Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A

Percentage Accurate: 92.9% → 99.7%

Time: 1.7s

Alternatives: 4

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{z} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{z} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, N/A× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{z\_m} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-y\_m}{z\_m} \cdot \left(-x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x\_m}{z\_m} \cdot \left(-y\_m\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (/ (* x_m y_m) z_m) 5e-11)
      (* (/ (- y_m) z_m) (- x_m))
      (* (/ (- x_m) z_m) (- y_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((x_m * y_m) / z_m) <= 5e-11) {
		tmp = (-y_m / z_m) * -x_m;
	} else {
		tmp = (-x_m / z_m) * -y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, and z_m 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(z_s, y_s, x_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (((x_m * y_m) / z_m) <= 5d-11) then
        tmp = (-y_m / z_m) * -x_m
    else
        tmp = (-x_m / z_m) * -y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((x_m * y_m) / z_m) <= 5e-11) {
		tmp = (-y_m / z_m) * -x_m;
	} else {
		tmp = (-x_m / z_m) * -y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if ((x_m * y_m) / z_m) <= 5e-11:
		tmp = (-y_m / z_m) * -x_m
	else:
		tmp = (-x_m / z_m) * -y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / z_m) <= 5e-11)
		tmp = Float64(Float64(Float64(-y_m) / z_m) * Float64(-x_m));
	else
		tmp = Float64(Float64(Float64(-x_m) / z_m) * Float64(-y_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (((x_m * y_m) / z_m) <= 5e-11)
		tmp = (-y_m / z_m) * -x_m;
	else
		tmp = (-x_m / z_m) * -y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], 5e-11], N[(N[((-y$95$m) / z$95$m), $MachinePrecision] * (-x$95$m)), $MachinePrecision], N[(N[((-x$95$m) / z$95$m), $MachinePrecision] * (-y$95$m)), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) z) < 5.00000000000000018e-11

   1. Initial program 93.0%

      \[\frac{x \cdot y}{z} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. frac-2neg-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 99.7

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

   3. Applied rewrites 99.7%

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

   if 5.00000000000000018e-11 < (/.f64 (*.f64 x y) z)

   1. Initial program 92.8%

      \[\frac{x \cdot y}{z} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg-rev N/A

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

      8. remove-double-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 99.7

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

   3. Applied rewrites 99.7%

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

Alternative 2: 99.4% accurate, N/A× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{z\_m} \leq 2 \cdot 10^{-56}:\\ \;\;\;\;\left({z\_m}^{-1} \cdot \left(-1 \cdot y\_m\right)\right) \cdot \left(-x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x\_m}{z\_m} \cdot \left(-y\_m\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (/ (* x_m y_m) z_m) 2e-56)
      (* (* (pow z_m -1.0) (* -1.0 y_m)) (- x_m))
      (* (/ (- x_m) z_m) (- y_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((x_m * y_m) / z_m) <= 2e-56) {
		tmp = (pow(z_m, -1.0) * (-1.0 * y_m)) * -x_m;
	} else {
		tmp = (-x_m / z_m) * -y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, and z_m 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(z_s, y_s, x_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (((x_m * y_m) / z_m) <= 2d-56) then
        tmp = ((z_m ** (-1.0d0)) * ((-1.0d0) * y_m)) * -x_m
    else
        tmp = (-x_m / z_m) * -y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((x_m * y_m) / z_m) <= 2e-56) {
		tmp = (Math.pow(z_m, -1.0) * (-1.0 * y_m)) * -x_m;
	} else {
		tmp = (-x_m / z_m) * -y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if ((x_m * y_m) / z_m) <= 2e-56:
		tmp = (math.pow(z_m, -1.0) * (-1.0 * y_m)) * -x_m
	else:
		tmp = (-x_m / z_m) * -y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / z_m) <= 2e-56)
		tmp = Float64(Float64((z_m ^ -1.0) * Float64(-1.0 * y_m)) * Float64(-x_m));
	else
		tmp = Float64(Float64(Float64(-x_m) / z_m) * Float64(-y_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (((x_m * y_m) / z_m) <= 2e-56)
		tmp = ((z_m ^ -1.0) * (-1.0 * y_m)) * -x_m;
	else
		tmp = (-x_m / z_m) * -y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], 2e-56], N[(N[(N[Power[z$95$m, -1.0], $MachinePrecision] * N[(-1.0 * y$95$m), $MachinePrecision]), $MachinePrecision] * (-x$95$m)), $MachinePrecision], N[(N[((-x$95$m) / z$95$m), $MachinePrecision] * (-y$95$m)), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) z) < 2.0000000000000001e-56

