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

Percentage Accurate: 83.3% → 96.8%

Time: 2.6s

Alternatives: 11

Speedup: 1.0×

• 26.1% 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: 83.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   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. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 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

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

   10. lift-+.f64 N/A

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

   11. distribute-lft-in N/A

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

   12. *-rgt-identity N/A

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

   13. lower-fma.f64 94.5

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

3. Applied rewrites 94.5%

   \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]
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. frac-times N/A

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

   5. *-commutative N/A

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

   6. frac-times N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lift-fma.f64 N/A

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

   10. distribute-lft1-in N/A

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

   11. add-flip N/A

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

   12. metadata-eval N/A

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

   13. lift--.f64 N/A

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

   14. times-frac N/A

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

   15. lift-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 96.7

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

5. Applied rewrites 96.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-flip N/A

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

   5. mult-flip-rev N/A

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

   6. lower-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 96.8

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

7. Applied rewrites 96.8%

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

Alternative 2: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 2 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{\mathsf{fma}\left(z, z, z\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)} \cdot y\_m}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* x y_m) (* (* z z) (+ z 1.0))) 2e-158)
    (* (/ (/ y_m z) (fma z z z)) x)
    (/ (* (/ x (fma z z z)) y_m) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((x * y_m) / ((z * z) * (z + 1.0))) <= 2e-158) {
		tmp = ((y_m / z) / fma(z, z, z)) * x;
	} else {
		tmp = ((x / fma(z, z, z)) * y_m) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 2e-158)
		tmp = Float64(Float64(Float64(y_m / z) / fma(z, z, z)) * x);
	else
		tmp = Float64(Float64(Float64(x / fma(z, z, z)) * y_m) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(x * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-158], N[(N[(N[(y$95$m / z), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(x / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 2.00000000000000013e-158

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 84.2

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 84.2

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

   3. Applied rewrites 84.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-/.f64 89.4

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

   5. Applied rewrites 89.4%

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

   if 2.00000000000000013e-158 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 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

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

      10. lift-+.f64 N/A

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

      11. distribute-lft-in N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 94.5

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

   3. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]
   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. frac-times N/A

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

      5. *-commutative N/A

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

      6. frac-times N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 94.5

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

   5. Applied rewrites 94.5%

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

Alternative 3: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   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. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 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

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

   10. lift-+.f64 N/A

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

   11. distribute-lft-in N/A

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

   12. *-rgt-identity N/A

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

   13. lower-fma.f64 94.5

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

3. Applied rewrites 94.5%

   \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]
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. frac-times N/A

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

   5. *-commutative N/A

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

   6. frac-times N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lift-fma.f64 N/A

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

   10. distribute-lft1-in N/A

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

   11. add-flip N/A

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

   12. metadata-eval N/A

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

   13. lift--.f64 N/A

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

   14. times-frac N/A

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

   15. lift-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 96.7

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

5. Applied rewrites 96.7%

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

Alternative 4: 94.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   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. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 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

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

   10. lift-+.f64 N/A

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

   11. distribute-lft-in N/A

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

   12. *-rgt-identity N/A

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

   13. lower-fma.f64 94.5

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

3. Applied rewrites 94.5%

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

Alternative 5: 91.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, z, z\right) \cdot z\\ t_1 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -400:\\ \;\;\;\;\frac{y\_m}{t\_0} \cdot x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{x}{1 \cdot z} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x}{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (fma z z z) z)) (t_1 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (if (<= t_1 -400.0)
      (* (/ y_m t_0) x)
      (if (<= t_1 0.0) (/ (* (/ x (* 1.0 z)) y_m) z) (* y_m (/ x t_0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = fma(z, z, z) * z;
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -400.0) {
		tmp = (y_m / t_0) * x;
	} else if (t_1 <= 0.0) {
		tmp = ((x / (1.0 * z)) * y_m) / z;
	} else {
		tmp = y_m * (x / t_0);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	t_0 = Float64(fma(z, z, z) * z)
	t_1 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_1 <= -400.0)
		tmp = Float64(Float64(y_m / t_0) * x);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(x / Float64(1.0 * z)) * y_m) / z);
	else
		tmp = Float64(y_m * Float64(x / t_0));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, -400.0], N[(N[(y$95$m / t$95$0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(x / N[(1.0 * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -400

