Main:z from

Percentage Accurate: 91.1% → 96.9%

Time: 19.1s

Alternatives: 21

Speedup: 0.5×

Specification

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

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

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

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

Initial Program: 91.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

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

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

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

Alternative 1: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ t_2 := \sqrt{z - -1}\\ t_3 := \sqrt{t - -1}\\ t_4 := \sqrt{y - -1}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - t\_2\right)\right) - \left(\sqrt{t} - t\_3\right)\\ \mathbf{elif}\;t\_1 \leq 2.0001:\\ \;\;\;\;\left(\frac{0.5}{t \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{0.5}{z \cdot \sqrt{\frac{1}{z}}}\right) - t\_4\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\left(-\sqrt{x}\right) + t\_4\right) - \left(\left(\sqrt{y} - \left(t\_3 - \sqrt{t}\right)\right) - \left(t\_2 - \sqrt{z}\right)\right)\right)\\ \end{array} \end{array} \]

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
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t))))
        (t_2 (sqrt (- z -1.0)))
        (t_3 (sqrt (- t -1.0)))
        (t_4 (sqrt (- y -1.0))))
   (if (<= t_1 4e-8)
     (- (- (/ 0.5 (* x (sqrt (/ 1.0 x)))) (- (sqrt z) t_2)) (- (sqrt t) t_3))
     (if (<= t_1 2.0001)
       (-
        (- (/ 0.5 (* t (sqrt (/ 1.0 t)))) (- (sqrt x) (sqrt (- x -1.0))))
        (- (- (sqrt y) (/ 0.5 (* z (sqrt (/ 1.0 z))))) t_4))
       (+
        1.0
        (-
         (+ (- (sqrt x)) t_4)
         (- (- (sqrt y) (- t_3 (sqrt t))) (- t_2 (sqrt z)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double t_2 = sqrt((z - -1.0));
	double t_3 = sqrt((t - -1.0));
	double t_4 = sqrt((y - -1.0));
	double tmp;
	if (t_1 <= 4e-8) {
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) - (sqrt(z) - t_2)) - (sqrt(t) - t_3);
	} else if (t_1 <= 2.0001) {
		tmp = ((0.5 / (t * sqrt((1.0 / t)))) - (sqrt(x) - sqrt((x - -1.0)))) - ((sqrt(y) - (0.5 / (z * sqrt((1.0 / z))))) - t_4);
	} else {
		tmp = 1.0 + ((-sqrt(x) + t_4) - ((sqrt(y) - (t_3 - sqrt(t))) - (t_2 - sqrt(z))));
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    t_2 = sqrt((z - (-1.0d0)))
    t_3 = sqrt((t - (-1.0d0)))
    t_4 = sqrt((y - (-1.0d0)))
    if (t_1 <= 4d-8) then
        tmp = ((0.5d0 / (x * sqrt((1.0d0 / x)))) - (sqrt(z) - t_2)) - (sqrt(t) - t_3)
    else if (t_1 <= 2.0001d0) then
        tmp = ((0.5d0 / (t * sqrt((1.0d0 / t)))) - (sqrt(x) - sqrt((x - (-1.0d0))))) - ((sqrt(y) - (0.5d0 / (z * sqrt((1.0d0 / z))))) - t_4)
    else
        tmp = 1.0d0 + ((-sqrt(x) + t_4) - ((sqrt(y) - (t_3 - sqrt(t))) - (t_2 - sqrt(z))))
    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 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	double t_2 = Math.sqrt((z - -1.0));
	double t_3 = Math.sqrt((t - -1.0));
	double t_4 = Math.sqrt((y - -1.0));
	double tmp;
	if (t_1 <= 4e-8) {
		tmp = ((0.5 / (x * Math.sqrt((1.0 / x)))) - (Math.sqrt(z) - t_2)) - (Math.sqrt(t) - t_3);
	} else if (t_1 <= 2.0001) {
		tmp = ((0.5 / (t * Math.sqrt((1.0 / t)))) - (Math.sqrt(x) - Math.sqrt((x - -1.0)))) - ((Math.sqrt(y) - (0.5 / (z * Math.sqrt((1.0 / z))))) - t_4);
	} else {
		tmp = 1.0 + ((-Math.sqrt(x) + t_4) - ((Math.sqrt(y) - (t_3 - Math.sqrt(t))) - (t_2 - Math.sqrt(z))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	t_2 = math.sqrt((z - -1.0))
	t_3 = math.sqrt((t - -1.0))
	t_4 = math.sqrt((y - -1.0))
	tmp = 0
	if t_1 <= 4e-8:
		tmp = ((0.5 / (x * math.sqrt((1.0 / x)))) - (math.sqrt(z) - t_2)) - (math.sqrt(t) - t_3)
	elif t_1 <= 2.0001:
		tmp = ((0.5 / (t * math.sqrt((1.0 / t)))) - (math.sqrt(x) - math.sqrt((x - -1.0)))) - ((math.sqrt(y) - (0.5 / (z * math.sqrt((1.0 / z))))) - t_4)
	else:
		tmp = 1.0 + ((-math.sqrt(x) + t_4) - ((math.sqrt(y) - (t_3 - math.sqrt(t))) - (t_2 - math.sqrt(z))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	t_2 = sqrt(Float64(z - -1.0))
	t_3 = sqrt(Float64(t - -1.0))
	t_4 = sqrt(Float64(y - -1.0))
	tmp = 0.0
	if (t_1 <= 4e-8)
		tmp = Float64(Float64(Float64(0.5 / Float64(x * sqrt(Float64(1.0 / x)))) - Float64(sqrt(z) - t_2)) - Float64(sqrt(t) - t_3));
	elseif (t_1 <= 2.0001)
		tmp = Float64(Float64(Float64(0.5 / Float64(t * sqrt(Float64(1.0 / t)))) - Float64(sqrt(x) - sqrt(Float64(x - -1.0)))) - Float64(Float64(sqrt(y) - Float64(0.5 / Float64(z * sqrt(Float64(1.0 / z))))) - t_4));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(-sqrt(x)) + t_4) - Float64(Float64(sqrt(y) - Float64(t_3 - sqrt(t))) - Float64(t_2 - sqrt(z)))));
	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 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	t_2 = sqrt((z - -1.0));
	t_3 = sqrt((t - -1.0));
	t_4 = sqrt((y - -1.0));
	tmp = 0.0;
	if (t_1 <= 4e-8)
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) - (sqrt(z) - t_2)) - (sqrt(t) - t_3);
	elseif (t_1 <= 2.0001)
		tmp = ((0.5 / (t * sqrt((1.0 / t)))) - (sqrt(x) - sqrt((x - -1.0)))) - ((sqrt(y) - (0.5 / (z * sqrt((1.0 / z))))) - t_4);
	else
		tmp = 1.0 + ((-sqrt(x) + t_4) - ((sqrt(y) - (t_3 - sqrt(t))) - (t_2 - sqrt(z))));
	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[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 4e-8], N[(N[(N[(0.5 / N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0001], N[(N[(N[(0.5 / N[(t * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] - N[(0.5 / N[(z * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[((-N[Sqrt[x], $MachinePrecision]) + t$95$4), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] - N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 4.0000000000000001e-8

   1. Initial program 91.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.7

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

   4. Applied rewrites 36.7%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   6. Applied rewrites 36.7%

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

   7. Taylor expanded in x around inf

      \[\leadsto \left(\frac{\frac{1}{2}}{\color{blue}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \color{blue}{\sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      4. lower-/.f64 11.2

         \[\leadsto \left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

   9. Applied rewrites 11.2%

      \[\leadsto \left(\frac{0.5}{\color{blue}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

   if 4.0000000000000001e-8 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00010000000000021

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{\color{blue}{z \cdot \sqrt{\frac{1}{z}}}}\right) - \sqrt{y - -1}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{z \cdot \color{blue}{\sqrt{\frac{1}{z}}}}\right) - \sqrt{y - -1}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{z \cdot \sqrt{\frac{1}{z}}}\right) - \sqrt{y - -1}\right) \]

      4. lower-/.f64 63.2

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{0.5}{z \cdot \sqrt{\frac{1}{z}}}\right) - \sqrt{y - -1}\right) \]

   6. Applied rewrites 63.2%

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \color{blue}{\frac{0.5}{z \cdot \sqrt{\frac{1}{z}}}}\right) - \sqrt{y - -1}\right) \]

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{\color{blue}{t \cdot \sqrt{\frac{1}{t}}}} - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{z \cdot \sqrt{\frac{1}{z}}}\right) - \sqrt{y - -1}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{t \cdot \color{blue}{\sqrt{\frac{1}{t}}}} - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{z \cdot \sqrt{\frac{1}{z}}}\right) - \sqrt{y - -1}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{t \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{z \cdot \sqrt{\frac{1}{z}}}\right) - \sqrt{y - -1}\right) \]

      4. lower-/.f64 63.4

         \[\leadsto \left(\frac{0.5}{t \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{0.5}{z \cdot \sqrt{\frac{1}{z}}}\right) - \sqrt{y - -1}\right) \]

   9. Applied rewrites 63.4%

      \[\leadsto \left(\color{blue}{\frac{0.5}{t \cdot \sqrt{\frac{1}{t}}}} - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{0.5}{z \cdot \sqrt{\frac{1}{z}}}\right) - \sqrt{y - -1}\right) \]

   if 2.00010000000000021 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-+r- N/A

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

      7. associate-+l- N/A

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

   3. Applied rewrites 90.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

Alternative 2: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z - -1}\\ t_2 := \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \left(t\_2 + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + t\_3\\ t_5 := \sqrt{t - -1}\\ \mathbf{if}\;t\_4 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - t\_1\right)\right) - \left(\sqrt{t} - t\_5\right)\\ \mathbf{elif}\;t\_4 \leq 2.0001:\\ \;\;\;\;\left(t\_2 + \frac{0.5}{z \cdot \sqrt{\frac{1}{z}}}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\left(-\sqrt{x}\right) + \sqrt{y - -1}\right) - \left(\left(\sqrt{y} - \left(t\_5 - \sqrt{t}\right)\right) - \left(t\_1 - \sqrt{z}\right)\right)\right)\\ \end{array} \end{array} \]

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 (sqrt (- z -1.0)))
        (t_2 (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4 (+ (+ t_2 (- (sqrt (+ z 1.0)) (sqrt z))) t_3))
        (t_5 (sqrt (- t -1.0))))
   (if (<= t_4 4e-8)
     (- (- (/ 0.5 (* x (sqrt (/ 1.0 x)))) (- (sqrt z) t_1)) (- (sqrt t) t_5))
     (if (<= t_4 2.0001)
       (+ (+ t_2 (/ 0.5 (* z (sqrt (/ 1.0 z))))) t_3)
       (+
        1.0
        (-
         (+ (- (sqrt x)) (sqrt (- y -1.0)))
         (- (- (sqrt y) (- t_5 (sqrt t))) (- t_1 (sqrt z)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z - -1.0));
	double t_2 = (sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = (t_2 + (sqrt((z + 1.0)) - sqrt(z))) + t_3;
	double t_5 = sqrt((t - -1.0));
	double tmp;
	if (t_4 <= 4e-8) {
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) - (sqrt(z) - t_1)) - (sqrt(t) - t_5);
	} else if (t_4 <= 2.0001) {
		tmp = (t_2 + (0.5 / (z * sqrt((1.0 / z))))) + t_3;
	} else {
		tmp = 1.0 + ((-sqrt(x) + sqrt((y - -1.0))) - ((sqrt(y) - (t_5 - sqrt(t))) - (t_1 - sqrt(z))));
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt((z - (-1.0d0)))
    t_2 = (sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))
    t_3 = sqrt((t + 1.0d0)) - sqrt(t)
    t_4 = (t_2 + (sqrt((z + 1.0d0)) - sqrt(z))) + t_3
    t_5 = sqrt((t - (-1.0d0)))
    if (t_4 <= 4d-8) then
        tmp = ((0.5d0 / (x * sqrt((1.0d0 / x)))) - (sqrt(z) - t_1)) - (sqrt(t) - t_5)
    else if (t_4 <= 2.0001d0) then
        tmp = (t_2 + (0.5d0 / (z * sqrt((1.0d0 / z))))) + t_3
    else
        tmp = 1.0d0 + ((-sqrt(x) + sqrt((y - (-1.0d0)))) - ((sqrt(y) - (t_5 - sqrt(t))) - (t_1 - sqrt(z))))
    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 = Math.sqrt((z - -1.0));
	double t_2 = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y));
	double t_3 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_4 = (t_2 + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + t_3;
	double t_5 = Math.sqrt((t - -1.0));
	double tmp;
	if (t_4 <= 4e-8) {
		tmp = ((0.5 / (x * Math.sqrt((1.0 / x)))) - (Math.sqrt(z) - t_1)) - (Math.sqrt(t) - t_5);
	} else if (t_4 <= 2.0001) {
		tmp = (t_2 + (0.5 / (z * Math.sqrt((1.0 / z))))) + t_3;
	} else {
		tmp = 1.0 + ((-Math.sqrt(x) + Math.sqrt((y - -1.0))) - ((Math.sqrt(y) - (t_5 - Math.sqrt(t))) - (t_1 - Math.sqrt(z))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z - -1.0))
	t_2 = (math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))
	t_3 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_4 = (t_2 + (math.sqrt((z + 1.0)) - math.sqrt(z))) + t_3
	t_5 = math.sqrt((t - -1.0))
	tmp = 0
	if t_4 <= 4e-8:
		tmp = ((0.5 / (x * math.sqrt((1.0 / x)))) - (math.sqrt(z) - t_1)) - (math.sqrt(t) - t_5)
	elif t_4 <= 2.0001:
		tmp = (t_2 + (0.5 / (z * math.sqrt((1.0 / z))))) + t_3
	else:
		tmp = 1.0 + ((-math.sqrt(x) + math.sqrt((y - -1.0))) - ((math.sqrt(y) - (t_5 - math.sqrt(t))) - (t_1 - math.sqrt(z))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z - -1.0))
	t_2 = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = Float64(Float64(t_2 + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + t_3)
	t_5 = sqrt(Float64(t - -1.0))
	tmp = 0.0
	if (t_4 <= 4e-8)
		tmp = Float64(Float64(Float64(0.5 / Float64(x * sqrt(Float64(1.0 / x)))) - Float64(sqrt(z) - t_1)) - Float64(sqrt(t) - t_5));
	elseif (t_4 <= 2.0001)
		tmp = Float64(Float64(t_2 + Float64(0.5 / Float64(z * sqrt(Float64(1.0 / z))))) + t_3);
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(-sqrt(x)) + sqrt(Float64(y - -1.0))) - Float64(Float64(sqrt(y) - Float64(t_5 - sqrt(t))) - Float64(t_1 - sqrt(z)))));
	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 = sqrt((z - -1.0));
	t_2 = (sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	t_3 = sqrt((t + 1.0)) - sqrt(t);
	t_4 = (t_2 + (sqrt((z + 1.0)) - sqrt(z))) + t_3;
	t_5 = sqrt((t - -1.0));
	tmp = 0.0;
	if (t_4 <= 4e-8)
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) - (sqrt(z) - t_1)) - (sqrt(t) - t_5);
	elseif (t_4 <= 2.0001)
		tmp = (t_2 + (0.5 / (z * sqrt((1.0 / z))))) + t_3;
	else
		tmp = 1.0 + ((-sqrt(x) + sqrt((y - -1.0))) - ((sqrt(y) - (t_5 - sqrt(t))) - (t_1 - sqrt(z))));
	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[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 4e-8], N[(N[(N[(0.5 / N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0001], N[(N[(t$95$2 + N[(0.5 / N[(z * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(1.0 + N[(N[((-N[Sqrt[x], $MachinePrecision]) + N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] - N[(t$95$5 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 4.0000000000000001e-8