   1. Initial program 92.2%

      \[\frac{x \cdot y}{z} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. frac-2neg-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 99.7

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

   3. Applied rewrites 99.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. mul-1-neg N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. remove-double-neg N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 2.0000000000000001e-56 < (/.f64 (*.f64 x y) z)

   1. Initial program 93.4%

      \[\frac{x \cdot y}{z} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg-rev N/A

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

      8. remove-double-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 99.3

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

   3. Applied rewrites 99.3%

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

Alternative 3: 99.1% accurate, N/A× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8 \cdot 10^{-72}:\\ \;\;\;\;\left({z\_m}^{-1} \cdot \left(-1 \cdot x\_m\right)\right) \cdot \left(-y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left({z\_m}^{-1} \cdot \left(-1 \cdot y\_m\right)\right) \cdot \left(-x\_m\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 8e-72)
      (* (* (pow z_m -1.0) (* -1.0 x_m)) (- y_m))
      (* (* (pow z_m -1.0) (* -1.0 y_m)) (- x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 8e-72) {
		tmp = (pow(z_m, -1.0) * (-1.0 * x_m)) * -y_m;
	} else {
		tmp = (pow(z_m, -1.0) * (-1.0 * y_m)) * -x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, and z_m 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(z_s, y_s, x_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 8d-72) then
        tmp = ((z_m ** (-1.0d0)) * ((-1.0d0) * x_m)) * -y_m
    else
        tmp = ((z_m ** (-1.0d0)) * ((-1.0d0) * y_m)) * -x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 8e-72) {
		tmp = (Math.pow(z_m, -1.0) * (-1.0 * x_m)) * -y_m;
	} else {
		tmp = (Math.pow(z_m, -1.0) * (-1.0 * y_m)) * -x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 8e-72:
		tmp = (math.pow(z_m, -1.0) * (-1.0 * x_m)) * -y_m
	else:
		tmp = (math.pow(z_m, -1.0) * (-1.0 * y_m)) * -x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 8e-72)
		tmp = Float64(Float64((z_m ^ -1.0) * Float64(-1.0 * x_m)) * Float64(-y_m));
	else
		tmp = Float64(Float64((z_m ^ -1.0) * Float64(-1.0 * y_m)) * Float64(-x_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 8e-72)
		tmp = ((z_m ^ -1.0) * (-1.0 * x_m)) * -y_m;
	else
		tmp = ((z_m ^ -1.0) * (-1.0 * y_m)) * -x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 8e-72], N[(N[(N[Power[z$95$m, -1.0], $MachinePrecision] * N[(-1.0 * x$95$m), $MachinePrecision]), $MachinePrecision] * (-y$95$m)), $MachinePrecision], N[(N[(N[Power[z$95$m, -1.0], $MachinePrecision] * N[(-1.0 * y$95$m), $MachinePrecision]), $MachinePrecision] * (-x$95$m)), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.9999999999999997e-72

   1. Initial program 91.8%

      \[\frac{x \cdot y}{z} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg-rev N/A

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

      8. remove-double-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 99.7

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

   3. Applied rewrites 99.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. remove-double-neg N/A

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

      12. inv-pow N/A

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

      13. lower-pow.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 7.9999999999999997e-72 < z

   1. Initial program 93.5%

      \[\frac{x \cdot y}{z} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. frac-2neg-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 98.9

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

   3. Applied rewrites 98.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. mul-1-neg N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. remove-double-neg N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

Alternative 4: 92.1% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

   \[\frac{x \cdot y}{z} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. associate-*l/ N/A

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

   6. lower-*.f64 N/A

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

   7. frac-2neg-rev N/A

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

   8. remove-double-neg N/A

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

   9. lower-/.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-neg.f64 92.2

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

3. Applied rewrites 92.2%

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

   1. lift-neg.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. associate-*r/ N/A

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

   5. frac-2neg N/A

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

   6. associate-*r/ N/A

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

   7. associate-*l/ N/A

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

   8. lower-*.f64 N/A

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

   9. frac-2neg N/A

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

   10. metadata-eval N/A

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

   11. remove-double-neg N/A

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

   12. inv-pow N/A

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

   13. lower-pow.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-*.f64 92.1

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

5. Applied rewrites 92.1%

   \[\leadsto \color{blue}{\left({z}^{-1} \cdot \left(-1 \cdot x\right)\right)} \cdot \left(-y\right) \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64
  (/ (* x y) z))