Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.3% → 99.3%

Time: 6.3s

Alternatives: 8

Speedup: 0.9×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot y - z \cdot y\right) \cdot t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (- (* x y) (* z y)) t))

C

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

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

Java

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

Python

def code(x, y, z, t):
	return ((x * y) - (z * y)) * t

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot y - z \cdot y\right) \cdot t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (- (* x y) (* z y)) t))

C

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

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

Java

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

Python

def code(x, y, z, t):
	return ((x * y) - (z * y)) * t

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := x \cdot y\_m - z \cdot y\_m\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-273} \lor \neg \left(t\_2 \leq 2 \cdot 10^{-129}\right):\\ \;\;\;\;\left(\left(x - z\right) \cdot y\_m\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - z\right) \cdot t\_m\right) \cdot y\_m\\ \end{array}\right) \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (- (* x y_m) (* z y_m))))
   (*
    y_s
    (*
     t_s
     (if (or (<= t_2 -5e-273) (not (<= t_2 2e-129)))
       (* (* (- x z) y_m) t_m)
       (* (* (- x z) t_m) y_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (x * y_m) - (z * y_m);
	double tmp;
	if ((t_2 <= -5e-273) || !(t_2 <= 2e-129)) {
		tmp = ((x - z) * y_m) * t_m;
	} else {
		tmp = ((x - z) * t_m) * y_m;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m =     private
t\_s =     private
y\_m =     private
y\_s =     private
NOTE: x, y_m, z, and t_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(y_s, t_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x * y_m) - (z * y_m)
    if ((t_2 <= (-5d-273)) .or. (.not. (t_2 <= 2d-129))) then
        tmp = ((x - z) * y_m) * t_m
    else
        tmp = ((x - z) * t_m) * y_m
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (x * y_m) - (z * y_m);
	double tmp;
	if ((t_2 <= -5e-273) || !(t_2 <= 2e-129)) {
		tmp = ((x - z) * y_m) * t_m;
	} else {
		tmp = ((x - z) * t_m) * y_m;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	t_2 = (x * y_m) - (z * y_m)
	tmp = 0
	if (t_2 <= -5e-273) or not (t_2 <= 2e-129):
		tmp = ((x - z) * y_m) * t_m
	else:
		tmp = ((x - z) * t_m) * y_m
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	t_2 = Float64(Float64(x * y_m) - Float64(z * y_m))
	tmp = 0.0
	if ((t_2 <= -5e-273) || !(t_2 <= 2e-129))
		tmp = Float64(Float64(Float64(x - z) * y_m) * t_m);
	else
		tmp = Float64(Float64(Float64(x - z) * t_m) * y_m);
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	t_2 = (x * y_m) - (z * y_m);
	tmp = 0.0;
	if ((t_2 <= -5e-273) || ~((t_2 <= 2e-129)))
		tmp = ((x - z) * y_m) * t_m;
	else
		tmp = ((x - z) * t_m) * y_m;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x * y$95$m), $MachinePrecision] - N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(t$95$s * If[Or[LessEqual[t$95$2, -5e-273], N[Not[LessEqual[t$95$2, 2e-129]], $MachinePrecision]], N[(N[(N[(x - z), $MachinePrecision] * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -4.99999999999999965e-273 or 1.9999999999999999e-129 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 90.0%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 93.2

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

   4. Applied rewrites 93.2%

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

   if -4.99999999999999965e-273 < (-.f64 (*.f64 x y) (*.f64 z y)) < 1.9999999999999999e-129