   1. Initial program 91.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.7

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

   4. Applied rewrites 36.7%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   6. Applied rewrites 36.7%

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

   7. Taylor expanded in x around inf

      \[\leadsto \left(\frac{\frac{1}{2}}{\color{blue}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \color{blue}{\sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      4. lower-/.f64 11.2

         \[\leadsto \left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

   9. Applied rewrites 11.2%

      \[\leadsto \left(\frac{0.5}{\color{blue}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

   if 4.0000000000000001e-8 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00010000000000021

   1. Initial program 91.1%

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

   2. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\frac{1}{2}}{z \cdot \sqrt{\frac{1}{z}}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\frac{1}{2}}{z \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 63.5

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{0.5}{z \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 63.5%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{0.5}{z \cdot \sqrt{\frac{1}{z}}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2.00010000000000021 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-+r- N/A

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

      7. associate-+l- N/A

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

   3. Applied rewrites 90.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

Alternative 3: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ t_2 := \sqrt{z - -1}\\ t_3 := \sqrt{y - -1}\\ t_4 := \sqrt{t - -1}\\ t_5 := t\_4 - \sqrt{t}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - t\_2\right)\right) - \left(\sqrt{t} - t\_4\right)\\ \mathbf{elif}\;t\_1 \leq 2.0001:\\ \;\;\;\;\left(t\_5 - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{0.5}{\sqrt{z}}\right) - t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\left(-\sqrt{x}\right) + t\_3\right) - \left(\left(\sqrt{y} - t\_5\right) - \left(t\_2 - \sqrt{z}\right)\right)\right)\\ \end{array} \end{array} \]

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
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t))))
        (t_2 (sqrt (- z -1.0)))
        (t_3 (sqrt (- y -1.0)))
        (t_4 (sqrt (- t -1.0)))
        (t_5 (- t_4 (sqrt t))))
   (if (<= t_1 4e-8)
     (- (- (/ 0.5 (* x (sqrt (/ 1.0 x)))) (- (sqrt z) t_2)) (- (sqrt t) t_4))
     (if (<= t_1 2.0001)
       (-
        (- t_5 (- (sqrt x) (sqrt (- x -1.0))))
        (- (- (sqrt y) (/ 0.5 (sqrt z))) t_3))
       (+
        1.0
        (- (+ (- (sqrt x)) t_3) (- (- (sqrt y) t_5) (- t_2 (sqrt z)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double t_2 = sqrt((z - -1.0));
	double t_3 = sqrt((y - -1.0));
	double t_4 = sqrt((t - -1.0));
	double t_5 = t_4 - sqrt(t);
	double tmp;
	if (t_1 <= 4e-8) {
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) - (sqrt(z) - t_2)) - (sqrt(t) - t_4);
	} else if (t_1 <= 2.0001) {
		tmp = (t_5 - (sqrt(x) - sqrt((x - -1.0)))) - ((sqrt(y) - (0.5 / sqrt(z))) - t_3);
	} else {
		tmp = 1.0 + ((-sqrt(x) + t_3) - ((sqrt(y) - t_5) - (t_2 - sqrt(z))));
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    t_2 = sqrt((z - (-1.0d0)))
    t_3 = sqrt((y - (-1.0d0)))
    t_4 = sqrt((t - (-1.0d0)))
    t_5 = t_4 - sqrt(t)
    if (t_1 <= 4d-8) then
        tmp = ((0.5d0 / (x * sqrt((1.0d0 / x)))) - (sqrt(z) - t_2)) - (sqrt(t) - t_4)
    else if (t_1 <= 2.0001d0) then
        tmp = (t_5 - (sqrt(x) - sqrt((x - (-1.0d0))))) - ((sqrt(y) - (0.5d0 / sqrt(z))) - t_3)
    else
        tmp = 1.0d0 + ((-sqrt(x) + t_3) - ((sqrt(y) - t_5) - (t_2 - sqrt(z))))
    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 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	double t_2 = Math.sqrt((z - -1.0));
	double t_3 = Math.sqrt((y - -1.0));
	double t_4 = Math.sqrt((t - -1.0));
	double t_5 = t_4 - Math.sqrt(t);
	double tmp;
	if (t_1 <= 4e-8) {
		tmp = ((0.5 / (x * Math.sqrt((1.0 / x)))) - (Math.sqrt(z) - t_2)) - (Math.sqrt(t) - t_4);
	} else if (t_1 <= 2.0001) {
		tmp = (t_5 - (Math.sqrt(x) - Math.sqrt((x - -1.0)))) - ((Math.sqrt(y) - (0.5 / Math.sqrt(z))) - t_3);
	} else {
		tmp = 1.0 + ((-Math.sqrt(x) + t_3) - ((Math.sqrt(y) - t_5) - (t_2 - Math.sqrt(z))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	t_2 = math.sqrt((z - -1.0))
	t_3 = math.sqrt((y - -1.0))
	t_4 = math.sqrt((t - -1.0))
	t_5 = t_4 - math.sqrt(t)
	tmp = 0
	if t_1 <= 4e-8:
		tmp = ((0.5 / (x * math.sqrt((1.0 / x)))) - (math.sqrt(z) - t_2)) - (math.sqrt(t) - t_4)
	elif t_1 <= 2.0001:
		tmp = (t_5 - (math.sqrt(x) - math.sqrt((x - -1.0)))) - ((math.sqrt(y) - (0.5 / math.sqrt(z))) - t_3)
	else:
		tmp = 1.0 + ((-math.sqrt(x) + t_3) - ((math.sqrt(y) - t_5) - (t_2 - math.sqrt(z))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	t_2 = sqrt(Float64(z - -1.0))
	t_3 = sqrt(Float64(y - -1.0))
	t_4 = sqrt(Float64(t - -1.0))
	t_5 = Float64(t_4 - sqrt(t))
	tmp = 0.0
	if (t_1 <= 4e-8)
		tmp = Float64(Float64(Float64(0.5 / Float64(x * sqrt(Float64(1.0 / x)))) - Float64(sqrt(z) - t_2)) - Float64(sqrt(t) - t_4));
	elseif (t_1 <= 2.0001)
		tmp = Float64(Float64(t_5 - Float64(sqrt(x) - sqrt(Float64(x - -1.0)))) - Float64(Float64(sqrt(y) - Float64(0.5 / sqrt(z))) - t_3));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(-sqrt(x)) + t_3) - Float64(Float64(sqrt(y) - t_5) - Float64(t_2 - sqrt(z)))));
	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 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	t_2 = sqrt((z - -1.0));
	t_3 = sqrt((y - -1.0));
	t_4 = sqrt((t - -1.0));
	t_5 = t_4 - sqrt(t);
	tmp = 0.0;
	if (t_1 <= 4e-8)
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) - (sqrt(z) - t_2)) - (sqrt(t) - t_4);
	elseif (t_1 <= 2.0001)
		tmp = (t_5 - (sqrt(x) - sqrt((x - -1.0)))) - ((sqrt(y) - (0.5 / sqrt(z))) - t_3);
	else
		tmp = 1.0 + ((-sqrt(x) + t_3) - ((sqrt(y) - t_5) - (t_2 - sqrt(z))));
	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[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-8], N[(N[(N[(0.5 / N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0001], N[(N[(t$95$5 - N[(N[Sqrt[x], $MachinePrecision] - N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] - N[(0.5 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[((-N[Sqrt[x], $MachinePrecision]) + t$95$3), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] - t$95$5), $MachinePrecision] - N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 4.0000000000000001e-8

   1. Initial program 91.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.7

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

   4. Applied rewrites 36.7%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   6. Applied rewrites 36.7%

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

   7. Taylor expanded in x around inf

      \[\leadsto \left(\frac{\frac{1}{2}}{\color{blue}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \color{blue}{\sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      4. lower-/.f64 11.2

         \[\leadsto \left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

   9. Applied rewrites 11.2%

      \[\leadsto \left(\frac{0.5}{\color{blue}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

   if 4.0000000000000001e-8 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00010000000000021

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{\color{blue}{z \cdot \sqrt{\frac{1}{z}}}}\right) - \sqrt{y - -1}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{z \cdot \color{blue}{\sqrt{\frac{1}{z}}}}\right) - \sqrt{y - -1}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{z \cdot \sqrt{\frac{1}{z}}}\right) - \sqrt{y - -1}\right) \]

      4. lower-/.f64 63.2

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{0.5}{z \cdot \sqrt{\frac{1}{z}}}\right) - \sqrt{y - -1}\right) \]

   6. Applied rewrites 63.2%

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \color{blue}{\frac{0.5}{z \cdot \sqrt{\frac{1}{z}}}}\right) - \sqrt{y - -1}\right) \]

   7. Taylor expanded in z around 0

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{\color{blue}{\sqrt{z}}}\right) - \sqrt{y - -1}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{\sqrt{z}}\right) - \sqrt{y - -1}\right) \]

      2. lower-sqrt.f64 63.2

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{0.5}{\sqrt{z}}\right) - \sqrt{y - -1}\right) \]

   9. Applied rewrites 63.2%

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{0.5}{\color{blue}{\sqrt{z}}}\right) - \sqrt{y - -1}\right) \]

   if 2.00010000000000021 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-+r- N/A

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

      7. associate-+l- N/A

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

   3. Applied rewrites 90.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

Alternative 4: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ t_2 := \sqrt{z - -1}\\ t_3 := \sqrt{y - -1}\\ t_4 := \sqrt{t - -1}\\ t_5 := t\_4 - \sqrt{t}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - t\_2\right)\right) - \left(\sqrt{t} - t\_4\right)\\ \mathbf{elif}\;t\_1 \leq 2.0001:\\ \;\;\;\;\left(t\_5 - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{0.5}{\sqrt{z}}\right) - t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_5 - \left(\sqrt{x} - 1\right)\right) - \left(\left(\sqrt{y} - \left(t\_2 - \sqrt{z}\right)\right) - t\_3\right)\\ \end{array} \end{array} \]

