Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, A

Percentage Accurate: 99.2% → 99.2%

Time: 3.6s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return 1.0 - (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 = 1.0d0 - (x / ((y - z) * (y - t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return 1.0 - (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 = 1.0d0 - (x / ((y - z) * (y - t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 - \frac{\frac{x}{y - t}}{y - z} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- 1.0 (/ (/ x (- y t)) (- y z))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 - ((x / (y - t)) / (y - z));
}

Fortran

NOTE: x, y, z, and t 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(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 = 1.0d0 - ((x / (y - t)) / (y - z))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 - ((x / (y - t)) / (y - z));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 - ((x / (y - t)) / (y - z))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 - Float64(Float64(x / Float64(y - t)) / Float64(y - z)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 - ((x / (y - t)) / (y - z));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 - N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift--.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift--.f64 98.3

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

3. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- 1.0 (/ x (* (- y z) (- y t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - z) * (y - t)));
}

Fortran

NOTE: x, y, z, and t 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(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 = 1.0d0 - (x / ((y - z) * (y - t)))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - z) * (y - t)));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 - (x / ((y - z) * (y - t)))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 - Float64(x / Float64(Float64(y - z) * Float64(y - t))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 - (x / ((y - z) * (y - t)));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 - N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
2. Add Preprocessing

Alternative 3: 92.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} + 1\\ \mathbf{elif}\;t \leq 10^{-114}:\\ \;\;\;\;1 - \frac{\frac{x}{y - z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z} + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t -8.2e-95)
   (+ (/ x (* (- y t) z)) 1.0)
   (if (<= t 1e-114) (- 1.0 (/ (/ x (- y z)) y)) (+ (/ (/ x t) (- y z)) 1.0))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.2e-95) {
		tmp = (x / ((y - t) * z)) + 1.0;
	} else if (t <= 1e-114) {
		tmp = 1.0 - ((x / (y - z)) / y);
	} else {
		tmp = ((x / t) / (y - z)) + 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t 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(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 <= (-8.2d-95)) then
        tmp = (x / ((y - t) * z)) + 1.0d0
    else if (t <= 1d-114) then
        tmp = 1.0d0 - ((x / (y - z)) / y)
    else
        tmp = ((x / t) / (y - z)) + 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.2e-95) {
		tmp = (x / ((y - t) * z)) + 1.0;
	} else if (t <= 1e-114) {
		tmp = 1.0 - ((x / (y - z)) / y);
	} else {
		tmp = ((x / t) / (y - z)) + 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= -8.2e-95:
		tmp = (x / ((y - t) * z)) + 1.0
	elif t <= 1e-114:
		tmp = 1.0 - ((x / (y - z)) / y)
	else:
		tmp = ((x / t) / (y - z)) + 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= -8.2e-95)
		tmp = Float64(Float64(x / Float64(Float64(y - t) * z)) + 1.0);
	elseif (t <= 1e-114)
		tmp = Float64(1.0 - Float64(Float64(x / Float64(y - z)) / y));
	else
		tmp = Float64(Float64(Float64(x / t) / Float64(y - z)) + 1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -8.2e-95)
		tmp = (x / ((y - t) * z)) + 1.0;
	elseif (t <= 1e-114)
		tmp = 1.0 - ((x / (y - z)) / y);
	else
		tmp = ((x / t) / (y - z)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, -8.2e-95], N[(N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t, 1e-114], N[(1.0 - N[(N[(x / N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.1999999999999995e-95

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 97.2

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

   4. Applied rewrites 97.2%

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

   if -8.1999999999999995e-95 < t < 1.0000000000000001e-114

   1. Initial program 97.9%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 88.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 89.8

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

   6. Applied rewrites 89.8%

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

   if 1.0000000000000001e-114 < t

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 93.5

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

   4. Applied rewrites 93.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{t}}{y - z} + 1 \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{t}}{y - z} + 1 \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{t}}{y - z} + 1 \]

      8. lift--.f64 93.3

         \[\leadsto \frac{\frac{x}{t}}{y - z} + 1 \]

