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

Percentage Accurate: 83.2% → 98.3%

Time: 3.6s

Alternatives: 13

Speedup: 1.0×

• 25.2% 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: 83.2% 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: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{x\_m}{z} \cdot \frac{\frac{y\_m}{z}}{z}\\ t_1 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-218}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+215}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 (* (/ x_m z) (/ (/ y_m z) z))) (t_1 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 -1e+19)
       t_0
       (if (<= t_1 2e-218)
         (/ (* (/ x_m z) y_m) z)
         (if (<= t_1 5e+215) (/ (* y_m x_m) (* (fma z z 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 = (x_m / z) * ((y_m / z) / z);
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -1e+19) {
		tmp = t_0;
	} else if (t_1 <= 2e-218) {
		tmp = ((x_m / z) * y_m) / z;
	} else if (t_1 <= 5e+215) {
		tmp = (y_m * x_m) / (fma(z, z, 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(x_m / z) * Float64(Float64(y_m / z) / z))
	t_1 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_1 <= -1e+19)
		tmp = t_0;
	elseif (t_1 <= 2e-218)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / z);
	elseif (t_1 <= 5e+215)
		tmp = Float64(Float64(y_m * x_m) / Float64(fma(z, z, z) * z));
	else
		tmp = t_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_] := Block[{t$95$0 = N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / 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, -1e+19], t$95$0, If[LessEqual[t$95$1, 2e-218], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5e+215], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z * z + z), $MachinePrecision] * z), $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))) < -1e19 or 5.0000000000000001e215 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.2%

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

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

      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

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

      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-lft-in N/A

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

      12. pow2 N/A

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

      13. *-rgt-identity N/A

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

      14. pow2 N/A

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

      15. metadata-eval N/A

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

      16. unswap-sqr N/A

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

      17. *-commutative N/A

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

      18. *-rgt-identity N/A

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

      19. unswap-sqr N/A

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

      20. pow2 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

      23. pow2 N/A

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

      24. *-rgt-identity N/A

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

      25. *-lft-identity N/A

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

      26. lower-fma.f64 95.0

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

   3. Applied rewrites 95.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. times-frac N/A

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

      8. associate-/r* N/A

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

      9. associate-/l* N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 62.1%

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

   if -1e19 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 2.0000000000000001e-218

   1. Initial program 83.2%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      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-lft-in N/A

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

      14. pow2 N/A

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

      15. *-rgt-identity N/A

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

      16. pow2 N/A

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

      17. metadata-eval N/A

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

      18. unswap-sqr N/A

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

      19. *-commutative N/A

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

      20. *-rgt-identity N/A

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

      21. unswap-sqr N/A

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

      22. pow2 N/A

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

      23. metadata-eval N/A

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

      24. *-rgt-identity N/A

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

      25. pow2 N/A

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

      26. *-rgt-identity N/A

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

      27. *-lft-identity N/A

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

      28. lower-fma.f64 85.3

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

   3. Applied rewrites 85.3%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 75.4

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

   8. Applied rewrites 75.4%

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

   if 2.0000000000000001e-218 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.0000000000000001e215