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 84.2

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 84.2

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

   3. Applied rewrites 84.2%

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

   if -400 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0

   1. Initial program 83.3%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 70.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.8

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.8

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

   6. Applied rewrites 73.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.9

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

   8. Applied rewrites 74.9%

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

   if 0.0 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 85.2

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 85.2

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

   3. Applied rewrites 85.2%

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

Alternative 6: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 10^{+277}:\\ \;\;\;\;y\_m \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 \cdot z} \cdot y\_m}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* x y_m) (* (* z z) (+ z 1.0))) 1e+277)
    (* y_m (/ x (* (fma z z z) z)))
    (/ (* (/ x (* 1.0 z)) y_m) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((x * y_m) / ((z * z) * (z + 1.0))) <= 1e+277) {
		tmp = y_m * (x / (fma(z, z, z) * z));
	} else {
		tmp = ((x / (1.0 * z)) * y_m) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 1e+277)
		tmp = Float64(y_m * Float64(x / Float64(fma(z, z, z) * z)));
	else
		tmp = Float64(Float64(Float64(x / Float64(1.0 * z)) * y_m) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(x * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+277], N[(y$95$m * N[(x / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(1.0 * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 1e277

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 85.2

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 85.2

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

   3. Applied rewrites 85.2%

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

   if 1e277 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 83.3%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 70.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.8

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.8

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

   6. Applied rewrites 73.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.9

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

   8. Applied rewrites 74.9%

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

Alternative 7: 77.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

2. Taylor expanded in z around 0

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

4. Applied rewrites 70.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 73.8

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 73.8

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

6. Applied rewrites 73.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 77.0

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

8. Applied rewrites 77.0%

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

Alternative 8: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y\_m \leq 2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x}{\left(1 \cdot z\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* x y_m) 2e-190)
    (/ (/ x z) (/ z y_m))
    (* y_m (/ x (* (* 1.0 z) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((x * y_m) <= 2e-190) {
		tmp = (x / z) / (z / y_m);
	} else {
		tmp = y_m * (x / ((1.0 * z) * z));
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * y_m) <= 2d-190) then
        tmp = (x / z) / (z / y_m)
    else
        tmp = y_m * (x / ((1.0d0 * z) * z))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((x * y_m) <= 2e-190) {
		tmp = (x / z) / (z / y_m);
	} else {
		tmp = y_m * (x / ((1.0 * z) * z));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (x * y_m) <= 2e-190:
		tmp = (x / z) / (z / y_m)
	else:
		tmp = y_m * (x / ((1.0 * z) * z))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(x * y_m) <= 2e-190)
		tmp = Float64(Float64(x / z) / Float64(z / y_m));
	else
		tmp = Float64(y_m * Float64(x / Float64(Float64(1.0 * z) * z)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((x * y_m) <= 2e-190)
		tmp = (x / z) / (z / y_m);
	else
		tmp = y_m * (x / ((1.0 * z) * z));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(x * y$95$m), $MachinePrecision], 2e-190], N[(N[(x / z), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x / N[(N[(1.0 * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 2e-190

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 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

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

      10. lift-+.f64 N/A

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

      11. distribute-lft-in N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 94.5

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

   3. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]
   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. frac-times N/A

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

      5. *-commutative N/A

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

      6. frac-times N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-fma.f64 N/A

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

      10. distribute-lft1-in N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lift--.f64 N/A

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

      14. times-frac N/A

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

      15. lift-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 96.8

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in z around 0

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

      1. lower-/.f64 74.5

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

   10. Applied rewrites 74.5%

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

   if 2e-190 < (*.f64 x y)

   1. Initial program 83.3%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 70.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.8