   6. Applied rewrites 93.3%

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

Alternative 4: 92.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} + 1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-119}:\\ \;\;\;\;1 - \frac{\frac{x}{y - z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t} - -1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t -8.2e-95)
   (+ (/ x (* (- y t) z)) 1.0)
   (if (<= t 2.1e-119)
     (- 1.0 (/ (/ x (- y z)) y))
     (- (/ x (* (- y z) t)) -1.0))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.2e-95) {
		tmp = (x / ((y - t) * z)) + 1.0;
	} else if (t <= 2.1e-119) {
		tmp = 1.0 - ((x / (y - z)) / y);
	} else {
		tmp = (x / ((y - z) * t)) - -1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t 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(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 <= (-8.2d-95)) then
        tmp = (x / ((y - t) * z)) + 1.0d0
    else if (t <= 2.1d-119) then
        tmp = 1.0d0 - ((x / (y - z)) / y)
    else
        tmp = (x / ((y - z) * t)) - (-1.0d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.2e-95) {
		tmp = (x / ((y - t) * z)) + 1.0;
	} else if (t <= 2.1e-119) {
		tmp = 1.0 - ((x / (y - z)) / y);
	} else {
		tmp = (x / ((y - z) * t)) - -1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= -8.2e-95:
		tmp = (x / ((y - t) * z)) + 1.0
	elif t <= 2.1e-119:
		tmp = 1.0 - ((x / (y - z)) / y)
	else:
		tmp = (x / ((y - z) * t)) - -1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= -8.2e-95)
		tmp = Float64(Float64(x / Float64(Float64(y - t) * z)) + 1.0);
	elseif (t <= 2.1e-119)
		tmp = Float64(1.0 - Float64(Float64(x / Float64(y - z)) / y));
	else
		tmp = Float64(Float64(x / Float64(Float64(y - z) * t)) - -1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -8.2e-95)
		tmp = (x / ((y - t) * z)) + 1.0;
	elseif (t <= 2.1e-119)
		tmp = 1.0 - ((x / (y - z)) / y);
	else
		tmp = (x / ((y - z) * t)) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, -8.2e-95], N[(N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t, 2.1e-119], N[(1.0 - N[(N[(x / N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.1999999999999995e-95

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 97.2

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

   4. Applied rewrites 97.2%

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

   if -8.1999999999999995e-95 < t < 2.1e-119

   1. Initial program 97.9%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 89.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 90.0

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

   6. Applied rewrites 90.0%

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

   if 2.1e-119 < t

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 93.3

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

   4. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l/ N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 93.3

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

   6. Applied rewrites 93.3%

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

Alternative 5: 92.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} + 1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-119}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t} - -1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t -8.2e-95)
   (+ (/ x (* (- y t) z)) 1.0)
   (if (<= t 2.1e-119)
     (- 1.0 (/ (/ x y) (- y z)))
     (- (/ x (* (- y z) t)) -1.0))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.2e-95) {
		tmp = (x / ((y - t) * z)) + 1.0;
	} else if (t <= 2.1e-119) {
		tmp = 1.0 - ((x / y) / (y - z));
	} else {
		tmp = (x / ((y - z) * t)) - -1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t 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(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 <= (-8.2d-95)) then
        tmp = (x / ((y - t) * z)) + 1.0d0
    else if (t <= 2.1d-119) then
        tmp = 1.0d0 - ((x / y) / (y - z))
    else
        tmp = (x / ((y - z) * t)) - (-1.0d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.2e-95) {
		tmp = (x / ((y - t) * z)) + 1.0;
	} else if (t <= 2.1e-119) {
		tmp = 1.0 - ((x / y) / (y - z));
	} else {
		tmp = (x / ((y - z) * t)) - -1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= -8.2e-95:
		tmp = (x / ((y - t) * z)) + 1.0
	elif t <= 2.1e-119:
		tmp = 1.0 - ((x / y) / (y - z))
	else:
		tmp = (x / ((y - z) * t)) - -1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= -8.2e-95)
		tmp = Float64(Float64(x / Float64(Float64(y - t) * z)) + 1.0);
	elseif (t <= 2.1e-119)
		tmp = Float64(1.0 - Float64(Float64(x / y) / Float64(y - z)));
	else
		tmp = Float64(Float64(x / Float64(Float64(y - z) * t)) - -1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -8.2e-95)
		tmp = (x / ((y - t) * z)) + 1.0;
	elseif (t <= 2.1e-119)
		tmp = 1.0 - ((x / y) / (y - z));
	else
		tmp = (x / ((y - z) * t)) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, -8.2e-95], N[(N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t, 2.1e-119], N[(1.0 - N[(N[(x / y), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.1999999999999995e-95

