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

Percentage Accurate: 90.5% → 98.5%

Time: 4.2s

Alternatives: 13

Speedup: 0.8×

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: 90.5% 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: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 6.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m \cdot z, z \cdot x, x \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(x \cdot y\_m\right) \cdot z, z, x \cdot y\_m\right)}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 6.5e-65)
    (/ 1.0 (fma (* y_m z) (* z x) (* x y_m)))
    (/ 1.0 (fma (* (* x y_m) z) z (* x y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 6.5e-65) {
		tmp = 1.0 / fma((y_m * z), (z * x), (x * y_m));
	} else {
		tmp = 1.0 / fma(((x * y_m) * z), z, (x * y_m));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 6.5e-65)
		tmp = Float64(1.0 / fma(Float64(y_m * z), Float64(z * x), Float64(x * y_m)));
	else
		tmp = Float64(1.0 / fma(Float64(Float64(x * y_m) * z), z, Float64(x * y_m)));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 6.5e-65], N[(1.0 / N[(N[(y$95$m * z), $MachinePrecision] * N[(z * x), $MachinePrecision] + N[(x * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x * y$95$m), $MachinePrecision] * z), $MachinePrecision] * z + N[(x * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.5e-65

   1. Initial program 84.6%

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

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

      5. lift-+.f64 N/A

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

      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 84.6

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

   3. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. *-lft-identity N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if 6.5e-65 < y

   1. Initial program 92.8%

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

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

      5. lift-+.f64 N/A

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

      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 92.4

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

   3. Applied rewrites 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-lft-identity N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 2: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{+118}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m \cdot z, z \cdot x, x \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x \cdot y\_m}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 5e+118)
    (/ 1.0 (fma (* y_m z) (* z x) (* x y_m)))
    (/ (/ 1.0 (fma z z 1.0)) (* x y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5e+118) {
		tmp = 1.0 / fma((y_m * z), (z * x), (x * y_m));
	} else {
		tmp = (1.0 / fma(z, z, 1.0)) / (x * y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 5e+118)
		tmp = Float64(1.0 / fma(Float64(y_m * z), Float64(z * x), Float64(x * y_m)));
	else
		tmp = Float64(Float64(1.0 / fma(z, z, 1.0)) / Float64(x * y_m));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 5e+118], N[(1.0 / N[(N[(y$95$m * z), $MachinePrecision] * N[(z * x), $MachinePrecision] + N[(x * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999972e118

   1. Initial program 88.4%

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

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

      5. lift-+.f64 N/A

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

      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 88.3

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

   3. Applied rewrites 88.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. *-lft-identity N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 4.99999999999999972e118 < y

   1. Initial program 93.4%

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

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

      5. lift-+.f64 N/A

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

      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 93.0

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

   3. Applied rewrites 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

Alternative 3: 96.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 10^{+154}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x \cdot \left(y\_m \cdot z\right), z, x \cdot y\_m\right)}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 1e+154)
    (/ (/ (/ 1.0 x) (fma z z 1.0)) y_m)
    (/ 1.0 (fma (* x (* y_m z)) z (* x y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 1e+154) {
		tmp = ((1.0 / x) / fma(z, z, 1.0)) / y_m;
	} else {
		tmp = 1.0 / fma((x * (y_m * z)), z, (x * y_m));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 1e+154)
		tmp = Float64(Float64(Float64(1.0 / x) / fma(z, z, 1.0)) / y_m);
	else
		tmp = Float64(1.0 / fma(Float64(x * Float64(y_m * z)), z, Float64(x * y_m)));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 1e+154], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(N[(x * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision] * z + N[(x * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.00000000000000004e154

   1. Initial program 92.8%

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

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

      5. lift-+.f64 N/A

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

      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. lift-/.f64 N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 95.1

