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

Percentage Accurate: 88.2% → 98.6%

Time: 6.2s

Alternatives: 13

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

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: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 10500000000:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot y\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{z\_m}^{-1}}{x\_m \cdot z\_m}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 10500000000.0)
     (/ (/ 1.0 (* (fma z_m z_m 1.0) y_m)) x_m)
     (/ (/ (pow z_m -1.0) (* x_m z_m)) y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 10500000000.0) {
		tmp = (1.0 / (fma(z_m, z_m, 1.0) * y_m)) / x_m;
	} else {
		tmp = (pow(z_m, -1.0) / (x_m * z_m)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 10500000000.0)
		tmp = Float64(Float64(1.0 / Float64(fma(z_m, z_m, 1.0) * y_m)) / x_m);
	else
		tmp = Float64(Float64((z_m ^ -1.0) / Float64(x_m * z_m)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 10500000000.0], N[(N[(1.0 / N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[Power[z$95$m, -1.0], $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.05e10

   1. Initial program 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 95.7

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 95.7

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 95.7

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

   4. Applied rewrites 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. inv-pow N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 95.6

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

   6. Applied rewrites 95.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.7

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

   8. Applied rewrites 95.7%

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

   if 1.05e10 < z

   1. Initial program 74.8%

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

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 74.4

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 74.4

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 74.3%

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

   9. Applied rewrites 84.0%

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

Alternative 2: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot \left(1 + z\_m \cdot z\_m\right) \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(x\_m \cdot z\_m\right) \cdot z\_m}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* y_m (+ 1.0 (* z_m z_m))) 2e+303)
     (/ (/ 1.0 y_m) (* (fma z_m z_m 1.0) x_m))
     (/ (/ 1.0 (* (* x_m z_m) z_m)) y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((y_m * (1.0 + (z_m * z_m))) <= 2e+303) {
		tmp = (1.0 / y_m) / (fma(z_m, z_m, 1.0) * x_m);
	} else {
		tmp = (1.0 / ((x_m * z_m) * z_m)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(y_m * Float64(1.0 + Float64(z_m * z_m))) <= 2e+303)
		tmp = Float64(Float64(1.0 / y_m) / Float64(fma(z_m, z_m, 1.0) * x_m));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(x_m * z_m) * z_m)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+303], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(x$95$m * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 2e303

   1. Initial program 92.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 91.3

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 91.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 91.3

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

   4. Applied rewrites 91.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. inv-pow N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 91.3

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

   6. Applied rewrites 91.3%

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

   if 2e303 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 75.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 80.6

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 80.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 80.6

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 80.5%

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

   9. Applied rewrites 91.8%

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

Alternative 3: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot \left(1 + z\_m \cdot z\_m\right) \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(y\_m \cdot z\_m, z\_m, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(x\_m \cdot z\_m\right) \cdot z\_m}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* y_m (+ 1.0 (* z_m z_m))) 2e+303)
     (/ (/ 1.0 x_m) (fma (* y_m z_m) z_m y_m))
     (/ (/ 1.0 (* (* x_m z_m) z_m)) y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((y_m * (1.0 + (z_m * z_m))) <= 2e+303) {
		tmp = (1.0 / x_m) / fma((y_m * z_m), z_m, y_m);
	} else {
		tmp = (1.0 / ((x_m * z_m) * z_m)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(y_m * Float64(1.0 + Float64(z_m * z_m))) <= 2e+303)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(y_m * z_m), z_m, y_m));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(x_m * z_m) * z_m)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+303], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(y$95$m * z$95$m), $MachinePrecision] * z$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(x$95$m * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 2e303

   1. Initial program 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 96.1

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

   4. Applied rewrites 96.1%

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

   if 2e303 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 75.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 80.6

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 80.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 80.6

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 80.5%

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

   9. Applied rewrites 91.8%

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

Alternative 4: 91.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;1 + z\_m \cdot z\_m \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\left(y\_m \cdot \mathsf{fma}\left(z\_m, z\_m, 1\right)\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z\_m \cdot z\_m\right) \cdot x\_m\right) \cdot y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (+ 1.0 (* z_m z_m)) 5e+18)
     (/ 1.0 (* (* y_m (fma z_m z_m 1.0)) x_m))
     (/ 1.0 (* (* (* z_m z_m) x_m) y_m))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((1.0 + (z_m * z_m)) <= 5e+18) {
		tmp = 1.0 / ((y_m * fma(z_m, z_m, 1.0)) * x_m);
	} else {
		tmp = 1.0 / (((z_m * z_m) * x_m) * y_m);
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(1.0 + Float64(z_m * z_m)) <= 5e+18)
		tmp = Float64(1.0 / Float64(Float64(y_m * fma(z_m, z_m, 1.0)) * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z_m * z_m) * x_m) * y_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], 5e+18], N[(1.0 / N[(N[(y$95$m * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 z z)) < 5e18