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 97.2

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

   4. Applied rewrites 97.2%

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

   if -8.1999999999999995e-95 < t < 2.1e-119

   1. Initial program 97.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 95.7

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

   3. Applied rewrites 95.7%

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

   4. Taylor expanded in y around inf

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

      1. lower-/.f64 87.5

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

   6. Applied rewrites 87.5%

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

   if 2.1e-119 < t

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 93.3

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

   4. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l/ N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 93.3

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

   6. Applied rewrites 93.3%

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

Alternative 6: 91.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} + 1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-119}:\\ \;\;\;\;1 - \frac{x}{\left(y - z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t} - -1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t -8.2e-95)
   (+ (/ x (* (- y t) z)) 1.0)
   (if (<= t 2.1e-119)
     (- 1.0 (/ x (* (- y z) y)))
     (- (/ x (* (- y z) t)) -1.0))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.2e-95) {
		tmp = (x / ((y - t) * z)) + 1.0;
	} else if (t <= 2.1e-119) {
		tmp = 1.0 - (x / ((y - z) * y));
	} else {
		tmp = (x / ((y - z) * t)) - -1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t 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(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 <= (-8.2d-95)) then
        tmp = (x / ((y - t) * z)) + 1.0d0
    else if (t <= 2.1d-119) then
        tmp = 1.0d0 - (x / ((y - z) * y))
    else
        tmp = (x / ((y - z) * t)) - (-1.0d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.2e-95) {
		tmp = (x / ((y - t) * z)) + 1.0;
	} else if (t <= 2.1e-119) {
		tmp = 1.0 - (x / ((y - z) * y));
	} else {
		tmp = (x / ((y - z) * t)) - -1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= -8.2e-95:
		tmp = (x / ((y - t) * z)) + 1.0
	elif t <= 2.1e-119:
		tmp = 1.0 - (x / ((y - z) * y))
	else:
		tmp = (x / ((y - z) * t)) - -1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= -8.2e-95)
		tmp = Float64(Float64(x / Float64(Float64(y - t) * z)) + 1.0);
	elseif (t <= 2.1e-119)
		tmp = Float64(1.0 - Float64(x / Float64(Float64(y - z) * y)));
	else
		tmp = Float64(Float64(x / Float64(Float64(y - z) * t)) - -1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -8.2e-95)
		tmp = (x / ((y - t) * z)) + 1.0;
	elseif (t <= 2.1e-119)
		tmp = 1.0 - (x / ((y - z) * y));
	else
		tmp = (x / ((y - z) * t)) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, -8.2e-95], N[(N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t, 2.1e-119], N[(1.0 - N[(x / N[(N[(y - z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.1999999999999995e-95

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 97.2

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

   4. Applied rewrites 97.2%

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

   if -8.1999999999999995e-95 < t < 2.1e-119

   1. Initial program 97.9%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 89.1%

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

   if 2.1e-119 < t

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 93.3

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

   4. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l/ N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 93.3

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

   6. Applied rewrites 93.3%

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

Alternative 7: 90.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-119}:\\ \;\;\;\;1 - \frac{x}{\left(y - z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t} - -1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 2.1e-119) (- 1.0 (/ x (* (- y z) y))) (- (/ x (* (- y z) t)) -1.0)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.1e-119) {
		tmp = 1.0 - (x / ((y - z) * y));
	} else {
		tmp = (x / ((y - z) * t)) - -1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t 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(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 <= 2.1d-119) then
        tmp = 1.0d0 - (x / ((y - z) * y))
    else
        tmp = (x / ((y - z) * t)) - (-1.0d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.1e-119) {
		tmp = 1.0 - (x / ((y - z) * y));
	} else {
		tmp = (x / ((y - z) * t)) - -1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 2.1e-119:
		tmp = 1.0 - (x / ((y - z) * y))
	else:
		tmp = (x / ((y - z) * t)) - -1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 2.1e-119)
		tmp = Float64(1.0 - Float64(x / Float64(Float64(y - z) * y)));
	else
		tmp = Float64(Float64(x / Float64(Float64(y - z) * t)) - -1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 2.1e-119)
		tmp = 1.0 - (x / ((y - z) * y));
	else
		tmp = (x / ((y - z) * t)) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 2.1e-119], N[(1.0 - N[(x / N[(N[(y - z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.1e-119