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

   3. Applied rewrites 95.1%

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

   if 1.00000000000000004e154 < z

   1. Initial program 74.6%

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

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

      5. lift-+.f64 N/A

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

      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 74.6

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

   3. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-lft-identity N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 97.4

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

   5. Applied rewrites 97.4%

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

Alternative 4: 96.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y\_m \cdot z, z, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x \cdot y\_m}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 2e-26)
    (/ (/ 1.0 x) (fma (* y_m z) z y_m))
    (/ (/ 1.0 (fma z z 1.0)) (* x y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2e-26) {
		tmp = (1.0 / x) / fma((y_m * z), z, y_m);
	} else {
		tmp = (1.0 / fma(z, z, 1.0)) / (x * y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 2e-26)
		tmp = Float64(Float64(1.0 / x) / fma(Float64(y_m * z), z, y_m));
	else
		tmp = Float64(Float64(1.0 / fma(z, z, 1.0)) / Float64(x * y_m));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 2e-26], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y$95$m * z), $MachinePrecision] * z + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.0000000000000001e-26

   1. Initial program 85.1%

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

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

      3. lift-+.f64 N/A

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

      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. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 93.5

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

   3. Applied rewrites 93.5%

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

   if 2.0000000000000001e-26 < y

   1. Initial program 93.2%

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

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

      5. lift-+.f64 N/A

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

      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 92.8

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

   3. Applied rewrites 92.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lower-*.f64 97.2

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

   5. Applied rewrites 97.2%

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

Alternative 5: 95.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 8500000000000:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y\_m \cdot z, z, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 8500000000000.0)
    (/ (/ 1.0 x) (fma (* y_m z) z y_m))
    (/ (/ 1.0 (* y_m x)) (fma z z 1.0)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 8500000000000.0) {
		tmp = (1.0 / x) / fma((y_m * z), z, y_m);
	} else {
		tmp = (1.0 / (y_m * x)) / fma(z, z, 1.0);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 8500000000000.0)
		tmp = Float64(Float64(1.0 / x) / fma(Float64(y_m * z), z, y_m));
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * x)) / fma(z, z, 1.0));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 8500000000000.0], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y$95$m * z), $MachinePrecision] * z + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 8.5e12

   1. Initial program 86.2%

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

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

      3. lift-+.f64 N/A

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

      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. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 93.4

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

   3. Applied rewrites 93.4%

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

   if 8.5e12 < y

   1. Initial program 93.3%

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

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

      5. lift-+.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 97.6

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

   3. Applied rewrites 97.6%

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

Alternative 6: 92.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;1 + z \cdot z \leq 2000000:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\_m\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(z \cdot z\right) \cdot x}}{y\_m}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (+ 1.0 (* z z)) 2000000.0)
    (/ 1.0 (* (* (fma z z 1.0) y_m) x))
    (/ (/ 1.0 (* (* z z) x)) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((1.0 + (z * z)) <= 2000000.0) {
		tmp = 1.0 / ((fma(z, z, 1.0) * y_m) * x);
	} else {
		tmp = (1.0 / ((z * z) * x)) / y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(1.0 + Float64(z * z)) <= 2000000.0)
		tmp = Float64(1.0 / Float64(Float64(fma(z, z, 1.0) * y_m) * x));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(z * z) * x)) / y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision], 2000000.0], N[(1.0 / N[(N[(N[(z * z + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(z * z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 z z)) < 2e6

   1. Initial program 99.6%

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

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

      5. lift-+.f64 N/A

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

      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 99.2

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

   3. Applied rewrites 99.2%

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

   if 2e6 < (+.f64 #s(literal 1 binary64) (*.f64 z z))

   1. Initial program 80.8%

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

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

      5. lift-+.f64 N/A

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

      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. lift-/.f64 N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 85.1

