Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Percentage Accurate: 82.5% → 96.7%

Time: 6.1s

Alternatives: 13

Speedup: 0.9×

• 25.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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 = (x * y) / ((z * z) * (z + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = (x * y) / ((z * z) * (z + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+17}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y\_m}{z}}{z}\\ \end{array} \end{array} \]

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 -5e+19)
    (* (/ y_m z) (/ (/ x z) z))
    (if (<= z 1.05e+17)
      (/ (* y_m (/ x (fma z z z))) z)
      (/ (* (/ x z) (/ y_m z)) z)))))

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 <= -5e+19) {
		tmp = (y_m / z) * ((x / z) / z);
	} else if (z <= 1.05e+17) {
		tmp = (y_m * (x / fma(z, z, z))) / z;
	} else {
		tmp = ((x / z) * (y_m / z)) / z;
	}
	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 <= -5e+19)
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x / z) / z));
	elseif (z <= 1.05e+17)
		tmp = Float64(Float64(y_m * Float64(x / fma(z, z, z))) / z);
	else
		tmp = Float64(Float64(Float64(x / z) * Float64(y_m / z)) / z);
	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, -5e+19], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e+17], N[(N[(y$95$m * N[(x / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -5e19

   1. Initial program 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. distribute-lft1-in N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. distribute-lft1-in N/A

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

      18. lift-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-fma.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. lift-fma.f64 N/A

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

      23. lift-*.f64 81.6

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

   6. Applied rewrites 81.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l/ N/A

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

      9. associate-/l* N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 98.3

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 98.3%

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

   if -5e19 < z < 1.05e17

   1. Initial program 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 84.8

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

   4. Applied rewrites 84.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. frac-times N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-/.f64 97.4

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

   6. Applied rewrites 97.4%

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

   if 1.05e17 < z

   1. Initial program 83.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 98.1%

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

Alternative 2: 93.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 2 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{\mathsf{fma}\left(z, z, z\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \end{array} \end{array} \]

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 (<= (/ (* x y_m) (* (* z z) (+ z 1.0))) 2e+24)
    (* (/ (/ y_m z) (fma z z z)) x)
    (/ (* y_m (/ x (fma z z z))) z))))

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 (((x * y_m) / ((z * z) * (z + 1.0))) <= 2e+24) {
		tmp = ((y_m / z) / fma(z, z, z)) * x;
	} else {
		tmp = (y_m * (x / fma(z, z, z))) / z;
	}
	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(Float64(x * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 2e+24)
		tmp = Float64(Float64(Float64(y_m / z) / fma(z, z, z)) * x);
	else
		tmp = Float64(Float64(y_m * Float64(x / fma(z, z, z))) / z);
	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[(N[(x * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+24], N[(N[(N[(y$95$m / z), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(y$95$m * N[(x / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 2e24

   1. Initial program 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 95.5

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

   4. Applied rewrites 95.5%

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

   if 2e24 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 66.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 70.1

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

   4. Applied rewrites 70.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. frac-times N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-/.f64 88.1

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

   6. Applied rewrites 88.1%

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

Alternative 3: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1000000000 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-82}\right):\\ \;\;\;\;x \cdot \frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y\_m}{z}\\ \end{array} \end{array} \end{array} \]

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
 (let* ((t_0 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (if (or (<= t_0 -1000000000.0) (not (<= t_0 5e-82)))
      (* x (/ y_m (* (fma z z z) z)))
      (* (/ x z) (/ y_m z))))))

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 t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -1000000000.0) || !(t_0 <= 5e-82)) {
		tmp = x * (y_m / (fma(z, z, z) * z));
	} else {
		tmp = (x / z) * (y_m / z);
	}
	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)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if ((t_0 <= -1000000000.0) || !(t_0 <= 5e-82))
		tmp = Float64(x * Float64(y_m / Float64(fma(z, z, z) * z)));
	else
		tmp = Float64(Float64(x / z) * Float64(y_m / z));
	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_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[Or[LessEqual[t$95$0, -1000000000.0], N[Not[LessEqual[t$95$0, 5e-82]], $MachinePrecision]], N[(x * N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1e9 or 4.9999999999999998e-82 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 84.9

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

   4. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. distribute-lft1-in N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. distribute-lft1-in N/A

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

      18. lift-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-fma.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. lift-fma.f64 N/A

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

      23. lift-*.f64 86.2

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

   6. Applied rewrites 86.2%

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

   if -1e9 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.9999999999999998e-82