   1. Initial program 98.4%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 86.1%

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

   if 2.1e-119 < t

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 93.3

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

   4. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l/ N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 93.3

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

   6. Applied rewrites 93.3%

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

Alternative 8: 89.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;\frac{\frac{x}{y - t}}{z}\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t} - -1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))))
   (if (<= t_1 -1000000.0)
     (/ (/ x (- y t)) z)
     (if (<= t_1 0.002) 1.0 (- (/ x (* (- y z) t)) -1.0)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = (x / (y - t)) / z;
	} else if (t_1 <= 0.002) {
		tmp = 1.0;
	} else {
		tmp = (x / ((y - z) * t)) - -1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t 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(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) :: t_1
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    if (t_1 <= (-1000000.0d0)) then
        tmp = (x / (y - t)) / z
    else if (t_1 <= 0.002d0) then
        tmp = 1.0d0
    else
        tmp = (x / ((y - z) * t)) - (-1.0d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = (x / (y - t)) / z;
	} else if (t_1 <= 0.002) {
		tmp = 1.0;
	} else {
		tmp = (x / ((y - z) * t)) - -1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	tmp = 0
	if t_1 <= -1000000.0:
		tmp = (x / (y - t)) / z
	elif t_1 <= 0.002:
		tmp = 1.0
	else:
		tmp = (x / ((y - z) * t)) - -1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = Float64(Float64(x / Float64(y - t)) / z);
	elseif (t_1 <= 0.002)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x / Float64(Float64(y - z) * t)) - -1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	tmp = 0.0;
	if (t_1 <= -1000000.0)
		tmp = (x / (y - t)) / z;
	elseif (t_1 <= 0.002)
		tmp = 1.0;
	else
		tmp = (x / ((y - z) * t)) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 0.002], 1.0, N[(N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -1e6

   1. Initial program 96.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 93.2

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

   3. Applied rewrites 93.2%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 61.6

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

   6. Applied rewrites 61.6%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{z + \frac{x}{y - t}}{z} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z + \frac{x}{y - t}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{z + \frac{x}{y - t}}{z} \]

      4. lift--.f64 56.6

         \[\leadsto \frac{z + \frac{x}{y - t}}{z} \]

   9. Applied rewrites 56.6%

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

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{x}{y - t}}{z} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{x}{y - t}}{z} \]

       2. lift--.f64 56.0

          \[\leadsto \frac{\frac{x}{y - t}}{z} \]

   12. Applied rewrites 56.0%

       \[\leadsto \frac{\frac{x}{y - t}}{z} \]

   if -1e6 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 2e-3

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 98.9%

      \[\leadsto \color{blue}{1} \]

   if 2e-3 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 97.1%

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

   2. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 59.5

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

   4. Applied rewrites 59.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l/ N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 59.5

