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

Percentage Accurate: 91.6% → 97.0%

Time: 3.1s

Alternatives: 7

Speedup: N/A×

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: 91.6% 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: 97.0% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot \left(-z\right), -z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{-1}}{\mathsf{fma}\left(z, z, 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.12e-18)
   (/ (/ 1.0 x) (fma (* y (- z)) (- z) y))
   (/ (/ (pow x -1.0) (fma z z 1.0)) y)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.12e-18) {
		tmp = (1.0 / x) / fma((y * -z), -z, y);
	} else {
		tmp = (pow(x, -1.0) / fma(z, z, 1.0)) / y;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.12e-18)
		tmp = Float64(Float64(1.0 / x) / fma(Float64(y * Float64(-z)), Float64(-z), y));
	else
		tmp = Float64(Float64((x ^ -1.0) / fma(z, z, 1.0)) / y);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1.12e-18], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y * (-z)), $MachinePrecision] * (-z) + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, -1.0], $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.12000000000000001e-18

   1. Initial program 90.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 96.7

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

   4. Applied rewrites 96.7%

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

   if 1.12000000000000001e-18 < y

   1. Initial program 93.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 97.2

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

   4. Applied rewrites 97.2%

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

Alternative 2: 96.8% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-35}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot \left(-z\right), -z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(y \cdot x\right)}^{-1} \cdot {\left(\mathsf{fma}\left(z, z, 1\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1e-35)
   (/ (/ 1.0 x) (fma (* y (- z)) (- z) y))
   (* (pow (* y x) -1.0) (pow (fma z z 1.0) -1.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e-35) {
		tmp = (1.0 / x) / fma((y * -z), -z, y);
	} else {
		tmp = pow((y * x), -1.0) * pow(fma(z, z, 1.0), -1.0);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1e-35)
		tmp = Float64(Float64(1.0 / x) / fma(Float64(y * Float64(-z)), Float64(-z), y));
	else
		tmp = Float64((Float64(y * x) ^ -1.0) * (fma(z, z, 1.0) ^ -1.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1e-35], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y * (-z)), $MachinePrecision] * (-z) + y), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(y * x), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(z * z + 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000001e-35

   1. Initial program 90.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 96.6

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

   4. Applied rewrites 96.6%

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

   if 1.00000000000000001e-35 < y

   1. Initial program 93.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. associate-*r* N/A

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

      10. pow2 N/A

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

      11. unpow-prod-down N/A

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

      12. inv-pow N/A

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

      13. lower-*.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. pow2 N/A

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

      20. +-commutative N/A

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

      21. pow2 N/A

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

      22. lower-fma.f64 97.0

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

   4. Applied rewrites 97.0%

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

Alternative 3: 93.4% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+279}:\\ \;\;\;\;\frac{\frac{1}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;{\left(y \cdot x\right)}^{-1} \cdot {\left(\mathsf{fma}\left(z, z, 1\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (if (<= t_0 2e+279)
     (/ (/ 1.0 x) t_0)
     (* (pow (* y x) -1.0) (pow (fma z z 1.0) -1.0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+279) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = pow((y * x), -1.0) * pow(fma(z, z, 1.0), -1.0);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 2e+279)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64((Float64(y * x) ^ -1.0) * (fma(z, z, 1.0) ^ -1.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+279], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Power[N[(y * x), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(z * z + 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 2.00000000000000012e279

   1. Initial program 95.1%

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

   if 2.00000000000000012e279 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 77.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. associate-*r* N/A

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

      10. pow2 N/A

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

      11. unpow-prod-down N/A

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

      12. inv-pow N/A

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

      13. lower-*.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. pow2 N/A

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

      20. +-commutative N/A

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

      21. pow2 N/A

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

      22. lower-fma.f64 81.9

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

   4. Applied rewrites 81.9%

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

Alternative 4: 93.3% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+275}:\\ \;\;\;\;{\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right)}^{-1} \cdot {x}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(y \cdot x\right)}^{-1} \cdot {\left(\mathsf{fma}\left(z, z, 1\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* y (+ 1.0 (* z z))) 2e+275)
   (* (pow (* (fma z z 1.0) y) -1.0) (pow x -1.0))
   (* (pow (* y x) -1.0) (pow (fma z z 1.0) -1.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((y * (1.0 + (z * z))) <= 2e+275) {
		tmp = pow((fma(z, z, 1.0) * y), -1.0) * pow(x, -1.0);
	} else {
		tmp = pow((y * x), -1.0) * pow(fma(z, z, 1.0), -1.0);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 2e+275)
		tmp = Float64((Float64(fma(z, z, 1.0) * y) ^ -1.0) * (x ^ -1.0));
	else
		tmp = Float64((Float64(y * x) ^ -1.0) * (fma(z, z, 1.0) ^ -1.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+275], N[(N[Power[N[(N[(z * z + 1.0), $MachinePrecision] * y), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(y * x), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(z * z + 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 1.99999999999999992e275

