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

Percentage Accurate: 88.3% → 99.1%

Time: 7.4s

Alternatives: 10

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: 88.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{+152}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x\_m}}{\mathsf{fma}\left(z\_m, z\_m, 1\right)}}{-y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m \cdot z\_m, z\_m \cdot x\_m, y\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

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 (<= z_m 1e+152)
     (/ (/ (/ -1.0 x_m) (fma z_m z_m 1.0)) (- y_m))
     (/ 1.0 (fma (* y_m z_m) (* z_m x_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 <= 1e+152) {
		tmp = ((-1.0 / x_m) / fma(z_m, z_m, 1.0)) / -y_m;
	} else {
		tmp = 1.0 / fma((y_m * z_m), (z_m * x_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 <= 1e+152)
		tmp = Float64(Float64(Float64(-1.0 / x_m) / fma(z_m, z_m, 1.0)) / Float64(-y_m));
	else
		tmp = Float64(1.0 / fma(Float64(y_m * z_m), Float64(z_m * x_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[z$95$m, 1e+152], N[(N[(N[(-1.0 / x$95$m), $MachinePrecision] / N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision] / (-y$95$m)), $MachinePrecision], N[(1.0 / N[(N[(y$95$m * z$95$m), $MachinePrecision] * N[(z$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1e152

   1. Initial program 93.7%

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

   4. Applied rewrites 91.2%

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

   if 1e152 < z

   1. Initial program 77.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 77.6

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 77.6

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

   4. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-lft-identity N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

Alternative 2: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.46 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot x\_m\right) \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m \cdot z\_m, z\_m \cdot x\_m, y\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

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 (<= z_m 1.46e+153)
     (/ 1.0 (* (* (fma z_m z_m 1.0) x_m) y_m))
     (/ 1.0 (fma (* y_m z_m) (* z_m x_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 <= 1.46e+153) {
		tmp = 1.0 / ((fma(z_m, z_m, 1.0) * x_m) * y_m);
	} else {
		tmp = 1.0 / fma((y_m * z_m), (z_m * x_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 <= 1.46e+153)
		tmp = Float64(1.0 / Float64(Float64(fma(z_m, z_m, 1.0) * x_m) * y_m));
	else
		tmp = Float64(1.0 / fma(Float64(y_m * z_m), Float64(z_m * x_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[z$95$m, 1.46e+153], N[(1.0 / N[(N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y$95$m * z$95$m), $MachinePrecision] * N[(z$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.4600000000000001e153

   1. Initial program 93.7%

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

   4. Applied rewrites 91.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. distribute-neg-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-frac N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. frac-2neg-rev N/A

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

      12. associate-/r* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. associate-/l/ N/A

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

      20. inv-pow N/A

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

   6. Applied rewrites 89.9%

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

   if 1.4600000000000001e153 < z

   1. Initial program 77.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 77.6

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 77.6

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

   4. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-lft-identity N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

Alternative 3: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{+87}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x\_m \cdot \left(y\_m \cdot z\_m\right), z\_m, y\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot x\_m\right) \cdot y\_m}\\ \end{array}\right) \end{array} \]

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 (<= y_m 2e+87)
     (/ 1.0 (fma (* x_m (* y_m z_m)) z_m (* y_m x_m)))
     (/ 1.0 (* (* (fma z_m z_m 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) {
	double tmp;
	if (y_m <= 2e+87) {
		tmp = 1.0 / fma((x_m * (y_m * z_m)), z_m, (y_m * x_m));
	} else {
		tmp = 1.0 / ((fma(z_m, z_m, 1.0) * 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 (y_m <= 2e+87)
		tmp = Float64(1.0 / fma(Float64(x_m * Float64(y_m * z_m)), z_m, Float64(y_m * x_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(fma(z_m, z_m, 1.0) * x_m) * y_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, 2e+87], N[(1.0 / N[(N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] * z$95$m + N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.9999999999999999e87

   1. Initial program 90.8%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 90.6

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 90.6

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

   4. Applied rewrites 90.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-*.f64 N/A

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

      10. *-lft-identity N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 99.0

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

   6. Applied rewrites 99.0%

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

   if 1.9999999999999999e87 < y

   1. Initial program 93.7%

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. distribute-neg-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-frac N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. frac-2neg-rev N/A

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

      12. associate-/r* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. associate-/l/ N/A

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

      20. inv-pow N/A

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

   6. Applied rewrites 99.7%

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

Alternative 4: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{+68}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot x\_m\right) \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x\_m \cdot z\_m\right) \cdot z\_m\right) \cdot y\_m}\\ \end{array}\right) \end{array} \]

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 1e+68)
     (/ 1.0 (* (* (fma z_m z_m 1.0) x_m) y_m))
     (/ 1.0 (* (* (* x_m z_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 <= 1e+68) {
		tmp = 1.0 / ((fma(z_m, z_m, 1.0) * x_m) * y_m);
	} else {
		tmp = 1.0 / (((x_m * z_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 <= 1e+68)
		tmp = Float64(1.0 / Float64(Float64(fma(z_m, z_m, 1.0) * x_m) * y_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x_m * z_m) * z_m) * y_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, 1e+68], N[(1.0 / N[(N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 9.99999999999999953e67

   1. Initial program 94.5%

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

   4. Applied rewrites 92.9%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. distribute-neg-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-frac N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. frac-2neg-rev N/A

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

      12. associate-/r* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. associate-/l/ N/A

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

      20. inv-pow N/A

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

   6. Applied rewrites 91.4%

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

   if 9.99999999999999953e67 < z

   1. Initial program 79.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 79.5

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 79.5

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

   4. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-lft-identity N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 96.4