   1. Initial program 83.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 83.2

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. pow2 N/A

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

      12. *-rgt-identity N/A

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

      13. pow2 N/A

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

      14. metadata-eval N/A

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

      15. unswap-sqr N/A

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

      16. *-commutative N/A

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

      17. *-rgt-identity N/A

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

      18. unswap-sqr N/A

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

      19. pow2 N/A

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

      20. metadata-eval N/A

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

      21. *-rgt-identity N/A

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

      22. pow2 N/A

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

      23. *-rgt-identity N/A

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

      24. *-lft-identity N/A

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

      25. lower-fma.f64 83.2

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

   3. Applied rewrites 83.2%

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

Alternative 2: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{x\_m}{z} \cdot \frac{\frac{y\_m}{z}}{z}\\ t_1 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-91}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+215}:\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 (* (/ x_m z) (/ (/ y_m z) z))) (t_1 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 -1e+19)
       t_0
       (if (<= t_1 1e-91)
         (/ (* (/ x_m z) y_m) z)
         (if (<= t_1 5e+215) (* (/ y_m (* (fma z z z) z)) x_m) 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 = (x_m / z) * ((y_m / z) / z);
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -1e+19) {
		tmp = t_0;
	} else if (t_1 <= 1e-91) {
		tmp = ((x_m / z) * y_m) / z;
	} else if (t_1 <= 5e+215) {
		tmp = (y_m / (fma(z, z, z) * z)) * x_m;
	} 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(x_m / z) * Float64(Float64(y_m / z) / z))
	t_1 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_1 <= -1e+19)
		tmp = t_0;
	elseif (t_1 <= 1e-91)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / z);
	elseif (t_1 <= 5e+215)
		tmp = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x_m);
	else
		tmp = t_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_] := Block[{t$95$0 = N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / 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, -1e+19], t$95$0, If[LessEqual[t$95$1, 1e-91], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5e+215], N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $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))) < -1e19 or 5.0000000000000001e215 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.2%

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

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

      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

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

      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-lft-in N/A

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

      12. pow2 N/A

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

      13. *-rgt-identity N/A

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

      14. pow2 N/A

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

      15. metadata-eval N/A

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

      16. unswap-sqr N/A

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

      17. *-commutative N/A

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

      18. *-rgt-identity N/A

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

      19. unswap-sqr N/A

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

      20. pow2 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

      23. pow2 N/A

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

      24. *-rgt-identity N/A

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

      25. *-lft-identity N/A

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

      26. lower-fma.f64 95.0

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

   3. Applied rewrites 95.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. times-frac N/A

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

      8. associate-/r* N/A

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

      9. associate-/l* N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 62.1%

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

   if -1e19 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.00000000000000002e-91

   1. Initial program 83.2%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      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-lft-in N/A

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

      14. pow2 N/A

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

      15. *-rgt-identity N/A

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

      16. pow2 N/A

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

      17. metadata-eval N/A

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

      18. unswap-sqr N/A

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

      19. *-commutative N/A

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

      20. *-rgt-identity N/A

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

      21. unswap-sqr N/A

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

      22. pow2 N/A

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

      23. metadata-eval N/A

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

      24. *-rgt-identity N/A

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

      25. pow2 N/A

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

      26. *-rgt-identity N/A

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

      27. *-lft-identity N/A

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

      28. lower-fma.f64 85.3

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

   3. Applied rewrites 85.3%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 75.4

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

   8. Applied rewrites 75.4%

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

   if 1.00000000000000002e-91 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.0000000000000001e215

   1. Initial program 83.2%

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

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

      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

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

      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

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

   3. Applied rewrites 83.5%

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

Alternative 3: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-139}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{\frac{y\_m}{z}}{1 + z}\\ \end{array}\right) \end{array} \]

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 1e-139)
     (/ (* (/ x_m z) y_m) (fma z z z))
     (* (/ x_m z) (/ (/ y_m z) (+ 1.0 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 <= 1e-139) {
		tmp = ((x_m / z) * y_m) / fma(z, z, z);
	} else {
		tmp = (x_m / z) * ((y_m / z) / (1.0 + 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 (x_m <= 1e-139)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / fma(z, z, z));
	else
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_m / z) / Float64(1.0 + 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[x$95$m, 1e-139], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000003e-139