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
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t))))
        (t_2 (sqrt (- z -1.0)))
        (t_3 (sqrt (- y -1.0)))
        (t_4 (sqrt (- t -1.0)))
        (t_5 (- t_4 (sqrt t))))
   (if (<= t_1 4e-8)
     (- (- (/ 0.5 (* x (sqrt (/ 1.0 x)))) (- (sqrt z) t_2)) (- (sqrt t) t_4))
     (if (<= t_1 2.0001)
       (-
        (- t_5 (- (sqrt x) (sqrt (- x -1.0))))
        (- (- (sqrt y) (/ 0.5 (sqrt z))) t_3))
       (- (- t_5 (- (sqrt x) 1.0)) (- (- (sqrt y) (- t_2 (sqrt z))) t_3))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double t_2 = sqrt((z - -1.0));
	double t_3 = sqrt((y - -1.0));
	double t_4 = sqrt((t - -1.0));
	double t_5 = t_4 - sqrt(t);
	double tmp;
	if (t_1 <= 4e-8) {
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) - (sqrt(z) - t_2)) - (sqrt(t) - t_4);
	} else if (t_1 <= 2.0001) {
		tmp = (t_5 - (sqrt(x) - sqrt((x - -1.0)))) - ((sqrt(y) - (0.5 / sqrt(z))) - t_3);
	} else {
		tmp = (t_5 - (sqrt(x) - 1.0)) - ((sqrt(y) - (t_2 - sqrt(z))) - t_3);
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    t_2 = sqrt((z - (-1.0d0)))
    t_3 = sqrt((y - (-1.0d0)))
    t_4 = sqrt((t - (-1.0d0)))
    t_5 = t_4 - sqrt(t)
    if (t_1 <= 4d-8) then
        tmp = ((0.5d0 / (x * sqrt((1.0d0 / x)))) - (sqrt(z) - t_2)) - (sqrt(t) - t_4)
    else if (t_1 <= 2.0001d0) then
        tmp = (t_5 - (sqrt(x) - sqrt((x - (-1.0d0))))) - ((sqrt(y) - (0.5d0 / sqrt(z))) - t_3)
    else
        tmp = (t_5 - (sqrt(x) - 1.0d0)) - ((sqrt(y) - (t_2 - sqrt(z))) - t_3)
    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 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	double t_2 = Math.sqrt((z - -1.0));
	double t_3 = Math.sqrt((y - -1.0));
	double t_4 = Math.sqrt((t - -1.0));
	double t_5 = t_4 - Math.sqrt(t);
	double tmp;
	if (t_1 <= 4e-8) {
		tmp = ((0.5 / (x * Math.sqrt((1.0 / x)))) - (Math.sqrt(z) - t_2)) - (Math.sqrt(t) - t_4);
	} else if (t_1 <= 2.0001) {
		tmp = (t_5 - (Math.sqrt(x) - Math.sqrt((x - -1.0)))) - ((Math.sqrt(y) - (0.5 / Math.sqrt(z))) - t_3);
	} else {
		tmp = (t_5 - (Math.sqrt(x) - 1.0)) - ((Math.sqrt(y) - (t_2 - Math.sqrt(z))) - t_3);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	t_2 = math.sqrt((z - -1.0))
	t_3 = math.sqrt((y - -1.0))
	t_4 = math.sqrt((t - -1.0))
	t_5 = t_4 - math.sqrt(t)
	tmp = 0
	if t_1 <= 4e-8:
		tmp = ((0.5 / (x * math.sqrt((1.0 / x)))) - (math.sqrt(z) - t_2)) - (math.sqrt(t) - t_4)
	elif t_1 <= 2.0001:
		tmp = (t_5 - (math.sqrt(x) - math.sqrt((x - -1.0)))) - ((math.sqrt(y) - (0.5 / math.sqrt(z))) - t_3)
	else:
		tmp = (t_5 - (math.sqrt(x) - 1.0)) - ((math.sqrt(y) - (t_2 - math.sqrt(z))) - t_3)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	t_2 = sqrt(Float64(z - -1.0))
	t_3 = sqrt(Float64(y - -1.0))
	t_4 = sqrt(Float64(t - -1.0))
	t_5 = Float64(t_4 - sqrt(t))
	tmp = 0.0
	if (t_1 <= 4e-8)
		tmp = Float64(Float64(Float64(0.5 / Float64(x * sqrt(Float64(1.0 / x)))) - Float64(sqrt(z) - t_2)) - Float64(sqrt(t) - t_4));
	elseif (t_1 <= 2.0001)
		tmp = Float64(Float64(t_5 - Float64(sqrt(x) - sqrt(Float64(x - -1.0)))) - Float64(Float64(sqrt(y) - Float64(0.5 / sqrt(z))) - t_3));
	else
		tmp = Float64(Float64(t_5 - Float64(sqrt(x) - 1.0)) - Float64(Float64(sqrt(y) - Float64(t_2 - sqrt(z))) - t_3));
	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 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	t_2 = sqrt((z - -1.0));
	t_3 = sqrt((y - -1.0));
	t_4 = sqrt((t - -1.0));
	t_5 = t_4 - sqrt(t);
	tmp = 0.0;
	if (t_1 <= 4e-8)
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) - (sqrt(z) - t_2)) - (sqrt(t) - t_4);
	elseif (t_1 <= 2.0001)
		tmp = (t_5 - (sqrt(x) - sqrt((x - -1.0)))) - ((sqrt(y) - (0.5 / sqrt(z))) - t_3);
	else
		tmp = (t_5 - (sqrt(x) - 1.0)) - ((sqrt(y) - (t_2 - sqrt(z))) - t_3);
	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[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-8], N[(N[(N[(0.5 / N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0001], N[(N[(t$95$5 - N[(N[Sqrt[x], $MachinePrecision] - N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] - N[(0.5 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$5 - N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] - N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 4.0000000000000001e-8

   1. Initial program 91.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.7

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

   4. Applied rewrites 36.7%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   6. Applied rewrites 36.7%

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

   7. Taylor expanded in x around inf

      \[\leadsto \left(\frac{\frac{1}{2}}{\color{blue}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \color{blue}{\sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      4. lower-/.f64 11.2

         \[\leadsto \left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

   9. Applied rewrites 11.2%

      \[\leadsto \left(\frac{0.5}{\color{blue}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

   if 4.0000000000000001e-8 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00010000000000021

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{\color{blue}{z \cdot \sqrt{\frac{1}{z}}}}\right) - \sqrt{y - -1}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{z \cdot \color{blue}{\sqrt{\frac{1}{z}}}}\right) - \sqrt{y - -1}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{z \cdot \sqrt{\frac{1}{z}}}\right) - \sqrt{y - -1}\right) \]

      4. lower-/.f64 63.2

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{0.5}{z \cdot \sqrt{\frac{1}{z}}}\right) - \sqrt{y - -1}\right) \]

   6. Applied rewrites 63.2%

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \color{blue}{\frac{0.5}{z \cdot \sqrt{\frac{1}{z}}}}\right) - \sqrt{y - -1}\right) \]

   7. Taylor expanded in z around 0

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{\color{blue}{\sqrt{z}}}\right) - \sqrt{y - -1}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{\frac{1}{2}}{\sqrt{z}}\right) - \sqrt{y - -1}\right) \]

      2. lower-sqrt.f64 63.2

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{0.5}{\sqrt{z}}\right) - \sqrt{y - -1}\right) \]

   9. Applied rewrites 63.2%

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \frac{0.5}{\color{blue}{\sqrt{z}}}\right) - \sqrt{y - -1}\right) \]

   if 2.00010000000000021 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

Alternative 5: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z - -1}\\ t_2 := \sqrt{t - -1}\\ \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 0.2:\\ \;\;\;\;\left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - t\_1\right)\right) - \left(\sqrt{t} - t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_2 - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\left(\sqrt{y} - \left(t\_1 - \sqrt{z}\right)\right) - \sqrt{y - -1}\right)\\ \end{array} \end{array} \]

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 (sqrt (- z -1.0))) (t_2 (sqrt (- t -1.0))))
   (if (<=
        (+
         (+
          (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
          (- (sqrt (+ z 1.0)) (sqrt z)))
         (- (sqrt (+ t 1.0)) (sqrt t)))
        0.2)
     (- (- (/ 0.5 (* x (sqrt (/ 1.0 x)))) (- (sqrt z) t_1)) (- (sqrt t) t_2))
     (-
      (- (- t_2 (sqrt t)) (- (sqrt x) 1.0))
      (- (- (sqrt y) (- t_1 (sqrt z))) (sqrt (- y -1.0)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z - -1.0));
	double t_2 = sqrt((t - -1.0));
	double tmp;
	if (((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t))) <= 0.2) {
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) - (sqrt(z) - t_1)) - (sqrt(t) - t_2);
	} else {
		tmp = ((t_2 - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) - (t_1 - sqrt(z))) - sqrt((y - -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) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z - (-1.0d0)))
    t_2 = sqrt((t - (-1.0d0)))
    if (((((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))) <= 0.2d0) then
        tmp = ((0.5d0 / (x * sqrt((1.0d0 / x)))) - (sqrt(z) - t_1)) - (sqrt(t) - t_2)
    else
        tmp = ((t_2 - sqrt(t)) - (sqrt(x) - 1.0d0)) - ((sqrt(y) - (t_1 - sqrt(z))) - sqrt((y - (-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 = Math.sqrt((z - -1.0));
	double t_2 = Math.sqrt((t - -1.0));
	double tmp;
	if (((((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t))) <= 0.2) {
		tmp = ((0.5 / (x * Math.sqrt((1.0 / x)))) - (Math.sqrt(z) - t_1)) - (Math.sqrt(t) - t_2);
	} else {
		tmp = ((t_2 - Math.sqrt(t)) - (Math.sqrt(x) - 1.0)) - ((Math.sqrt(y) - (t_1 - Math.sqrt(z))) - Math.sqrt((y - -1.0)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z - -1.0))
	t_2 = math.sqrt((t - -1.0))
	tmp = 0
	if ((((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))) <= 0.2:
		tmp = ((0.5 / (x * math.sqrt((1.0 / x)))) - (math.sqrt(z) - t_1)) - (math.sqrt(t) - t_2)
	else:
		tmp = ((t_2 - math.sqrt(t)) - (math.sqrt(x) - 1.0)) - ((math.sqrt(y) - (t_1 - math.sqrt(z))) - math.sqrt((y - -1.0)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z - -1.0))
	t_2 = sqrt(Float64(t - -1.0))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t))) <= 0.2)
		tmp = Float64(Float64(Float64(0.5 / Float64(x * sqrt(Float64(1.0 / x)))) - Float64(sqrt(z) - t_1)) - Float64(sqrt(t) - t_2));
	else
		tmp = Float64(Float64(Float64(t_2 - sqrt(t)) - Float64(sqrt(x) - 1.0)) - Float64(Float64(sqrt(y) - Float64(t_1 - sqrt(z))) - sqrt(Float64(y - -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 = sqrt((z - -1.0));
	t_2 = sqrt((t - -1.0));
	tmp = 0.0;
	if (((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t))) <= 0.2)
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) - (sqrt(z) - t_1)) - (sqrt(t) - t_2);
	else
		tmp = ((t_2 - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) - (t_1 - sqrt(z))) - sqrt((y - -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[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.2], N[(N[(N[(0.5 / N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] - N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 0.20000000000000001

   1. Initial program 91.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.7

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

   4. Applied rewrites 36.7%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   6. Applied rewrites 36.7%

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

   7. Taylor expanded in x around inf

      \[\leadsto \left(\frac{\frac{1}{2}}{\color{blue}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \color{blue}{\sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      4. lower-/.f64 11.2

         \[\leadsto \left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

   9. Applied rewrites 11.2%

      \[\leadsto \left(\frac{0.5}{\color{blue}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

   if 0.20000000000000001 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

Alternative 6: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x - -1}\\ t_2 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ t_3 := \sqrt{z - -1}\\ t_4 := \sqrt{t - -1}\\ \mathbf{if}\;t\_2 \leq 0.2:\\ \;\;\;\;\left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - t\_3\right)\right) - \left(\sqrt{t} - t\_4\right)\\ \mathbf{elif}\;t\_2 \leq 2.98:\\ \;\;\;\;t\_1 - \left(\left(-\sqrt{y - -1}\right) - \left(\left(\left(t\_3 - \sqrt{z}\right) - \sqrt{y}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_4 - \sqrt{t}\right) - \left(\sqrt{x} - t\_1\right)\right) - \left(\left(\sqrt{y} + \left(\sqrt{z} + -0.5 \cdot z\right)\right) - 2\right)\\ \end{array} \end{array} \]

