7. Applied rewrites 79.9%

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

   9. Applied rewrites 92.7%

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

4. Final simplification 91.8%

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

Alternative 2: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{{x\_m}^{-1}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;{\left(z \cdot \left(\left(z \cdot x\_m\right) \cdot y\_m\right)\right)}^{-1}\\ \end{array}\right) \end{array} \end{array} \]

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
 (let* ((t_0 (* y_m (+ 1.0 (* z z)))))
   (*
    x_s
    (*
     y_s
     (if (<= t_0 5e+302)
       (/ (pow x_m -1.0) t_0)
       (pow (* z (* (* z x_m) y_m)) -1.0))))))

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 t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 5e+302) {
		tmp = pow(x_m, -1.0) / t_0;
	} else {
		tmp = pow((z * ((z * x_m) * y_m)), -1.0);
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 5d+302) then
        tmp = (x_m ** (-1.0d0)) / t_0
    else
        tmp = (z * ((z * x_m) * y_m)) ** (-1.0d0)
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 5e+302) {
		tmp = Math.pow(x_m, -1.0) / t_0;
	} else {
		tmp = Math.pow((z * ((z * x_m) * y_m)), -1.0);
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 5e+302:
		tmp = math.pow(x_m, -1.0) / t_0
	else:
		tmp = math.pow((z * ((z * x_m) * y_m)), -1.0)
	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)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 5e+302)
		tmp = Float64((x_m ^ -1.0) / t_0);
	else
		tmp = Float64(z * Float64(Float64(z * x_m) * y_m)) ^ -1.0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 5e+302)
		tmp = (x_m ^ -1.0) / t_0;
	else
		tmp = (z * ((z * x_m) * y_m)) ^ -1.0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 5e302

   1. Initial program 91.6%

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

   if 5e302 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 73.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 79.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 79.9

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

   7. Applied rewrites 79.9%

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

   9. Applied rewrites 92.7%

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

4. Final simplification 91.8%

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

Alternative 3: 98.5% accurate, 0.3× speedup

Math

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

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 (+ 1.0 (* z z))) 5e+302)
     (/ (pow x_m -1.0) (fma (* y_m z) z y_m))
     (pow (* z (* (* z x_m) y_m)) -1.0)))))

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 * (1.0 + (z * z))) <= 5e+302) {
		tmp = pow(x_m, -1.0) / fma((y_m * z), z, y_m);
	} else {
		tmp = pow((z * ((z * x_m) * y_m)), -1.0);
	}
	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(1.0 + Float64(z * z))) <= 5e+302)
		tmp = Float64((x_m ^ -1.0) / fma(Float64(y_m * z), z, y_m));
	else
		tmp = Float64(z * Float64(Float64(z * x_m) * y_m)) ^ -1.0;
	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[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[(N[Power[x$95$m, -1.0], $MachinePrecision] / N[(N[(y$95$m * z), $MachinePrecision] * z + y$95$m), $MachinePrecision]), $MachinePrecision], N[Power[N[(z * N[(N[(z * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 5e302

   1. Initial program 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 95.1

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

   4. Applied rewrites 95.1%

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

   if 5e302 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 73.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 79.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 79.9

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

   7. Applied rewrites 79.9%

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

   9. Applied rewrites 92.7%

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

4. Final simplification 94.7%

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

Alternative 4: 98.2% accurate, 0.3× speedup

Math

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

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 (+ 1.0 (* z z))) 5e+302)
     (pow (* (* (fma z z 1.0) y_m) x_m) -1.0)
     (pow (* z (* (* z x_m) y_m)) -1.0)))))

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 * (1.0 + (z * z))) <= 5e+302) {
		tmp = pow(((fma(z, z, 1.0) * y_m) * x_m), -1.0);
	} else {
		tmp = pow((z * ((z * x_m) * y_m)), -1.0);
	}
	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(1.0 + Float64(z * z))) <= 5e+302)
		tmp = Float64(Float64(fma(z, z, 1.0) * y_m) * x_m) ^ -1.0;
	else
		tmp = Float64(z * Float64(Float64(z * x_m) * y_m)) ^ -1.0;
	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[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[Power[N[(N[(N[(z * z + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(z * N[(N[(z * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 5e302

