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

Percentage Accurate: 89.4% → 99.0%

Time: 4.0s

Alternatives: 8

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: 89.4% 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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.4 \cdot 10^{+148}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m \cdot z, z \cdot x\_m, y\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x\_m}^{-1}}{\mathsf{fma}\left(z, z, 1\right)}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 1.4e+148)
     (/ 1.0 (fma (* y_m z) (* z x_m) (* y_m x_m)))
     (/ (/ (pow x_m -1.0) (fma z z 1.0)) y_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1.4e+148) {
		tmp = 1.0 / fma((y_m * z), (z * x_m), (y_m * x_m));
	} else {
		tmp = (pow(x_m, -1.0) / fma(z, z, 1.0)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 1.4e+148)
		tmp = Float64(1.0 / fma(Float64(y_m * z), Float64(z * x_m), Float64(y_m * x_m)));
	else
		tmp = Float64(Float64((x_m ^ -1.0) / fma(z, z, 1.0)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1.4e+148], N[(1.0 / N[(N[(y$95$m * z), $MachinePrecision] * N[(z * x$95$m), $MachinePrecision] + N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x$95$m, -1.0], $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.3999999999999999e148

   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. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 90.0

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

   4. Applied rewrites 90.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*l* N/A

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

      13. associate-*l* N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 97.4

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

   6. Applied rewrites 97.4%

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

   if 1.3999999999999999e148 < y

   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

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 2: 91.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := 1 + z \cdot z\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{y\_m \cdot x\_m}\\ \mathbf{elif}\;t\_0 \leq 10^{+261}:\\ \;\;\;\;\frac{1}{\left(x\_m \cdot y\_m\right) \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y\_m \cdot z\right) \cdot z\right) \cdot x\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z z))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 2.0)
       (/ (fma (- z) z 1.0) (* y_m x_m))
       (if (<= t_0 1e+261)
         (/ 1.0 (* (* x_m y_m) (* z z)))
         (/ 1.0 (* (* (* y_m z) z) x_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = 1.0 + (z * z);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = fma(-z, z, 1.0) / (y_m * x_m);
	} else if (t_0 <= 1e+261) {
		tmp = 1.0 / ((x_m * y_m) * (z * z));
	} else {
		tmp = 1.0 / (((y_m * z) * z) * x_m);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(1.0 + Float64(z * z))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(y_m * x_m));
	elseif (t_0 <= 1e+261)
		tmp = Float64(1.0 / Float64(Float64(x_m * y_m) * Float64(z * z)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(y_m * z) * z) * x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 2.0], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+261], N[(1.0 / N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(y$95$m * z), $MachinePrecision] * z), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 2 < (+.f64 #s(literal 1 binary64) (*.f64 z z)) < 9.9999999999999993e260

   1. Initial program 89.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 87.6

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

   7. Applied rewrites 87.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 92.3

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

   9. Applied rewrites 92.3%

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

   if 9.9999999999999993e260 < (+.f64 #s(literal 1 binary64) (*.f64 z z))

   1. Initial program 72.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 72.6

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

   4. Applied rewrites 72.6%

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

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

Alternative 3: 91.8% accurate, 0.9× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (+ 1.0 (* z z)) 2.0)
     (/ (fma (- z) z 1.0) (* y_m x_m))
     (/ 1.0 (* (* (* z z) x_m) y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((1.0 + (z * z)) <= 2.0) {
		tmp = fma(-z, z, 1.0) / (y_m * x_m);
	} else {
		tmp = 1.0 / (((z * z) * x_m) * y_m);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(1.0 + Float64(z * z)) <= 2.0)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(y_m * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z * z) * x_m) * y_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z * z), $MachinePrecision] * 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)) < 2

