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

Percentage Accurate: 88.7% → 98.0%

Time: 5.2s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x_s, y_s, x_m, y_m, z_m)
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_m
    real(8) :: tmp
    if (z_m <= 1.0d0) then
        tmp = (1.0d0 / y_m) / x_m
    else
        tmp = ((1.0d0 / z_m) / (z_m * x_m)) / y_m
    end if
    code = x_s * (y_s * tmp)
end function

Java

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

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(x_s, y_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 1.0:
		tmp = (1.0 / y_m) / x_m
	else:
		tmp = ((1.0 / z_m) / (z_m * x_m)) / y_m
	return x_s * (y_s * tmp)

Julia

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

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 1.0)
		tmp = (1.0 / y_m) / x_m;
	else
		tmp = ((1.0 / z_m) / (z_m * x_m)) / y_m;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 95.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 93.2

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 93.2

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 93.2

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

   4. Applied rewrites 93.2%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unpow2 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 72.3

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

   7. Applied rewrites 72.3%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 78.0%

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

   if 1 < z

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 82.4

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 82.4

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 82.4

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

   4. Applied rewrites 82.4%

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

   5. Taylor expanded in z around inf

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

      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 82.2

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

   7. Applied rewrites 82.2%

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

   9. Applied rewrites 85.4%

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

4. Final simplification 79.8%

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

Alternative 2: 91.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \begin{array}{l} t_0 := 1 + z\_m \cdot z\_m\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\frac{1}{\left(z\_m \cdot z\_m\right) \cdot \left(y\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y\_m \cdot z\_m\right) \cdot z\_m\right) \cdot x\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z_m z_m))))
   (*
    x_s
    (*
     y_s
     (if (<= t_0 2.0)
       (/ (/ 1.0 y_m) x_m)
       (if (<= t_0 2e+291)
         (/ 1.0 (* (* z_m z_m) (* y_m x_m)))
         (/ 1.0 (* (* (* y_m z_m) z_m) x_m))))))))

C

z_m = fabs(z);
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_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double t_0 = 1.0 + (z_m * z_m);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = (1.0 / y_m) / x_m;
	} else if (t_0 <= 2e+291) {
		tmp = 1.0 / ((z_m * z_m) * (y_m * x_m));
	} else {
		tmp = 1.0 / (((y_m * z_m) * z_m) * x_m);
	}
	return x_s * (y_s * tmp);
}

Fortran

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

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

real(8) function code(x_s, y_s, x_m, y_m, z_m)
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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (z_m * z_m)
    if (t_0 <= 2.0d0) then
        tmp = (1.0d0 / y_m) / x_m
    else if (t_0 <= 2d+291) then
        tmp = 1.0d0 / ((z_m * z_m) * (y_m * x_m))
    else
        tmp = 1.0d0 / (((y_m * z_m) * z_m) * x_m)
    end if
    code = x_s * (y_s * tmp)
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double t_0 = 1.0 + (z_m * z_m);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = (1.0 / y_m) / x_m;
	} else if (t_0 <= 2e+291) {
		tmp = 1.0 / ((z_m * z_m) * (y_m * x_m));
	} else {
		tmp = 1.0 / (((y_m * z_m) * z_m) * x_m);
	}
	return x_s * (y_s * tmp);
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(x_s, y_s, x_m, y_m, z_m):
	t_0 = 1.0 + (z_m * z_m)
	tmp = 0
	if t_0 <= 2.0:
		tmp = (1.0 / y_m) / x_m
	elif t_0 <= 2e+291:
		tmp = 1.0 / ((z_m * z_m) * (y_m * x_m))
	else:
		tmp = 1.0 / (((y_m * z_m) * z_m) * x_m)
	return x_s * (y_s * tmp)

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	t_0 = Float64(1.0 + Float64(z_m * z_m))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = Float64(Float64(1.0 / y_m) / x_m);
	elseif (t_0 <= 2e+291)
		tmp = Float64(1.0 / Float64(Float64(z_m * z_m) * Float64(y_m * x_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(y_m * z_m) * z_m) * x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z_m)
	t_0 = 1.0 + (z_m * z_m);
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = (1.0 / y_m) / x_m;
	elseif (t_0 <= 2e+291)
		tmp = 1.0 / ((z_m * z_m) * (y_m * x_m));
	else
		tmp = 1.0 / (((y_m * z_m) * z_m) * x_m);
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$0, 2.0], N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], If[LessEqual[t$95$0, 2e+291], N[(1.0 / N[(N[(z$95$m * z$95$m), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(y$95$m * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.7

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.7

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unpow2 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 99.7%

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

   if 2 < (+.f64 #s(literal 1 binary64) (*.f64 z z)) < 1.9999999999999999e291

   1. Initial program 94.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 93.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 93.4

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 93.4

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

   4. Applied rewrites 93.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 95.4

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

   6. Applied rewrites 95.4%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 95.1

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

   9. Applied rewrites 95.1%

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

   if 1.9999999999999999e291 < (+.f64 #s(literal 1 binary64) (*.f64 z z))