   1. Initial program 83.2%

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

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

      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

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

      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-lft-in N/A

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

      12. pow2 N/A

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

      13. *-rgt-identity N/A

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

      14. pow2 N/A

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

      15. metadata-eval N/A

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

      16. unswap-sqr N/A

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

      17. *-commutative N/A

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

      18. *-rgt-identity N/A

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

      19. unswap-sqr N/A

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

      20. pow2 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

      23. pow2 N/A

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

      24. *-rgt-identity N/A

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

      25. *-lft-identity N/A

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

      26. lower-fma.f64 95.0

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

   3. Applied rewrites 95.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-fma.f64 95.1

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

   5. Applied rewrites 95.1%

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

   if 1.00000000000000003e-139 < x

   1. Initial program 83.2%

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

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

      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

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

      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-lft-in N/A

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

      12. pow2 N/A

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

      13. *-rgt-identity N/A

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

      14. pow2 N/A

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

      15. metadata-eval N/A

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

      16. unswap-sqr N/A

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

      17. *-commutative N/A

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

      18. *-rgt-identity N/A

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

      19. unswap-sqr N/A

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

      20. pow2 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

      23. pow2 N/A

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

      24. *-rgt-identity N/A

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

      25. *-lft-identity N/A

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

      26. lower-fma.f64 95.0

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

   3. Applied rewrites 95.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. times-frac N/A

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

      8. associate-/r* N/A

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

      9. associate-/l* N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

Alternative 4: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{\frac{y\_m}{z}}{z}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{y\_m}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \end{array}\right) \end{array} \end{array} \]

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 -1e+19)
       (* (/ x_m z) (/ (/ y_m z) z))
       (if (<= t_0 2e-161)
         (/ (* (/ x_m z) y_m) z)
         (/ (* x_m (/ y_m z)) (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) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if (t_0 <= -1e+19) {
		tmp = (x_m / z) * ((y_m / z) / z);
	} else if (t_0 <= 2e-161) {
		tmp = ((x_m / z) * y_m) / z;
	} else {
		tmp = (x_m * (y_m / z)) / fma(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 <= -1e+19)
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_m / z) / z));
	elseif (t_0 <= 2e-161)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / z);
	else
		tmp = Float64(Float64(x_m * Float64(y_m / z)) / fma(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, -1e+19], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-161], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1e19

   1. Initial program 83.2%

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

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

      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

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

      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-lft-in N/A

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

      12. pow2 N/A

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

      13. *-rgt-identity N/A

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

      14. pow2 N/A

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

      15. metadata-eval N/A

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

      16. unswap-sqr N/A

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

      17. *-commutative N/A

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

      18. *-rgt-identity N/A

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

      19. unswap-sqr N/A

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

      20. pow2 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

      23. pow2 N/A

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

      24. *-rgt-identity N/A

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

      25. *-lft-identity N/A

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

      26. lower-fma.f64 95.0

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

   3. Applied rewrites 95.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. times-frac N/A

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

      8. associate-/r* N/A

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

      9. associate-/l* N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 62.1%

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

   if -1e19 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 2.00000000000000006e-161

   1. Initial program 83.2%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      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-lft-in N/A

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

      14. pow2 N/A

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

      15. *-rgt-identity N/A

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

      16. pow2 N/A

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

      17. metadata-eval N/A

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

      18. unswap-sqr N/A

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

      19. *-commutative N/A

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

      20. *-rgt-identity N/A

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

      21. unswap-sqr N/A

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

      22. pow2 N/A

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

      23. metadata-eval N/A

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

      24. *-rgt-identity N/A

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

      25. pow2 N/A

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

      26. *-rgt-identity N/A

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

      27. *-lft-identity N/A

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

      28. lower-fma.f64 85.3

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

   3. Applied rewrites 85.3%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 75.4

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

   8. Applied rewrites 75.4%

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

   if 2.00000000000000006e-161 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.2%

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

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

      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

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

      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-lft-in N/A

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

      12. pow2 N/A

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

      13. *-rgt-identity N/A

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

      14. pow2 N/A

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

      15. metadata-eval N/A

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

      16. unswap-sqr N/A

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

      17. *-commutative N/A

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

      18. *-rgt-identity N/A

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

      19. unswap-sqr N/A

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

      20. pow2 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

      23. pow2 N/A

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

      24. *-rgt-identity N/A

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

      25. *-lft-identity N/A

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

      26. lower-fma.f64 95.0

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

   3. Applied rewrites 95.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. times-frac N/A

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

      8. associate-/r* N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 93.6

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

   5. Applied rewrites 93.6%

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

Alternative 5: 96.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 1.5 \cdot 10^{-278}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+215}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{\frac{y\_m}{z}}{z}\\ \end{array}\right) \end{array} \end{array} \]

