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

Percentage Accurate: 81.9% → 99.1%

Time: 4.2s

Alternatives: 15

Speedup: 0.9×

• 25.8% 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: 81.9% 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: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 5 \cdot 10^{-315}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{elif}\;x\_m \cdot y\_m \leq 2 \cdot 10^{+171}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{-z} \cdot \frac{-1}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{z} \cdot \frac{x\_m}{z - -1}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* x_m y_m) 5e-315)
     (* y_m (/ (/ x_m (fma z z z)) z))
     (if (<= (* x_m y_m) 2e+171)
       (* (/ (* y_m x_m) (- z)) (/ -1.0 (fma z z z)))
       (* (/ (/ y_m z) z) (/ x_m (- z -1.0))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((x_m * y_m) <= 5e-315) {
		tmp = y_m * ((x_m / fma(z, z, z)) / z);
	} else if ((x_m * y_m) <= 2e+171) {
		tmp = ((y_m * x_m) / -z) * (-1.0 / fma(z, z, z));
	} else {
		tmp = ((y_m / z) / z) * (x_m / (z - -1.0));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(x_m * y_m) <= 5e-315)
		tmp = Float64(y_m * Float64(Float64(x_m / fma(z, z, z)) / z));
	elseif (Float64(x_m * y_m) <= 2e+171)
		tmp = Float64(Float64(Float64(y_m * x_m) / Float64(-z)) * Float64(-1.0 / fma(z, z, z)));
	else
		tmp = Float64(Float64(Float64(y_m / z) / z) * Float64(x_m / Float64(z - -1.0)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 5e-315], N[(y$95$m * N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 2e+171], N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] / (-z)), $MachinePrecision] * N[(-1.0 / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision] * N[(x$95$m / N[(z - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 5.0000000023e-315

   1. Initial program 81.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 82.2

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

   4. Applied rewrites 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 91.0

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

   6. Applied rewrites 91.0%

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

   if 5.0000000023e-315 < (*.f64 x y) < 1.99999999999999991e171

   1. Initial program 91.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. 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 95.7

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

   4. Applied rewrites 95.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. pow2 N/A

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

      14. frac-2neg N/A

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

      15. mul-1-neg N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. distribute-lft1-in N/A

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

      20. distribute-lft-neg-out N/A

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

      21. times-frac N/A

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

      22. lower-*.f64 N/A

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

   6. Applied rewrites 99.7%

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

   if 1.99999999999999991e171 < (*.f64 x y)

   1. Initial program 67.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. pow2 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 96.3

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

   4. Applied rewrites 96.3%

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

Alternative 2: 95.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{y\_m}{z \cdot z}\\ t_1 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+53}:\\ \;\;\;\;\frac{t\_0}{z} \cdot x\_m\\ \mathbf{elif}\;t\_1 \leq 10^{-321}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+26}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ y_m (* z z))) (t_1 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 -2e+53)
       (* (/ t_0 z) x_m)
       (if (<= t_1 1e-321)
         (* (/ x_m z) (/ y_m z))
         (if (<= t_1 5e+26)
           (* y_m (/ x_m (* (fma z z z) z)))
           (* t_0 (/ x_m z)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m / (z * z);
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -2e+53) {
		tmp = (t_0 / z) * x_m;
	} else if (t_1 <= 1e-321) {
		tmp = (x_m / z) * (y_m / z);
	} else if (t_1 <= 5e+26) {
		tmp = y_m * (x_m / (fma(z, z, z) * z));
	} else {
		tmp = t_0 * (x_m / z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(y_m / Float64(z * z))
	t_1 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_1 <= -2e+53)
		tmp = Float64(Float64(t_0 / z) * x_m);
	elseif (t_1 <= 1e-321)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (t_1 <= 5e+26)
		tmp = Float64(y_m * Float64(x_m / Float64(fma(z, z, z) * z)));
	else
		tmp = Float64(t_0 * Float64(x_m / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, -2e+53], N[(N[(t$95$0 / z), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[t$95$1, 1e-321], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+26], N[(y$95$m * N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e53