   1. Initial program 77.6%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \leq -5 \cdot 10^{-273} \lor \neg \left(x \cdot y - z \cdot y \leq 2 \cdot 10^{-129}\right):\\ \;\;\;\;\left(\left(x - z\right) \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - z\right) \cdot t\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+203} \lor \neg \left(z \leq 1.35 \cdot 10^{+201}\right):\\ \;\;\;\;\left(\left(-z\right) \cdot y\_m\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - z\right) \cdot t\_m\right) \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (*
  y_s
  (*
   t_s
   (if (or (<= z -5.8e+203) (not (<= z 1.35e+201)))
     (* (* (- z) y_m) t_m)
     (* (* (- x z) t_m) y_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((z <= -5.8e+203) || !(z <= 1.35e+201)) {
		tmp = (-z * y_m) * t_m;
	} else {
		tmp = ((x - z) * t_m) * y_m;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m =     private
t\_s =     private
y\_m =     private
y\_s =     private
NOTE: x, y_m, z, and t_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(y_s, t_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((z <= (-5.8d+203)) .or. (.not. (z <= 1.35d+201))) then
        tmp = (-z * y_m) * t_m
    else
        tmp = ((x - z) * t_m) * y_m
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((z <= -5.8e+203) || !(z <= 1.35e+201)) {
		tmp = (-z * y_m) * t_m;
	} else {
		tmp = ((x - z) * t_m) * y_m;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if (z <= -5.8e+203) or not (z <= 1.35e+201):
		tmp = (-z * y_m) * t_m
	else:
		tmp = ((x - z) * t_m) * y_m
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if ((z <= -5.8e+203) || !(z <= 1.35e+201))
		tmp = Float64(Float64(Float64(-z) * y_m) * t_m);
	else
		tmp = Float64(Float64(Float64(x - z) * t_m) * y_m);
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if ((z <= -5.8e+203) || ~((z <= 1.35e+201)))
		tmp = (-z * y_m) * t_m;
	else
		tmp = ((x - z) * t_m) * y_m;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[Or[LessEqual[z, -5.8e+203], N[Not[LessEqual[z, 1.35e+201]], $MachinePrecision]], N[(N[((-z) * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -5.80000000000000021e203 or 1.35e201 < z

   1. Initial program 84.5%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if -5.80000000000000021e203 < z < 1.35e201

   1. Initial program 89.3%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 93.7

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

   4. Applied rewrites 93.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+203} \lor \neg \left(z \leq 1.35 \cdot 10^{+201}\right):\\ \;\;\;\;\left(\left(-z\right) \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - z\right) \cdot t\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+107} \lor \neg \left(x \leq 3.15 \cdot 10^{-43}\right):\\ \;\;\;\;\left(y\_m \cdot x\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(t\_m \cdot y\_m\right) \cdot \left(-z\right)\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (*
  y_s
  (*
   t_s
   (if (or (<= x -1.6e+107) (not (<= x 3.15e-43)))
     (* (* y_m x) t_m)
     (* (* t_m y_m) (- z))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((x <= -1.6e+107) || !(x <= 3.15e-43)) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = (t_m * y_m) * -z;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m =     private
t\_s =     private
y\_m =     private
y\_s =     private
NOTE: x, y_m, z, and t_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(y_s, t_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((x <= (-1.6d+107)) .or. (.not. (x <= 3.15d-43))) then
        tmp = (y_m * x) * t_m
    else
        tmp = (t_m * y_m) * -z
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((x <= -1.6e+107) || !(x <= 3.15e-43)) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = (t_m * y_m) * -z;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if (x <= -1.6e+107) or not (x <= 3.15e-43):
		tmp = (y_m * x) * t_m
	else:
		tmp = (t_m * y_m) * -z
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if ((x <= -1.6e+107) || !(x <= 3.15e-43))
		tmp = Float64(Float64(y_m * x) * t_m);
	else
		tmp = Float64(Float64(t_m * y_m) * Float64(-z));
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if ((x <= -1.6e+107) || ~((x <= 3.15e-43)))
		tmp = (y_m * x) * t_m;
	else
		tmp = (t_m * y_m) * -z;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[Or[LessEqual[x, -1.6e+107], N[Not[LessEqual[x, 3.15e-43]], $MachinePrecision]], N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(t$95$m * y$95$m), $MachinePrecision] * (-z)), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -1.60000000000000015e107 or 3.1500000000000001e-43 < x