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 84.7

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

   6. Applied rewrites 84.7%

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

Alternative 7: 91.6% accurate, 0.7× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (+ 1.0 (* z z)) 2.0)
    (/ (- 1.0 (* z z)) (* y_m x))
    (/ (/ 1.0 (* (* z z) x)) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((1.0 + (z * z)) <= 2.0) {
		tmp = (1.0 - (z * z)) / (y_m * x);
	} else {
		tmp = (1.0 / ((z * z) * x)) / y_m;
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((1.0d0 + (z * z)) <= 2.0d0) then
        tmp = (1.0d0 - (z * z)) / (y_m * x)
    else
        tmp = (1.0d0 / ((z * z) * x)) / y_m
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((1.0 + (z * z)) <= 2.0) {
		tmp = (1.0 - (z * z)) / (y_m * x);
	} else {
		tmp = (1.0 / ((z * z) * x)) / y_m;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (1.0 + (z * z)) <= 2.0:
		tmp = (1.0 - (z * z)) / (y_m * x)
	else:
		tmp = (1.0 / ((z * z) * x)) / y_m
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(1.0 + Float64(z * z)) <= 2.0)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y_m * x));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(z * z) * x)) / y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((1.0 + (z * z)) <= 2.0)
		tmp = (1.0 - (z * z)) / (y_m * x);
	else
		tmp = (1.0 / ((z * z) * x)) / y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

          \[\leadsto \frac{1 - z \cdot z}{x \cdot y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1 - z \cdot z}{x \cdot y} \]

      13. *-commutative N/A

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

      14. lower-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

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

   1. Initial program 81.0%

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

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

      5. lift-+.f64 N/A

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

      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. lift-/.f64 N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 85.3

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

   3. Applied rewrites 85.3%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 84.2

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

   6. Applied rewrites 84.2%

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

Alternative 8: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.88:\\ \;\;\;\;\frac{1 - z \cdot z}{y\_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot x\right) \cdot y\_m}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 0.88)
    (/ (- 1.0 (* z z)) (* y_m x))
    (/ 1.0 (* (* (* z z) x) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 0.88) {
		tmp = (1.0 - (z * z)) / (y_m * x);
	} else {
		tmp = 1.0 / (((z * z) * x) * y_m);
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.88d0) then
        tmp = (1.0d0 - (z * z)) / (y_m * x)
    else
        tmp = 1.0d0 / (((z * z) * x) * y_m)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 0.88) {
		tmp = (1.0 - (z * z)) / (y_m * x);
	} else {
		tmp = 1.0 / (((z * z) * x) * y_m);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 0.88:
		tmp = (1.0 - (z * z)) / (y_m * x)
	else:
		tmp = 1.0 / (((z * z) * x) * y_m)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 0.88)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y_m * x));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z * z) * x) * y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 0.88)
		tmp = (1.0 - (z * z)) / (y_m * x);
	else
		tmp = 1.0 / (((z * z) * x) * y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 0.88], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z * z), $MachinePrecision] * x), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.880000000000000004

   1. Initial program 93.5%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

          \[\leadsto \frac{1 - z \cdot z}{x \cdot y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1 - z \cdot z}{x \cdot y} \]

      13. *-commutative N/A

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

      14. lower-*.f64 69.7

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

   4. Applied rewrites 69.7%

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

   if 0.880000000000000004 < z

   1. Initial program 81.5%

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

   2. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 80.5

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

   4. Applied rewrites 80.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 81.7

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

      10. pow2 81.7

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

      11. +-commutative 81.7

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

      12. pow2 81.7

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

   6. Applied rewrites 81.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 84.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 84.0

