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

Percentage Accurate: 88.4% → 99.1%

Time: 9.3s

Alternatives: 11

Speedup: 0.3×

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: 88.4% 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: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+306}:\\ \;\;\;\;\frac{{x\_m}^{-1}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1 - {\left(z \cdot z\right)}^{-1}}{y\_m}}{z \cdot x\_m}}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z z)))))
   (*
    x_s
    (*
     y_s
     (if (<= t_0 1e+306)
       (/ (pow x_m -1.0) t_0)
       (/ (/ (/ (- 1.0 (pow (* z z) -1.0)) y_m) (* z x_m)) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+306) {
		tmp = pow(x_m, -1.0) / t_0;
	} else {
		tmp = (((1.0 - pow((z * z), -1.0)) / y_m) / (z * x_m)) / z;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
x\_m =     private
x\_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(x_s, y_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 1d+306) then
        tmp = (x_m ** (-1.0d0)) / t_0
    else
        tmp = (((1.0d0 - ((z * z) ** (-1.0d0))) / y_m) / (z * x_m)) / z
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+306) {
		tmp = Math.pow(x_m, -1.0) / t_0;
	} else {
		tmp = (((1.0 - Math.pow((z * z), -1.0)) / y_m) / (z * x_m)) / z;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 1e+306:
		tmp = math.pow(x_m, -1.0) / t_0
	else:
		tmp = (((1.0 - math.pow((z * z), -1.0)) / y_m) / (z * x_m)) / z
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 1e+306)
		tmp = Float64((x_m ^ -1.0) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - (Float64(z * z) ^ -1.0)) / y_m) / Float64(z * x_m)) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 1e+306)
		tmp = (x_m ^ -1.0) / t_0;
	else
		tmp = (((1.0 - ((z * z) ^ -1.0)) / y_m) / (z * x_m)) / z;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$0, 1e+306], N[(N[Power[x$95$m, -1.0], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Power[N[(z * z), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 1.00000000000000002e306

   1. Initial program 93.3%

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

   if 1.00000000000000002e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 74.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in z around -inf

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

   10. Applied rewrites 95.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.0%

       \[\leadsto \frac{\frac{\frac{1 - \frac{1}{z \cdot z}}{y}}{z \cdot x}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+306}:\\ \;\;\;\;\frac{{x}^{-1}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1 - {\left(z \cdot z\right)}^{-1}}{y}}{z \cdot x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+306}:\\ \;\;\;\;\frac{{x\_m}^{-1}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(z \cdot x\_m\right) \cdot z\right)}^{-1}}{y\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z z)))))
   (*
    x_s
    (*
     y_s
     (if (<= t_0 1e+306)
       (/ (pow x_m -1.0) t_0)
       (/ (pow (* (* z x_m) z) -1.0) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+306) {
		tmp = pow(x_m, -1.0) / t_0;
	} else {
		tmp = pow(((z * x_m) * z), -1.0) / y_m;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
x\_m =     private
x\_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(x_s, y_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 1d+306) then
        tmp = (x_m ** (-1.0d0)) / t_0
    else
        tmp = (((z * x_m) * z) ** (-1.0d0)) / y_m
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+306) {
		tmp = Math.pow(x_m, -1.0) / t_0;
	} else {
		tmp = Math.pow(((z * x_m) * z), -1.0) / y_m;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 1e+306:
		tmp = math.pow(x_m, -1.0) / t_0
	else:
		tmp = math.pow(((z * x_m) * z), -1.0) / y_m
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 1e+306)
		tmp = Float64((x_m ^ -1.0) / t_0);
	else
		tmp = Float64((Float64(Float64(z * x_m) * z) ^ -1.0) / y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 1e+306)
		tmp = (x_m ^ -1.0) / t_0;
	else
		tmp = (((z * x_m) * z) ^ -1.0) / y_m;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$0, 1e+306], N[(N[Power[x$95$m, -1.0], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Power[N[(N[(z * x$95$m), $MachinePrecision] * z), $MachinePrecision], -1.0], $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))) < 1.00000000000000002e306

   1. Initial program 93.3%

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

   if 1.00000000000000002e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 74.3%

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

   4. Applied rewrites 83.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 83.9

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

   7. Applied rewrites 83.9%

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

   9. Applied rewrites 91.0%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+306}:\\ \;\;\;\;\frac{{x}^{-1}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(z \cdot x\right) \cdot z\right)}^{-1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot \left(1 + z \cdot z\right) \leq 10^{+306}:\\ \;\;\;\;\frac{{x\_m}^{-1}}{\mathsf{fma}\left(z \cdot z, y\_m, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(z \cdot x\_m\right) \cdot z\right)}^{-1}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m (+ 1.0 (* z z))) 1e+306)
     (/ (pow x_m -1.0) (fma (* z z) y_m y_m))
     (/ (pow (* (* z x_m) z) -1.0) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * (1.0 + (z * z))) <= 1e+306) {
		tmp = pow(x_m, -1.0) / fma((z * z), y_m, y_m);
	} else {
		tmp = pow(((z * x_m) * z), -1.0) / y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * Float64(1.0 + Float64(z * z))) <= 1e+306)
		tmp = Float64((x_m ^ -1.0) / fma(Float64(z * z), y_m, y_m));
	else
		tmp = Float64((Float64(Float64(z * x_m) * z) ^ -1.0) / y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+306], N[(N[Power[x$95$m, -1.0], $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(z * x$95$m), $MachinePrecision] * z), $MachinePrecision], -1.0], $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))) < 1.00000000000000002e306