   1. Initial program 91.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 91.1

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

   5. Applied rewrites 91.1%

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

   if 5e302 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 73.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 79.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 79.9

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

   7. Applied rewrites 79.9%

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

   9. Applied rewrites 92.7%

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

4. Final simplification 91.3%

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

Alternative 5: 93.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(z, z, 1\right)\right)}^{-1}}{y\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;{\left(z \cdot \left(\left(z \cdot x\_m\right) \cdot y\_m\right)\right)}^{-1}\\ \end{array}\right) \end{array} \]

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 5.2e+79)
     (/ (pow (fma z z 1.0) -1.0) (* y_m x_m))
     (pow (* z (* (* z x_m) y_m)) -1.0)))))

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 <= 5.2e+79) {
		tmp = pow(fma(z, z, 1.0), -1.0) / (y_m * x_m);
	} else {
		tmp = pow((z * ((z * x_m) * y_m)), -1.0);
	}
	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 (z <= 5.2e+79)
		tmp = Float64((fma(z, z, 1.0) ^ -1.0) / Float64(y_m * x_m));
	else
		tmp = Float64(z * Float64(Float64(z * x_m) * y_m)) ^ -1.0;
	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[z, 5.2e+79], N[(N[Power[N[(z * z + 1.0), $MachinePrecision], -1.0], $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[Power[N[(z * N[(N[(z * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.20000000000000029e79

   1. Initial program 91.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. distribute-frac-neg2 N/A

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

      16. lift-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. lift-/.f64 N/A

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

      20. frac-2neg N/A

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

      21. lift-/.f64 N/A

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

      22. lower-/.f64 N/A

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

   6. Applied rewrites 94.4%

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

   7. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 94.2

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

   9. Applied rewrites 94.2%

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

   if 5.20000000000000029e79 < z

   1. Initial program 75.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 75.0

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

   7. Applied rewrites 75.0%

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

   9. Applied rewrites 93.1%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(z, z, 1\right)\right)}^{-1}}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{\left(z \cdot \left(\left(z \cdot x\right) \cdot y\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{\frac{-1}{y\_m}}{x\_m}}{\left(-z\right) \cdot z - 1}\\ \mathbf{else}:\\ \;\;\;\;{\left(z \cdot \left(\left(z \cdot x\_m\right) \cdot y\_m\right)\right)}^{-1}\\ \end{array}\right) \end{array} \]

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 8.5e+76)
     (/ (/ (/ -1.0 y_m) x_m) (- (* (- z) z) 1.0))
     (pow (* z (* (* z x_m) y_m)) -1.0)))))

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 <= 8.5e+76) {
		tmp = ((-1.0 / y_m) / x_m) / ((-z * z) - 1.0);
	} else {
		tmp = pow((z * ((z * x_m) * y_m)), -1.0);
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 8.5d+76) then
        tmp = (((-1.0d0) / y_m) / x_m) / ((-z * z) - 1.0d0)
    else
        tmp = (z * ((z * x_m) * y_m)) ** (-1.0d0)
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 8.5e+76) {
		tmp = ((-1.0 / y_m) / x_m) / ((-z * z) - 1.0);
	} else {
		tmp = Math.pow((z * ((z * x_m) * y_m)), -1.0);
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if z <= 8.5e+76:
		tmp = ((-1.0 / y_m) / x_m) / ((-z * z) - 1.0)
	else:
		tmp = math.pow((z * ((z * x_m) * y_m)), -1.0)
	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 (z <= 8.5e+76)
		tmp = Float64(Float64(Float64(-1.0 / y_m) / x_m) / Float64(Float64(Float64(-z) * z) - 1.0));
	else
		tmp = Float64(z * Float64(Float64(z * x_m) * y_m)) ^ -1.0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 8.5e+76)
		tmp = ((-1.0 / y_m) / x_m) / ((-z * z) - 1.0);
	else
		tmp = (z * ((z * x_m) * y_m)) ^ -1.0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z, 8.5e+76], N[(N[(N[(-1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / N[(N[((-z) * z), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(z * N[(N[(z * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.49999999999999992e76

   1. Initial program 91.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

   4. Applied rewrites 49.5%

      \[\leadsto \color{blue}{{\left(\left(\mathsf{fma}\left(y, z \cdot z, y\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(y, z \cdot z, y\right) \cdot x\right)\right)}^{-0.5}} \]