   1. Initial program 80.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. 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 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l/ N/A

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

      5. distribute-lft1-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      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. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-fma.f64 83.6

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

   6. Applied rewrites 83.6%

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

   7. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 83.6

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

   9. Applied rewrites 83.6%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 89.2

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

   11. Applied rewrites 89.2%

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

   if -2e53 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 9.98013e-322

   1. Initial program 70.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

   if 9.98013e-322 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.0000000000000001e26

   1. Initial program 90.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 88.9

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

   4. Applied rewrites 88.9%

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

   if 5.0000000000000001e26 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 93.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. pow2 N/A

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

      14. associate-/l* 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. associate-/l/ N/A

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

      19. *-commutative N/A

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

      20. associate-*l/ N/A

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

      21. distribute-lft1-in N/A

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

      22. times-frac N/A

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

   6. Applied rewrites 98.0%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 98.0%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 96.7

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

   11. Applied rewrites 96.7%

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

Alternative 3: 95.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{z \cdot z} \cdot x\_m\\ \mathbf{elif}\;t\_0 \leq 10^{-321}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+26}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot z} \cdot \frac{x\_m}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 -2e+53)
       (* (/ (/ y_m z) (* z z)) x_m)
       (if (<= t_0 1e-321)
         (* (/ x_m z) (/ y_m z))
         (if (<= t_0 5e+26)
           (* y_m (/ x_m (* (fma z z z) z)))
           (* (/ y_m (* z z)) (/ x_m z)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if (t_0 <= -2e+53) {
		tmp = ((y_m / z) / (z * z)) * x_m;
	} else if (t_0 <= 1e-321) {
		tmp = (x_m / z) * (y_m / z);
	} else if (t_0 <= 5e+26) {
		tmp = y_m * (x_m / (fma(z, z, z) * z));
	} else {
		tmp = (y_m / (z * z)) * (x_m / z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_0 <= -2e+53)
		tmp = Float64(Float64(Float64(y_m / z) / Float64(z * z)) * x_m);
	elseif (t_0 <= 1e-321)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (t_0 <= 5e+26)
		tmp = Float64(y_m * Float64(x_m / Float64(fma(z, z, z) * z)));
	else
		tmp = Float64(Float64(y_m / Float64(z * z)) * Float64(x_m / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, -2e+53], N[(N[(N[(y$95$m / z), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-321], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+26], N[(y$95$m * N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e53

   1. Initial program 80.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. 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 89.2

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 89.2

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

   7. Applied rewrites 89.2%

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

   if -2e53 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 9.98013e-322

   1. Initial program 70.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

   if 9.98013e-322 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.0000000000000001e26

   1. Initial program 90.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 88.9

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

   4. Applied rewrites 88.9%

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

   if 5.0000000000000001e26 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 93.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. pow2 N/A

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

      14. associate-/l* 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. associate-/l/ N/A

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

      19. *-commutative N/A

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

      20. associate-*l/ N/A

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

      21. distribute-lft1-in N/A

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

      22. times-frac N/A

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

   6. Applied rewrites 98.0%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 98.0%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 96.7