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 (sqrt (- x -1.0)))
        (t_2
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t))))
        (t_3 (sqrt (- z -1.0)))
        (t_4 (sqrt (- t -1.0))))
   (if (<= t_2 0.2)
     (- (- (/ 0.5 (* x (sqrt (/ 1.0 x)))) (- (sqrt z) t_3)) (- (sqrt t) t_4))
     (if (<= t_2 2.98)
       (-
        t_1
        (- (- (sqrt (- y -1.0))) (- (- (- t_3 (sqrt z)) (sqrt y)) (sqrt x))))
       (-
        (- (- t_4 (sqrt t)) (- (sqrt x) t_1))
        (- (+ (sqrt y) (+ (sqrt z) (* -0.5 z))) 2.0))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x - -1.0));
	double t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double t_3 = sqrt((z - -1.0));
	double t_4 = sqrt((t - -1.0));
	double tmp;
	if (t_2 <= 0.2) {
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) - (sqrt(z) - t_3)) - (sqrt(t) - t_4);
	} else if (t_2 <= 2.98) {
		tmp = t_1 - (-sqrt((y - -1.0)) - (((t_3 - sqrt(z)) - sqrt(y)) - sqrt(x)));
	} else {
		tmp = ((t_4 - sqrt(t)) - (sqrt(x) - t_1)) - ((sqrt(y) + (sqrt(z) + (-0.5 * z))) - 2.0);
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((x - (-1.0d0)))
    t_2 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    t_3 = sqrt((z - (-1.0d0)))
    t_4 = sqrt((t - (-1.0d0)))
    if (t_2 <= 0.2d0) then
        tmp = ((0.5d0 / (x * sqrt((1.0d0 / x)))) - (sqrt(z) - t_3)) - (sqrt(t) - t_4)
    else if (t_2 <= 2.98d0) then
        tmp = t_1 - (-sqrt((y - (-1.0d0))) - (((t_3 - sqrt(z)) - sqrt(y)) - sqrt(x)))
    else
        tmp = ((t_4 - sqrt(t)) - (sqrt(x) - t_1)) - ((sqrt(y) + (sqrt(z) + ((-0.5d0) * z))) - 2.0d0)
    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 = Math.sqrt((x - -1.0));
	double t_2 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	double t_3 = Math.sqrt((z - -1.0));
	double t_4 = Math.sqrt((t - -1.0));
	double tmp;
	if (t_2 <= 0.2) {
		tmp = ((0.5 / (x * Math.sqrt((1.0 / x)))) - (Math.sqrt(z) - t_3)) - (Math.sqrt(t) - t_4);
	} else if (t_2 <= 2.98) {
		tmp = t_1 - (-Math.sqrt((y - -1.0)) - (((t_3 - Math.sqrt(z)) - Math.sqrt(y)) - Math.sqrt(x)));
	} else {
		tmp = ((t_4 - Math.sqrt(t)) - (Math.sqrt(x) - t_1)) - ((Math.sqrt(y) + (Math.sqrt(z) + (-0.5 * z))) - 2.0);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x - -1.0))
	t_2 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	t_3 = math.sqrt((z - -1.0))
	t_4 = math.sqrt((t - -1.0))
	tmp = 0
	if t_2 <= 0.2:
		tmp = ((0.5 / (x * math.sqrt((1.0 / x)))) - (math.sqrt(z) - t_3)) - (math.sqrt(t) - t_4)
	elif t_2 <= 2.98:
		tmp = t_1 - (-math.sqrt((y - -1.0)) - (((t_3 - math.sqrt(z)) - math.sqrt(y)) - math.sqrt(x)))
	else:
		tmp = ((t_4 - math.sqrt(t)) - (math.sqrt(x) - t_1)) - ((math.sqrt(y) + (math.sqrt(z) + (-0.5 * z))) - 2.0)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x - -1.0))
	t_2 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	t_3 = sqrt(Float64(z - -1.0))
	t_4 = sqrt(Float64(t - -1.0))
	tmp = 0.0
	if (t_2 <= 0.2)
		tmp = Float64(Float64(Float64(0.5 / Float64(x * sqrt(Float64(1.0 / x)))) - Float64(sqrt(z) - t_3)) - Float64(sqrt(t) - t_4));
	elseif (t_2 <= 2.98)
		tmp = Float64(t_1 - Float64(Float64(-sqrt(Float64(y - -1.0))) - Float64(Float64(Float64(t_3 - sqrt(z)) - sqrt(y)) - sqrt(x))));
	else
		tmp = Float64(Float64(Float64(t_4 - sqrt(t)) - Float64(sqrt(x) - t_1)) - Float64(Float64(sqrt(y) + Float64(sqrt(z) + Float64(-0.5 * z))) - 2.0));
	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 = sqrt((x - -1.0));
	t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	t_3 = sqrt((z - -1.0));
	t_4 = sqrt((t - -1.0));
	tmp = 0.0;
	if (t_2 <= 0.2)
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) - (sqrt(z) - t_3)) - (sqrt(t) - t_4);
	elseif (t_2 <= 2.98)
		tmp = t_1 - (-sqrt((y - -1.0)) - (((t_3 - sqrt(z)) - sqrt(y)) - sqrt(x)));
	else
		tmp = ((t_4 - sqrt(t)) - (sqrt(x) - t_1)) - ((sqrt(y) + (sqrt(z) + (-0.5 * z))) - 2.0);
	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[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.2], N[(N[(N[(0.5 / N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.98], N[(t$95$1 - N[((-N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]) - N[(N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 0.20000000000000001

   1. Initial program 91.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.7

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

   4. Applied rewrites 36.7%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   6. Applied rewrites 36.7%

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

   7. Taylor expanded in x around inf

      \[\leadsto \left(\frac{\frac{1}{2}}{\color{blue}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \color{blue}{\sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      4. lower-/.f64 11.2

         \[\leadsto \left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

   9. Applied rewrites 11.2%

      \[\leadsto \left(\frac{0.5}{\color{blue}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

   if 0.20000000000000001 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.97999999999999998

   1. Initial program 91.1%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 32.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 60.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 60.8

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate--r+ N/A

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

      11. lower--.f64 N/A

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

   8. Applied rewrites 84.2%

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

   if 2.97999999999999998 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in z around 0

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \color{blue}{\left(\left(\sqrt{y} + \left(\sqrt{z} + \frac{-1}{2} \cdot z\right)\right) - \left(1 + \sqrt{1 + y}\right)\right)} \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} + \left(\sqrt{z} + \frac{-1}{2} \cdot z\right)\right) - \left(1 + \sqrt{1 + y}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} + \left(\sqrt{z} + \frac{-1}{2} \cdot z\right)\right) - \left(1 + \sqrt{1 + y}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} + \left(\sqrt{z} + \frac{-1}{2} \cdot z\right)\right) - \left(1 + \sqrt{1 + y}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} + \left(\sqrt{z} + \frac{-1}{2} \cdot z\right)\right) - \left(1 + \sqrt{1 + y}\right)\right) \]

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} + \left(\sqrt{z} + \frac{-1}{2} \cdot z\right)\right) - \left(1 + \sqrt{1 + y}\right)\right) \]

      9. lower-+.f64 34.1

         \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} + \left(\sqrt{z} + -0.5 \cdot z\right)\right) - \left(1 + \sqrt{1 + y}\right)\right) \]

   6. Applied rewrites 34.1%

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \color{blue}{\left(\left(\sqrt{y} + \left(\sqrt{z} + -0.5 \cdot z\right)\right) - \left(1 + \sqrt{1 + y}\right)\right)} \]

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 34.1%

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} + \left(\sqrt{z} + -0.5 \cdot z\right)\right) - 2\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y - -1}\\ t_2 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ t_3 := \sqrt{z - -1}\\ t_4 := \sqrt{t - -1}\\ \mathbf{if}\;t\_2 \leq 0.2:\\ \;\;\;\;\left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - t\_3\right)\right) - \left(\sqrt{t} - t\_4\right)\\ \mathbf{elif}\;t\_2 \leq 2.99999999999:\\ \;\;\;\;\sqrt{x - -1} - \left(\left(-t\_1\right) - \left(\left(\left(t\_3 - \sqrt{z}\right) - \sqrt{y}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_4 - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\left(\sqrt{y} - \left(1 - \sqrt{z}\right)\right) - t\_1\right)\\ \end{array} \end{array} \]

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 (sqrt (- y -1.0)))
        (t_2
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t))))
        (t_3 (sqrt (- z -1.0)))
        (t_4 (sqrt (- t -1.0))))
   (if (<= t_2 0.2)
     (- (- (/ 0.5 (* x (sqrt (/ 1.0 x)))) (- (sqrt z) t_3)) (- (sqrt t) t_4))
     (if (<= t_2 2.99999999999)
       (-
        (sqrt (- x -1.0))
        (- (- t_1) (- (- (- t_3 (sqrt z)) (sqrt y)) (sqrt x))))
       (-
        (- (- t_4 (sqrt t)) (- (sqrt x) 1.0))
        (- (- (sqrt y) (- 1.0 (sqrt z))) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y - -1.0));
	double t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double t_3 = sqrt((z - -1.0));
	double t_4 = sqrt((t - -1.0));
	double tmp;
	if (t_2 <= 0.2) {
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) - (sqrt(z) - t_3)) - (sqrt(t) - t_4);
	} else if (t_2 <= 2.99999999999) {
		tmp = sqrt((x - -1.0)) - (-t_1 - (((t_3 - sqrt(z)) - sqrt(y)) - sqrt(x)));
	} else {
		tmp = ((t_4 - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) - (1.0 - sqrt(z))) - t_1);
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((y - (-1.0d0)))
    t_2 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    t_3 = sqrt((z - (-1.0d0)))
    t_4 = sqrt((t - (-1.0d0)))
    if (t_2 <= 0.2d0) then
        tmp = ((0.5d0 / (x * sqrt((1.0d0 / x)))) - (sqrt(z) - t_3)) - (sqrt(t) - t_4)
    else if (t_2 <= 2.99999999999d0) then
        tmp = sqrt((x - (-1.0d0))) - (-t_1 - (((t_3 - sqrt(z)) - sqrt(y)) - sqrt(x)))
    else
        tmp = ((t_4 - sqrt(t)) - (sqrt(x) - 1.0d0)) - ((sqrt(y) - (1.0d0 - sqrt(z))) - t_1)
    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 = Math.sqrt((y - -1.0));
	double t_2 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	double t_3 = Math.sqrt((z - -1.0));
	double t_4 = Math.sqrt((t - -1.0));
	double tmp;
	if (t_2 <= 0.2) {
		tmp = ((0.5 / (x * Math.sqrt((1.0 / x)))) - (Math.sqrt(z) - t_3)) - (Math.sqrt(t) - t_4);
	} else if (t_2 <= 2.99999999999) {
		tmp = Math.sqrt((x - -1.0)) - (-t_1 - (((t_3 - Math.sqrt(z)) - Math.sqrt(y)) - Math.sqrt(x)));
	} else {
		tmp = ((t_4 - Math.sqrt(t)) - (Math.sqrt(x) - 1.0)) - ((Math.sqrt(y) - (1.0 - Math.sqrt(z))) - t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y - -1.0))
	t_2 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	t_3 = math.sqrt((z - -1.0))
	t_4 = math.sqrt((t - -1.0))
	tmp = 0
	if t_2 <= 0.2:
		tmp = ((0.5 / (x * math.sqrt((1.0 / x)))) - (math.sqrt(z) - t_3)) - (math.sqrt(t) - t_4)
	elif t_2 <= 2.99999999999:
		tmp = math.sqrt((x - -1.0)) - (-t_1 - (((t_3 - math.sqrt(z)) - math.sqrt(y)) - math.sqrt(x)))
	else:
		tmp = ((t_4 - math.sqrt(t)) - (math.sqrt(x) - 1.0)) - ((math.sqrt(y) - (1.0 - math.sqrt(z))) - t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y - -1.0))
	t_2 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	t_3 = sqrt(Float64(z - -1.0))
	t_4 = sqrt(Float64(t - -1.0))
	tmp = 0.0
	if (t_2 <= 0.2)
		tmp = Float64(Float64(Float64(0.5 / Float64(x * sqrt(Float64(1.0 / x)))) - Float64(sqrt(z) - t_3)) - Float64(sqrt(t) - t_4));
	elseif (t_2 <= 2.99999999999)
		tmp = Float64(sqrt(Float64(x - -1.0)) - Float64(Float64(-t_1) - Float64(Float64(Float64(t_3 - sqrt(z)) - sqrt(y)) - sqrt(x))));
	else
		tmp = Float64(Float64(Float64(t_4 - sqrt(t)) - Float64(sqrt(x) - 1.0)) - Float64(Float64(sqrt(y) - Float64(1.0 - sqrt(z))) - t_1));
	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 = sqrt((y - -1.0));
	t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	t_3 = sqrt((z - -1.0));
	t_4 = sqrt((t - -1.0));
	tmp = 0.0;
	if (t_2 <= 0.2)
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) - (sqrt(z) - t_3)) - (sqrt(t) - t_4);
	elseif (t_2 <= 2.99999999999)
		tmp = sqrt((x - -1.0)) - (-t_1 - (((t_3 - sqrt(z)) - sqrt(y)) - sqrt(x)));
	else
		tmp = ((t_4 - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) - (1.0 - sqrt(z))) - t_1);
	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[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.2], N[(N[(N[(0.5 / N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.99999999999], N[(N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] - N[((-t$95$1) - N[(N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] - N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 0.20000000000000001

   1. Initial program 91.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.7

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

   4. Applied rewrites 36.7%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   6. Applied rewrites 36.7%

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

   7. Taylor expanded in x around inf

      \[\leadsto \left(\frac{\frac{1}{2}}{\color{blue}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \color{blue}{\sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

      4. lower-/.f64 11.2

         \[\leadsto \left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

   9. Applied rewrites 11.2%

      \[\leadsto \left(\frac{0.5}{\color{blue}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{z} - \sqrt{z - -1}\right)\right) - \left(\sqrt{t} - \sqrt{t - -1}\right) \]

   if 0.20000000000000001 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.99999999999

   1. Initial program 91.1%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 32.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 60.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 60.8

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate--r+ N/A

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

      11. lower--.f64 N/A

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

   8. Applied rewrites 84.2%

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

   if 2.99999999999 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 30.5

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

   9. Applied rewrites 30.5%

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

Alternative 8: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ t_2 := \sqrt{x - -1}\\ \mathbf{if}\;t\_1 \leq 1.996:\\ \;\;\;\;\left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - t\_2\right)\right) - \left(\sqrt{y} - \sqrt{1 + y}\right)\\ \mathbf{elif}\;t\_1 \leq 3.5:\\ \;\;\;\;\left(\left(t\_2 + \sqrt{y - -1}\right) + \left(\left(\sqrt{z - -1} - \sqrt{z}\right) - \sqrt{y}\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - \left(1 + \sqrt{1 + z}\right)\right)\\ \end{array} \end{array} \]