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

   6. Applied rewrites 59.5%

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

Alternative 9: 88.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;\frac{\frac{x}{y - t}}{z}\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))))
   (if (<= t_1 -1000000.0)
     (/ (/ x (- y t)) z)
     (if (<= t_1 0.002) 1.0 (/ x (* (- y z) t))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = (x / (y - t)) / z;
	} else if (t_1 <= 0.002) {
		tmp = 1.0;
	} else {
		tmp = x / ((y - z) * t);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t 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(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) :: t_1
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    if (t_1 <= (-1000000.0d0)) then
        tmp = (x / (y - t)) / z
    else if (t_1 <= 0.002d0) then
        tmp = 1.0d0
    else
        tmp = x / ((y - z) * t)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = (x / (y - t)) / z;
	} else if (t_1 <= 0.002) {
		tmp = 1.0;
	} else {
		tmp = x / ((y - z) * t);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	tmp = 0
	if t_1 <= -1000000.0:
		tmp = (x / (y - t)) / z
	elif t_1 <= 0.002:
		tmp = 1.0
	else:
		tmp = x / ((y - z) * t)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = Float64(Float64(x / Float64(y - t)) / z);
	elseif (t_1 <= 0.002)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(Float64(y - z) * t));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	tmp = 0.0;
	if (t_1 <= -1000000.0)
		tmp = (x / (y - t)) / z;
	elseif (t_1 <= 0.002)
		tmp = 1.0;
	else
		tmp = x / ((y - z) * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 0.002], 1.0, N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -1e6

   1. Initial program 96.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 93.2

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

   3. Applied rewrites 93.2%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 61.6

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

   6. Applied rewrites 61.6%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{z + \frac{x}{y - t}}{z} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z + \frac{x}{y - t}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{z + \frac{x}{y - t}}{z} \]

      4. lift--.f64 56.6

         \[\leadsto \frac{z + \frac{x}{y - t}}{z} \]

   9. Applied rewrites 56.6%

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

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{x}{y - t}}{z} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{x}{y - t}}{z} \]

       2. lift--.f64 56.0

          \[\leadsto \frac{\frac{x}{y - t}}{z} \]

   12. Applied rewrites 56.0%

       \[\leadsto \frac{\frac{x}{y - t}}{z} \]

   if -1e6 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 2e-3

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 98.9%

      \[\leadsto \color{blue}{1} \]

   if 2e-3 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 97.1%

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

   2. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 59.5

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 57.6

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

   7. Applied rewrites 57.6%

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

Alternative 10: 88.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;\frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))))
   (if (<= t_1 -1000000.0)
     (/ x (* z (- y t)))
     (if (<= t_1 0.002) 1.0 (/ x (* (- y z) t))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = x / (z * (y - t));
	} else if (t_1 <= 0.002) {
		tmp = 1.0;
	} else {
		tmp = x / ((y - z) * t);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t 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(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) :: t_1
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    if (t_1 <= (-1000000.0d0)) then
        tmp = x / (z * (y - t))
    else if (t_1 <= 0.002d0) then
        tmp = 1.0d0
    else
        tmp = x / ((y - z) * t)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = x / (z * (y - t));
	} else if (t_1 <= 0.002) {
		tmp = 1.0;
	} else {
		tmp = x / ((y - z) * t);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	tmp = 0
	if t_1 <= -1000000.0:
		tmp = x / (z * (y - t))
	elif t_1 <= 0.002:
		tmp = 1.0
	else:
		tmp = x / ((y - z) * t)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = Float64(x / Float64(z * Float64(y - t)));
	elseif (t_1 <= 0.002)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(Float64(y - z) * t));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	tmp = 0.0;
	if (t_1 <= -1000000.0)
		tmp = x / (z * (y - t));
	elseif (t_1 <= 0.002)
		tmp = 1.0;
	else
		tmp = x / ((y - z) * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], N[(x / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.002], 1.0, N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -1e6

   1. Initial program 96.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 93.2

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

   3. Applied rewrites 93.2%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 61.6

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

   6. Applied rewrites 61.6%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 61.1

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

   9. Applied rewrites 61.1%

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

   if -1e6 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 2e-3

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 98.9%

      \[\leadsto \color{blue}{1} \]

   if 2e-3 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 97.1%

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

   2. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 59.5

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 57.6

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

   7. Applied rewrites 57.6%

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

Alternative 11: 88.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ t_2 := \frac{x}{\left(y - z\right) \cdot t}\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))) (t_2 (/ x (* (- y z) t))))
   (if (<= t_1 -1000000.0) t_2 (if (<= t_1 0.002) 1.0 t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / ((y - z) * t);
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.002) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t 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(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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    t_2 = x / ((y - z) * t)
    if (t_1 <= (-1000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 0.002d0) then
        tmp = 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / ((y - z) * t);
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.002) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	t_2 = x / ((y - z) * t)
	tmp = 0
	if t_1 <= -1000000.0:
		tmp = t_2
	elif t_1 <= 0.002:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	t_2 = Float64(x / Float64(Float64(y - z) * t))
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.002)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	t_2 = x / ((y - z) * t);
	tmp = 0.0;
	if (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.002)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], t$95$2, If[LessEqual[t$95$1, 0.002], 1.0, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -1e6 or 2e-3 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 96.9%