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

   11. Applied rewrites 96.7%

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

Alternative 4: 91.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{y\_m}{\left(z \cdot z\right) \cdot z} \cdot x\_m\\ t_1 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-321}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (/ y_m (* (* z z) z)) x_m)) (t_1 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 -2e+53)
       t_0
       (if (<= t_1 1e-321)
         (* (/ x_m z) (/ y_m z))
         (if (<= t_1 5e-5) (* y_m (/ x_m (* z z))) t_0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (y_m / ((z * z) * z)) * x_m;
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -2e+53) {
		tmp = t_0;
	} else if (t_1 <= 1e-321) {
		tmp = (x_m / z) * (y_m / z);
	} else if (t_1 <= 5e-5) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, 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_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y_m / ((z * z) * z)) * x_m
    t_1 = (z * z) * (z + 1.0d0)
    if (t_1 <= (-2d+53)) then
        tmp = t_0
    else if (t_1 <= 1d-321) then
        tmp = (x_m / z) * (y_m / z)
    else if (t_1 <= 5d-5) then
        tmp = y_m * (x_m / (z * z))
    else
        tmp = t_0
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (y_m / ((z * z) * z)) * x_m;
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -2e+53) {
		tmp = t_0;
	} else if (t_1 <= 1e-321) {
		tmp = (x_m / z) * (y_m / z);
	} else if (t_1 <= 5e-5) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = (y_m / ((z * z) * z)) * x_m
	t_1 = (z * z) * (z + 1.0)
	tmp = 0
	if t_1 <= -2e+53:
		tmp = t_0
	elif t_1 <= 1e-321:
		tmp = (x_m / z) * (y_m / z)
	elif t_1 <= 5e-5:
		tmp = y_m * (x_m / (z * z))
	else:
		tmp = t_0
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(y_m / Float64(Float64(z * z) * z)) * x_m)
	t_1 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_1 <= -2e+53)
		tmp = t_0;
	elseif (t_1 <= 1e-321)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (t_1 <= 5e-5)
		tmp = Float64(y_m * Float64(x_m / Float64(z * z)));
	else
		tmp = t_0;
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = (y_m / ((z * z) * z)) * x_m;
	t_1 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if (t_1 <= -2e+53)
		tmp = t_0;
	elseif (t_1 <= 1e-321)
		tmp = (x_m / z) * (y_m / z);
	elseif (t_1 <= 5e-5)
		tmp = y_m * (x_m / (z * z));
	else
		tmp = t_0;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(y$95$m / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, -2e+53], t$95$0, If[LessEqual[t$95$1, 1e-321], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-5], N[(y$95$m * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e53 or 5.00000000000000024e-5 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 87.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. 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 92.3

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

   4. Applied rewrites 92.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l/ N/A

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

      5. distribute-lft1-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      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. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-fma.f64 88.1

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 86.9

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

   9. Applied rewrites 86.9%

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

   if -2e53 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 9.98013e-322

   1. Initial program 70.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

   if 9.98013e-322 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.00000000000000024e-5

   1. Initial program 89.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 88.3

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

   4. Applied rewrites 88.3%

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

   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 85.7%

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

Alternative 5: 96.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{y\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (/ (* x_m y_m) (* (* z z) (+ z 1.0))) 2e-42)
     (* (/ (/ y_m (fma z z z)) z) x_m)
     (* y_m (/ (/ x_m (fma z z z)) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 2e-42) {
		tmp = ((y_m / fma(z, z, z)) / z) * x_m;
	} else {
		tmp = y_m * ((x_m / fma(z, z, z)) / z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 2e-42)
		tmp = Float64(Float64(Float64(y_m / fma(z, z, z)) / z) * x_m);
	else
		tmp = Float64(y_m * Float64(Float64(x_m / fma(z, z, z)) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-42], N[(N[(N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision], N[(y$95$m * N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $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)))) < 2.00000000000000008e-42

   1. Initial program 90.0%

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

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l/ N/A

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

      5. distribute-lft1-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      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. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-fma.f64 93.0

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

   6. Applied rewrites 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 95.5

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

   8. Applied rewrites 95.5%

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

   if 2.00000000000000008e-42 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 65.2

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

   4. Applied rewrites 65.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 81.5

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

   6. Applied rewrites 81.5%

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

Alternative 6: 96.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{\mathsf{fma}\left(z, z, z\right)} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (/ (* x_m y_m) (* (* z z) (+ z 1.0))) 2e-42)
     (* (/ (/ y_m z) (fma z z z)) x_m)
     (* y_m (/ (/ x_m (fma z z z)) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 2e-42) {
		tmp = ((y_m / z) / fma(z, z, z)) * x_m;
	} else {
		tmp = y_m * ((x_m / fma(z, z, z)) / z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 2e-42)
		tmp = Float64(Float64(Float64(y_m / z) / fma(z, z, z)) * x_m);
	else
		tmp = Float64(y_m * Float64(Float64(x_m / fma(z, z, z)) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-42], N[(N[(N[(y$95$m / z), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(y$95$m * N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $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)))) < 2.00000000000000008e-42