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
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t))))
        (t_2 (sqrt (- x -1.0))))
   (if (<= t_1 1.996)
     (-
      (- (- (sqrt (- t -1.0)) (sqrt t)) (- (sqrt x) t_2))
      (- (sqrt y) (sqrt (+ 1.0 y))))
     (if (<= t_1 3.5)
       (-
        (+
         (+ t_2 (sqrt (- y -1.0)))
         (- (- (sqrt (- z -1.0)) (sqrt z)) (sqrt y)))
        (sqrt x))
       (-
        (- (- 1.0 (sqrt t)) (- (sqrt x) 1.0))
        (- (+ (sqrt y) (sqrt z)) (+ 1.0 (sqrt (+ 1.0 z)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double t_2 = sqrt((x - -1.0));
	double tmp;
	if (t_1 <= 1.996) {
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - (sqrt(x) - t_2)) - (sqrt(y) - sqrt((1.0 + y)));
	} else if (t_1 <= 3.5) {
		tmp = ((t_2 + sqrt((y - -1.0))) + ((sqrt((z - -1.0)) - sqrt(z)) - sqrt(y))) - sqrt(x);
	} else {
		tmp = ((1.0 - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) + sqrt(z)) - (1.0 + sqrt((1.0 + z))));
	}
	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 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    t_2 = sqrt((x - (-1.0d0)))
    if (t_1 <= 1.996d0) then
        tmp = ((sqrt((t - (-1.0d0))) - sqrt(t)) - (sqrt(x) - t_2)) - (sqrt(y) - sqrt((1.0d0 + y)))
    else if (t_1 <= 3.5d0) then
        tmp = ((t_2 + sqrt((y - (-1.0d0)))) + ((sqrt((z - (-1.0d0))) - sqrt(z)) - sqrt(y))) - sqrt(x)
    else
        tmp = ((1.0d0 - sqrt(t)) - (sqrt(x) - 1.0d0)) - ((sqrt(y) + sqrt(z)) - (1.0d0 + sqrt((1.0d0 + z))))
    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 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	double t_2 = Math.sqrt((x - -1.0));
	double tmp;
	if (t_1 <= 1.996) {
		tmp = ((Math.sqrt((t - -1.0)) - Math.sqrt(t)) - (Math.sqrt(x) - t_2)) - (Math.sqrt(y) - Math.sqrt((1.0 + y)));
	} else if (t_1 <= 3.5) {
		tmp = ((t_2 + Math.sqrt((y - -1.0))) + ((Math.sqrt((z - -1.0)) - Math.sqrt(z)) - Math.sqrt(y))) - Math.sqrt(x);
	} else {
		tmp = ((1.0 - Math.sqrt(t)) - (Math.sqrt(x) - 1.0)) - ((Math.sqrt(y) + Math.sqrt(z)) - (1.0 + Math.sqrt((1.0 + z))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	t_2 = math.sqrt((x - -1.0))
	tmp = 0
	if t_1 <= 1.996:
		tmp = ((math.sqrt((t - -1.0)) - math.sqrt(t)) - (math.sqrt(x) - t_2)) - (math.sqrt(y) - math.sqrt((1.0 + y)))
	elif t_1 <= 3.5:
		tmp = ((t_2 + math.sqrt((y - -1.0))) + ((math.sqrt((z - -1.0)) - math.sqrt(z)) - math.sqrt(y))) - math.sqrt(x)
	else:
		tmp = ((1.0 - math.sqrt(t)) - (math.sqrt(x) - 1.0)) - ((math.sqrt(y) + math.sqrt(z)) - (1.0 + math.sqrt((1.0 + z))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	t_2 = sqrt(Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= 1.996)
		tmp = Float64(Float64(Float64(sqrt(Float64(t - -1.0)) - sqrt(t)) - Float64(sqrt(x) - t_2)) - Float64(sqrt(y) - sqrt(Float64(1.0 + y))));
	elseif (t_1 <= 3.5)
		tmp = Float64(Float64(Float64(t_2 + sqrt(Float64(y - -1.0))) + Float64(Float64(sqrt(Float64(z - -1.0)) - sqrt(z)) - sqrt(y))) - sqrt(x));
	else
		tmp = Float64(Float64(Float64(1.0 - sqrt(t)) - Float64(sqrt(x) - 1.0)) - Float64(Float64(sqrt(y) + sqrt(z)) - Float64(1.0 + sqrt(Float64(1.0 + z)))));
	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 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	t_2 = sqrt((x - -1.0));
	tmp = 0.0;
	if (t_1 <= 1.996)
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - (sqrt(x) - t_2)) - (sqrt(y) - sqrt((1.0 + y)));
	elseif (t_1 <= 3.5)
		tmp = ((t_2 + sqrt((y - -1.0))) + ((sqrt((z - -1.0)) - sqrt(z)) - sqrt(y))) - sqrt(x);
	else
		tmp = ((1.0 - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) + sqrt(z)) - (1.0 + sqrt((1.0 + z))));
	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[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 1.996], N[(N[(N[(N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 3.5], N[(N[(N[(t$95$2 + N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(1.0 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.996

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 65.1

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

   6. Applied rewrites 65.1%

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

   if 1.996 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.5

   1. Initial program 91.1%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 32.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 60.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. associate-+r- N/A

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

      6. lower--.f64 N/A

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

   8. Applied rewrites 66.3%

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

   if 3.5 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 7.3

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

   9. Applied rewrites 7.3%

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

   10. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-+.f64 7.3

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

   12. Applied rewrites 7.3%

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

Alternative 9: 90.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ t_2 := \sqrt{x - -1}\\ \mathbf{if}\;t\_1 \leq 2.000000015:\\ \;\;\;\;\left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - t\_2\right)\right) - \left(\sqrt{y} - \sqrt{1 + y}\right)\\ \mathbf{elif}\;t\_1 \leq 3.5:\\ \;\;\;\;\sqrt{y - -1} + \left(t\_2 - \left(\sqrt{x} - \left(\left(\sqrt{z - -1} - \sqrt{z}\right) - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - \left(1 + \sqrt{1 + z}\right)\right)\\ \end{array} \end{array} \]

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
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t))))
        (t_2 (sqrt (- x -1.0))))
   (if (<= t_1 2.000000015)
     (-
      (- (- (sqrt (- t -1.0)) (sqrt t)) (- (sqrt x) t_2))
      (- (sqrt y) (sqrt (+ 1.0 y))))
     (if (<= t_1 3.5)
       (+
        (sqrt (- y -1.0))
        (- t_2 (- (sqrt x) (- (- (sqrt (- z -1.0)) (sqrt z)) (sqrt y)))))
       (-
        (- (- 1.0 (sqrt t)) (- (sqrt x) 1.0))
        (- (+ (sqrt y) (sqrt z)) (+ 1.0 (sqrt (+ 1.0 z)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double t_2 = sqrt((x - -1.0));
	double tmp;
	if (t_1 <= 2.000000015) {
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - (sqrt(x) - t_2)) - (sqrt(y) - sqrt((1.0 + y)));
	} else if (t_1 <= 3.5) {
		tmp = sqrt((y - -1.0)) + (t_2 - (sqrt(x) - ((sqrt((z - -1.0)) - sqrt(z)) - sqrt(y))));
	} else {
		tmp = ((1.0 - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) + sqrt(z)) - (1.0 + sqrt((1.0 + z))));
	}
	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 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    t_2 = sqrt((x - (-1.0d0)))
    if (t_1 <= 2.000000015d0) then
        tmp = ((sqrt((t - (-1.0d0))) - sqrt(t)) - (sqrt(x) - t_2)) - (sqrt(y) - sqrt((1.0d0 + y)))
    else if (t_1 <= 3.5d0) then
        tmp = sqrt((y - (-1.0d0))) + (t_2 - (sqrt(x) - ((sqrt((z - (-1.0d0))) - sqrt(z)) - sqrt(y))))
    else
        tmp = ((1.0d0 - sqrt(t)) - (sqrt(x) - 1.0d0)) - ((sqrt(y) + sqrt(z)) - (1.0d0 + sqrt((1.0d0 + z))))
    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 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	double t_2 = Math.sqrt((x - -1.0));
	double tmp;
	if (t_1 <= 2.000000015) {
		tmp = ((Math.sqrt((t - -1.0)) - Math.sqrt(t)) - (Math.sqrt(x) - t_2)) - (Math.sqrt(y) - Math.sqrt((1.0 + y)));
	} else if (t_1 <= 3.5) {
		tmp = Math.sqrt((y - -1.0)) + (t_2 - (Math.sqrt(x) - ((Math.sqrt((z - -1.0)) - Math.sqrt(z)) - Math.sqrt(y))));
	} else {
		tmp = ((1.0 - Math.sqrt(t)) - (Math.sqrt(x) - 1.0)) - ((Math.sqrt(y) + Math.sqrt(z)) - (1.0 + Math.sqrt((1.0 + z))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	t_2 = math.sqrt((x - -1.0))
	tmp = 0
	if t_1 <= 2.000000015:
		tmp = ((math.sqrt((t - -1.0)) - math.sqrt(t)) - (math.sqrt(x) - t_2)) - (math.sqrt(y) - math.sqrt((1.0 + y)))
	elif t_1 <= 3.5:
		tmp = math.sqrt((y - -1.0)) + (t_2 - (math.sqrt(x) - ((math.sqrt((z - -1.0)) - math.sqrt(z)) - math.sqrt(y))))
	else:
		tmp = ((1.0 - math.sqrt(t)) - (math.sqrt(x) - 1.0)) - ((math.sqrt(y) + math.sqrt(z)) - (1.0 + math.sqrt((1.0 + z))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	t_2 = sqrt(Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= 2.000000015)
		tmp = Float64(Float64(Float64(sqrt(Float64(t - -1.0)) - sqrt(t)) - Float64(sqrt(x) - t_2)) - Float64(sqrt(y) - sqrt(Float64(1.0 + y))));
	elseif (t_1 <= 3.5)
		tmp = Float64(sqrt(Float64(y - -1.0)) + Float64(t_2 - Float64(sqrt(x) - Float64(Float64(sqrt(Float64(z - -1.0)) - sqrt(z)) - sqrt(y)))));
	else
		tmp = Float64(Float64(Float64(1.0 - sqrt(t)) - Float64(sqrt(x) - 1.0)) - Float64(Float64(sqrt(y) + sqrt(z)) - Float64(1.0 + sqrt(Float64(1.0 + z)))));
	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 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	t_2 = sqrt((x - -1.0));
	tmp = 0.0;
	if (t_1 <= 2.000000015)
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - (sqrt(x) - t_2)) - (sqrt(y) - sqrt((1.0 + y)));
	elseif (t_1 <= 3.5)
		tmp = sqrt((y - -1.0)) + (t_2 - (sqrt(x) - ((sqrt((z - -1.0)) - sqrt(z)) - sqrt(y))));
	else
		tmp = ((1.0 - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) + sqrt(z)) - (1.0 + sqrt((1.0 + z))));
	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[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 2.000000015], N[(N[(N[(N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 3.5], N[(N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision] + N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] - N[(N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(1.0 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00000001499999991

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 65.1

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

   6. Applied rewrites 65.1%

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

   if 2.00000001499999991 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.5

   1. Initial program 91.1%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 32.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 60.7%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift--.f64 N/A

         \[\leadsto \sqrt{y - -1} + \left(\sqrt{x - -1} - \left(\mathsf{neg}\left(\left(\sqrt{z - -1} - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)\right)\right)\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \sqrt{y - -1} + \left(\sqrt{x - -1} - \left(\mathsf{neg}\left(\left(\sqrt{z - -1} - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \sqrt{y - -1} + \left(\sqrt{x - -1} - \left(\mathsf{neg}\left(\left(\left(\sqrt{z - -1} - \left(\sqrt{z} + \sqrt{y}\right)\right) - \sqrt{x}\right)\right)\right)\right) \]

   8. Applied rewrites 66.6%

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

   if 3.5 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 7.3

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

   9. Applied rewrites 7.3%

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

   10. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-+.f64 7.3

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

   12. Applied rewrites 7.3%

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

Alternative 10: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ t_2 := \sqrt{x - -1}\\ \mathbf{if}\;t\_1 \leq 1:\\ \;\;\;\;t\_2 + \left(-\sqrt{x}\right)\\ \mathbf{elif}\;t\_1 \leq 3.5:\\ \;\;\;\;\sqrt{y - -1} + \left(t\_2 - \left(\sqrt{x} - \left(\left(\sqrt{z - -1} - \sqrt{z}\right) - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - \left(1 + \sqrt{1 + z}\right)\right)\\ \end{array} \end{array} \]