   1. Initial program 76.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 93.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 93.2

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

   9. Applied rewrites 93.2%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + z \cdot z \leq 2:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{elif}\;1 + z \cdot z \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\frac{1}{\left(z \cdot z\right) \cdot \left(y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y \cdot z\right) \cdot z\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot \left(1 + z\_m \cdot z\_m\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z\_m \cdot z\_m\right) \cdot \left(y\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m (+ 1.0 (* z_m z_m))) 2e+307)
     (/ 1.0 (* (* (fma z_m z_m 1.0) y_m) x_m))
     (/ 1.0 (* (* z_m z_m) (* y_m x_m)))))))

C

z_m = fabs(z);
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_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((y_m * (1.0 + (z_m * z_m))) <= 2e+307) {
		tmp = 1.0 / ((fma(z_m, z_m, 1.0) * y_m) * x_m);
	} else {
		tmp = 1.0 / ((z_m * z_m) * (y_m * x_m));
	}
	return x_s * (y_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(y_m * Float64(1.0 + Float64(z_m * z_m))) <= 2e+307)
		tmp = Float64(1.0 / Float64(Float64(fma(z_m, z_m, 1.0) * y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(z_m * z_m) * Float64(y_m * x_m)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+307], N[(1.0 / N[(N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z$95$m * z$95$m), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 1.99999999999999997e307

   1. Initial program 95.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 94.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 94.5

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 94.5

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

   4. Applied rewrites 94.5%

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

   if 1.99999999999999997e307 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 77.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 77.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 77.4

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 77.4

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

   4. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 77.1

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

   6. Applied rewrites 77.1%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 77.1

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

   9. Applied rewrites 77.1%

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

Alternative 4: 89.8% accurate, 0.9× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (+ 1.0 (* z_m z_m)) 2.0)
     (/ (/ 1.0 y_m) x_m)
     (/ 1.0 (* (* (* y_m z_m) z_m) x_m))))))

C

z_m = fabs(z);
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_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((1.0 + (z_m * z_m)) <= 2.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = 1.0 / (((y_m * z_m) * z_m) * x_m);
	}
	return x_s * (y_s * tmp);
}

Fortran

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

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

real(8) function code(x_s, y_s, x_m, y_m, z_m)
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_m
    real(8) :: tmp
    if ((1.0d0 + (z_m * z_m)) <= 2.0d0) then
        tmp = (1.0d0 / y_m) / x_m
    else
        tmp = 1.0d0 / (((y_m * z_m) * z_m) * x_m)
    end if
    code = x_s * (y_s * tmp)
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((1.0 + (z_m * z_m)) <= 2.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = 1.0 / (((y_m * z_m) * z_m) * x_m);
	}
	return x_s * (y_s * tmp);
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(x_s, y_s, x_m, y_m, z_m):
	tmp = 0
	if (1.0 + (z_m * z_m)) <= 2.0:
		tmp = (1.0 / y_m) / x_m
	else:
		tmp = 1.0 / (((y_m * z_m) * z_m) * x_m)
	return x_s * (y_s * tmp)

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(1.0 + Float64(z_m * z_m)) <= 2.0)
		tmp = Float64(Float64(1.0 / y_m) / x_m);
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(y_m * z_m) * z_m) * x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0;
	if ((1.0 + (z_m * z_m)) <= 2.0)
		tmp = (1.0 / y_m) / x_m;
	else
		tmp = 1.0 / (((y_m * z_m) * z_m) * x_m);
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.7

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.7

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unpow2 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 99.7%

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

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

   1. Initial program 84.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 84.7

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

   5. Applied rewrites 84.7%

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

   7. Applied rewrites 93.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 93.2

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

   9. Applied rewrites 93.2%

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

4. Final simplification 96.7%

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

Alternative 5: 97.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x_s, y_s, x_m, y_m, z_m)
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_m
    real(8) :: tmp
    if (z_m <= 1.0d0) then
        tmp = (1.0d0 / y_m) / x_m
    else
        tmp = (1.0d0 / ((z_m * x_m) * z_m)) / y_m
    end if
    code = x_s * (y_s * tmp)
end function

Java

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

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(x_s, y_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 1.0:
		tmp = (1.0 / y_m) / x_m
	else:
		tmp = (1.0 / ((z_m * x_m) * z_m)) / y_m
	return x_s * (y_s * tmp)

Julia

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

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 1.0)
		tmp = (1.0 / y_m) / x_m;
	else
		tmp = (1.0 / ((z_m * x_m) * z_m)) / y_m;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 95.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 93.2

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 93.2

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 93.2

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

   4. Applied rewrites 93.2%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unpow2 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 72.3

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

   7. Applied rewrites 72.3%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 78.0%

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

   if 1 < z

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 82.4

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 82.4

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 82.4

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

   4. Applied rewrites 82.4%

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

   5. Taylor expanded in z around inf

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

      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 82.2

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

   7. Applied rewrites 82.2%

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

   9. Applied rewrites 84.7%

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

4. Final simplification 79.7%

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

Alternative 6: 93.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.95 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m \cdot z\_m, z\_m, y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot \left(y\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 1.95e-90)
     (/ 1.0 (* (fma (* y_m z_m) z_m y_m) x_m))
     (/ 1.0 (* (fma z_m z_m 1.0) (* y_m x_m)))))))