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

   8. Applied rewrites 84.0%

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

Alternative 9: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.88:\\ \;\;\;\;\frac{1 - z \cdot z}{y\_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y\_m \cdot z\right) \cdot z\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 0.88)
    (/ (- 1.0 (* z z)) (* y_m x))
    (/ 1.0 (* (* (* y_m z) z) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 0.88) {
		tmp = (1.0 - (z * z)) / (y_m * x);
	} else {
		tmp = 1.0 / (((y_m * z) * z) * x);
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.88d0) then
        tmp = (1.0d0 - (z * z)) / (y_m * x)
    else
        tmp = 1.0d0 / (((y_m * z) * z) * x)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 0.88) {
		tmp = (1.0 - (z * z)) / (y_m * x);
	} else {
		tmp = 1.0 / (((y_m * z) * z) * x);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 0.88:
		tmp = (1.0 - (z * z)) / (y_m * x)
	else:
		tmp = 1.0 / (((y_m * z) * z) * x)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 0.88)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y_m * x));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(y_m * z) * z) * x));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 0.88)
		tmp = (1.0 - (z * z)) / (y_m * x);
	else
		tmp = 1.0 / (((y_m * z) * z) * x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 0.88], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(y$95$m * z), $MachinePrecision] * z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.880000000000000004

   1. Initial program 93.5%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

          \[\leadsto \frac{1 - z \cdot z}{x \cdot y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1 - z \cdot z}{x \cdot y} \]

      13. *-commutative N/A

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

      14. lower-*.f64 69.7

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

   4. Applied rewrites 69.7%

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

   if 0.880000000000000004 < z

   1. Initial program 81.5%

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

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

      5. lift-+.f64 N/A

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

      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. lift-/.f64 N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 85.6

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 86.2

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

   6. Applied rewrites 86.2%

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

Alternative 10: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.88:\\ \;\;\;\;\frac{1 - z \cdot z}{y\_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot y\_m\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 0.88)
    (/ (- 1.0 (* z z)) (* y_m x))
    (/ 1.0 (* (* (* z z) y_m) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 0.88) {
		tmp = (1.0 - (z * z)) / (y_m * x);
	} else {
		tmp = 1.0 / (((z * z) * y_m) * x);
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.88d0) then
        tmp = (1.0d0 - (z * z)) / (y_m * x)
    else
        tmp = 1.0d0 / (((z * z) * y_m) * x)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 0.88) {
		tmp = (1.0 - (z * z)) / (y_m * x);
	} else {
		tmp = 1.0 / (((z * z) * y_m) * x);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 0.88:
		tmp = (1.0 - (z * z)) / (y_m * x)
	else:
		tmp = 1.0 / (((z * z) * y_m) * x)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 0.88)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y_m * x));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z * z) * y_m) * x));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 0.88)
		tmp = (1.0 - (z * z)) / (y_m * x);
	else
		tmp = 1.0 / (((z * z) * y_m) * x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 0.88], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z * z), $MachinePrecision] * y$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.880000000000000004

   1. Initial program 93.5%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

          \[\leadsto \frac{1 - z \cdot z}{x \cdot y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1 - z \cdot z}{x \cdot y} \]

      13. *-commutative N/A

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

      14. lower-*.f64 69.7

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

   4. Applied rewrites 69.7%

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

   if 0.880000000000000004 < z

   1. Initial program 81.5%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 80.3

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

   4. Applied rewrites 80.3%

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

Alternative 11: 64.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.98:\\ \;\;\;\;\frac{1 - z \cdot z}{y\_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\left(y\_m \cdot y\_m\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 0.98) (/ (- 1.0 (* z z)) (* y_m x)) (/ y_m (* (* y_m y_m) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 0.98) {
		tmp = (1.0 - (z * z)) / (y_m * x);
	} else {
		tmp = y_m / ((y_m * y_m) * x);
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.98d0) then
        tmp = (1.0d0 - (z * z)) / (y_m * x)
    else
        tmp = y_m / ((y_m * y_m) * x)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 0.98) {
		tmp = (1.0 - (z * z)) / (y_m * x);
	} else {
		tmp = y_m / ((y_m * y_m) * x);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 0.98:
		tmp = (1.0 - (z * z)) / (y_m * x)
	else:
		tmp = y_m / ((y_m * y_m) * x)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 0.98)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y_m * x));
	else
		tmp = Float64(y_m / Float64(Float64(y_m * y_m) * x));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 0.98)
		tmp = (1.0 - (z * z)) / (y_m * x);
	else
		tmp = y_m / ((y_m * y_m) * x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 0.98], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(N[(y$95$m * y$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.97999999999999998