   1. Initial program 82.7%

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

   3. Taylor expanded in z around 0

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

      1. pow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -1000000000 \lor \neg \left(\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 5 \cdot 10^{-82}\right):\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1000000000:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;t\_0 \leq 10^{-312}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \end{array} \end{array} \end{array} \]

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
 (let* ((t_0 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (if (<= t_0 -1000000000.0)
      (* (/ y_m z) (/ x (* z z)))
      (if (<= t_0 1e-312)
        (/ (* y_m (/ x z)) z)
        (* y_m (/ x (* (fma z z z) z))))))))

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 t_0 = (z * z) * (z + 1.0);
	double tmp;
	if (t_0 <= -1000000000.0) {
		tmp = (y_m / z) * (x / (z * z));
	} else if (t_0 <= 1e-312) {
		tmp = (y_m * (x / z)) / z;
	} else {
		tmp = y_m * (x / (fma(z, z, z) * z));
	}
	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)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_0 <= -1000000000.0)
		tmp = Float64(Float64(y_m / z) * Float64(x / Float64(z * z)));
	elseif (t_0 <= 1e-312)
		tmp = Float64(Float64(y_m * Float64(x / z)) / z);
	else
		tmp = Float64(y_m * Float64(x / Float64(fma(z, z, z) * z)));
	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_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -1000000000.0], N[(N[(y$95$m / z), $MachinePrecision] * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-312], N[(N[(y$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(x / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1e9

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-lft-identity N/A

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

      14. lower-fma.f64 88.0

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 86.8

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

   7. Applied rewrites 86.8%

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

   if -1e9 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 9.9999999999847e-313

   1. Initial program 77.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. frac-times N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 98.9

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

   9. Applied rewrites 98.9%

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

   if 9.9999999999847e-313 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 90.7

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

   4. Applied rewrites 90.7%

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

Alternative 5: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+19} \lor \neg \left(z \leq 1.35 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \end{array} \end{array} \]

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 (or (<= z -5e+19) (not (<= z 1.35e+17)))
    (* (/ y_m z) (/ (/ x z) z))
    (/ (* y_m (/ x (fma z z z))) z))))

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 <= -5e+19) || !(z <= 1.35e+17)) {
		tmp = (y_m / z) * ((x / z) / z);
	} else {
		tmp = (y_m * (x / fma(z, z, z))) / z;
	}
	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 <= -5e+19) || !(z <= 1.35e+17))
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x / z) / z));
	else
		tmp = Float64(Float64(y_m * Float64(x / fma(z, z, z))) / z);
	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[Or[LessEqual[z, -5e+19], N[Not[LessEqual[z, 1.35e+17]], $MachinePrecision]], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -5e19 or 1.35e17 < z

   1. Initial program 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 82.0

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

   4. Applied rewrites 82.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. distribute-lft1-in N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. distribute-lft1-in N/A

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

      18. lift-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-fma.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. lift-fma.f64 N/A

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

      23. lift-*.f64 83.7

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

   6. Applied rewrites 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l/ N/A

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

      9. associate-/l* N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 98.2

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 98.2%

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

   if -5e19 < z < 1.35e17

   1. Initial program 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 84.8

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

   4. Applied rewrites 84.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. frac-times N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-/.f64 97.4

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

   6. Applied rewrites 97.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+19} \lor \neg \left(z \leq 1.35 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, z, z\right) \cdot z\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y\_m \leq 2 \cdot 10^{-196}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;x \cdot y\_m \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\frac{y\_m \cdot x}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x}{t\_0}\\ \end{array} \end{array} \end{array} \]

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
 (let* ((t_0 (* (fma z z z) z)))
   (*
    y_s
    (if (<= (* x y_m) 2e-196)
      (* (/ x z) (/ y_m z))
      (if (<= (* x y_m) 2e+297) (/ (* y_m x) t_0) (* y_m (/ x t_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 t_0 = fma(z, z, z) * z;
	double tmp;
	if ((x * y_m) <= 2e-196) {
		tmp = (x / z) * (y_m / z);
	} else if ((x * y_m) <= 2e+297) {
		tmp = (y_m * x) / t_0;
	} else {
		tmp = y_m * (x / t_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)
	t_0 = Float64(fma(z, z, z) * z)
	tmp = 0.0
	if (Float64(x * y_m) <= 2e-196)
		tmp = Float64(Float64(x / z) * Float64(y_m / z));
	elseif (Float64(x * y_m) <= 2e+297)
		tmp = Float64(Float64(y_m * x) / t_0);
	else
		tmp = Float64(y_m * Float64(x / t_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_] := Block[{t$95$0 = N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]}, N[(y$95$s * If[LessEqual[N[(x * y$95$m), $MachinePrecision], 2e-196], N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y$95$m), $MachinePrecision], 2e+297], N[(N[(y$95$m * x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(y$95$m * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 2.0000000000000001e-196