   1. Initial program 95.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. unpow-prod-down N/A

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

      12. inv-pow N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 94.9%

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

   if 1.99999999999999992e275 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 78.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. associate-*r* N/A

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

      10. pow2 N/A

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

      11. unpow-prod-down N/A

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

      12. inv-pow N/A

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

      13. lower-*.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. pow2 N/A

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

      20. +-commutative N/A

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

      21. pow2 N/A

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

      22. lower-fma.f64 82.5

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

   4. Applied rewrites 82.5%

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

Alternative 5: 74.6% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right)}^{-0.5}\\ \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+279}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot {x}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(y \cdot x\right)}^{-1} \cdot {\left(\mathsf{fma}\left(z, z, 1\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (pow (* (fma z z 1.0) y) -0.5)))
   (if (<= (* y (+ 1.0 (* z z))) 2e+279)
     (* (* t_0 t_0) (pow x -1.0))
     (* (pow (* y x) -1.0) (pow (fma z z 1.0) -1.0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = pow((fma(z, z, 1.0) * y), -0.5);
	double tmp;
	if ((y * (1.0 + (z * z))) <= 2e+279) {
		tmp = (t_0 * t_0) * pow(x, -1.0);
	} else {
		tmp = pow((y * x), -1.0) * pow(fma(z, z, 1.0), -1.0);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(fma(z, z, 1.0) * y) ^ -0.5
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 2e+279)
		tmp = Float64(Float64(t_0 * t_0) * (x ^ -1.0));
	else
		tmp = Float64((Float64(y * x) ^ -1.0) * (fma(z, z, 1.0) ^ -1.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[Power[N[(N[(z * z + 1.0), $MachinePrecision] * y), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+279], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(y * x), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(z * z + 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 2.00000000000000012e279

   1. Initial program 95.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. unpow-prod-down N/A

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

      12. inv-pow N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 94.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. sqr-pow N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

   6. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\left({\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right)}^{-0.5}\right)} \cdot {x}^{-1} \]

   if 2.00000000000000012e279 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 77.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. associate-*r* N/A

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

      10. pow2 N/A

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

      11. unpow-prod-down N/A

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

      12. inv-pow N/A

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

      13. lower-*.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. pow2 N/A

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

      20. +-commutative N/A

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

      21. pow2 N/A

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

      22. lower-fma.f64 81.9

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

   4. Applied rewrites 81.9%

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

Alternative 6: 74.5% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right)}^{-0.5}\\ \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+260}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot {x}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot x\right) \cdot \left(z \cdot z - -1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (pow (* (fma z z 1.0) y) -0.5)))
   (if (<= (* y (+ 1.0 (* z z))) 1e+260)
     (* (* t_0 t_0) (pow x -1.0))
     (/ 1.0 (* (* y x) (- (* z z) -1.0))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = pow((fma(z, z, 1.0) * y), -0.5);
	double tmp;
	if ((y * (1.0 + (z * z))) <= 1e+260) {
		tmp = (t_0 * t_0) * pow(x, -1.0);
	} else {
		tmp = 1.0 / ((y * x) * ((z * z) - -1.0));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(fma(z, z, 1.0) * y) ^ -0.5
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 1e+260)
		tmp = Float64(Float64(t_0 * t_0) * (x ^ -1.0));
	else
		tmp = Float64(1.0 / Float64(Float64(y * x) * Float64(Float64(z * z) - -1.0)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[Power[N[(N[(z * z + 1.0), $MachinePrecision] * y), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+260], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y * x), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 1.00000000000000007e260

   1. Initial program 95.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. unpow-prod-down N/A

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

      12. inv-pow N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 94.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. sqr-pow N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

   6. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\left({\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right)}^{-0.5}\right)} \cdot {x}^{-1} \]

   if 1.00000000000000007e260 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 78.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 87.6

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

   4. Applied rewrites 87.6%

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

   5. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 82.9

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

   7. Applied rewrites 82.9%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+260}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right)}^{-0.5}\right) \cdot {x}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot x\right) \cdot \left(z \cdot z - -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.5% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \frac{1}{\left(y \cdot x\right) \cdot \left(z \cdot z - -1\right)} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (/ 1.0 (* (* y x) (- (* z z) -1.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return 1.0 / ((y * x) * ((z * z) - -1.0));
}

Fortran

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

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

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

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return 1.0 / ((y * x) * ((z * z) - -1.0));
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return 1.0 / ((y * x) * ((z * z) - -1.0))

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(1.0 / Float64(Float64(y * x) * Float64(Float64(z * z) - -1.0)))
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = 1.0 / ((y * x) * ((z * z) - -1.0));
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(1.0 / N[(N[(y * x), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. +-commutative N/A

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

   8. pow2 N/A

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

   9. sqr-neg-rev N/A

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

   10. associate-*r* N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-neg.f64 95.7

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

4. Applied rewrites 95.7%

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

5. Taylor expanded in y around -inf

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

   1. lower-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. lower--.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 88.6

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

7. Applied rewrites 88.6%

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

8. Final simplification 88.6%

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

Reproduce

herbie shell --seed 2025057 
(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)))))