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
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t))))
        (t_2 (sqrt (- x -1.0))))
   (if (<= t_1 1.0)
     (+ t_2 (- (sqrt x)))
     (if (<= t_1 3.5)
       (+
        (sqrt (- y -1.0))
        (- t_2 (- (sqrt x) (- (- (sqrt (- z -1.0)) (sqrt z)) (sqrt y)))))
       (-
        (- (- 1.0 (sqrt t)) (- (sqrt x) 1.0))
        (- (+ (sqrt y) (sqrt z)) (+ 1.0 (sqrt (+ 1.0 z)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double t_2 = sqrt((x - -1.0));
	double tmp;
	if (t_1 <= 1.0) {
		tmp = t_2 + -sqrt(x);
	} else if (t_1 <= 3.5) {
		tmp = sqrt((y - -1.0)) + (t_2 - (sqrt(x) - ((sqrt((z - -1.0)) - sqrt(z)) - sqrt(y))));
	} else {
		tmp = ((1.0 - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) + sqrt(z)) - (1.0 + sqrt((1.0 + z))));
	}
	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 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    t_2 = sqrt((x - (-1.0d0)))
    if (t_1 <= 1.0d0) then
        tmp = t_2 + -sqrt(x)
    else if (t_1 <= 3.5d0) then
        tmp = sqrt((y - (-1.0d0))) + (t_2 - (sqrt(x) - ((sqrt((z - (-1.0d0))) - sqrt(z)) - sqrt(y))))
    else
        tmp = ((1.0d0 - sqrt(t)) - (sqrt(x) - 1.0d0)) - ((sqrt(y) + sqrt(z)) - (1.0d0 + sqrt((1.0d0 + z))))
    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 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	double t_2 = Math.sqrt((x - -1.0));
	double tmp;
	if (t_1 <= 1.0) {
		tmp = t_2 + -Math.sqrt(x);
	} else if (t_1 <= 3.5) {
		tmp = Math.sqrt((y - -1.0)) + (t_2 - (Math.sqrt(x) - ((Math.sqrt((z - -1.0)) - Math.sqrt(z)) - Math.sqrt(y))));
	} else {
		tmp = ((1.0 - Math.sqrt(t)) - (Math.sqrt(x) - 1.0)) - ((Math.sqrt(y) + Math.sqrt(z)) - (1.0 + Math.sqrt((1.0 + z))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	t_2 = math.sqrt((x - -1.0))
	tmp = 0
	if t_1 <= 1.0:
		tmp = t_2 + -math.sqrt(x)
	elif t_1 <= 3.5:
		tmp = math.sqrt((y - -1.0)) + (t_2 - (math.sqrt(x) - ((math.sqrt((z - -1.0)) - math.sqrt(z)) - math.sqrt(y))))
	else:
		tmp = ((1.0 - math.sqrt(t)) - (math.sqrt(x) - 1.0)) - ((math.sqrt(y) + math.sqrt(z)) - (1.0 + math.sqrt((1.0 + z))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	t_2 = sqrt(Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= 1.0)
		tmp = Float64(t_2 + Float64(-sqrt(x)));
	elseif (t_1 <= 3.5)
		tmp = Float64(sqrt(Float64(y - -1.0)) + Float64(t_2 - Float64(sqrt(x) - Float64(Float64(sqrt(Float64(z - -1.0)) - sqrt(z)) - sqrt(y)))));
	else
		tmp = Float64(Float64(Float64(1.0 - sqrt(t)) - Float64(sqrt(x) - 1.0)) - Float64(Float64(sqrt(y) + sqrt(z)) - Float64(1.0 + sqrt(Float64(1.0 + z)))));
	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 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	t_2 = sqrt((x - -1.0));
	tmp = 0.0;
	if (t_1 <= 1.0)
		tmp = t_2 + -sqrt(x);
	elseif (t_1 <= 3.5)
		tmp = sqrt((y - -1.0)) + (t_2 - (sqrt(x) - ((sqrt((z - -1.0)) - sqrt(z)) - sqrt(y))));
	else
		tmp = ((1.0 - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) + sqrt(z)) - (1.0 + sqrt((1.0 + z))));
	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[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 1.0], N[(t$95$2 + (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$1, 3.5], N[(N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision] + N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] - N[(N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(1.0 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   1. Initial program 91.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-+r- N/A

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

      7. associate-+l- N/A

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

   3. Applied rewrites 90.0%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 35.9

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

   6. Applied rewrites 35.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \sqrt{x - -1} + -1 \cdot \color{blue}{\sqrt{x}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{x - -1} + -1 \cdot \sqrt{x} \]

      2. lower-sqrt.f64 35.9

         \[\leadsto \sqrt{x - -1} + -1 \cdot \sqrt{x} \]

   9. Applied rewrites 35.9%

      \[\leadsto \sqrt{x - -1} + -1 \cdot \color{blue}{\sqrt{x}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{x - -1} + -1 \cdot \sqrt{x} \]

       2. mul-1-neg N/A

          \[\leadsto \sqrt{x - -1} + \left(\mathsf{neg}\left(\sqrt{x}\right)\right) \]

       3. lift-neg.f64 35.9

          \[\leadsto \sqrt{x - -1} + \left(-\sqrt{x}\right) \]

   11. Applied rewrites 35.9%

       \[\leadsto \sqrt{x - -1} + \color{blue}{\left(-\sqrt{x}\right)} \]

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.5

   1. Initial program 91.1%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 32.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 60.7%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift--.f64 N/A

         \[\leadsto \sqrt{y - -1} + \left(\sqrt{x - -1} - \left(\mathsf{neg}\left(\left(\sqrt{z - -1} - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)\right)\right)\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \sqrt{y - -1} + \left(\sqrt{x - -1} - \left(\mathsf{neg}\left(\left(\sqrt{z - -1} - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \sqrt{y - -1} + \left(\sqrt{x - -1} - \left(\mathsf{neg}\left(\left(\left(\sqrt{z - -1} - \left(\sqrt{z} + \sqrt{y}\right)\right) - \sqrt{x}\right)\right)\right)\right) \]

   8. Applied rewrites 66.6%

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

   if 3.5 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 7.3

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

   9. Applied rewrites 7.3%

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

   10. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-+.f64 7.3

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

   12. Applied rewrites 7.3%

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

Alternative 11: 89.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y - -1}\\ \mathbf{if}\;t \leq 5100000:\\ \;\;\;\;\left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\left(\sqrt{y} - \left(1 - \sqrt{z}\right)\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - -1} - \left(\left(-t\_1\right) - \left(\left(\left(\sqrt{z - -1} - \sqrt{z}\right) - \sqrt{y}\right) - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

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 (sqrt (- y -1.0))))
   (if (<= t 5100000.0)
     (-
      (- (- (sqrt (- t -1.0)) (sqrt t)) (- (sqrt x) 1.0))
      (- (- (sqrt y) (- 1.0 (sqrt z))) t_1))
     (-
      (sqrt (- x -1.0))
      (- (- t_1) (- (- (- (sqrt (- z -1.0)) (sqrt z)) (sqrt y)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y - -1.0));
	double tmp;
	if (t <= 5100000.0) {
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) - (1.0 - sqrt(z))) - t_1);
	} else {
		tmp = sqrt((x - -1.0)) - (-t_1 - (((sqrt((z - -1.0)) - sqrt(z)) - sqrt(y)) - sqrt(x)));
	}
	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 = sqrt((y - (-1.0d0)))
    if (t <= 5100000.0d0) then
        tmp = ((sqrt((t - (-1.0d0))) - sqrt(t)) - (sqrt(x) - 1.0d0)) - ((sqrt(y) - (1.0d0 - sqrt(z))) - t_1)
    else
        tmp = sqrt((x - (-1.0d0))) - (-t_1 - (((sqrt((z - (-1.0d0))) - sqrt(z)) - sqrt(y)) - sqrt(x)))
    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 = Math.sqrt((y - -1.0));
	double tmp;
	if (t <= 5100000.0) {
		tmp = ((Math.sqrt((t - -1.0)) - Math.sqrt(t)) - (Math.sqrt(x) - 1.0)) - ((Math.sqrt(y) - (1.0 - Math.sqrt(z))) - t_1);
	} else {
		tmp = Math.sqrt((x - -1.0)) - (-t_1 - (((Math.sqrt((z - -1.0)) - Math.sqrt(z)) - Math.sqrt(y)) - Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y - -1.0))
	tmp = 0
	if t <= 5100000.0:
		tmp = ((math.sqrt((t - -1.0)) - math.sqrt(t)) - (math.sqrt(x) - 1.0)) - ((math.sqrt(y) - (1.0 - math.sqrt(z))) - t_1)
	else:
		tmp = math.sqrt((x - -1.0)) - (-t_1 - (((math.sqrt((z - -1.0)) - math.sqrt(z)) - math.sqrt(y)) - math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y - -1.0))
	tmp = 0.0
	if (t <= 5100000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(t - -1.0)) - sqrt(t)) - Float64(sqrt(x) - 1.0)) - Float64(Float64(sqrt(y) - Float64(1.0 - sqrt(z))) - t_1));
	else
		tmp = Float64(sqrt(Float64(x - -1.0)) - Float64(Float64(-t_1) - Float64(Float64(Float64(sqrt(Float64(z - -1.0)) - sqrt(z)) - sqrt(y)) - sqrt(x))));
	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 = sqrt((y - -1.0));
	tmp = 0.0;
	if (t <= 5100000.0)
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) - (1.0 - sqrt(z))) - t_1);
	else
		tmp = sqrt((x - -1.0)) - (-t_1 - (((sqrt((z - -1.0)) - sqrt(z)) - sqrt(y)) - sqrt(x)));
	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[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 5100000.0], N[(N[(N[(N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] - N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] - N[((-t$95$1) - N[(N[(N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 5.1e6

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 30.5

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

   9. Applied rewrites 30.5%

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

   if 5.1e6 < t

   1. Initial program 91.1%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 32.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 60.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 60.8

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate--r+ N/A

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

      11. lower--.f64 N/A

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

   8. Applied rewrites 84.2%

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

Alternative 12: 89.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y - -1}\\ t_2 := \sqrt{z - -1} - \sqrt{z}\\ \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 3.5:\\ \;\;\;\;\sqrt{x - -1} - \left(\left(-t\_1\right) - \left(\left(t\_2 - \sqrt{y}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\left(\sqrt{y} - t\_2\right) - t\_1\right)\\ \end{array} \end{array} \]

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 (sqrt (- y -1.0))) (t_2 (- (sqrt (- z -1.0)) (sqrt z))))
   (if (<=
        (+
         (+
          (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
          (- (sqrt (+ z 1.0)) (sqrt z)))
         (- (sqrt (+ t 1.0)) (sqrt t)))
        3.5)
     (- (sqrt (- x -1.0)) (- (- t_1) (- (- t_2 (sqrt y)) (sqrt x))))
     (- (- (- 1.0 (sqrt t)) (- (sqrt x) 1.0)) (- (- (sqrt y) t_2) t_1)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y - -1.0));
	double t_2 = sqrt((z - -1.0)) - sqrt(z);
	double tmp;
	if (((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t))) <= 3.5) {
		tmp = sqrt((x - -1.0)) - (-t_1 - ((t_2 - sqrt(y)) - sqrt(x)));
	} else {
		tmp = ((1.0 - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) - t_2) - t_1);
	}
	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 = sqrt((y - (-1.0d0)))
    t_2 = sqrt((z - (-1.0d0))) - sqrt(z)
    if (((((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))) <= 3.5d0) then
        tmp = sqrt((x - (-1.0d0))) - (-t_1 - ((t_2 - sqrt(y)) - sqrt(x)))
    else
        tmp = ((1.0d0 - sqrt(t)) - (sqrt(x) - 1.0d0)) - ((sqrt(y) - t_2) - t_1)
    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 = Math.sqrt((y - -1.0));
	double t_2 = Math.sqrt((z - -1.0)) - Math.sqrt(z);
	double tmp;
	if (((((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t))) <= 3.5) {
		tmp = Math.sqrt((x - -1.0)) - (-t_1 - ((t_2 - Math.sqrt(y)) - Math.sqrt(x)));
	} else {
		tmp = ((1.0 - Math.sqrt(t)) - (Math.sqrt(x) - 1.0)) - ((Math.sqrt(y) - t_2) - t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y - -1.0))
	t_2 = math.sqrt((z - -1.0)) - math.sqrt(z)
	tmp = 0
	if ((((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))) <= 3.5:
		tmp = math.sqrt((x - -1.0)) - (-t_1 - ((t_2 - math.sqrt(y)) - math.sqrt(x)))
	else:
		tmp = ((1.0 - math.sqrt(t)) - (math.sqrt(x) - 1.0)) - ((math.sqrt(y) - t_2) - t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y - -1.0))
	t_2 = Float64(sqrt(Float64(z - -1.0)) - sqrt(z))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t))) <= 3.5)
		tmp = Float64(sqrt(Float64(x - -1.0)) - Float64(Float64(-t_1) - Float64(Float64(t_2 - sqrt(y)) - sqrt(x))));
	else
		tmp = Float64(Float64(Float64(1.0 - sqrt(t)) - Float64(sqrt(x) - 1.0)) - Float64(Float64(sqrt(y) - t_2) - t_1));
	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 = sqrt((y - -1.0));
	t_2 = sqrt((z - -1.0)) - sqrt(z);
	tmp = 0.0;
	if (((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t))) <= 3.5)
		tmp = sqrt((x - -1.0)) - (-t_1 - ((t_2 - sqrt(y)) - sqrt(x)));
	else
		tmp = ((1.0 - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) - t_2) - t_1);
	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[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.5], N[(N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] - N[((-t$95$1) - N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] - t$95$2), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.5

   1. Initial program 91.1%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 32.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 60.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 60.8

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate--r+ N/A

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

      11. lower--.f64 N/A

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

   8. Applied rewrites 84.2%

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

   if 3.5 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 7.3

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

   9. Applied rewrites 7.3%

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

Alternative 13: 89.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 3.5:\\ \;\;\;\;\sqrt{x - -1} - \left(\left(-\sqrt{y - -1}\right) - \left(\left(\left(\sqrt{z - -1} - \sqrt{z}\right) - \sqrt{y}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - \left(1 + \sqrt{1 + z}\right)\right)\\ \end{array} \end{array} \]