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.8

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

   6. Applied rewrites 73.8%

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

Alternative 9: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y\_m \leq 2 \cdot 10^{-301}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot y\_m}{z}}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* x y_m) 2e-301) (/ (/ x z) (/ z y_m)) (/ (/ (* x y_m) z) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((x * y_m) <= 2e-301) {
		tmp = (x / z) / (z / y_m);
	} else {
		tmp = ((x * y_m) / z) / z;
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * y_m) <= 2d-301) then
        tmp = (x / z) / (z / y_m)
    else
        tmp = ((x * y_m) / z) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((x * y_m) <= 2e-301) {
		tmp = (x / z) / (z / y_m);
	} else {
		tmp = ((x * y_m) / z) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (x * y_m) <= 2e-301:
		tmp = (x / z) / (z / y_m)
	else:
		tmp = ((x * y_m) / z) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(x * y_m) <= 2e-301)
		tmp = Float64(Float64(x / z) / Float64(z / y_m));
	else
		tmp = Float64(Float64(Float64(x * y_m) / z) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((x * y_m) <= 2e-301)
		tmp = (x / z) / (z / y_m);
	else
		tmp = ((x * y_m) / z) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(x * y$95$m), $MachinePrecision], 2e-301], N[(N[(x / z), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y$95$m), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 2.00000000000000013e-301

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 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

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

      10. lift-+.f64 N/A

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

      11. distribute-lft-in N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 94.5

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

   3. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]
   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. frac-times N/A

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

      5. *-commutative N/A

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

      6. frac-times N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-fma.f64 N/A

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

      10. distribute-lft1-in N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lift--.f64 N/A

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

      14. times-frac N/A

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

      15. lift-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 96.8

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in z around 0

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

      1. lower-/.f64 74.5

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

   10. Applied rewrites 74.5%

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

   if 2.00000000000000013e-301 < (*.f64 x y)

   1. Initial program 83.3%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 70.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.8

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.8

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

   6. Applied rewrites 73.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.9

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

   8. Applied rewrites 74.9%

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

   9. Taylor expanded in z around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 70.6

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

   11. Applied rewrites 70.6%

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

Alternative 10: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y\_m \leq 2 \cdot 10^{-194}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot y\_m}{z}}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* x y_m) 2e-194) (* (/ y_m z) (/ x z)) (/ (/ (* x y_m) z) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((x * y_m) <= 2e-194) {
		tmp = (y_m / z) * (x / z);
	} else {
		tmp = ((x * y_m) / z) / z;
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * y_m) <= 2d-194) then
        tmp = (y_m / z) * (x / z)
    else
        tmp = ((x * y_m) / z) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((x * y_m) <= 2e-194) {
		tmp = (y_m / z) * (x / z);
	} else {
		tmp = ((x * y_m) / z) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (x * y_m) <= 2e-194:
		tmp = (y_m / z) * (x / z)
	else:
		tmp = ((x * y_m) / z) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(x * y_m) <= 2e-194)
		tmp = Float64(Float64(y_m / z) * Float64(x / z));
	else
		tmp = Float64(Float64(Float64(x * y_m) / z) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((x * y_m) <= 2e-194)
		tmp = (y_m / z) * (x / z);
	else
		tmp = ((x * y_m) / z) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(x * y$95$m), $MachinePrecision], 2e-194], N[(N[(y$95$m / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y$95$m), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 2.00000000000000004e-194

   1. Initial program 83.3%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 70.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.8

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.8

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

   6. Applied rewrites 73.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-/l/ N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 74.2

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

   8. Applied rewrites 74.2%

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

   9. Taylor expanded in z around 0

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

       1. lower-/.f64 74.2

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

   11. Applied rewrites 74.2%

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

   if 2.00000000000000004e-194 < (*.f64 x y)

   1. Initial program 83.3%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 70.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.8

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.8

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

   6. Applied rewrites 73.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.9

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

   8. Applied rewrites 74.9%

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

   9. Taylor expanded in z around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 70.6

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

   11. Applied rewrites 70.6%

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

Alternative 11: 74.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

2. Taylor expanded in z around 0

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

4. Applied rewrites 70.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 73.8

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 73.8

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

6. Applied rewrites 73.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/l/ N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. associate-/l/ N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-/.f64 74.2

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

8. Applied rewrites 74.2%

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

9. Taylor expanded in z around 0

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

    1. lower-/.f64 74.2

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

11. Applied rewrites 74.2%

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

Reproduce

herbie shell --seed 2025156 
(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))))