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

Percentage Accurate: 88.9% → 98.3%

Time: 3.4s

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.9%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. 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. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. pow2 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. pow2 N/A

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

   15. +-commutative N/A

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

   16. pow2 N/A

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

   17. lower-fma.f64 88.6

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

3. Applied rewrites 88.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. pow2 N/A

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

   8. distribute-lft-in N/A

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

   9. associate-*r* N/A

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

   10. *-rgt-identity N/A

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

   11. pow2 N/A

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

   12. associate-*l* N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 96.2

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

5. Applied rewrites 96.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 98.3

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

7. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.55 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x\_m \cdot \left(z \cdot y\_m\right), z, y\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(y\_m \cdot x\_m\right) \cdot z, z, y\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 1.55e+24)
     (/ 1.0 (fma (* x_m (* z y_m)) z (* y_m x_m)))
     (/ 1.0 (fma (* (* y_m x_m) z) z (* y_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1.55e+24) {
		tmp = 1.0 / fma((x_m * (z * y_m)), z, (y_m * x_m));
	} else {
		tmp = 1.0 / fma(((y_m * x_m) * z), z, (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 1.55e+24)
		tmp = Float64(1.0 / fma(Float64(x_m * Float64(z * y_m)), z, Float64(y_m * x_m)));
	else
		tmp = Float64(1.0 / fma(Float64(Float64(y_m * x_m) * z), z, Float64(y_m * x_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1.55e+24], N[(1.0 / N[(N[(x$95$m * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision] * z + N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] * z), $MachinePrecision] * z + N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.55000000000000005e24

   1. Initial program 80.8%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. 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. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 80.8

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

   3. Applied rewrites 80.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. *-rgt-identity N/A

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

      11. pow2 N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if 1.55000000000000005e24 < y

   1. Initial program 92.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. 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. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 91.7

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. distribute-lft-in N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 3: 96.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.9%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. 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. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. pow2 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. pow2 N/A

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

   15. +-commutative N/A

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

   16. pow2 N/A

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

   17. lower-fma.f64 88.6

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

3. Applied rewrites 88.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. pow2 N/A

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

   8. distribute-lft-in N/A

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

   9. associate-*r* N/A

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

   10. *-rgt-identity N/A

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

   11. pow2 N/A

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

   12. associate-*l* N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 96.2

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

5. Applied rewrites 96.2%

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

Alternative 4: 94.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z \cdot y\_m, z, y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(y\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 2e+14)
     (/ 1.0 (* (fma (* z y_m) z y_m) x_m))
     (/ 1.0 (* (fma z z 1.0) (* y_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 2e+14) {
		tmp = 1.0 / (fma((z * y_m), z, y_m) * x_m);
	} else {
		tmp = 1.0 / (fma(z, z, 1.0) * (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 2e+14)
		tmp = Float64(1.0 / Float64(fma(Float64(z * y_m), z, y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(fma(z, z, 1.0) * Float64(y_m * x_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 2e+14], N[(1.0 / N[(N[(N[(z * y$95$m), $MachinePrecision] * z + y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z * z + 1.0), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2e14

   1. Initial program 80.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. 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. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 80.3

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

   3. Applied rewrites 80.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. pow2 N/A

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

      4. distribute-lft1-in N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if 2e14 < y

   1. Initial program 92.1%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. 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. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 91.7

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 96.8

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

   5. Applied rewrites 96.8%

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

Alternative 5: 92.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(y\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 5e-64)
     (/ 1.0 (* (* (fma z z 1.0) y_m) x_m))
     (/ 1.0 (* (fma z z 1.0) (* y_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 5e-64) {
		tmp = 1.0 / ((fma(z, z, 1.0) * y_m) * x_m);
	} else {
		tmp = 1.0 / (fma(z, z, 1.0) * (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 5e-64)
		tmp = Float64(1.0 / Float64(Float64(fma(z, z, 1.0) * y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(fma(z, z, 1.0) * Float64(y_m * x_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 5e-64], N[(1.0 / N[(N[(N[(z * z + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z * z + 1.0), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.00000000000000033e-64

   1. Initial program 76.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. 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. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 76.3

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

   3. Applied rewrites 76.3%

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

   if 5.00000000000000033e-64 < y

   1. Initial program 91.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. 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. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 90.9

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

   3. Applied rewrites 90.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

Alternative 6: 91.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y\_m \cdot x\_m\right) \cdot \left(z \cdot z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* y_m (+ 1.0 (* z z))) 5e+303)
     (/ 1.0 (* (* (fma z z 1.0) y_m) x_m))
     (/ 1.0 (* (* y_m x_m) (* z z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * (1.0 + (z * z))) <= 5e+303) {
		tmp = 1.0 / ((fma(z, z, 1.0) * y_m) * x_m);
	} else {
		tmp = 1.0 / ((y_m * x_m) * (z * z));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * Float64(1.0 + Float64(z * z))) <= 5e+303)
		tmp = Float64(1.0 / Float64(Float64(fma(z, z, 1.0) * y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(y_m * x_m) * Float64(z * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+303], N[(1.0 / N[(N[(N[(z * z + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(z * 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))) < 4.9999999999999997e303

   1. Initial program 99.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. 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. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 99.2

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

   3. Applied rewrites 99.2%

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

   if 4.9999999999999997e303 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 70.1%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. 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. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 70.1

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

   3. Applied rewrites 70.1%

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

   4. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 68.4

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

   6. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 78.8

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

   8. Applied rewrites 78.8%

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

Alternative 7: 71.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.000205:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{y\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y\_m \cdot x\_m\right) \cdot \left(z \cdot z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 0.000205)
     (/ (fma (- z) z 1.0) (* y_m x_m))
     (/ 1.0 (* (* y_m x_m) (* z z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 0.000205) {
		tmp = fma(-z, z, 1.0) / (y_m * x_m);
	} else {
		tmp = 1.0 / ((y_m * x_m) * (z * z));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 0.000205)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(y_m * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(y_m * x_m) * Float64(z * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 0.000205], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.05e-4

   1. Initial program 92.2%

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

   2. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

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

   if 2.05e-4 < z

   1. Initial program 78.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. 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. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 78.6

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

   3. Applied rewrites 78.6%

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

   4. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 76.9

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

   6. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 81.2

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

   8. Applied rewrites 81.2%

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

Alternative 8: 70.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.000205:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{y\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot y\_m\right) \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 0.000205)
     (/ (fma (- z) z 1.0) (* y_m x_m))
     (/ 1.0 (* (* (* z z) y_m) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 0.000205) {
		tmp = fma(-z, z, 1.0) / (y_m * x_m);
	} else {
		tmp = 1.0 / (((z * z) * y_m) * x_m);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 0.000205)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(y_m * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z * z) * y_m) * x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 0.000205], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z * z), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.05e-4

   1. Initial program 92.2%

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

   2. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

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

   if 2.05e-4 < z

   1. Initial program 78.7%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

Alternative 9: 59.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(y_s, x_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * (1.0d0 / (y_m * x_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.9%

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

2. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 59.6

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

4. Applied rewrites 59.6%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))