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 (<=
      (+
       (+
        (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
        (- (sqrt (+ z 1.0)) (sqrt z)))
       (- (sqrt (+ t 1.0)) (sqrt t)))
      3.5)
   (-
    (sqrt (- x -1.0))
    (-
     (- (sqrt (- y -1.0)))
     (- (- (- (sqrt (- z -1.0)) (sqrt z)) (sqrt y)) (sqrt x))))
   (-
    (- (- 1.0 (sqrt t)) (- (sqrt x) 1.0))
    (- (+ (sqrt y) (sqrt z)) (+ 1.0 (sqrt (+ 1.0 z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t))) <= 3.5) {
		tmp = sqrt((x - -1.0)) - (-sqrt((y - -1.0)) - (((sqrt((z - -1.0)) - sqrt(z)) - sqrt(y)) - sqrt(x)));
	} else {
		tmp = ((1.0 - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) + sqrt(z)) - (1.0 + sqrt((1.0 + z))));
	}
	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 (((((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))) <= 3.5d0) then
        tmp = sqrt((x - (-1.0d0))) - (-sqrt((y - (-1.0d0))) - (((sqrt((z - (-1.0d0))) - sqrt(z)) - sqrt(y)) - sqrt(x)))
    else
        tmp = ((1.0d0 - sqrt(t)) - (sqrt(x) - 1.0d0)) - ((sqrt(y) + sqrt(z)) - (1.0d0 + sqrt((1.0d0 + z))))
    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 (((((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t))) <= 3.5) {
		tmp = Math.sqrt((x - -1.0)) - (-Math.sqrt((y - -1.0)) - (((Math.sqrt((z - -1.0)) - Math.sqrt(z)) - Math.sqrt(y)) - Math.sqrt(x)));
	} else {
		tmp = ((1.0 - Math.sqrt(t)) - (Math.sqrt(x) - 1.0)) - ((Math.sqrt(y) + Math.sqrt(z)) - (1.0 + Math.sqrt((1.0 + z))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if ((((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))) <= 3.5:
		tmp = math.sqrt((x - -1.0)) - (-math.sqrt((y - -1.0)) - (((math.sqrt((z - -1.0)) - math.sqrt(z)) - math.sqrt(y)) - math.sqrt(x)))
	else:
		tmp = ((1.0 - math.sqrt(t)) - (math.sqrt(x) - 1.0)) - ((math.sqrt(y) + math.sqrt(z)) - (1.0 + math.sqrt((1.0 + z))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t))) <= 3.5)
		tmp = Float64(sqrt(Float64(x - -1.0)) - Float64(Float64(-sqrt(Float64(y - -1.0))) - Float64(Float64(Float64(sqrt(Float64(z - -1.0)) - sqrt(z)) - sqrt(y)) - sqrt(x))));
	else
		tmp = Float64(Float64(Float64(1.0 - sqrt(t)) - Float64(sqrt(x) - 1.0)) - Float64(Float64(sqrt(y) + sqrt(z)) - Float64(1.0 + sqrt(Float64(1.0 + z)))));
	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 (((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t))) <= 3.5)
		tmp = sqrt((x - -1.0)) - (-sqrt((y - -1.0)) - (((sqrt((z - -1.0)) - sqrt(z)) - sqrt(y)) - sqrt(x)));
	else
		tmp = ((1.0 - sqrt(t)) - (sqrt(x) - 1.0)) - ((sqrt(y) + sqrt(z)) - (1.0 + sqrt((1.0 + z))));
	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[N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.5], N[(N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] - N[((-N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]) - N[(N[(N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(1.0 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.5

   1. Initial program 91.1%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 32.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 60.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 60.8

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate--r+ N/A

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

      11. lower--.f64 N/A

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

   8. Applied rewrites 84.2%

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

   if 3.5 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 7.3

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

   9. Applied rewrites 7.3%

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

   10. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-+.f64 7.3

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

   12. Applied rewrites 7.3%

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

Alternative 14: 88.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} - 1\\ t_2 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ t_3 := \sqrt{1 + y}\\ \mathbf{if}\;t\_2 \leq 2.000000015:\\ \;\;\;\;\left(\left(\sqrt{t - -1} - \sqrt{t}\right) - t\_1\right) - \left(\sqrt{y} - t\_3\right)\\ \mathbf{elif}\;t\_2 \leq 3.5:\\ \;\;\;\;\left(1 + t\_3\right) + \left(\sqrt{z - -1} - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{t}\right) - t\_1\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - \left(1 + \sqrt{1 + z}\right)\right)\\ \end{array} \end{array} \]

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 (- (sqrt x) 1.0))
        (t_2
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t))))
        (t_3 (sqrt (+ 1.0 y))))
   (if (<= t_2 2.000000015)
     (- (- (- (sqrt (- t -1.0)) (sqrt t)) t_1) (- (sqrt y) t_3))
     (if (<= t_2 3.5)
       (+ (+ 1.0 t_3) (- (sqrt (- z -1.0)) (+ (+ (sqrt z) (sqrt y)) (sqrt x))))
       (-
        (- (- 1.0 (sqrt t)) t_1)
        (- (+ (sqrt y) (sqrt z)) (+ 1.0 (sqrt (+ 1.0 z)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(x) - 1.0;
	double t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double t_3 = sqrt((1.0 + y));
	double tmp;
	if (t_2 <= 2.000000015) {
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - t_1) - (sqrt(y) - t_3);
	} else if (t_2 <= 3.5) {
		tmp = (1.0 + t_3) + (sqrt((z - -1.0)) - ((sqrt(z) + sqrt(y)) + sqrt(x)));
	} else {
		tmp = ((1.0 - sqrt(t)) - t_1) - ((sqrt(y) + sqrt(z)) - (1.0 + sqrt((1.0 + z))));
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = sqrt(x) - 1.0d0
    t_2 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    t_3 = sqrt((1.0d0 + y))
    if (t_2 <= 2.000000015d0) then
        tmp = ((sqrt((t - (-1.0d0))) - sqrt(t)) - t_1) - (sqrt(y) - t_3)
    else if (t_2 <= 3.5d0) then
        tmp = (1.0d0 + t_3) + (sqrt((z - (-1.0d0))) - ((sqrt(z) + sqrt(y)) + sqrt(x)))
    else
        tmp = ((1.0d0 - sqrt(t)) - t_1) - ((sqrt(y) + sqrt(z)) - (1.0d0 + sqrt((1.0d0 + z))))
    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 = Math.sqrt(x) - 1.0;
	double t_2 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	double t_3 = Math.sqrt((1.0 + y));
	double tmp;
	if (t_2 <= 2.000000015) {
		tmp = ((Math.sqrt((t - -1.0)) - Math.sqrt(t)) - t_1) - (Math.sqrt(y) - t_3);
	} else if (t_2 <= 3.5) {
		tmp = (1.0 + t_3) + (Math.sqrt((z - -1.0)) - ((Math.sqrt(z) + Math.sqrt(y)) + Math.sqrt(x)));
	} else {
		tmp = ((1.0 - Math.sqrt(t)) - t_1) - ((Math.sqrt(y) + Math.sqrt(z)) - (1.0 + Math.sqrt((1.0 + z))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(x) - 1.0
	t_2 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	t_3 = math.sqrt((1.0 + y))
	tmp = 0
	if t_2 <= 2.000000015:
		tmp = ((math.sqrt((t - -1.0)) - math.sqrt(t)) - t_1) - (math.sqrt(y) - t_3)
	elif t_2 <= 3.5:
		tmp = (1.0 + t_3) + (math.sqrt((z - -1.0)) - ((math.sqrt(z) + math.sqrt(y)) + math.sqrt(x)))
	else:
		tmp = ((1.0 - math.sqrt(t)) - t_1) - ((math.sqrt(y) + math.sqrt(z)) - (1.0 + math.sqrt((1.0 + z))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) - 1.0)
	t_2 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	t_3 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (t_2 <= 2.000000015)
		tmp = Float64(Float64(Float64(sqrt(Float64(t - -1.0)) - sqrt(t)) - t_1) - Float64(sqrt(y) - t_3));
	elseif (t_2 <= 3.5)
		tmp = Float64(Float64(1.0 + t_3) + Float64(sqrt(Float64(z - -1.0)) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x))));
	else
		tmp = Float64(Float64(Float64(1.0 - sqrt(t)) - t_1) - Float64(Float64(sqrt(y) + sqrt(z)) - Float64(1.0 + sqrt(Float64(1.0 + z)))));
	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 = sqrt(x) - 1.0;
	t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	t_3 = sqrt((1.0 + y));
	tmp = 0.0;
	if (t_2 <= 2.000000015)
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - t_1) - (sqrt(y) - t_3);
	elseif (t_2 <= 3.5)
		tmp = (1.0 + t_3) + (sqrt((z - -1.0)) - ((sqrt(z) + sqrt(y)) + sqrt(x)));
	else
		tmp = ((1.0 - sqrt(t)) - t_1) - ((sqrt(y) + sqrt(z)) - (1.0 + sqrt((1.0 + z))));
	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[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 2.000000015], N[(N[(N[(N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 3.5], N[(N[(1.0 + t$95$3), $MachinePrecision] + N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(1.0 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00000001499999991

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in z around inf

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \color{blue}{\left(\sqrt{y} - \sqrt{1 + y}\right)} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 63.9

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

   9. Applied rewrites 63.9%

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

   if 2.00000001499999991 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.5

   1. Initial program 91.1%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 32.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 60.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \left(1 + \sqrt{1 + y}\right) + \left(\color{blue}{\sqrt{z - -1}} - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 60.7

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

   9. Applied rewrites 60.7%

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

   if 3.5 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 7.3

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

   9. Applied rewrites 7.3%

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

   10. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-+.f64 7.3

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

   12. Applied rewrites 7.3%

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

Alternative 15: 88.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} - 1\\ t_2 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ t_3 := \sqrt{1 + y}\\ \mathbf{if}\;t\_2 \leq 2.000000015:\\ \;\;\;\;\left(\left(\sqrt{t - -1} - \sqrt{t}\right) - t\_1\right) - \left(\sqrt{y} - t\_3\right)\\ \mathbf{elif}\;t\_2 \leq 3.5:\\ \;\;\;\;\left(1 + t\_3\right) + \left(\sqrt{z - -1} - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{t}\right) - t\_1\right) - \left(\left(\sqrt{y} - \left(1 - \sqrt{z}\right)\right) - \sqrt{y - -1}\right)\\ \end{array} \end{array} \]

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 (- (sqrt x) 1.0))
        (t_2
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t))))
        (t_3 (sqrt (+ 1.0 y))))
   (if (<= t_2 2.000000015)
     (- (- (- (sqrt (- t -1.0)) (sqrt t)) t_1) (- (sqrt y) t_3))
     (if (<= t_2 3.5)
       (+ (+ 1.0 t_3) (- (sqrt (- z -1.0)) (+ (+ (sqrt z) (sqrt y)) (sqrt x))))
       (-
        (- (- 1.0 (sqrt t)) t_1)
        (- (- (sqrt y) (- 1.0 (sqrt z))) (sqrt (- y -1.0))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(x) - 1.0;
	double t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double t_3 = sqrt((1.0 + y));
	double tmp;
	if (t_2 <= 2.000000015) {
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - t_1) - (sqrt(y) - t_3);
	} else if (t_2 <= 3.5) {
		tmp = (1.0 + t_3) + (sqrt((z - -1.0)) - ((sqrt(z) + sqrt(y)) + sqrt(x)));
	} else {
		tmp = ((1.0 - sqrt(t)) - t_1) - ((sqrt(y) - (1.0 - sqrt(z))) - sqrt((y - -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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt(x) - 1.0d0
    t_2 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    t_3 = sqrt((1.0d0 + y))
    if (t_2 <= 2.000000015d0) then
        tmp = ((sqrt((t - (-1.0d0))) - sqrt(t)) - t_1) - (sqrt(y) - t_3)
    else if (t_2 <= 3.5d0) then
        tmp = (1.0d0 + t_3) + (sqrt((z - (-1.0d0))) - ((sqrt(z) + sqrt(y)) + sqrt(x)))
    else
        tmp = ((1.0d0 - sqrt(t)) - t_1) - ((sqrt(y) - (1.0d0 - sqrt(z))) - sqrt((y - (-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 = Math.sqrt(x) - 1.0;
	double t_2 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	double t_3 = Math.sqrt((1.0 + y));
	double tmp;
	if (t_2 <= 2.000000015) {
		tmp = ((Math.sqrt((t - -1.0)) - Math.sqrt(t)) - t_1) - (Math.sqrt(y) - t_3);
	} else if (t_2 <= 3.5) {
		tmp = (1.0 + t_3) + (Math.sqrt((z - -1.0)) - ((Math.sqrt(z) + Math.sqrt(y)) + Math.sqrt(x)));
	} else {
		tmp = ((1.0 - Math.sqrt(t)) - t_1) - ((Math.sqrt(y) - (1.0 - Math.sqrt(z))) - Math.sqrt((y - -1.0)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(x) - 1.0
	t_2 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	t_3 = math.sqrt((1.0 + y))
	tmp = 0
	if t_2 <= 2.000000015:
		tmp = ((math.sqrt((t - -1.0)) - math.sqrt(t)) - t_1) - (math.sqrt(y) - t_3)
	elif t_2 <= 3.5:
		tmp = (1.0 + t_3) + (math.sqrt((z - -1.0)) - ((math.sqrt(z) + math.sqrt(y)) + math.sqrt(x)))
	else:
		tmp = ((1.0 - math.sqrt(t)) - t_1) - ((math.sqrt(y) - (1.0 - math.sqrt(z))) - math.sqrt((y - -1.0)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) - 1.0)
	t_2 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	t_3 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (t_2 <= 2.000000015)
		tmp = Float64(Float64(Float64(sqrt(Float64(t - -1.0)) - sqrt(t)) - t_1) - Float64(sqrt(y) - t_3));
	elseif (t_2 <= 3.5)
		tmp = Float64(Float64(1.0 + t_3) + Float64(sqrt(Float64(z - -1.0)) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x))));
	else
		tmp = Float64(Float64(Float64(1.0 - sqrt(t)) - t_1) - Float64(Float64(sqrt(y) - Float64(1.0 - sqrt(z))) - sqrt(Float64(y - -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 = sqrt(x) - 1.0;
	t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	t_3 = sqrt((1.0 + y));
	tmp = 0.0;
	if (t_2 <= 2.000000015)
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - t_1) - (sqrt(y) - t_3);
	elseif (t_2 <= 3.5)
		tmp = (1.0 + t_3) + (sqrt((z - -1.0)) - ((sqrt(z) + sqrt(y)) + sqrt(x)));
	else
		tmp = ((1.0 - sqrt(t)) - t_1) - ((sqrt(y) - (1.0 - sqrt(z))) - sqrt((y - -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[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 2.000000015], N[(N[(N[(N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 3.5], N[(N[(1.0 + t$95$3), $MachinePrecision] + N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] - N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00000001499999991