   1. Initial program 90.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 2.00000000000000008e-42 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 65.2

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

   4. Applied rewrites 65.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 81.5

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

   6. Applied rewrites 81.5%

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

Alternative 7: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+53} \lor \neg \left(t\_0 \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{y\_m}{\left(z \cdot z\right) \cdot z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (*
     x_s
     (if (or (<= t_0 -2e+53) (not (<= t_0 5e-5)))
       (* (/ y_m (* (* z z) z)) x_m)
       (/ (* (/ x_m z) y_m) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -2e+53) || !(t_0 <= 5e-5)) {
		tmp = (y_m / ((z * z) * z)) * x_m;
	} else {
		tmp = ((x_m / z) * y_m) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, 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_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z * z) * (z + 1.0d0)
    if ((t_0 <= (-2d+53)) .or. (.not. (t_0 <= 5d-5))) then
        tmp = (y_m / ((z * z) * z)) * x_m
    else
        tmp = ((x_m / z) * y_m) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -2e+53) || !(t_0 <= 5e-5)) {
		tmp = (y_m / ((z * z) * z)) * x_m;
	} else {
		tmp = ((x_m / z) * y_m) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = (z * z) * (z + 1.0)
	tmp = 0
	if (t_0 <= -2e+53) or not (t_0 <= 5e-5):
		tmp = (y_m / ((z * z) * z)) * x_m
	else:
		tmp = ((x_m / z) * y_m) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if ((t_0 <= -2e+53) || !(t_0 <= 5e-5))
		tmp = Float64(Float64(y_m / Float64(Float64(z * z) * z)) * x_m);
	else
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if ((t_0 <= -2e+53) || ~((t_0 <= 5e-5)))
		tmp = (y_m / ((z * z) * z)) * x_m;
	else
		tmp = ((x_m / z) * y_m) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[Or[LessEqual[t$95$0, -2e+53], N[Not[LessEqual[t$95$0, 5e-5]], $MachinePrecision]], N[(N[(y$95$m / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e53 or 5.00000000000000024e-5 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 87.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. 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 92.3

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

   4. Applied rewrites 92.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l/ N/A

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

      5. distribute-lft1-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      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. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-fma.f64 88.1

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 86.9

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

   9. Applied rewrites 86.9%

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

   if -2e53 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.00000000000000024e-5

   1. Initial program 79.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 94.6

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

   5. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 93.1

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

   7. Applied rewrites 93.1%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -2 \cdot 10^{+53} \lor \neg \left(\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{y}{\left(z \cdot z\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+53}:\\ \;\;\;\;\frac{y\_m}{\left(z \cdot z\right) \cdot z} \cdot x\_m\\ \mathbf{elif}\;t\_0 \leq 10^{-321}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 -2e+53)
       (* (/ y_m (* (* z z) z)) x_m)
       (if (<= t_0 1e-321)
         (* (/ x_m z) (/ y_m z))
         (* y_m (/ x_m (* (fma z z z) z)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if (t_0 <= -2e+53) {
		tmp = (y_m / ((z * z) * z)) * x_m;
	} else if (t_0 <= 1e-321) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = y_m * (x_m / (fma(z, z, z) * z));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_0 <= -2e+53)
		tmp = Float64(Float64(y_m / Float64(Float64(z * z) * z)) * x_m);
	elseif (t_0 <= 1e-321)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(fma(z, z, z) * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, -2e+53], N[(N[(y$95$m / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-321], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e53

   1. Initial program 80.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. 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 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l/ N/A

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

      5. distribute-lft1-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      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. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-fma.f64 83.6