   1. Initial program 83.9%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. mul-1-neg N/A

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

      17. remove-double-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

   if -1.60000000000000015e107 < x < 3.1500000000000001e-43

   1. Initial program 91.9%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 91.0

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.2

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

   7. Applied rewrites 71.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 76.8

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

   9. Applied rewrites 76.8%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+107} \lor \neg \left(x \leq 3.15 \cdot 10^{-43}\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.28 \cdot 10^{+20} \lor \neg \left(z \leq 1.3 \cdot 10^{-37}\right):\\ \;\;\;\;\left(\left(-z\right) \cdot y\_m\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot x\right) \cdot t\_m\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (*
  y_s
  (*
   t_s
   (if (or (<= z -1.28e+20) (not (<= z 1.3e-37)))
     (* (* (- z) y_m) t_m)
     (* (* y_m x) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((z <= -1.28e+20) || !(z <= 1.3e-37)) {
		tmp = (-z * y_m) * t_m;
	} else {
		tmp = (y_m * x) * t_m;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m =     private
t\_s =     private
y\_m =     private
y\_s =     private
NOTE: x, y_m, z, and t_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(y_s, t_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((z <= (-1.28d+20)) .or. (.not. (z <= 1.3d-37))) then
        tmp = (-z * y_m) * t_m
    else
        tmp = (y_m * x) * t_m
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((z <= -1.28e+20) || !(z <= 1.3e-37)) {
		tmp = (-z * y_m) * t_m;
	} else {
		tmp = (y_m * x) * t_m;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if (z <= -1.28e+20) or not (z <= 1.3e-37):
		tmp = (-z * y_m) * t_m
	else:
		tmp = (y_m * x) * t_m
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if ((z <= -1.28e+20) || !(z <= 1.3e-37))
		tmp = Float64(Float64(Float64(-z) * y_m) * t_m);
	else
		tmp = Float64(Float64(y_m * x) * t_m);
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if ((z <= -1.28e+20) || ~((z <= 1.3e-37)))
		tmp = (-z * y_m) * t_m;
	else
		tmp = (y_m * x) * t_m;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[Or[LessEqual[z, -1.28e+20], N[Not[LessEqual[z, 1.3e-37]], $MachinePrecision]], N[(N[((-z) * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.28e20 or 1.2999999999999999e-37 < z

   1. Initial program 87.6%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 76.6

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

   5. Applied rewrites 76.6%

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

   if -1.28e20 < z < 1.2999999999999999e-37

   1. Initial program 89.5%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. mul-1-neg N/A

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

      17. remove-double-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.28 \cdot 10^{+20} \lor \neg \left(z \leq 1.3 \cdot 10^{-37}\right):\\ \;\;\;\;\left(\left(-z\right) \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+25} \lor \neg \left(x \leq 1.7 \cdot 10^{-45}\right):\\ \;\;\;\;\left(y\_m \cdot x\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-t\_m\right) \cdot z\right) \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (*
  y_s
  (*
   t_s
   (if (or (<= x -2.8e+25) (not (<= x 1.7e-45)))
     (* (* y_m x) t_m)
     (* (* (- t_m) z) y_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((x <= -2.8e+25) || !(x <= 1.7e-45)) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = (-t_m * z) * y_m;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m =     private
t\_s =     private
y\_m =     private
y\_s =     private
NOTE: x, y_m, z, and t_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(y_s, t_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((x <= (-2.8d+25)) .or. (.not. (x <= 1.7d-45))) then
        tmp = (y_m * x) * t_m
    else
        tmp = (-t_m * z) * y_m
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((x <= -2.8e+25) || !(x <= 1.7e-45)) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = (-t_m * z) * y_m;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if (x <= -2.8e+25) or not (x <= 1.7e-45):
		tmp = (y_m * x) * t_m
	else:
		tmp = (-t_m * z) * y_m
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if ((x <= -2.8e+25) || !(x <= 1.7e-45))
		tmp = Float64(Float64(y_m * x) * t_m);
	else
		tmp = Float64(Float64(Float64(-t_m) * z) * y_m);
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if ((x <= -2.8e+25) || ~((x <= 1.7e-45)))
		tmp = (y_m * x) * t_m;
	else
		tmp = (-t_m * z) * y_m;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[Or[LessEqual[x, -2.8e+25], N[Not[LessEqual[x, 1.7e-45]], $MachinePrecision]], N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[((-t$95$m) * z), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -2.8000000000000002e25 or 1.70000000000000002e-45 < x

   1. Initial program 85.3%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. mul-1-neg N/A

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

      17. remove-double-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 71.6

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

   5. Applied rewrites 71.6%

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

   if -2.8000000000000002e25 < x < 1.70000000000000002e-45

   1. Initial program 91.9%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 75.3

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

   5. Applied rewrites 75.3%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+25} \lor \neg \left(x \leq 1.7 \cdot 10^{-45}\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-t\right) \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.35:\\ \;\;\;\;\left(\left(x - z\right) \cdot y\_m\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t\_m \cdot y\_m\right)\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (*
  y_s
  (* t_s (if (<= t_m 3.35) (* (* (- x z) y_m) t_m) (* (- x z) (* t_m y_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 3.35) {
		tmp = ((x - z) * y_m) * t_m;
	} else {
		tmp = (x - z) * (t_m * y_m);
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m =     private
t\_s =     private
y\_m =     private
y\_s =     private
NOTE: x, y_m, z, and t_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(y_s, t_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.35d0) then
        tmp = ((x - z) * y_m) * t_m
    else
        tmp = (x - z) * (t_m * y_m)
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 3.35) {
		tmp = ((x - z) * y_m) * t_m;
	} else {
		tmp = (x - z) * (t_m * y_m);
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 3.35:
		tmp = ((x - z) * y_m) * t_m
	else:
		tmp = (x - z) * (t_m * y_m)
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 3.35)
		tmp = Float64(Float64(Float64(x - z) * y_m) * t_m);
	else
		tmp = Float64(Float64(x - z) * Float64(t_m * y_m));
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 3.35)
		tmp = ((x - z) * y_m) * t_m;
	else
		tmp = (x - z) * (t_m * y_m);
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[LessEqual[t$95$m, 3.35], N[(N[(N[(x - z), $MachinePrecision] * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.35000000000000009