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 1.5e-278)
       (* (/ x_m z) (/ y_m (fma z z z)))
       (if (<= t_0 5e+215)
         (/ (* y_m x_m) (* (fma z z z) z))
         (* (/ x_m z) (/ (/ y_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 <= 1.5e-278) {
		tmp = (x_m / z) * (y_m / fma(z, z, z));
	} else if (t_0 <= 5e+215) {
		tmp = (y_m * x_m) / (fma(z, z, z) * z);
	} else {
		tmp = (x_m / z) * ((y_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 <= 1.5e-278)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / fma(z, z, z)));
	elseif (t_0 <= 5e+215)
		tmp = Float64(Float64(y_m * x_m) / Float64(fma(z, z, z) * z));
	else
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_m / 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, 1.5e-278], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+215], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.5e-278

   1. Initial program 83.2%

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

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

      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

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

      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-lft-in N/A

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

      12. pow2 N/A

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

      13. *-rgt-identity N/A

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

      14. pow2 N/A

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

      15. metadata-eval N/A

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

      16. unswap-sqr N/A

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

      17. *-commutative N/A

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

      18. *-rgt-identity N/A

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

      19. unswap-sqr N/A

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

      20. pow2 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

      23. pow2 N/A

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

      24. *-rgt-identity N/A

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

      25. *-lft-identity N/A

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

      26. lower-fma.f64 95.0

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

   3. Applied rewrites 95.0%

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

   if 1.5e-278 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.0000000000000001e215

   1. Initial program 83.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 83.2

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. pow2 N/A

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

      12. *-rgt-identity N/A

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

      13. pow2 N/A

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

      14. metadata-eval N/A

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

      15. unswap-sqr N/A

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

      16. *-commutative N/A

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

      17. *-rgt-identity N/A

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

      18. unswap-sqr N/A

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

      19. pow2 N/A

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

      20. metadata-eval N/A

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

      21. *-rgt-identity N/A

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

      22. pow2 N/A

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

      23. *-rgt-identity N/A

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

      24. *-lft-identity N/A

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

      25. lower-fma.f64 83.2

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

   3. Applied rewrites 83.2%

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

   if 5.0000000000000001e215 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.2%