   1. Initial program 80.8%

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

   3. Taylor expanded in z around 0

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

      1. pow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.1

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

   5. Applied rewrites 80.1%

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

   if 2.0000000000000001e-196 < (*.f64 x y) < 2e297

   1. Initial program 96.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 96.2

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f64 96.2

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

   4. Applied rewrites 96.2%

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

   if 2e297 < (*.f64 x y)

   1. Initial program 53.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 69.5

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

   4. Applied rewrites 69.5%

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

Alternative 7: 94.4% accurate, 0.7× speedup

Math

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

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 (* (/ y_m (+ 1.0 z)) (/ (/ x z) z))))

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 * ((y_m / (1.0 + z)) * ((x / z) / z));
}

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 * ((y_m / (1.0d0 + z)) * ((x / z) / z))
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 * ((y_m / (1.0 + z)) * ((x / z) / z));
}

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 * ((y_m / (1.0 + z)) * ((x / z) / z))

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(Float64(y_m / Float64(1.0 + z)) * Float64(Float64(x / z) / z)))
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 * ((y_m / (1.0 + z)) * ((x / z) / z));
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[(N[(y$95$m / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. distribute-rgt-in N/A

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

   14. *-lft-identity N/A

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

   15. lower-fma.f64 83.7

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

4. Applied rewrites 83.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lift-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lift-fma.f64 N/A

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

   10. distribute-lft1-in N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

   17. distribute-lft1-in N/A

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

   18. lift-fma.f64 N/A

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

   19. *-commutative N/A

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

   20. lift-fma.f64 N/A

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

   21. lower-/.f64 N/A

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

   22. lift-fma.f64 N/A

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

   23. lift-*.f64 82.6

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

6. Applied rewrites 82.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-/l/ N/A

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

   9. associate-/l* N/A

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

   10. distribute-lft1-in N/A

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

   11. +-commutative N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-+.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 94.4

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

8. Applied rewrites 94.4%

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

Alternative 8: 88.4% accurate, 0.7× speedup

Math

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

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 (or (<= z -1.0) (not (<= z 1.0)))
    (* y_m (/ x (* (* z z) z)))
    (* (/ x z) (/ y_m z)))))

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 <= -1.0) || !(z <= 1.0)) {
		tmp = y_m * (x / ((z * z) * z));
	} else {
		tmp = (x / z) * (y_m / z);
	}
	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 <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = y_m * (x / ((z * z) * z))
    else
        tmp = (x / z) * (y_m / z)
    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 <= -1.0) || !(z <= 1.0)) {
		tmp = y_m * (x / ((z * z) * z));
	} else {
		tmp = (x / z) * (y_m / z);
	}
	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 <= -1.0) or not (z <= 1.0):
		tmp = y_m * (x / ((z * z) * z))
	else:
		tmp = (x / z) * (y_m / z)
	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 <= -1.0) || !(z <= 1.0))
		tmp = Float64(y_m * Float64(x / Float64(Float64(z * z) * z)));
	else
		tmp = Float64(Float64(x / z) * Float64(y_m / z));
	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 <= -1.0) || ~((z <= 1.0)))
		tmp = y_m * (x / ((z * z) * z));
	else
		tmp = (x / z) * (y_m / z);
	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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(y$95$m * N[(x / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 83.1

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

   4. Applied rewrites 83.1%

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

   5. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 80.6

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

   7. Applied rewrites 80.6%

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

   if -1 < z < 1

   1. Initial program 84.2%

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

   3. Taylor expanded in z around 0

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

      1. pow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

4. Final simplification 88.6%

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

Alternative 9: 82.3% accurate, 0.8× speedup

Math

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

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 (or (<= z -1.0) (not (<= z 1.0)))
    (* y_m (/ x (* (* z z) z)))
    (* y_m (/ x (* z z))))))