   1. Initial program 88.7%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 91.0

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

   4. Applied rewrites 91.0%

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

   if 3.35000000000000009 < t

   1. Initial program 87.8%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 7: 56.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5:\\ \;\;\;\;\left(y\_m \cdot x\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(t\_m \cdot y\_m\right) \cdot x\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (* y_s (* t_s (if (<= t_m 5.0) (* (* y_m x) t_m) (* (* t_m y_m) x)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 5.0) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = (t_m * y_m) * x;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m =     private
t\_s =     private
y\_m =     private
y\_s =     private
NOTE: x, y_m, z, and t_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(y_s, t_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 5.0d0) then
        tmp = (y_m * x) * t_m
    else
        tmp = (t_m * y_m) * x
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 5.0) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = (t_m * y_m) * x;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 5.0:
		tmp = (y_m * x) * t_m
	else:
		tmp = (t_m * y_m) * x
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 5.0)
		tmp = Float64(Float64(y_m * x) * t_m);
	else
		tmp = Float64(Float64(t_m * y_m) * x);
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 5.0)
		tmp = (y_m * x) * t_m;
	else
		tmp = (t_m * y_m) * x;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[LessEqual[t$95$m, 5.0], N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(t$95$m * y$95$m), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5

   1. Initial program 88.7%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. mul-1-neg N/A

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

      17. remove-double-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 51.1

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

   5. Applied rewrites 51.1%

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

   if 5 < t

   1. Initial program 87.8%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. mul-1-neg N/A

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

      17. remove-double-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 51.8

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

   5. Applied rewrites 51.8%

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

   7. Applied rewrites 58.2%

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

Alternative 8: 53.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m =     private
t\_s =     private
y\_m =     private
y\_s =     private
NOTE: x, y_m, z, and t_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(y_s, t_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = y_s * (t_s * ((t_m * y_m) * x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.5%

   \[\left(x \cdot y - z \cdot y\right) \cdot t \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. mul-1-neg N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. mul-1-neg N/A

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

   16. mul-1-neg N/A

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

   17. remove-double-neg N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 51.3

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

5. Applied rewrites 51.3%

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

7. Applied rewrites 54.2%

   \[\leadsto \left(t \cdot y\right) \cdot \color{blue}{x} \]
8. Add Preprocessing

Developer Target 1: 95.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t < -9.231879582886777 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t < 2.543067051564877 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (< t -9.231879582886777e-80)
   (* (* y t) (- x z))
   (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t < -9.231879582886777e-80) {
		tmp = (y * t) * (x - z);
	} else if (t < 2.543067051564877e+83) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (y * (x - z)) * t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t < (-9.231879582886777d-80)) then
        tmp = (y * t) * (x - z)
    else if (t < 2.543067051564877d+83) then
        tmp = y * (t * (x - z))
    else
        tmp = (y * (x - z)) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t < -9.231879582886777e-80) {
		tmp = (y * t) * (x - z);
	} else if (t < 2.543067051564877e+83) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (y * (x - z)) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t < -9.231879582886777e-80:
		tmp = (y * t) * (x - z)
	elif t < 2.543067051564877e+83:
		tmp = y * (t * (x - z))
	else:
		tmp = (y * (x - z)) * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t < -9.231879582886777e-80)
		tmp = Float64(Float64(y * t) * Float64(x - z));
	elseif (t < 2.543067051564877e+83)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	else
		tmp = Float64(Float64(y * Float64(x - z)) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t < -9.231879582886777e-80)
		tmp = (y * t) * (x - z);
	elseif (t < 2.543067051564877e+83)
		tmp = y * (t * (x - z));
	else
		tmp = (y * (x - z)) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Less[t, -9.231879582886777e-80], N[(N[(y * t), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision], If[Less[t, 2.543067051564877e+83], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025015 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -9231879582886777/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (* y t) (- x z)) (if (< t 254306705156487700000000000000000000000000000000000000000000000000000000000000000000) (* y (* t (- x z))) (* (* y (- x z)) t))))

  (* (- (* x y) (* z y)) t))