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

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

      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

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

      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-lft-in N/A

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

      12. pow2 N/A

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

      13. *-rgt-identity N/A

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

      14. pow2 N/A

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

      15. metadata-eval N/A

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

      16. unswap-sqr N/A

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

      17. *-commutative N/A

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

      18. *-rgt-identity N/A

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

      19. unswap-sqr N/A

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

      20. pow2 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

      23. pow2 N/A

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

      24. *-rgt-identity N/A

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

      25. *-lft-identity N/A

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

      26. lower-fma.f64 95.0

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

   3. Applied rewrites 95.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. times-frac N/A

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

      8. associate-/r* N/A

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

      9. associate-/l* N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 62.1%

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

Alternative 6: 95.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{x\_m}{z} \cdot \frac{\frac{y\_m}{z}}{z}\\ t_1 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 (* (/ x_m z) (/ (/ y_m z) z))) (t_1 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 -1e+19)
       t_0
       (if (<= t_1 5e-9) (/ (* (/ x_m z) y_m) 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 = (x_m / z) * ((y_m / z) / z);
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -1e+19) {
		tmp = t_0;
	} else if (t_1 <= 5e-9) {
		tmp = ((x_m / z) * y_m) / 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 = (x_m / z) * ((y_m / z) / z)
    t_1 = (z * z) * (z + 1.0d0)
    if (t_1 <= (-1d+19)) then
        tmp = t_0
    else if (t_1 <= 5d-9) then
        tmp = ((x_m / z) * y_m) / 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 = (x_m / z) * ((y_m / z) / z);
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -1e+19) {
		tmp = t_0;
	} else if (t_1 <= 5e-9) {
		tmp = ((x_m / z) * y_m) / 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 = (x_m / z) * ((y_m / z) / z)
	t_1 = (z * z) * (z + 1.0)
	tmp = 0
	if t_1 <= -1e+19:
		tmp = t_0
	elif t_1 <= 5e-9:
		tmp = ((x_m / z) * y_m) / 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(x_m / z) * Float64(Float64(y_m / z) / z))
	t_1 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_1 <= -1e+19)
		tmp = t_0;
	elseif (t_1 <= 5e-9)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / 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 = (x_m / z) * ((y_m / z) / z);
	t_1 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if (t_1 <= -1e+19)
		tmp = t_0;
	elseif (t_1 <= 5e-9)
		tmp = ((x_m / z) * y_m) / 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[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / 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, -1e+19], t$95$0, If[LessEqual[t$95$1, 5e-9], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1e19 or 5.0000000000000001e-9 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.2%

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

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

      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

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

      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-lft-in N/A

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

      12. pow2 N/A

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

      13. *-rgt-identity N/A

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

      14. pow2 N/A

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

      15. metadata-eval N/A

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

      16. unswap-sqr N/A

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

      17. *-commutative N/A

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

      18. *-rgt-identity N/A

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

      19. unswap-sqr N/A

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

      20. pow2 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

      23. pow2 N/A

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

      24. *-rgt-identity N/A

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

      25. *-lft-identity N/A

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

      26. lower-fma.f64 95.0

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

   3. Applied rewrites 95.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. times-frac N/A

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

      8. associate-/r* N/A

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

      9. associate-/l* N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 62.1%

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

   if -1e19 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.0000000000000001e-9

   1. Initial program 83.2%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      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-lft-in N/A

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

      14. pow2 N/A

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

      15. *-rgt-identity N/A

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

      16. pow2 N/A

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

      17. metadata-eval N/A

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

      18. unswap-sqr N/A

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

      19. *-commutative N/A

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

      20. *-rgt-identity N/A

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

      21. unswap-sqr N/A

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

      22. pow2 N/A

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

      23. metadata-eval N/A

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

      24. *-rgt-identity N/A

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

      25. pow2 N/A

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

      26. *-rgt-identity N/A

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

      27. *-lft-identity N/A

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

      28. lower-fma.f64 85.3

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

   3. Applied rewrites 85.3%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 75.4

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

   8. Applied rewrites 75.4%

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

Alternative 7: 95.1% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 83.2%

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

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

   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

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

   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-lft-in N/A

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

   12. pow2 N/A

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

   13. *-rgt-identity N/A

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

   14. pow2 N/A

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

   15. metadata-eval N/A

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

   16. unswap-sqr N/A

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

   17. *-commutative N/A

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

   18. *-rgt-identity N/A

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

   19. unswap-sqr N/A

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

   20. pow2 N/A

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

   21. metadata-eval N/A

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

   22. *-rgt-identity N/A

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

   23. pow2 N/A

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

   24. *-rgt-identity N/A

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

   25. *-lft-identity N/A

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

   26. lower-fma.f64 95.0

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

3. Applied rewrites 95.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-fma.f64 95.1

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

5. Applied rewrites 95.1%

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

Alternative 8: 95.0% accurate, 0.8× speedup

Math

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

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)))
   (*
    y_s
    (* x_s (if (<= z -1.0) t_0 (if (<= z 1.0) (/ (* (/ x_m z) y_m) 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 tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = ((x_m / z) * y_m) / 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) :: tmp
    t_0 = ((y_m / (z * z)) / z) * x_m
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = ((x_m / z) * y_m) / 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 tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = ((x_m / z) * y_m) / 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
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 1.0:
		tmp = ((x_m / z) * y_m) / 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(Float64(y_m / Float64(z * z)) / z) * x_m)
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / 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;
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = ((x_m / z) * y_m) / 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[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 1.0], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 83.2%