   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

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 5e18 < (+.f64 #s(literal 1 binary64) (*.f64 z z))

   1. Initial program 80.3%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 80.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 78.5

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

   7. Applied rewrites 78.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 80.0

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

   9. Applied rewrites 80.0%

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

Alternative 5: 91.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;1 + z\_m \cdot z\_m \leq 1.00005:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z\_m, z\_m, 1\right)}{x\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z\_m \cdot z\_m\right) \cdot x\_m\right) \cdot y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (+ 1.0 (* z_m z_m)) 1.00005)
     (/ (fma (- z_m) z_m 1.0) (* x_m y_m))
     (/ 1.0 (* (* (* z_m z_m) x_m) y_m))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((1.0 + (z_m * z_m)) <= 1.00005) {
		tmp = fma(-z_m, z_m, 1.0) / (x_m * y_m);
	} else {
		tmp = 1.0 / (((z_m * z_m) * x_m) * y_m);
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(1.0 + Float64(z_m * z_m)) <= 1.00005)
		tmp = Float64(fma(Float64(-z_m), z_m, 1.0) / Float64(x_m * y_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z_m * z_m) * x_m) * y_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], 1.00005], N[(N[((-z$95$m) * z$95$m + 1.0), $MachinePrecision] / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 z z)) < 1.00005000000000011

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.3%

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

   if 1.00005000000000011 < (+.f64 #s(literal 1 binary64) (*.f64 z z))

   1. Initial program 81.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 78.5

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

   7. Applied rewrites 78.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 79.9

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

   9. Applied rewrites 79.9%

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

Alternative 6: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4300000000000:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot y\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(x\_m \cdot z\_m\right) \cdot z\_m}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 4300000000000.0)
     (/ (/ 1.0 (* (fma z_m z_m 1.0) y_m)) x_m)
     (/ (/ 1.0 (* (* x_m z_m) z_m)) y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 4300000000000.0) {
		tmp = (1.0 / (fma(z_m, z_m, 1.0) * y_m)) / x_m;
	} else {
		tmp = (1.0 / ((x_m * z_m) * z_m)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 4300000000000.0)
		tmp = Float64(Float64(1.0 / Float64(fma(z_m, z_m, 1.0) * y_m)) / x_m);
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(x_m * z_m) * z_m)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 4300000000000.0], N[(N[(1.0 / N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(1.0 / N[(N[(x$95$m * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.3e12

   1. Initial program 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 95.7

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 95.7

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 95.7

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

   4. Applied rewrites 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. inv-pow N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 95.7

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

   6. Applied rewrites 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.7

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

   8. Applied rewrites 95.7%

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

   if 4.3e12 < z

   1. Initial program 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 74.0

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 74.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 74.0

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 73.9%

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

   9. Applied rewrites 83.7%

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

Alternative 7: 90.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 0.007:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z\_m, z\_m, 1\right)}{x\_m \cdot y\_m}\\ \mathbf{elif}\;z\_m \leq 3.8 \cdot 10^{+116}:\\ \;\;\;\;\frac{1}{\left(x\_m \cdot y\_m\right) \cdot \left(z\_m \cdot z\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y\_m \cdot z\_m\right) \cdot z\_m\right) \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 0.007)
     (/ (fma (- z_m) z_m 1.0) (* x_m y_m))
     (if (<= z_m 3.8e+116)
       (/ 1.0 (* (* x_m y_m) (* z_m z_m)))
       (/ 1.0 (* (* (* y_m z_m) z_m) x_m)))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.007) {
		tmp = fma(-z_m, z_m, 1.0) / (x_m * y_m);
	} else if (z_m <= 3.8e+116) {
		tmp = 1.0 / ((x_m * y_m) * (z_m * z_m));
	} else {
		tmp = 1.0 / (((y_m * z_m) * z_m) * x_m);
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 0.007)
		tmp = Float64(fma(Float64(-z_m), z_m, 1.0) / Float64(x_m * y_m));
	elseif (z_m <= 3.8e+116)
		tmp = Float64(1.0 / Float64(Float64(x_m * y_m) * Float64(z_m * z_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(y_m * z_m) * z_m) * x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 0.007], N[(N[((-z$95$m) * z$95$m + 1.0), $MachinePrecision] / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 3.8e+116], N[(1.0 / N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(y$95$m * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 0.00700000000000000015