   1. Initial program 93.5%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

          \[\leadsto \frac{1 - z \cdot z}{x \cdot y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1 - z \cdot z}{x \cdot y} \]

      13. *-commutative N/A

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

      14. lower-*.f64 69.7

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

   4. Applied rewrites 69.7%

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

   if 0.97999999999999998 < z

   1. Initial program 81.5%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

          \[\leadsto \frac{1 - z \cdot z}{x \cdot y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1 - z \cdot z}{x \cdot y} \]

      13. *-commutative N/A

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

      14. lower-*.f64 8.3

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

   4. Applied rewrites 8.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

         \[\leadsto \frac{1 - z \cdot z}{y \cdot x} \]

      4. lift--.f64 N/A

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

      5. pow2 N/A

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

      6. div-sub N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. frac-sub N/A

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

      12. lower-/.f64 N/A

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

   6. Applied rewrites 1.0%

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

   7. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg2 N/A

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

      3. distribute-lft-out-- N/A

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

      4. mul-1-neg N/A

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

      5. frac-2neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 3.7

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

   9. Applied rewrites 3.7%

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

   10. Taylor expanded in z around 0

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

   12. Applied rewrites 37.4%

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

Alternative 12: 61.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{+42}:\\ \;\;\;\;\frac{1}{y\_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\left(y\_m \cdot y\_m\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= z 2.3e+42) (/ 1.0 (* y_m x)) (/ y_m (* (* y_m y_m) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 2.3e+42) {
		tmp = 1.0 / (y_m * x);
	} else {
		tmp = y_m / ((y_m * y_m) * x);
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2.3d+42) then
        tmp = 1.0d0 / (y_m * x)
    else
        tmp = y_m / ((y_m * y_m) * x)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 2.3e+42) {
		tmp = 1.0 / (y_m * x);
	} else {
		tmp = y_m / ((y_m * y_m) * x);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 2.3e+42:
		tmp = 1.0 / (y_m * x)
	else:
		tmp = y_m / ((y_m * y_m) * x)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 2.3e+42)
		tmp = Float64(1.0 / Float64(y_m * x));
	else
		tmp = Float64(y_m / Float64(Float64(y_m * y_m) * x));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 2.3e+42)
		tmp = 1.0 / (y_m * x);
	else
		tmp = y_m / ((y_m * y_m) * x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 2.3e+42], N[(1.0 / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(N[(y$95$m * y$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.3e42

   1. Initial program 93.7%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 71.2

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

   4. Applied rewrites 71.2%

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

   if 2.3e42 < z

   1. Initial program 78.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

          \[\leadsto \frac{1 - z \cdot z}{x \cdot y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1 - z \cdot z}{x \cdot y} \]

      13. *-commutative N/A

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

      14. lower-*.f64 7.0

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

   4. Applied rewrites 7.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

         \[\leadsto \frac{1 - z \cdot z}{y \cdot x} \]

      4. lift--.f64 N/A

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

      5. pow2 N/A

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

      6. div-sub N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. frac-sub N/A

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

      12. lower-/.f64 N/A

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

   6. Applied rewrites 0.9%

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

   7. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg2 N/A

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

      3. distribute-lft-out-- N/A

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

      4. mul-1-neg N/A

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

      5. frac-2neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 2.1

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

   9. Applied rewrites 2.1%

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

   10. Taylor expanded in z around 0

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

   12. Applied rewrites 38.4%

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

Alternative 13: 59.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

2. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 59.5

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

4. Applied rewrites 59.5%

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

Reproduce

herbie shell --seed 2025112 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))