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

   6. Applied rewrites 83.6%

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

   7. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 83.6

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

   9. Applied rewrites 83.6%

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

   if -2e53 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 9.98013e-322

   1. Initial program 70.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

   if 9.98013e-322 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 92.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. *-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 89.9

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

   4. Applied rewrites 89.9%

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

Alternative 9: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+53} \lor \neg \left(t\_0 \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{y\_m}{\left(z \cdot z\right) \cdot z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (*
     x_s
     (if (or (<= t_0 -2e+53) (not (<= t_0 5e-5)))
       (* (/ y_m (* (* z z) z)) x_m)
       (* y_m (/ x_m (* z z))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -2e+53) || !(t_0 <= 5e-5)) {
		tmp = (y_m / ((z * z) * z)) * x_m;
	} else {
		tmp = y_m * (x_m / (z * z));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, 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_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z * z) * (z + 1.0d0)
    if ((t_0 <= (-2d+53)) .or. (.not. (t_0 <= 5d-5))) then
        tmp = (y_m / ((z * z) * z)) * x_m
    else
        tmp = y_m * (x_m / (z * z))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -2e+53) || !(t_0 <= 5e-5)) {
		tmp = (y_m / ((z * z) * z)) * x_m;
	} else {
		tmp = y_m * (x_m / (z * z));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = (z * z) * (z + 1.0)
	tmp = 0
	if (t_0 <= -2e+53) or not (t_0 <= 5e-5):
		tmp = (y_m / ((z * z) * z)) * x_m
	else:
		tmp = y_m * (x_m / (z * z))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if ((t_0 <= -2e+53) || !(t_0 <= 5e-5))
		tmp = Float64(Float64(y_m / Float64(Float64(z * z) * z)) * x_m);
	else
		tmp = Float64(y_m * Float64(x_m / Float64(z * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if ((t_0 <= -2e+53) || ~((t_0 <= 5e-5)))
		tmp = (y_m / ((z * z) * z)) * x_m;
	else
		tmp = y_m * (x_m / (z * z));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[Or[LessEqual[t$95$0, -2e+53], N[Not[LessEqual[t$95$0, 5e-5]], $MachinePrecision]], N[(N[(y$95$m / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e53 or 5.00000000000000024e-5 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 87.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. 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 92.3

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

   4. Applied rewrites 92.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l/ N/A

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

      5. distribute-lft1-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      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. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-fma.f64 88.1

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 86.9

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

   9. Applied rewrites 86.9%

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

   if -2e53 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.00000000000000024e-5

   1. Initial program 79.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 79.3

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

   4. Applied rewrites 79.3%

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

   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 78.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -2 \cdot 10^{+53} \lor \neg \left(\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{y}{\left(z \cdot z\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 10^{+48}:\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (/ (* x_m y_m) (* (* z z) (+ z 1.0))) 1e+48)
     (* (/ y_m (* (fma z z z) z)) x_m)
     (/ (* (/ x_m z) y_m) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 1e+48) {
		tmp = (y_m / (fma(z, z, z) * z)) * x_m;
	} else {
		tmp = ((x_m / z) * y_m) / z;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 1e+48)
		tmp = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x_m);
	else
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+48], N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision]]), $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)))) < 1.00000000000000004e48

   1. Initial program 90.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. 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 95.7

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

   4. Applied rewrites 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l/ N/A

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

      5. distribute-lft1-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      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. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-fma.f64 93.3

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

   6. Applied rewrites 93.3%

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

   if 1.00000000000000004e48 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 65.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 79.1

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

   5. Applied rewrites 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 80.3

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

   7. Applied rewrites 80.3%

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

Alternative 11: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{+24} \lor \neg \left(z \leq 160000000000\right):\\ \;\;\;\;\frac{\frac{y\_m}{z}}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (or (<= z -1.16e+24) (not (<= z 160000000000.0)))
     (* (/ (/ y_m z) z) (/ x_m z))
     (* y_m (/ (/ x_m (fma z z z)) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z <= -1.16e+24) || !(z <= 160000000000.0)) {
		tmp = ((y_m / z) / z) * (x_m / z);
	} else {
		tmp = y_m * ((x_m / fma(z, z, z)) / z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if ((z <= -1.16e+24) || !(z <= 160000000000.0))
		tmp = Float64(Float64(Float64(y_m / z) / z) * Float64(x_m / z));
	else
		tmp = Float64(y_m * Float64(Float64(x_m / fma(z, z, z)) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -1.16e+24], N[Not[LessEqual[z, 160000000000.0]], $MachinePrecision]], N[(N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.16000000000000005e24 or 1.6e11 < z