   1. Initial program 95.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 68.8%

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

   if 0.00700000000000000015 < z < 3.7999999999999999e116

   1. Initial program 94.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 89.6

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

   7. Applied rewrites 89.6%

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

   if 3.7999999999999999e116 < z

   1. Initial program 62.7%

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

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 62.7

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 87.2%

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

Alternative 8: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 9.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\left(y\_m \cdot \mathsf{fma}\left(z\_m, z\_m, 1\right)\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(x\_m \cdot z\_m\right) \cdot z\_m}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 9.8e+35)
     (/ 1.0 (* (* y_m (fma z_m z_m 1.0)) x_m))
     (/ (/ 1.0 (* (* x_m z_m) z_m)) y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 9.8e+35) {
		tmp = 1.0 / ((y_m * fma(z_m, z_m, 1.0)) * x_m);
	} else {
		tmp = (1.0 / ((x_m * z_m) * z_m)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 9.8e+35)
		tmp = Float64(1.0 / Float64(Float64(y_m * fma(z_m, z_m, 1.0)) * x_m));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(x_m * z_m) * z_m)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 9.8e+35], N[(1.0 / N[(N[(y$95$m * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(x$95$m * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 9.8000000000000005e35

   1. Initial program 95.8%

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

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 95.7

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

   4. Applied rewrites 95.7%

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

   if 9.8000000000000005e35 < z

   1. Initial program 72.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 72.0

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 72.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 71.9%

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

   9. Applied rewrites 82.5%

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

Alternative 9: 89.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 0.007:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z\_m, z\_m, 1\right)}{x\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y\_m \cdot z\_m\right) \cdot z\_m\right) \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 0.007)
     (/ (fma (- z_m) z_m 1.0) (* x_m y_m))
     (/ 1.0 (* (* (* y_m z_m) z_m) x_m))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.007) {
		tmp = fma(-z_m, z_m, 1.0) / (x_m * y_m);
	} else {
		tmp = 1.0 / (((y_m * z_m) * z_m) * x_m);
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 0.007)
		tmp = Float64(fma(Float64(-z_m), z_m, 1.0) / Float64(x_m * y_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(y_m * z_m) * z_m) * x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 0.007], N[(N[((-z$95$m) * z$95$m + 1.0), $MachinePrecision] / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(y$95$m * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.00700000000000000015

   1. Initial program 95.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 68.8%

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

   if 0.00700000000000000015 < z

   1. Initial program 76.7%

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

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 76.3

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 76.3

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 88.2%

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

Alternative 10: 87.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 0.007:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z\_m, z\_m, 1\right)}{x\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z\_m \cdot z\_m\right) \cdot y\_m\right) \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 0.007)
     (/ (fma (- z_m) z_m 1.0) (* x_m y_m))
     (/ 1.0 (* (* (* z_m z_m) y_m) x_m))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.007) {
		tmp = fma(-z_m, z_m, 1.0) / (x_m * y_m);
	} else {
		tmp = 1.0 / (((z_m * z_m) * y_m) * x_m);
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 0.007)
		tmp = Float64(fma(Float64(-z_m), z_m, 1.0) / Float64(x_m * y_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z_m * z_m) * y_m) * x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 0.007], N[(N[((-z$95$m) * z$95$m + 1.0), $MachinePrecision] / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.00700000000000000015

   1. Initial program 95.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 68.8%

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

   if 0.00700000000000000015 < z

   1. Initial program 76.7%

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

   5. Applied rewrites 74.7%

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

Alternative 11: 91.8% accurate, 1.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* (* (fma z_m z_m 1.0) x_m) y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (1.0 / ((fma(z_m, z_m, 1.0) * x_m) * y_m)));
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(x_s * Float64(1.0 / Float64(Float64(fma(z_m, z_m, 1.0) * x_m) * y_m))))
end

Wolfram

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

Derivation

1. Initial program 90.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 89.8

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 89.8

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

4. Applied rewrites 89.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 89.8

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

6. Applied rewrites 89.8%

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

Alternative 12: 57.7% accurate, 1.6× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ (/ 1.0 y_m) x_m))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * ((1.0 / y_m) / x_m));
}

Fortran

z_m =     private
x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, y_m, and z_m 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_m)
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_m
    code = y_s * (x_s * ((1.0d0 / y_m) / x_m))
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * ((1.0 / y_m) / x_m));
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	return y_s * (x_s * ((1.0 / y_m) / x_m))

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(x_s * Float64(Float64(1.0 / y_m) / x_m)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 89.9

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

   7. lift-/.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 89.9

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 89.9

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

4. Applied rewrites 89.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. inv-pow N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/l/ N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. associate-/r* N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-*.f64 89.8

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

6. Applied rewrites 89.8%

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

7. Taylor expanded in z around 0

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

9. Applied rewrites 59.0%

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

Alternative 13: 57.7% accurate, 2.1× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* y_m x_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (1.0 / (y_m * x_m)));
}

Fortran

z_m =     private
x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, y_m, and z_m 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_m)
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_m
    code = y_s * (x_s * (1.0d0 / (y_m * x_m)))
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (1.0 / (y_m * x_m)));
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	return y_s * (x_s * (1.0 / (y_m * x_m)))

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(x_s * Float64(1.0 / Float64(y_m * x_m))))
end

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := 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 90.0%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 59.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 59.0

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

7. Applied rewrites 59.0%

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

Developer Target 1: 92.2% 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 2025021 
(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)))))