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

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

   1. Initial program 80.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 80.6

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 79.7

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

   7. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 81.2

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

   9. Applied rewrites 81.2%

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

Alternative 4: 90.5% accurate, 0.9× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (+ 1.0 (* z z)) 2.0)
     (/ (fma (- z) z 1.0) (* y_m x_m))
     (/ 1.0 (* (* (* y_m z) z) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((1.0 + (z * z)) <= 2.0) {
		tmp = fma(-z, z, 1.0) / (y_m * x_m);
	} else {
		tmp = 1.0 / (((y_m * z) * z) * x_m);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(1.0 + Float64(z * z)) <= 2.0)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(y_m * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(y_m * z) * z) * x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(y$95$m * z), $MachinePrecision] * z), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

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

   1. Initial program 80.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 80.6

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

   4. Applied rewrites 80.6%

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

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 88.0

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

   7. Applied rewrites 88.0%

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

Alternative 5: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 0.35:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m \cdot z, z \cdot x\_m, y\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(y\_m \cdot x\_m\right) \cdot z, z, y\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 0.35)
     (/ 1.0 (fma (* y_m z) (* z x_m) (* y_m x_m)))
     (/ 1.0 (fma (* (* y_m x_m) z) z (* y_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 0.35) {
		tmp = 1.0 / fma((y_m * z), (z * x_m), (y_m * x_m));
	} else {
		tmp = 1.0 / fma(((y_m * x_m) * z), z, (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 0.35)
		tmp = Float64(1.0 / fma(Float64(y_m * z), Float64(z * x_m), Float64(y_m * x_m)));
	else
		tmp = Float64(1.0 / fma(Float64(Float64(y_m * x_m) * z), z, Float64(y_m * x_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 0.35], N[(1.0 / N[(N[(y$95$m * z), $MachinePrecision] * N[(z * x$95$m), $MachinePrecision] + N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] * z), $MachinePrecision] * z + N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 0.34999999999999998

   1. Initial program 88.8%

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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 88.8

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

   4. Applied rewrites 88.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*l* N/A

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

      13. associate-*l* N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 96.8

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

   6. Applied rewrites 96.8%

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

   if 0.34999999999999998 < y

   1. Initial program 93.7%

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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 93.7

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

   4. Applied rewrites 93.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. distribute-lft-in N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 99.3

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

   6. Applied rewrites 99.3%

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

Alternative 6: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{+188}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m \cdot z, z \cdot x\_m, y\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot x\_m\right) \cdot y\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 1e+188)
     (/ 1.0 (fma (* y_m z) (* z x_m) (* y_m x_m)))
     (/ 1.0 (* (* (fma z z 1.0) x_m) y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1e+188) {
		tmp = 1.0 / fma((y_m * z), (z * x_m), (y_m * x_m));
	} else {
		tmp = 1.0 / ((fma(z, z, 1.0) * x_m) * y_m);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 1e+188)
		tmp = Float64(1.0 / fma(Float64(y_m * z), Float64(z * x_m), Float64(y_m * x_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(fma(z, z, 1.0) * x_m) * y_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1e+188], N[(1.0 / N[(N[(y$95$m * z), $MachinePrecision] * N[(z * x$95$m), $MachinePrecision] + N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z * z + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1e188

   1. Initial program 89.7%

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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 89.7

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

   4. Applied rewrites 89.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*l* N/A

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

      13. associate-*l* N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 97.3

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

   6. Applied rewrites 97.3%

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

   if 1e188 < y

   1. Initial program 94.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 94.7

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

   4. Applied rewrites 94.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-fma.f64 99.9

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

   6. Applied rewrites 99.9%

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

Alternative 7: 92.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. pow2 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. pow2 N/A

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

   15. +-commutative N/A

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

   16. pow2 N/A

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

   17. lower-fma.f64 90.4

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

4. Applied rewrites 90.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. pow2 N/A

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

   7. +-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. pow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-fma.f64 91.1

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

6. Applied rewrites 91.1%

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

Alternative 8: 58.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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 62.3%

   \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y}} \]
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 62.4

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

   7. pow2 62.4

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

   8. +-commutative 62.4

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

   9. distribute-lft-in 62.4

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

   10. *-rgt-identity 62.4

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

   11. pow2 62.4

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

   12. associate-*l* 62.4

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

7. Applied rewrites 62.4%

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

Developer Target 1: 92.8% 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 2025051 
(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)))))