   5. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 94.5

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

   7. Applied rewrites 94.5%

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

   if 8.49999999999999992e76 < z

   1. Initial program 75.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 73.4%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 73.3

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

   7. Applied rewrites 73.3%

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

   9. Applied rewrites 93.3%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{\frac{-1}{y}}{x}}{\left(-z\right) \cdot z - 1}\\ \mathbf{else}:\\ \;\;\;\;{\left(z \cdot \left(\left(z \cdot x\right) \cdot y\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.03:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-z, z, 1\right)}{y\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;{\left(z \cdot \left(\left(z \cdot x\_m\right) \cdot y\_m\right)\right)}^{-1}\\ \end{array}\right) \end{array} \]

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 0.03)
     (/ (/ (fma (- z) z 1.0) y_m) x_m)
     (pow (* z (* (* z x_m) y_m)) -1.0)))))

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 <= 0.03) {
		tmp = (fma(-z, z, 1.0) / y_m) / x_m;
	} else {
		tmp = pow((z * ((z * x_m) * y_m)), -1.0);
	}
	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 (z <= 0.03)
		tmp = Float64(Float64(fma(Float64(-z), z, 1.0) / y_m) / x_m);
	else
		tmp = Float64(z * Float64(Float64(z * x_m) * y_m)) ^ -1.0;
	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[z, 0.03], N[(N[(N[((-z) * z + 1.0), $MachinePrecision] / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[Power[N[(z * N[(N[(z * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.029999999999999999

   1. Initial program 90.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unpow2 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 0.029999999999999999 < z

   1. Initial program 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 80.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 80.0

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

   7. Applied rewrites 80.0%

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

   9. Applied rewrites 94.7%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.03:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-z, z, 1\right)}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(z \cdot \left(\left(z \cdot x\right) \cdot y\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.03:\\ \;\;\;\;\frac{{y\_m}^{-1}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;{\left(z \cdot \left(\left(z \cdot x\_m\right) \cdot y\_m\right)\right)}^{-1}\\ \end{array}\right) \end{array} \]

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 0.03)
     (/ (pow y_m -1.0) x_m)
     (pow (* z (* (* z x_m) y_m)) -1.0)))))

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 <= 0.03) {
		tmp = pow(y_m, -1.0) / x_m;
	} else {
		tmp = pow((z * ((z * x_m) * y_m)), -1.0);
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.03d0) then
        tmp = (y_m ** (-1.0d0)) / x_m
    else
        tmp = (z * ((z * x_m) * y_m)) ** (-1.0d0)
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 0.03) {
		tmp = Math.pow(y_m, -1.0) / x_m;
	} else {
		tmp = Math.pow((z * ((z * x_m) * y_m)), -1.0);
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if z <= 0.03:
		tmp = math.pow(y_m, -1.0) / x_m
	else:
		tmp = math.pow((z * ((z * x_m) * y_m)), -1.0)
	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 (z <= 0.03)
		tmp = Float64((y_m ^ -1.0) / x_m);
	else
		tmp = Float64(z * Float64(Float64(z * x_m) * y_m)) ^ -1.0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 0.03)
		tmp = (y_m ^ -1.0) / x_m;
	else
		tmp = (z * ((z * x_m) * y_m)) ^ -1.0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z, 0.03], N[(N[Power[y$95$m, -1.0], $MachinePrecision] / x$95$m), $MachinePrecision], N[Power[N[(z * N[(N[(z * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.029999999999999999

   1. Initial program 90.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unpow2 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 71.0%

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

   if 0.029999999999999999 < z

   1. Initial program 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 80.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 80.0

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

   7. Applied rewrites 80.0%

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

   9. Applied rewrites 94.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.03:\\ \;\;\;\;\frac{{y}^{-1}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(z \cdot \left(\left(z \cdot x\right) \cdot y\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.03:\\ \;\;\;\;\frac{{y\_m}^{-1}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;{\left(z \cdot \left(z \cdot \left(y\_m \cdot x\_m\right)\right)\right)}^{-1}\\ \end{array}\right) \end{array} \]

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 0.03)
     (/ (pow y_m -1.0) x_m)
     (pow (* z (* z (* y_m x_m))) -1.0)))))