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

   2. Taylor expanded in z around inf

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

      1. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{3}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{{z}^{3}} \cdot \color{blue}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y}{{z}^{3}} \cdot \color{blue}{x} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{y}{{z}^{3}} \cdot x \]

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow3 N/A

         \[\leadsto \frac{y}{{z}^{3}} \cdot x \]

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 60.3

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

   6. Applied rewrites 60.3%

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

   if -1 < z < 1

   1. Initial program 83.2%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      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-lft-in N/A

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

      14. pow2 N/A

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

      15. *-rgt-identity N/A

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

      16. pow2 N/A

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

      17. metadata-eval N/A

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

      18. unswap-sqr N/A

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

      19. *-commutative N/A

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

      20. *-rgt-identity N/A

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

      21. unswap-sqr N/A

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

      22. pow2 N/A

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

      23. metadata-eval N/A

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

      24. *-rgt-identity N/A

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

      25. pow2 N/A

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

      26. *-rgt-identity N/A

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

      27. *-lft-identity N/A

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

      28. lower-fma.f64 85.3

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

   3. Applied rewrites 85.3%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 75.4

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

   8. Applied rewrites 75.4%

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

Alternative 9: 93.1% accurate, 0.8× speedup

Math

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

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) (/ x_m (* z z)))))
   (*
    y_s
    (* x_s (if (<= z -1.0) t_0 (if (<= z 1.0) (/ (* (/ x_m z) y_m) 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) * (x_m / (z * z));
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = ((x_m / z) * y_m) / 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) :: tmp
    t_0 = (y_m / z) * (x_m / (z * z))
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = ((x_m / z) * y_m) / 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) * (x_m / (z * z));
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = ((x_m / z) * y_m) / 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) * (x_m / (z * z))
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 1.0:
		tmp = ((x_m / z) * y_m) / 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 / z) * Float64(x_m / Float64(z * z)))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / 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) * (x_m / (z * z));
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = ((x_m / z) * y_m) / 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 / z), $MachinePrecision] * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 1.0], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 83.2%

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

   2. Taylor expanded in z around inf

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

      1. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{3}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{{z}^{3}} \cdot \color{blue}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y}{{z}^{3}} \cdot \color{blue}{x} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{y}{{z}^{3}} \cdot x \]

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

         \[\leadsto \frac{y}{{z}^{3}} \cdot x \]

      6. associate-*l/ N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{{z}^{3}}} \]

      7. cube-mult N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 58.3

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

   6. Applied rewrites 58.3%

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

   if -1 < z < 1

   1. Initial program 83.2%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      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-lft-in N/A

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

      14. pow2 N/A

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

      15. *-rgt-identity N/A

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

      16. pow2 N/A

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

      17. metadata-eval N/A

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

      18. unswap-sqr N/A

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

      19. *-commutative N/A

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

      20. *-rgt-identity N/A

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

      21. unswap-sqr N/A

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

      22. pow2 N/A

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

      23. metadata-eval N/A

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

      24. *-rgt-identity N/A

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

      25. pow2 N/A

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

      26. *-rgt-identity N/A

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

      27. *-lft-identity N/A

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

      28. lower-fma.f64 85.3

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

   3. Applied rewrites 85.3%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 75.4

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

   8. Applied rewrites 75.4%

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

Alternative 10: 91.9% accurate, 0.8× speedup

Math

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

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)))
   (*
    y_s
    (* x_s (if (<= z -1.0) t_0 (if (<= z 1.0) (/ (* (/ x_m z) y_m) 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 tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = ((x_m / z) * y_m) / 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) :: tmp
    t_0 = (y_m / ((z * z) * z)) * x_m
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = ((x_m / z) * y_m) / 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 tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = ((x_m / z) * y_m) / 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
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 1.0:
		tmp = ((x_m / z) * y_m) / 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)
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / 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;
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = ((x_m / z) * y_m) / 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]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 1.0], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 83.2%

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

   2. Taylor expanded in z around inf

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

      1. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{3}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{{z}^{3}} \cdot \color{blue}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y}{{z}^{3}} \cdot \color{blue}{x} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{y}{{z}^{3}} \cdot x \]

