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

Percentage Accurate: 88.2% → 98.3%

Time: 10.3s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.2% 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: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(1.0 / N[(N[(y$95$m * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision] * z + N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $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. 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 88.3

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

   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 88.3

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

4. Applied rewrites 88.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. distribute-lft1-in N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-*l* N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-fma.f64 N/A

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

6. Applied rewrites 98.5%

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

Alternative 2: 96.5% accurate, 0.8× speedup

Math

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

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 (+ 1.0 (* z z))) 1e+305)
     (/ 1.0 (* x_m (fma (* y_m z) z y_m)))
     (/ 1.0 (* (* y_m z) (* x_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 ((y_m * (1.0 + (z * z))) <= 1e+305) {
		tmp = 1.0 / (x_m * fma((y_m * z), z, y_m));
	} else {
		tmp = 1.0 / ((y_m * z) * (x_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(y_m * Float64(1.0 + Float64(z * z))) <= 1e+305)
		tmp = Float64(1.0 / Float64(x_m * fma(Float64(y_m * z), z, y_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(y_m * z) * Float64(x_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[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+305], N[(1.0 / N[(x$95$m * N[(N[(y$95$m * z), $MachinePrecision] * z + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y$95$m * z), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 9.9999999999999994e304

   1. Initial program 92.0%

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

      1. lift-*.f64 N/A

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

      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 91.9

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

      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 91.9

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

   4. Applied rewrites 91.9%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 94.8

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

   6. Applied rewrites 94.8%

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

   if 9.9999999999999994e304 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      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 62.7

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

      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 62.7

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 62.7

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

   7. Applied rewrites 62.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 62.7

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

   9. Applied rewrites 62.7%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 96.0

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 96.0

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 96.0

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

   11. Applied rewrites 96.0%

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

4. Final simplification 94.9%

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

Alternative 3: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\frac{1}{y\_m \cdot \mathsf{fma}\left(x\_m, z \cdot z, x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y\_m \cdot z\right) \cdot \left(x\_m \cdot z\right)}\\ \end{array}\right) \end{array} \]

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+262)
     (/ 1.0 (* y_m (fma x_m (* z z) x_m)))
     (/ 1.0 (* (* y_m z) (* x_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) <= 5e+262) {
		tmp = 1.0 / (y_m * fma(x_m, (z * z), x_m));
	} else {
		tmp = 1.0 / ((y_m * z) * (x_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) <= 5e+262)
		tmp = Float64(1.0 / Float64(y_m * fma(x_m, Float64(z * z), x_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(y_m * z) * Float64(x_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], 5e+262], 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[(N[(y$95$m * z), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000008e262

   1. Initial program 97.1%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. 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 97.0

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

      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 97.0

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

   4. Applied rewrites 97.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 98.5

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

      13. lift-*.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-in N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 98.5

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

   6. Applied rewrites 98.5%

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

   if 5.00000000000000008e262 < (*.f64 z z)

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      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 67.0

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

      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 67.0

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

   4. Applied rewrites 67.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 67.0

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

   7. Applied rewrites 67.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 67.0

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

   9. Applied rewrites 67.0%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 97.1

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 97.1

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 97.1

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

   11. Applied rewrites 97.1%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\frac{1}{y \cdot \mathsf{fma}\left(x, z \cdot z, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot z\right) \cdot \left(x \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 50000:\\ \;\;\;\;\frac{1}{x\_m \cdot \mathsf{fma}\left(y\_m, z \cdot z, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y\_m \cdot z\right) \cdot \left(x\_m \cdot z\right)}\\ \end{array}\right) \end{array} \]

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) 50000.0)
     (/ 1.0 (* x_m (fma y_m (* z z) y_m)))
     (/ 1.0 (* (* y_m z) (* x_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) <= 50000.0) {
		tmp = 1.0 / (x_m * fma(y_m, (z * z), y_m));
	} else {
		tmp = 1.0 / ((y_m * z) * (x_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) <= 50000.0)
		tmp = Float64(1.0 / Float64(x_m * fma(y_m, Float64(z * z), y_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(y_m * z) * Float64(x_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], 50000.0], 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[(N[(y$95$m * z), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e4

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      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 99.6

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

   if 5e4 < (*.f64 z z)

   1. Initial program 73.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 76.8

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

      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 76.8

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 76.8

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

   7. Applied rewrites 76.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 76.8

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

   9. Applied rewrites 76.8%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 96.2

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 96.2

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 96.2

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

   11. Applied rewrites 96.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 50000:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(y, z \cdot z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot z\right) \cdot \left(x \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{y\_m \cdot \left(-x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y\_m \cdot z\right) \cdot \left(x\_m \cdot z\right)}\\ \end{array}\right) \end{array} \]