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 <= 0.03) {
		tmp = pow(y_m, -1.0) / x_m;
	} else {
		tmp = pow((z * (z * (y_m * x_m))), -1.0);
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.03d0) then
        tmp = (y_m ** (-1.0d0)) / x_m
    else
        tmp = (z * (z * (y_m * x_m))) ** (-1.0d0)
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 0.03) {
		tmp = Math.pow(y_m, -1.0) / x_m;
	} else {
		tmp = Math.pow((z * (z * (y_m * x_m))), -1.0);
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if z <= 0.03:
		tmp = math.pow(y_m, -1.0) / x_m
	else:
		tmp = math.pow((z * (z * (y_m * x_m))), -1.0)
	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 (z <= 0.03)
		tmp = Float64((y_m ^ -1.0) / x_m);
	else
		tmp = Float64(z * Float64(z * Float64(y_m * x_m))) ^ -1.0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 0.03)
		tmp = (y_m ^ -1.0) / x_m;
	else
		tmp = (z * (z * (y_m * x_m))) ^ -1.0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z, 0.03], N[(N[Power[y$95$m, -1.0], $MachinePrecision] / x$95$m), $MachinePrecision], N[Power[N[(z * N[(z * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.029999999999999999

   1. Initial program 90.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unpow2 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 71.0%

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

   if 0.029999999999999999 < z

   1. Initial program 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 80.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 80.0

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

   7. Applied rewrites 80.0%

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

   9. Applied rewrites 89.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.03:\\ \;\;\;\;\frac{{y}^{-1}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(z \cdot \left(z \cdot \left(y \cdot x\right)\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.03:\\ \;\;\;\;\frac{{y\_m}^{-1}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(z \cdot z\right) \cdot y\_m\right) \cdot x\_m\right)}^{-1}\\ \end{array}\right) \end{array} \]

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 0.03)
     (/ (pow y_m -1.0) x_m)
     (pow (* (* (* z z) y_m) x_m) -1.0)))))

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 <= 0.03) {
		tmp = pow(y_m, -1.0) / x_m;
	} else {
		tmp = pow((((z * z) * y_m) * x_m), -1.0);
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.03d0) then
        tmp = (y_m ** (-1.0d0)) / x_m
    else
        tmp = (((z * z) * y_m) * x_m) ** (-1.0d0)
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 0.03) {
		tmp = Math.pow(y_m, -1.0) / x_m;
	} else {
		tmp = Math.pow((((z * z) * y_m) * x_m), -1.0);
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if z <= 0.03:
		tmp = math.pow(y_m, -1.0) / x_m
	else:
		tmp = math.pow((((z * z) * y_m) * x_m), -1.0)
	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 (z <= 0.03)
		tmp = Float64((y_m ^ -1.0) / x_m);
	else
		tmp = Float64(Float64(Float64(z * z) * y_m) * x_m) ^ -1.0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 0.03)
		tmp = (y_m ^ -1.0) / x_m;
	else
		tmp = (((z * z) * y_m) * x_m) ^ -1.0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z, 0.03], N[(N[Power[y$95$m, -1.0], $MachinePrecision] / x$95$m), $MachinePrecision], N[Power[N[(N[(N[(z * z), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], -1.0], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.029999999999999999

   1. Initial program 90.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unpow2 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 71.0%

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

   if 0.029999999999999999 < z

   1. Initial program 80.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 80.0

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

   5. Applied rewrites 80.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.03:\\ \;\;\;\;\frac{{y}^{-1}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(z \cdot z\right) \cdot y\right) \cdot x\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.9% accurate, 0.3× speedup

Math

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

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 (pow (* y_m x_m) -1.0))))

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 * pow((y_m * x_m), -1.0));
}

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 * ((y_m * x_m) ** (-1.0d0)))
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 * Math.pow((y_m * x_m), -1.0));
}

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 * math.pow((y_m * x_m), -1.0))

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(y_m * x_m) ^ -1.0)))
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 * ((y_m * x_m) ^ -1.0));
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[Power[N[(y$95$m * x$95$m), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 58.7

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

5. Applied rewrites 58.7%

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

7. Applied rewrites 58.8%

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

8. Final simplification 58.8%

   \[\leadsto {\left(y \cdot x\right)}^{-1} \]
9. Add Preprocessing

Developer Target 1: 92.0% 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 2024340 
(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)))))