   1. Initial program 93.3%

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

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-fma.f64 93.3

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

   4. Applied rewrites 93.3%

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

   if 1.00000000000000002e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 74.3%

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

   4. Applied rewrites 83.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 83.9

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

   7. Applied rewrites 83.9%

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

   9. Applied rewrites 91.0%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+306}:\\ \;\;\;\;\frac{{x}^{-1}}{\mathsf{fma}\left(z \cdot z, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(z \cdot x\right) \cdot z\right)}^{-1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{+135}:\\ \;\;\;\;\frac{{\left(x\_m \cdot \mathsf{fma}\left(z, z, 1\right)\right)}^{-1}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{{x\_m}^{-1}}{y\_m}}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z 4e+135)
     (/ (pow (* x_m (fma z z 1.0)) -1.0) y_m)
     (/ (/ (/ (pow x_m -1.0) y_m) z) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 4e+135) {
		tmp = pow((x_m * fma(z, z, 1.0)), -1.0) / y_m;
	} else {
		tmp = ((pow(x_m, -1.0) / y_m) / z) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 4e+135)
		tmp = Float64((Float64(x_m * fma(z, z, 1.0)) ^ -1.0) / y_m);
	else
		tmp = Float64(Float64(Float64((x_m ^ -1.0) / y_m) / z) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z, 4e+135], N[(N[Power[N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(N[Power[x$95$m, -1.0], $MachinePrecision] / y$95$m), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.99999999999999985e135

   1. Initial program 92.1%

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

   4. Applied rewrites 92.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 92.9

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.9

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

   6. Applied rewrites 92.9%

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

   if 3.99999999999999985e135 < z

   1. Initial program 79.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

   7. Applied rewrites 91.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 91.6%

       \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{y}}{z}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{+135}:\\ \;\;\;\;\frac{{\left(x \cdot \mathsf{fma}\left(z, z, 1\right)\right)}^{-1}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{{x}^{-1}}{y}}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot \left(1 + z \cdot z\right) \leq 10^{+306}:\\ \;\;\;\;{\left(\mathsf{fma}\left(y\_m, z \cdot z, y\_m\right) \cdot x\_m\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(\left(\left(-x\_m\right) \cdot y\_m\right) \cdot z\right) \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m (+ 1.0 (* z z))) 1e+306)
     (pow (* (fma y_m (* z z) y_m) x_m) -1.0)
     (/ -1.0 (* (* (* (- x_m) y_m) z) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * (1.0 + (z * z))) <= 1e+306) {
		tmp = pow((fma(y_m, (z * z), y_m) * x_m), -1.0);
	} else {
		tmp = -1.0 / (((-x_m * y_m) * z) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * Float64(1.0 + Float64(z * z))) <= 1e+306)
		tmp = Float64(fma(y_m, Float64(z * z), y_m) * x_m) ^ -1.0;
	else
		tmp = Float64(-1.0 / Float64(Float64(Float64(Float64(-x_m) * y_m) * z) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+306], N[Power[N[(N[(y$95$m * N[(z * z), $MachinePrecision] + y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], -1.0], $MachinePrecision], N[(-1.0 / N[(N[(N[((-x$95$m) * y$95$m), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 1.00000000000000002e306

   1. Initial program 93.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. 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 93.2

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 93.2

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

   4. Applied rewrites 93.2%

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

   if 1.00000000000000002e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 74.3%

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

   4. Applied rewrites 83.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 83.9

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 83.9

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

   6. Applied rewrites 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 83.9

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

   8. Applied rewrites 83.9%

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

   9. Taylor expanded in z around inf

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. mul-1-neg N/A

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

       5. distribute-rgt-neg-out N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. distribute-rgt-neg-in N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. associate-*r* N/A

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

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

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

       16. mul-1-neg N/A

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

       17. lower-*.f64 N/A

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

       18. associate-*r* N/A

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

       19. lower-*.f64 N/A

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

       20. mul-1-neg N/A

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

       21. lower-neg.f64 95.7

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

   11. Applied rewrites 95.7%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+306}:\\ \;\;\;\;{\left(\mathsf{fma}\left(y, z \cdot z, y\right) \cdot x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(\left(\left(-x\right) \cdot y\right) \cdot z\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.95 \cdot 10^{+129}:\\ \;\;\;\;{\left(\left(x\_m \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot y\_m\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(z \cdot x\_m\right) \cdot z\right)}^{-1}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z 1.95e+129)
     (pow (* (* x_m (fma z z 1.0)) y_m) -1.0)
     (/ (pow (* (* z x_m) z) -1.0) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.95e+129) {
		tmp = pow(((x_m * fma(z, z, 1.0)) * y_m), -1.0);
	} else {
		tmp = pow(((z * x_m) * z), -1.0) / y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 1.95e+129)
		tmp = Float64(Float64(x_m * fma(z, z, 1.0)) * y_m) ^ -1.0;
	else
		tmp = Float64((Float64(Float64(z * x_m) * z) ^ -1.0) / y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z, 1.95e+129], N[Power[N[(N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], -1.0], $MachinePrecision], N[(N[Power[N[(N[(z * x$95$m), $MachinePrecision] * z), $MachinePrecision], -1.0], $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.9499999999999999e129