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 57.1

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

   4. Applied rewrites 57.1%

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

   if -1 < z < 1

   1. Initial program 83.2%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      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-lft-in N/A

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

      14. pow2 N/A

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

      15. *-rgt-identity N/A

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

      16. pow2 N/A

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

      17. metadata-eval N/A

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

      18. unswap-sqr N/A

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

      19. *-commutative N/A

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

      20. *-rgt-identity N/A

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

      21. unswap-sqr N/A

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

      22. pow2 N/A

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

      23. metadata-eval N/A

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

      24. *-rgt-identity N/A

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

      25. pow2 N/A

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

      26. *-rgt-identity N/A

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

      27. *-lft-identity N/A

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

      28. lower-fma.f64 85.3

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

   3. Applied rewrites 85.3%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 75.4

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

   8. Applied rewrites 75.4%

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

Alternative 11: 80.3% accurate, 1.5× speedup

Math

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

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(Float64(x_m / 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[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.2%

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

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   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-lft-in N/A

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

   14. pow2 N/A

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

   15. *-rgt-identity N/A

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

   16. pow2 N/A

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

   17. metadata-eval N/A

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

   18. unswap-sqr N/A

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

   19. *-commutative N/A

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

   20. *-rgt-identity N/A

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

   21. unswap-sqr N/A

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

   22. pow2 N/A

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

   23. metadata-eval N/A

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

   24. *-rgt-identity N/A

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

   25. pow2 N/A

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

   26. *-rgt-identity N/A

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

   27. *-lft-identity N/A

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

   28. lower-fma.f64 85.3

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

3. Applied rewrites 85.3%

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

4. Taylor expanded in z around 0

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

6. Applied rewrites 75.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 80.3

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

8. Applied rewrites 80.3%

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

Alternative 12: 75.3% accurate, 1.6× speedup

Math

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

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 83.2%

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

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   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-lft-in N/A

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

   14. pow2 N/A

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

   15. *-rgt-identity N/A

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

   16. pow2 N/A

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

   17. metadata-eval N/A

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

   18. unswap-sqr N/A

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

   19. *-commutative N/A

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

   20. *-rgt-identity N/A

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

   21. unswap-sqr N/A

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

   22. pow2 N/A

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

   23. metadata-eval N/A

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

   24. *-rgt-identity N/A

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

   25. pow2 N/A

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

   26. *-rgt-identity N/A

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

   27. *-lft-identity N/A

       \[\leadsto y \cdot \frac{x}{\left(z \cdot z + \color{blue}{z}\right) \cdot z} \]

   28. lower-fma.f64 85.3

       \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(z, z, z\right)} \cdot z} \]

3. Applied rewrites 85.3%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z}} \]

4. Taylor expanded in z around 0

   \[\leadsto y \cdot \frac{x}{\color{blue}{z} \cdot z} \]
5. Step-by-step derivation

6. Applied rewrites 75.3%

   \[\leadsto y \cdot \frac{x}{\color{blue}{z} \cdot z} \]
7. Add Preprocessing

Alternative 13: 69.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \left(\frac{y\_m}{z \cdot z} \cdot x\_m\right)\right) \end{array} \]

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)) x_m))))

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)) * x_m));
}

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)) * x_m))
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)) * x_m));
}

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)) * x_m))

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(y_m / Float64(z * z)) * x_m)))
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)) * x_m));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.2%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
3. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{2}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{y}{{z}^{2}} \cdot \color{blue}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{y}{{z}^{2}} \cdot \color{blue}{x} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{y}{{z}^{2}} \cdot x \]

   5. pow2 N/A

      \[\leadsto \frac{y}{z \cdot z} \cdot x \]

   6. lift-*.f64 69.2

      \[\leadsto \frac{y}{z \cdot z} \cdot x \]

4. Applied rewrites 69.2%

   \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025123 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x y) (* (* z z) (+ z 1.0))))