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

Percentage Accurate: 89.9% → 98.7%

Time: 3.4s

Alternatives: 6

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: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot y}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* y (+ 1.0 (* z z))) 5e+304)
    (/ (/ 1.0 (* (fma z z 1.0) y)) x_m)
    (/ 1.0 (* z (* y (* z x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y * (1.0 + (z * z))) <= 5e+304) {
		tmp = (1.0 / (fma(z, z, 1.0) * y)) / x_m;
	} else {
		tmp = 1.0 / (z * (y * (z * x_m)));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 5e+304)
		tmp = Float64(Float64(1.0 / Float64(fma(z, z, 1.0) * y)) / x_m);
	else
		tmp = Float64(1.0 / Float64(z * Float64(y * Float64(z * x_m))));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+304], N[(N[(1.0 / N[(N[(z * z + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision], N[(1.0 / N[(z * N[(y * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 4.9999999999999997e304

   1. Initial program 93.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 93.3

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

   4. Applied rewrites 93.3%

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

   if 4.9999999999999997e304 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 67.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 67.6

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

   4. Applied rewrites 67.6%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 87.5

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

   7. Applied rewrites 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

Alternative 2: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* y (+ 1.0 (* z z))) 5e+304)
    (/ 1.0 (* (* (fma z z 1.0) y) x_m))
    (/ 1.0 (* z (* y (* z x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y * (1.0 + (z * z))) <= 5e+304) {
		tmp = 1.0 / ((fma(z, z, 1.0) * y) * x_m);
	} else {
		tmp = 1.0 / (z * (y * (z * x_m)));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 5e+304)
		tmp = Float64(1.0 / Float64(Float64(fma(z, z, 1.0) * y) * x_m));
	else
		tmp = Float64(1.0 / Float64(z * Float64(y * Float64(z * x_m))));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+304], N[(1.0 / N[(N[(N[(z * z + 1.0), $MachinePrecision] * y), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z * N[(y * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 4.9999999999999997e304

   1. Initial program 93.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 93.4

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

   4. Applied rewrites 93.4%

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

   if 4.9999999999999997e304 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 67.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 67.6

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

   4. Applied rewrites 67.6%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 87.5

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

   7. Applied rewrites 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

Alternative 3: 96.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   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 89.8

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

4. Applied rewrites 89.8%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. inv-pow N/A

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

   5. pow2 N/A

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

   6. +-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. pow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-fma.f64 89.5

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

6. Applied rewrites 89.5%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. pow2 N/A

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

   4. distribute-lft1-in N/A

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

   5. *-commutative N/A

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

   6. pow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 93.2

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

8. Applied rewrites 93.2%

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

Alternative 4: 74.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.85:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{y \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z 0.85)
    (/ (fma (- z) z 1.0) (* y x_m))
    (/ 1.0 (* z (* y (* z x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= 0.85) {
		tmp = fma(-z, z, 1.0) / (y * x_m);
	} else {
		tmp = 1.0 / (z * (y * (z * x_m)));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= 0.85)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(y * x_m));
	else
		tmp = Float64(1.0 / Float64(z * Float64(y * Float64(z * x_m))));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, 0.85], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(y * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z * N[(y * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.849999999999999978

   1. Initial program 93.6%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   if 0.849999999999999978 < z

   1. Initial program 78.3%

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

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 77.4

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 90.0

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

   7. Applied rewrites 90.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 94.3

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

   9. Applied rewrites 94.3%

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

Alternative 5: 58.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 59.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 59.1

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

7. Applied rewrites 59.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 58.7

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

   7. pow2 58.7

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

   8. distribute-lft-in 58.7

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

   9. *-rgt-identity 58.7

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

   10. +-commutative 58.7

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

   11. pow2 58.7

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

   12. associate-*l* 58.7

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

9. Applied rewrites 58.7%

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

Alternative 6: 58.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 59.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 59.1

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

7. Applied rewrites 59.1%

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

8. Final simplification 59.1%

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

Reproduce

herbie shell --seed 2025085 
(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)))))