   1. Initial program 92.1%

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

   4. Applied rewrites 93.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 93.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 93.3

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

   6. Applied rewrites 93.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 93.3

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

   8. Applied rewrites 93.3%

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

   if 1.9499999999999999e129 < z

   1. Initial program 80.3%

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.6

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

   7. Applied rewrites 77.6%

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

   9. Applied rewrites 85.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.95 \cdot 10^{+129}:\\ \;\;\;\;{\left(\left(x \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot y\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(z \cdot x\right) \cdot z\right)}^{-1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{+128}:\\ \;\;\;\;{\left(\left(x\_m \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot y\_m\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(\left(\left(-x\_m\right) \cdot y\_m\right) \cdot z\right) \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z 3.3e+128)
     (pow (* (* x_m (fma z z 1.0)) y_m) -1.0)
     (/ -1.0 (* (* (* (- x_m) y_m) z) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 3.3e+128) {
		tmp = pow(((x_m * fma(z, z, 1.0)) * y_m), -1.0);
	} else {
		tmp = -1.0 / (((-x_m * y_m) * z) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 3.3e+128)
		tmp = Float64(Float64(x_m * fma(z, z, 1.0)) * y_m) ^ -1.0;
	else
		tmp = Float64(-1.0 / Float64(Float64(Float64(Float64(-x_m) * y_m) * z) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z, 3.3e+128], N[Power[N[(N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], -1.0], $MachinePrecision], N[(-1.0 / N[(N[(N[((-x$95$m) * y$95$m), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.3000000000000001e128

   1. Initial program 92.1%

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

   4. Applied rewrites 93.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 93.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 93.3

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

   6. Applied rewrites 93.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 93.3

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

   8. Applied rewrites 93.3%

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

   if 3.3000000000000001e128 < z

   1. Initial program 80.3%

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

   4. Applied rewrites 77.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 77.6

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 77.6

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

   6. Applied rewrites 77.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 77.6

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

   8. Applied rewrites 77.6%

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

   9. Taylor expanded in z around inf

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. mul-1-neg N/A

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

       5. distribute-rgt-neg-out N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. distribute-rgt-neg-in N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. associate-*r* N/A

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

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

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

       16. mul-1-neg N/A

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

       17. lower-*.f64 N/A

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

       18. associate-*r* N/A

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

       19. lower-*.f64 N/A

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

       20. mul-1-neg N/A

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

       21. lower-neg.f64 89.9

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

   11. Applied rewrites 89.9%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{+128}:\\ \;\;\;\;{\left(\left(x \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot y\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(\left(\left(-x\right) \cdot y\right) \cdot z\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

y\_m =     private
y\_s =     private
x\_m =     private
x\_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(x_s, y_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.0d0) then
        tmp = (x_m ** (-1.0d0)) / y_m
    else
        tmp = (((y_m * z) * z) * x_m) ** (-1.0d0)
    end if
    code = x_s * (y_s * tmp)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.3%

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

   3. Taylor expanded in z around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if 1 < z

   1. Initial program 81.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 80.9

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

   7. Applied rewrites 80.9%

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

   9. Applied rewrites 83.9%

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

4. Final simplification 78.0%

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

Alternative 9: 73.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

y\_m =     private
y\_s =     private
x\_m =     private
x\_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(x_s, y_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.0d0) then
        tmp = (x_m ** (-1.0d0)) / y_m
    else
        tmp = (((z * z) * y_m) * x_m) ** (-1.0d0)
    end if
    code = x_s * (y_s * tmp)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.3%

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

   3. Taylor expanded in z around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if 1 < z

   1. Initial program 81.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 80.9

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

   5. Applied rewrites 80.9%

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

4. Final simplification 77.3%

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

Alternative 10: 88.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[Power[N[(N[(y$95$m * N[(z * z), $MachinePrecision] + y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], -1.0], $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. 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 90.3

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. *-rgt-identity N/A

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

   12. lower-fma.f64 90.3

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

4. Applied rewrites 90.3%

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

5. Final simplification 90.3%

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

Alternative 11: 58.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

y\_m =     private
y\_s =     private
x\_m =     private
x\_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(x_s, y_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * ((y_m * x_m) ** (-1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 61.9

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

5. Applied rewrites 61.9%

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

7. Applied rewrites 62.1%

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

8. Final simplification 62.1%

   \[\leadsto {\left(y \cdot x\right)}^{-1} \]
9. Add Preprocessing

Developer Target 1: 92.6% 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 2024346 
(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)))))