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

Percentage Accurate: 89.9% → 97.8%

Time: 10.2s

Alternatives: 9

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 89.1

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. *-rgt-identity N/A

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

   12. lower-fma.f64 89.1

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

4. Applied rewrites 89.1%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. remove-double-div N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*l/ N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. *-lft-identity N/A

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

   15. lift-*.f64 N/A

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

   16. un-div-inv N/A

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

   17. lift-/.f64 N/A

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

   18. remove-double-div N/A

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

   19. 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 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 98.3

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

8. Applied rewrites 98.3%

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

Alternative 2: 97.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+308}:\\ \;\;\;\;\frac{1}{x\_m \cdot \mathsf{fma}\left(y, z \cdot z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(x\_m \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* y (+ 1.0 (* z z))) 1e+308)
    (/ 1.0 (* x_m (fma y (* z z) y)))
    (/ 1.0 (* y (* z (* x_m z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y * (1.0 + (z * z))) <= 1e+308) {
		tmp = 1.0 / (x_m * fma(y, (z * z), y));
	} else {
		tmp = 1.0 / (y * (z * (x_m * z)));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 1e+308)
		tmp = Float64(1.0 / Float64(x_m * fma(y, Float64(z * z), y)));
	else
		tmp = Float64(1.0 / Float64(y * Float64(z * Float64(x_m * z))));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+308], N[(1.0 / N[(x$95$m * N[(y * N[(z * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(z * 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))) < 1e308

   1. Initial program 92.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.5

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 92.5

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

   4. Applied rewrites 92.5%

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

   if 1e308 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 74.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.5

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 74.5

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

   4. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 78.4

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

      14. lift-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-in N/A

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

      18. *-rgt-identity N/A

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

      19. lower-fma.f64 78.4

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

   6. Applied rewrites 78.4%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 78.4

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

   9. Applied rewrites 78.4%

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

   11. Applied rewrites 90.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+308}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(y, z \cdot z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+163}:\\ \;\;\;\;\frac{1}{y \cdot \mathsf{fma}\left(x\_m, z \cdot z, x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(x\_m \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* z z) 2e+163)
    (/ 1.0 (* y (fma x_m (* z z) x_m)))
    (/ 1.0 (* y (* z (* x_m z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+163) {
		tmp = 1.0 / (y * fma(x_m, (z * z), x_m));
	} else {
		tmp = 1.0 / (y * (z * (x_m * z)));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+163)
		tmp = Float64(1.0 / Float64(y * fma(x_m, Float64(z * z), x_m)));
	else
		tmp = Float64(1.0 / Float64(y * Float64(z * Float64(x_m * z))));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+163], N[(1.0 / N[(y * N[(x$95$m * N[(z * z), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(z * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e163

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.8

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 98.9

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

      14. lift-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-in N/A

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

      18. *-rgt-identity N/A

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

      19. lower-fma.f64 98.9

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

   6. Applied rewrites 98.9%

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

   if 1.9999999999999999e163 < (*.f64 z z)

   1. Initial program 74.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.2

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 74.2

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

   4. Applied rewrites 74.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 77.1

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

      14. lift-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-in N/A

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

      18. *-rgt-identity N/A

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

      19. lower-fma.f64 77.1

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

   6. Applied rewrites 77.1%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 77.1

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

   9. Applied rewrites 77.1%

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

   11. Applied rewrites 89.3%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+163}:\\ \;\;\;\;\frac{1}{y \cdot \mathsf{fma}\left(x, z \cdot z, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.02:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(y \cdot \left(z \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* z z) 0.02) (/ (/ 1.0 x_m) y) (/ 1.0 (* x_m (* y (* z z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * z) <= 0.02) {
		tmp = (1.0 / x_m) / y;
	} else {
		tmp = 1.0 / (x_m * (y * (z * z)));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 0.02d0) then
        tmp = (1.0d0 / x_m) / y
    else
        tmp = 1.0d0 / (x_m * (y * (z * z)))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * z) <= 0.02) {
		tmp = (1.0 / x_m) / y;
	} else {
		tmp = 1.0 / (x_m * (y * (z * z)));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	tmp = 0
	if (z * z) <= 0.02:
		tmp = (1.0 / x_m) / y
	else:
		tmp = 1.0 / (x_m * (y * (z * z)))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.02)
		tmp = Float64(Float64(1.0 / x_m) / y);
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(y * Float64(z * z))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z * z) <= 0.02)
		tmp = (1.0 / x_m) / y;
	else
		tmp = 1.0 / (x_m * (y * (z * z)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 0.02], N[(N[(1.0 / x$95$m), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.0200000000000000004

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. /-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.7

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 99.2

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

   7. Applied rewrites 99.2%

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

   if 0.0200000000000000004 < (*.f64 z z)

   1. Initial program 77.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

Alternative 5: 77.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. /-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 75.1

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

   7. Applied rewrites 75.1%

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

   if 1 < z

   1. Initial program 78.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 78.9

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 78.9

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

   4. Applied rewrites 78.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.8

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

      14. lift-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-in N/A

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

      18. *-rgt-identity N/A

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

      19. lower-fma.f64 81.8

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

   6. Applied rewrites 81.8%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 81.6

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

   9. Applied rewrites 81.6%

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

   11. Applied rewrites 91.4%

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

4. Final simplification 79.4%

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

Alternative 6: 75.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. /-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 75.1

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

   7. Applied rewrites 75.1%

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

   if 1 < z

   1. Initial program 78.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 78.9

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 78.9

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

   4. Applied rewrites 78.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.8

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

      14. lift-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-in N/A

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

      18. *-rgt-identity N/A

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

      19. lower-fma.f64 81.8

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

   6. Applied rewrites 81.8%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 81.6

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

   9. Applied rewrites 81.6%

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

4. Final simplification 76.8%

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

Alternative 7: 96.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 89.1

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. *-rgt-identity N/A

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

   12. lower-fma.f64 89.1

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

4. Applied rewrites 89.1%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. remove-double-div N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*l/ N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. *-lft-identity N/A

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

   15. lift-*.f64 N/A

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

   16. un-div-inv N/A

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

   17. lift-/.f64 N/A

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

   18. remove-double-div N/A

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

   19. 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 99.0%

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

Alternative 8: 96.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 89.1

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. *-rgt-identity N/A

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

   12. lower-fma.f64 89.1

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

4. Applied rewrites 89.1%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*l* N/A

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

   5. distribute-lft-in N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt1-in N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 90.3

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

   14. lift-*.f64 N/A

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

   15. lift-fma.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. distribute-lft-in N/A

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

   18. *-rgt-identity N/A

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

   19. lower-fma.f64 90.3

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

6. Applied rewrites 90.3%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 95.1

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

8. Applied rewrites 95.1%

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

9. Final simplification 95.1%

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

Alternative 9: 59.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 63.9

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

5. Applied rewrites 63.9%

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

Developer Target 1: 92.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(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)))))