C

z_m = fabs(z);
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_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1.95e-90) {
		tmp = 1.0 / (fma((y_m * z_m), z_m, y_m) * x_m);
	} else {
		tmp = 1.0 / (fma(z_m, z_m, 1.0) * (y_m * x_m));
	}
	return x_s * (y_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 1.95e-90)
		tmp = Float64(1.0 / Float64(fma(Float64(y_m * z_m), z_m, y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(fma(z_m, z_m, 1.0) * Float64(y_m * x_m)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 1.95e-90], N[(1.0 / N[(N[(N[(y$95$m * z$95$m), $MachinePrecision] * z$95$m + y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.95000000000000002e-90

   1. Initial program 91.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 90.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 90.9

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

   4. Applied rewrites 90.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 96.8

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

   6. Applied rewrites 96.8%

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

   if 1.95000000000000002e-90 < y

   1. Initial program 96.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 95.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 95.0

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

   4. Applied rewrites 95.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 95.0

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

   6. Applied rewrites 95.0%

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

Alternative 7: 92.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot \left(y\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y\_m \cdot z\_m\right) \cdot z\_m\right) \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z_m 1.8e+147)
     (/ 1.0 (* (fma z_m z_m 1.0) (* y_m x_m)))
     (/ 1.0 (* (* (* y_m z_m) z_m) x_m))))))

C

z_m = fabs(z);
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_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.8e+147) {
		tmp = 1.0 / (fma(z_m, z_m, 1.0) * (y_m * x_m));
	} else {
		tmp = 1.0 / (((y_m * z_m) * z_m) * x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 1.8e+147)
		tmp = Float64(1.0 / Float64(fma(z_m, z_m, 1.0) * Float64(y_m * x_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(y_m * z_m) * z_m) * x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z$95$m, 1.8e+147], N[(1.0 / N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(y$95$m * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.8000000000000001e147

   1. Initial program 95.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 94.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 94.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 95.0

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

   6. Applied rewrites 95.0%

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

   if 1.8000000000000001e147 < z

   1. Initial program 75.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 75.4

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

   5. Applied rewrites 75.4%

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

   7. Applied rewrites 95.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.5

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

   9. Applied rewrites 95.5%

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

Alternative 8: 88.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x_s, y_s, x_m, y_m, z_m)
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_m
    real(8) :: tmp
    if (z_m <= 1.0d0) then
        tmp = (1.0d0 / y_m) / x_m
    else
        tmp = 1.0d0 / (((z_m * z_m) * y_m) * x_m)
    end if
    code = x_s * (y_s * tmp)
end function

Java

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

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(x_s, y_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 1.0:
		tmp = (1.0 / y_m) / x_m
	else:
		tmp = 1.0 / (((z_m * z_m) * y_m) * x_m)
	return x_s * (y_s * tmp)

Julia

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

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 1.0)
		tmp = (1.0 / y_m) / x_m;
	else
		tmp = 1.0 / (((z_m * z_m) * y_m) * x_m);
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 95.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 93.2

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 93.2

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 93.2

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

   4. Applied rewrites 93.2%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unpow2 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 72.3

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

   7. Applied rewrites 72.3%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 78.0%

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

   if 1 < z

   1. Initial program 83.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 83.7

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

   5. Applied rewrites 83.7%

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

4. Final simplification 79.4%

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

Alternative 9: 57.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x_s, y_s, x_m, y_m, z_m)
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_m
    code = x_s * (y_s * ((1.0d0 / y_m) / x_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 90.5

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

   7. lift-/.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 90.5

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 90.5

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

4. Applied rewrites 90.5%

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

5. Taylor expanded in z around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. mul-1-neg N/A

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

   8. unpow2 N/A

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

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

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

   10. lower-fma.f64 N/A

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

   11. lower-neg.f64 55.2

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

7. Applied rewrites 55.2%

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

8. Taylor expanded in z around 0

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

10. Applied rewrites 63.6%

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

11. Final simplification 63.6%

    \[\leadsto \frac{\frac{1}{y}}{x} \]
12. Add Preprocessing

Alternative 10: 57.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x_s, y_s, x_m, y_m, z_m)
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_m
    code = x_s * (y_s * ((1.0d0 / x_m) / y_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.8%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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 63.6

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

5. Applied rewrites 63.6%

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

Alternative 11: 57.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x_s, y_s, x_m, y_m, z_m)
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_m
    code = x_s * (y_s * (1.0d0 / (y_m * x_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 92.1

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 92.1

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 92.1

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

4. Applied rewrites 92.1%

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

5. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 63.1

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

7. Applied rewrites 63.1%

   \[\leadsto \frac{1}{\color{blue}{y \cdot x}} \]
8. 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 2025017 
(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)))))