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-9)
     (/ (fma z z -1.0) (* y_m (- x_m)))
     (/ 1.0 (* (* y_m z) (* x_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) <= 5e-9) {
		tmp = fma(z, z, -1.0) / (y_m * -x_m);
	} else {
		tmp = 1.0 / ((y_m * z) * (x_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) <= 5e-9)
		tmp = Float64(fma(z, z, -1.0) / Float64(y_m * Float64(-x_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(y_m * z) * Float64(x_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], 5e-9], N[(N[(z * z + -1.0), $MachinePrecision] / N[(y$95$m * (-x$95$m)), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y$95$m * z), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e-9

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{z}^{2}}{x \cdot y} + \frac{1}{x \cdot y}} \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

   if 5.0000000000000001e-9 < (*.f64 z z)

   1. Initial program 73.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. 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 77.2

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

      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 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 76.0

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

   7. Applied rewrites 76.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 75.9

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

   9. Applied rewrites 75.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 95.0

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 95.0

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 95.0

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

   11. Applied rewrites 95.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{y \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot z\right) \cdot \left(x \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{y\_m \cdot \left(-x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(z \cdot \left(x\_m \cdot z\right)\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 5e-9)
     (/ (fma z z -1.0) (* y_m (- x_m)))
     (/ 1.0 (* y_m (* z (* x_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) <= 5e-9) {
		tmp = fma(z, z, -1.0) / (y_m * -x_m);
	} else {
		tmp = 1.0 / (y_m * (z * (x_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) <= 5e-9)
		tmp = Float64(fma(z, z, -1.0) / Float64(y_m * Float64(-x_m)));
	else
		tmp = Float64(1.0 / Float64(y_m * Float64(z * Float64(x_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], 5e-9], N[(N[(z * z + -1.0), $MachinePrecision] / N[(y$95$m * (-x$95$m)), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y$95$m * N[(z * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e-9

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{z}^{2}}{x \cdot y} + \frac{1}{x \cdot y}} \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

   if 5.0000000000000001e-9 < (*.f64 z z)

   1. Initial program 73.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. 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 77.2

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

      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 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 76.0

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

   7. Applied rewrites 76.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 75.9

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

   9. Applied rewrites 75.9%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. lower-*.f64 85.3

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 85.3

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

   11. Applied rewrites 85.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{y \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.88:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{y\_m \cdot \left(-x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x\_m \cdot \left(z \cdot z\right)\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z 0.88)
     (/ (fma z z -1.0) (* y_m (- x_m)))
     (/ 1.0 (* y_m (* x_m (* z z))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 0.88) {
		tmp = fma(z, z, -1.0) / (y_m * -x_m);
	} else {
		tmp = 1.0 / (y_m * (x_m * (z * z)));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 0.88)
		tmp = Float64(fma(z, z, -1.0) / Float64(y_m * Float64(-x_m)));
	else
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * Float64(z * z))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z, 0.88], N[(N[(z * z + -1.0), $MachinePrecision] / N[(y$95$m * (-x$95$m)), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.880000000000000004

   1. Initial program 91.7%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

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

      2. cancel-sign-sub N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. 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 73.8

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

   5. Applied rewrites 73.8%

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

   if 0.880000000000000004 < z

   1. Initial program 77.0%

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

      1. lift-*.f64 N/A

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

      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 78.7

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

      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 78.7

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

   4. Applied rewrites 78.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 76.3

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

   7. Applied rewrites 76.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 76.3

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

   9. Applied rewrites 76.3%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.88:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{y \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.88:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{y\_m \cdot \left(-x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(y\_m \cdot \left(z \cdot z\right)\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z 0.88)
     (/ (fma z z -1.0) (* y_m (- x_m)))
     (/ 1.0 (* x_m (* y_m (* z z))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 0.88) {
		tmp = fma(z, z, -1.0) / (y_m * -x_m);
	} else {
		tmp = 1.0 / (x_m * (y_m * (z * z)));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 0.88)
		tmp = Float64(fma(z, z, -1.0) / Float64(y_m * Float64(-x_m)));
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(y_m * Float64(z * z))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z, 0.88], N[(N[(z * z + -1.0), $MachinePrecision] / N[(y$95$m * (-x$95$m)), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(y$95$m * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.880000000000000004

   1. Initial program 91.7%

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

   3. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

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

      2. cancel-sign-sub N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. 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 73.8

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

   5. Applied rewrites 73.8%

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

   if 0.880000000000000004 < z

   1. Initial program 77.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. 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 74.6

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

   5. Applied rewrites 74.6%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.88:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{y \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.3% accurate, 2.1× speedup

Math

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

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 (* y_m x_m)))))

C

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

Julia

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

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(1.0 / N[(y$95$m * x$95$m), $MachinePrecision]), $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. lower-/.f64 N/A

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

   2. lower-*.f64 64.2

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

5. Applied rewrites 64.2%

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

6. Final simplification 64.2%

   \[\leadsto \frac{1}{y \cdot x} \]
7. Add Preprocessing

Developer Target 1: 92.1% 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 2024216 
(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)))))