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

   6. Applied rewrites 96.4%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 82.7

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

   9. Applied rewrites 82.7%

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

Alternative 5: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{\left(y\_m \cdot x\_m\right) \cdot \mathsf{fma}\left(z\_m, z\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x\_m \cdot z\_m\right) \cdot z\_m\right) \cdot y\_m}\\ \end{array}\right) \end{array} \]

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 5e+41)
     (/ 1.0 (* (* y_m x_m) (fma z_m z_m 1.0)))
     (/ 1.0 (* (* (* x_m z_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 <= 5e+41) {
		tmp = 1.0 / ((y_m * x_m) * fma(z_m, z_m, 1.0));
	} else {
		tmp = 1.0 / (((x_m * z_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 <= 5e+41)
		tmp = Float64(1.0 / Float64(Float64(y_m * x_m) * fma(z_m, z_m, 1.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x_m * z_m) * z_m) * y_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, 5e+41], N[(1.0 / N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.00000000000000022e41

   1. Initial program 94.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 94.1

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 94.1

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

   4. Applied rewrites 94.1%

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

   6. Applied rewrites 94.4%

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

   if 5.00000000000000022e41 < z

   1. Initial program 81.8%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 81.8

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 81.8

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

   4. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-lft-identity N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 95.3

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

   6. Applied rewrites 95.3%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 83.1

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

   9. Applied rewrites 83.1%

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

Alternative 6: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 0.0072:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x\_m \cdot z\_m\right) \cdot z\_m\right) \cdot y\_m}\\ \end{array}\right) \end{array} \]

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 0.0072)
     (/ (/ 1.0 x_m) y_m)
     (/ 1.0 (* (* (* x_m z_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 <= 0.0072) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (((x_m * z_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 <= 0.0072d0) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = 1.0d0 / (((x_m * z_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 <= 0.0072) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (((x_m * z_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 <= 0.0072:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = 1.0 / (((x_m * z_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 <= 0.0072)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x_m * z_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 <= 0.0072)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = 1.0 / (((x_m * z_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, 0.0072], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.0071999999999999998

   1. Initial program 94.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 67.7

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

   5. Applied rewrites 67.7%

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

   if 0.0071999999999999998 < z

   1. Initial program 83.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 83.6

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 83.6

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

   4. Applied rewrites 83.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-lft-identity N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 95.7

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

   6. Applied rewrites 95.7%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 82.5

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

   9. Applied rewrites 82.5%

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

Alternative 7: 87.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 0.0072:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z\_m \cdot z\_m\right) \cdot y\_m\right) \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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 0.0072)
     (/ (/ 1.0 x_m) y_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 <= 0.0072) {
		tmp = (1.0 / x_m) / y_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 <= 0.0072d0) then
        tmp = (1.0d0 / x_m) / y_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 <= 0.0072) {
		tmp = (1.0 / x_m) / y_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 <= 0.0072:
		tmp = (1.0 / x_m) / y_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 <= 0.0072)
		tmp = Float64(Float64(1.0 / x_m) / y_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 <= 0.0072)
		tmp = (1.0 / x_m) / y_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, 0.0072], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$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 < 0.0071999999999999998

   1. Initial program 94.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 67.7

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

   5. Applied rewrites 67.7%

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

   if 0.0071999999999999998 < z

   1. Initial program 83.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 82.7

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

   5. Applied rewrites 82.7%

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

Alternative 8: 58.0% 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 91.3%

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

3. Taylor expanded in z around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 54.7

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

5. Applied rewrites 54.7%

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

Alternative 9: 58.0% 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 91.3%

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

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{{z}^{2}}{x \cdot y} + \frac{1}{x \cdot y}} \]
4. 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. /-rgt-identity N/A

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

   9. /-rgt-identity N/A

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

   10. unpow2 N/A

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

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

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

   12. lower-fma.f64 N/A

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

   13. lower-neg.f64 48.4

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

5. Applied rewrites 48.4%

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

7. Applied rewrites 35.3%

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

8. Taylor expanded in z around 0

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

10. Applied rewrites 54.8%

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

Alternative 10: 58.0% 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 91.3%

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

3. Taylor expanded in z around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 54.7

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

5. Applied rewrites 54.7%

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

7. Applied rewrites 54.6%

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

Developer Target 1: 92.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + z \cdot z\\ t_1 := y \cdot t\_0\\ t_2 := \frac{\frac{1}{y}}{t\_0 \cdot x}\\ \mathbf{if}\;t\_1 < -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{t\_0 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z z))) (t_1 (* y t_0)) (t_2 (/ (/ 1.0 y) (* t_0 x))))
   (if (< t_1 (- INFINITY))
     t_2
     (if (< t_1 8.680743250567252e+305) (/ (/ 1.0 x) (* t_0 y)) t_2))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z * z)
	t_1 = y * t_0
	t_2 = (1.0 / y) / (t_0 * x)
	tmp = 0
	if t_1 < -math.inf:
		tmp = t_2
	elif t_1 < 8.680743250567252e+305:
		tmp = (1.0 / x) / (t_0 * y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z * z))
	t_1 = Float64(y * t_0)
	t_2 = Float64(Float64(1.0 / y) / Float64(t_0 * x))
	tmp = 0.0
	if (t_1 < Float64(-Inf))
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = Float64(Float64(1.0 / x) / Float64(t_0 * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z * z);
	t_1 = y * t_0;
	t_2 = (1.0 / y) / (t_0 * x);
	tmp = 0.0;
	if (t_1 < -Inf)
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = (1.0 / x) / (t_0 * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / y), $MachinePrecision] / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$1, (-Infinity)], t$95$2, If[Less[t$95$1, 8.680743250567252e+305], N[(N[(1.0 / x), $MachinePrecision] / N[(t$95$0 * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Reproduce

herbie shell --seed 2025006 
(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)))))