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

   2. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 60.0

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

   4. Applied rewrites 60.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 58.8

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

   7. Applied rewrites 58.8%

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

   if -1e6 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 2e-3

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 98.9%

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

Alternative 12: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ t_2 := \frac{x}{-t \cdot z}\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))) (t_2 (/ x (- (* t z)))))
   (if (<= t_1 -1000000.0) t_2 (if (<= t_1 0.002) 1.0 t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / -(t * z);
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.002) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t 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(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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    t_2 = x / -(t * z)
    if (t_1 <= (-1000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 0.002d0) then
        tmp = 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / -(t * z);
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.002) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	t_2 = x / -(t * z)
	tmp = 0
	if t_1 <= -1000000.0:
		tmp = t_2
	elif t_1 <= 0.002:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	t_2 = Float64(x / Float64(-Float64(t * z)))
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.002)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	t_2 = x / -(t * z);
	tmp = 0.0;
	if (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.002)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / (-N[(t * z), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], t$95$2, If[LessEqual[t$95$1, 0.002], 1.0, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -1e6 or 2e-3 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 96.9%

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

   2. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 60.0

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

   4. Applied rewrites 60.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 58.8

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

   7. Applied rewrites 58.8%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto \frac{x}{-t \cdot z} \]

      3. lower-*.f64 43.5

         \[\leadsto \frac{x}{-t \cdot z} \]

   10. Applied rewrites 43.5%

       \[\leadsto \frac{x}{-t \cdot z} \]

   if -1e6 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 2e-3

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 98.9%

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

Alternative 13: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ t_2 := \frac{x}{t \cdot y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+87}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))) (t_2 (/ x (* t y))))
   (if (<= t_1 -5e+87) t_2 (if (<= t_1 0.002) 1.0 t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / (t * y);
	double tmp;
	if (t_1 <= -5e+87) {
		tmp = t_2;
	} else if (t_1 <= 0.002) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t 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(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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    t_2 = x / (t * y)
    if (t_1 <= (-5d+87)) then
        tmp = t_2
    else if (t_1 <= 0.002d0) then
        tmp = 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / (t * y);
	double tmp;
	if (t_1 <= -5e+87) {
		tmp = t_2;
	} else if (t_1 <= 0.002) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	t_2 = x / (t * y)
	tmp = 0
	if t_1 <= -5e+87:
		tmp = t_2
	elif t_1 <= 0.002:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	t_2 = Float64(x / Float64(t * y))
	tmp = 0.0
	if (t_1 <= -5e+87)
		tmp = t_2;
	elseif (t_1 <= 0.002)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	t_2 = x / (t * y);
	tmp = 0.0;
	if (t_1 <= -5e+87)
		tmp = t_2;
	elseif (t_1 <= 0.002)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(t * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+87], t$95$2, If[LessEqual[t$95$1, 0.002], 1.0, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -4.9999999999999998e87 or 2e-3 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 96.4%

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

   2. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 61.0

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

   4. Applied rewrites 61.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 59.9

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

   7. Applied rewrites 59.9%

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

   8. Taylor expanded in y around inf

      \[\leadsto \frac{x}{t \cdot y} \]
   9. Step-by-step derivation

      1. lower-*.f64 28.6

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

   10. Applied rewrites 28.6%

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

   if -4.9999999999999998e87 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 2e-3

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 94.6%

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

Alternative 14: 74.9% accurate, 15.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 1.0)

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0;
}

Fortran

NOTE: x, y, z, and t 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(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 = 1.0d0
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return 1.0
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 74.9%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025128 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, A"
  :precision binary64
  (- 1.0 (/ x (* (- y z) (- y t)))))