   1. Initial program 87.3%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. 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 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. pow2 N/A

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

      14. associate-/l* 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. associate-/l/ N/A

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

      19. *-commutative N/A

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

      20. associate-*l/ N/A

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

      21. distribute-lft1-in N/A

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

      22. times-frac N/A

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

   6. Applied rewrites 95.8%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 95.8%

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

   if -1.16000000000000005e24 < z < 1.6e11

   1. Initial program 79.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 80.1

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

   4. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 89.9

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

   6. Applied rewrites 89.9%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{+24} \lor \neg \left(z \leq 160000000000\right):\\ \;\;\;\;\frac{\frac{y}{z}}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.6%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
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. 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 96.6

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

4. Applied rewrites 96.6%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. frac-times N/A

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

   7. pow2 N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

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

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

   11. metadata-eval N/A

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

   12. associate-/r* N/A

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

   13. pow2 N/A

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

   14. associate-/l* 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. associate-/l/ N/A

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

   19. *-commutative N/A

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

   20. associate-*l/ N/A

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

   21. distribute-lft1-in N/A

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

   22. times-frac N/A

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

6. Applied rewrites 95.9%

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

Alternative 13: 94.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{y\_m}{z \cdot z}}{z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z -9.6e+23)
     (* (/ (/ y_m (* z z)) z) x_m)
     (* y_m (/ (/ x_m (fma z z z)) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= -9.6e+23) {
		tmp = ((y_m / (z * z)) / z) * x_m;
	} else {
		tmp = y_m * ((x_m / fma(z, z, z)) / z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= -9.6e+23)
		tmp = Float64(Float64(Float64(y_m / Float64(z * z)) / z) * x_m);
	else
		tmp = Float64(y_m * Float64(Float64(x_m / fma(z, z, z)) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, -9.6e+23], N[(N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision], N[(y$95$m * N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -9.6e23

   1. Initial program 81.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. 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 88.9

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

   4. Applied rewrites 88.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l/ N/A

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

      5. distribute-lft1-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      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. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-fma.f64 83.1

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

   6. Applied rewrites 83.1%

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

   7. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 83.1

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

   9. Applied rewrites 83.1%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 88.9

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

   11. Applied rewrites 88.9%

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

   if -9.6e23 < z

   1. Initial program 84.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 91.7

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

   6. Applied rewrites 91.7%

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

Alternative 14: 94.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.6%

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

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. distribute-rgt-in N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 94.0

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

4. Applied rewrites 94.0%

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

Alternative 15: 74.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.6%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
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 83.0

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

4. Applied rewrites 83.0%

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

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 72.3%

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

Developer Target 1: 96.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z 249.6182814532307)
   (/ (* y (/ x z)) (+ z (* z z)))
   (/ (* (/ (/ y z) (+ 1.0 z)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (z < 249.6182814532307d0) then
        tmp = (y * (x / z)) / (z + (z * z))
    else
        tmp = (((y / z) / (1.0d0 + z)) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < 249.6182814532307:
		tmp = (y * (x / z)) / (z + (z * z))
	else:
		tmp = (((y / z) / (1.0 + z)) * x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < 249.6182814532307)
		tmp = Float64(Float64(y * Float64(x / z)) / Float64(z + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) / Float64(1.0 + z)) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < 249.6182814532307)
		tmp = (y * (x / z)) / (z + (z * z));
	else
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, 249.6182814532307], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(z + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]

Reproduce

herbie shell --seed 2025056 
(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))))