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 <= -1.0) || !(z <= 1.0)) {
		tmp = y_m * (x / ((z * z) * z));
	} else {
		tmp = y_m * (x / (z * z));
	}
	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 <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = y_m * (x / ((z * z) * z))
    else
        tmp = y_m * (x / (z * z))
    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 <= -1.0) || !(z <= 1.0)) {
		tmp = y_m * (x / ((z * z) * z));
	} else {
		tmp = y_m * (x / (z * z));
	}
	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 <= -1.0) or not (z <= 1.0):
		tmp = y_m * (x / ((z * z) * z))
	else:
		tmp = y_m * (x / (z * z))
	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 <= -1.0) || !(z <= 1.0))
		tmp = Float64(y_m * Float64(x / Float64(Float64(z * z) * z)));
	else
		tmp = Float64(y_m * Float64(x / Float64(z * z)));
	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 <= -1.0) || ~((z <= 1.0)))
		tmp = y_m * (x / ((z * z) * z));
	else
		tmp = y_m * (x / (z * z));
	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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(y$95$m * N[(x / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 83.1

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

   4. Applied rewrites 83.1%

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

   5. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 80.6

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

   7. Applied rewrites 80.6%

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

   if -1 < z < 1

   1. Initial program 84.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 84.1

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

   4. Applied rewrites 84.1%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 82.3%

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

4. Final simplification 81.6%

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

Alternative 10: 81.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y\_m \leq 2 \cdot 10^{-196}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \end{array} \end{array} \]

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 (<= (* x y_m) 2e-196)
    (* (/ x z) (/ y_m z))
    (* y_m (/ x (* (fma z z z) z))))))

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 ((x * y_m) <= 2e-196) {
		tmp = (x / z) * (y_m / z);
	} else {
		tmp = y_m * (x / (fma(z, z, z) * z));
	}
	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(x * y_m) <= 2e-196)
		tmp = Float64(Float64(x / z) * Float64(y_m / z));
	else
		tmp = Float64(y_m * Float64(x / Float64(fma(z, z, z) * z)));
	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[(x * y$95$m), $MachinePrecision], 2e-196], N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 2.0000000000000001e-196

   1. Initial program 80.8%

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

   3. Taylor expanded in z around 0

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

      1. pow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.1

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

   5. Applied rewrites 80.1%

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

   if 2.0000000000000001e-196 < (*.f64 x y)

   1. Initial program 85.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 87.6

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

   4. Applied rewrites 87.6%

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

Alternative 11: 93.8% accurate, 0.9× speedup

Math

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

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 (* (/ y_m z) (/ x (fma z z z)))))

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 * ((y_m / z) * (x / fma(z, z, z)));
}

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(Float64(y_m / z) * Float64(x / fma(z, z, z))))
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[(N[(y$95$m / z), $MachinePrecision] * N[(x / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. distribute-rgt-in N/A

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

   13. *-lft-identity N/A

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

   14. lower-fma.f64 93.3

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

4. Applied rewrites 93.3%

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

Alternative 12: 72.9% accurate, 1.4× speedup

Math

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

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 (* y_m (/ x (* z z)))))

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 * (y_m * (x / (z * z)));
}

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 * (y_m * (x / (z * z)))
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 * (y_m * (x / (z * z)));
}

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 * (y_m * (x / (z * z)))

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(y_m * Float64(x / Float64(z * z))))
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 * (y_m * (x / (z * z)));
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[(y$95$m * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. distribute-rgt-in N/A

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

   14. *-lft-identity N/A

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

   15. lower-fma.f64 83.7

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

4. Applied rewrites 83.7%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 73.9%

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

Alternative 13: 70.2% accurate, 1.4× speedup

Math

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

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 (* x (/ y_m (* z z)))))

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 * (x * (y_m / (z * z)));
}

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 * (x * (y_m / (z * z)))
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 * (x * (y_m / (z * z)));
}

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 * (x * (y_m / (z * z)))

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(x * Float64(y_m / Float64(z * z))))
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 * (x * (y_m / (z * z)));
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[(x * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

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

3. Taylor expanded in z around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 72.1

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

5. Applied rewrites 72.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 71.6

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

   6. associate-*l* 71.6

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

   7. *-commutative 71.6

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

   8. distribute-lft1-in 71.6

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

   9. lift-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-fma.f64 71.6

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

7. Applied rewrites 71.6%

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

Reproduce

herbie shell --seed 2025085 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z 2496182814532307/10000000000000) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z)))

  (/ (* x y) (* (* z z) (+ z 1.0))))