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in z around inf

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \color{blue}{\left(\sqrt{y} - \sqrt{1 + y}\right)} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 63.9

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

   9. Applied rewrites 63.9%

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

   if 2.00000001499999991 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.5

   1. Initial program 91.1%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 32.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 60.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \left(1 + \sqrt{1 + y}\right) + \left(\color{blue}{\sqrt{z - -1}} - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 60.7

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

   9. Applied rewrites 60.7%

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

   if 3.5 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 7.3

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

   9. Applied rewrites 7.3%

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

   10. Taylor expanded in z around 0

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

       1. lower--.f64 N/A

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

       2. lower-sqrt.f64 7.3

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

   12. Applied rewrites 7.3%

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

Alternative 16: 84.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 8.2 \cdot 10^{+14}:\\ \;\;\;\;\left(1 + t\_1\right) + \left(\sqrt{z - -1} - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\sqrt{y} - t\_1\right)\\ \end{array} \end{array} \]

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 (sqrt (+ 1.0 y))))
   (if (<= z 8.2e+14)
     (+ (+ 1.0 t_1) (- (sqrt (- z -1.0)) (+ (+ (sqrt z) (sqrt y)) (sqrt x))))
     (-
      (- (- (sqrt (- t -1.0)) (sqrt t)) (- (sqrt x) 1.0))
      (- (sqrt y) t_1)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double tmp;
	if (z <= 8.2e+14) {
		tmp = (1.0 + t_1) + (sqrt((z - -1.0)) - ((sqrt(z) + sqrt(y)) + sqrt(x)));
	} else {
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - (sqrt(x) - 1.0)) - (sqrt(y) - t_1);
	}
	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 = sqrt((1.0d0 + y))
    if (z <= 8.2d+14) then
        tmp = (1.0d0 + t_1) + (sqrt((z - (-1.0d0))) - ((sqrt(z) + sqrt(y)) + sqrt(x)))
    else
        tmp = ((sqrt((t - (-1.0d0))) - sqrt(t)) - (sqrt(x) - 1.0d0)) - (sqrt(y) - t_1)
    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 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 8.2e+14) {
		tmp = (1.0 + t_1) + (Math.sqrt((z - -1.0)) - ((Math.sqrt(z) + Math.sqrt(y)) + Math.sqrt(x)));
	} else {
		tmp = ((Math.sqrt((t - -1.0)) - Math.sqrt(t)) - (Math.sqrt(x) - 1.0)) - (Math.sqrt(y) - t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 8.2e+14:
		tmp = (1.0 + t_1) + (math.sqrt((z - -1.0)) - ((math.sqrt(z) + math.sqrt(y)) + math.sqrt(x)))
	else:
		tmp = ((math.sqrt((t - -1.0)) - math.sqrt(t)) - (math.sqrt(x) - 1.0)) - (math.sqrt(y) - t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 8.2e+14)
		tmp = Float64(Float64(1.0 + t_1) + Float64(sqrt(Float64(z - -1.0)) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x))));
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(t - -1.0)) - sqrt(t)) - Float64(sqrt(x) - 1.0)) - Float64(sqrt(y) - t_1));
	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 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 8.2e+14)
		tmp = (1.0 + t_1) + (sqrt((z - -1.0)) - ((sqrt(z) + sqrt(y)) + sqrt(x)));
	else
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - (sqrt(x) - 1.0)) - (sqrt(y) - t_1);
	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[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 8.2e+14], N[(N[(1.0 + t$95$1), $MachinePrecision] + N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 8.2e14

   1. Initial program 91.1%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 32.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 60.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \left(1 + \sqrt{1 + y}\right) + \left(\color{blue}{\sqrt{z - -1}} - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 60.7

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

   9. Applied rewrites 60.7%

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

   if 8.2e14 < z

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in z around inf

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \color{blue}{\left(\sqrt{y} - \sqrt{1 + y}\right)} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 63.9

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

   9. Applied rewrites 63.9%

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

Alternative 17: 83.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 8.2 \cdot 10^{+14}:\\ \;\;\;\;\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{z - -1} - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\sqrt{y} - \sqrt{1 + y}\right)\\ \end{array} \end{array} \]

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 (<= z 8.2e+14)
   (+
    (+ 1.0 (sqrt (+ 1.0 x)))
    (- (sqrt (- z -1.0)) (+ (+ (sqrt z) (sqrt y)) (sqrt x))))
   (-
    (- (- (sqrt (- t -1.0)) (sqrt t)) (- (sqrt x) 1.0))
    (- (sqrt y) (sqrt (+ 1.0 y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 8.2e+14) {
		tmp = (1.0 + sqrt((1.0 + x))) + (sqrt((z - -1.0)) - ((sqrt(z) + sqrt(y)) + sqrt(x)));
	} else {
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - (sqrt(x) - 1.0)) - (sqrt(y) - sqrt((1.0 + y)));
	}
	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 (z <= 8.2d+14) then
        tmp = (1.0d0 + sqrt((1.0d0 + x))) + (sqrt((z - (-1.0d0))) - ((sqrt(z) + sqrt(y)) + sqrt(x)))
    else
        tmp = ((sqrt((t - (-1.0d0))) - sqrt(t)) - (sqrt(x) - 1.0d0)) - (sqrt(y) - sqrt((1.0d0 + y)))
    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 (z <= 8.2e+14) {
		tmp = (1.0 + Math.sqrt((1.0 + x))) + (Math.sqrt((z - -1.0)) - ((Math.sqrt(z) + Math.sqrt(y)) + Math.sqrt(x)));
	} else {
		tmp = ((Math.sqrt((t - -1.0)) - Math.sqrt(t)) - (Math.sqrt(x) - 1.0)) - (Math.sqrt(y) - Math.sqrt((1.0 + y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 8.2e+14:
		tmp = (1.0 + math.sqrt((1.0 + x))) + (math.sqrt((z - -1.0)) - ((math.sqrt(z) + math.sqrt(y)) + math.sqrt(x)))
	else:
		tmp = ((math.sqrt((t - -1.0)) - math.sqrt(t)) - (math.sqrt(x) - 1.0)) - (math.sqrt(y) - math.sqrt((1.0 + y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 8.2e+14)
		tmp = Float64(Float64(1.0 + sqrt(Float64(1.0 + x))) + Float64(sqrt(Float64(z - -1.0)) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x))));
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(t - -1.0)) - sqrt(t)) - Float64(sqrt(x) - 1.0)) - Float64(sqrt(y) - sqrt(Float64(1.0 + y))));
	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 (z <= 8.2e+14)
		tmp = (1.0 + sqrt((1.0 + x))) + (sqrt((z - -1.0)) - ((sqrt(z) + sqrt(y)) + sqrt(x)));
	else
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - (sqrt(x) - 1.0)) - (sqrt(y) - sqrt((1.0 + y)));
	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[z, 8.2e+14], N[(N[(1.0 + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.2e14

   1. Initial program 91.1%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 32.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 60.7%

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

   7. Taylor expanded in y around 0

      \[\leadsto \left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{z - -1}} - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 61.9

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

   9. Applied rewrites 61.9%

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

   if 8.2e14 < z

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in z around inf

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \color{blue}{\left(\sqrt{y} - \sqrt{1 + y}\right)} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 63.9

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

   9. Applied rewrites 63.9%

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

Alternative 18: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 0.4:\\ \;\;\;\;\left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\sqrt{y} - \sqrt{1 + y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x - -1} + \sqrt{y - -1}\right) + \left(1 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= (- (sqrt (+ z 1.0)) (sqrt z)) 0.4)
   (-
    (- (- (sqrt (- t -1.0)) (sqrt t)) (- (sqrt x) 1.0))
    (- (sqrt y) (sqrt (+ 1.0 y))))
   (+
    (+ (sqrt (- x -1.0)) (sqrt (- y -1.0)))
    (- 1.0 (+ (sqrt x) (+ (sqrt y) (sqrt z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((sqrt((z + 1.0)) - sqrt(z)) <= 0.4) {
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - (sqrt(x) - 1.0)) - (sqrt(y) - sqrt((1.0 + y)));
	} else {
		tmp = (sqrt((x - -1.0)) + sqrt((y - -1.0))) + (1.0 - (sqrt(x) + (sqrt(y) + sqrt(z))));
	}
	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 ((sqrt((z + 1.0d0)) - sqrt(z)) <= 0.4d0) then
        tmp = ((sqrt((t - (-1.0d0))) - sqrt(t)) - (sqrt(x) - 1.0d0)) - (sqrt(y) - sqrt((1.0d0 + y)))
    else
        tmp = (sqrt((x - (-1.0d0))) + sqrt((y - (-1.0d0)))) + (1.0d0 - (sqrt(x) + (sqrt(y) + sqrt(z))))
    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 ((Math.sqrt((z + 1.0)) - Math.sqrt(z)) <= 0.4) {
		tmp = ((Math.sqrt((t - -1.0)) - Math.sqrt(t)) - (Math.sqrt(x) - 1.0)) - (Math.sqrt(y) - Math.sqrt((1.0 + y)));
	} else {
		tmp = (Math.sqrt((x - -1.0)) + Math.sqrt((y - -1.0))) + (1.0 - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (math.sqrt((z + 1.0)) - math.sqrt(z)) <= 0.4:
		tmp = ((math.sqrt((t - -1.0)) - math.sqrt(t)) - (math.sqrt(x) - 1.0)) - (math.sqrt(y) - math.sqrt((1.0 + y)))
	else:
		tmp = (math.sqrt((x - -1.0)) + math.sqrt((y - -1.0))) + (1.0 - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) <= 0.4)
		tmp = Float64(Float64(Float64(sqrt(Float64(t - -1.0)) - sqrt(t)) - Float64(sqrt(x) - 1.0)) - Float64(sqrt(y) - sqrt(Float64(1.0 + y))));
	else
		tmp = Float64(Float64(sqrt(Float64(x - -1.0)) + sqrt(Float64(y - -1.0))) + Float64(1.0 - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))));
	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 ((sqrt((z + 1.0)) - sqrt(z)) <= 0.4)
		tmp = ((sqrt((t - -1.0)) - sqrt(t)) - (sqrt(x) - 1.0)) - (sqrt(y) - sqrt((1.0 + y)));
	else
		tmp = (sqrt((x - -1.0)) + sqrt((y - -1.0))) + (1.0 - (sqrt(x) + (sqrt(y) + sqrt(z))));
	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[N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.4], N[(N[(N[(N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 0.40000000000000002

   1. Initial program 91.1%

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in z around inf

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \color{blue}{\left(\sqrt{y} - \sqrt{1 + y}\right)} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 63.9

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

   9. Applied rewrites 63.9%

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

   if 0.40000000000000002 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 91.1%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 32.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 60.7%

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

   7. Taylor expanded in z around 0

      \[\leadsto \left(\sqrt{x - -1} + \sqrt{y - -1}\right) + \left(1 - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 24.7

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

   9. Applied rewrites 24.7%

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

Alternative 19: 65.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - -1} + \left(-\sqrt{x}\right)\\ \end{array} \end{array} \]

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 (<= y 5e+15)
   (- (+ (sqrt (+ 1.0 x)) (sqrt (+ 1.0 y))) (+ (sqrt x) (sqrt y)))
   (+ (sqrt (- x -1.0)) (- (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5e+15) {
		tmp = (sqrt((1.0 + x)) + sqrt((1.0 + y))) - (sqrt(x) + sqrt(y));
	} else {
		tmp = sqrt((x - -1.0)) + -sqrt(x);
	}
	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 (y <= 5d+15) then
        tmp = (sqrt((1.0d0 + x)) + sqrt((1.0d0 + y))) - (sqrt(x) + sqrt(y))
    else
        tmp = sqrt((x - (-1.0d0))) + -sqrt(x)
    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 (y <= 5e+15) {
		tmp = (Math.sqrt((1.0 + x)) + Math.sqrt((1.0 + y))) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = Math.sqrt((x - -1.0)) + -Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 5e+15:
		tmp = (math.sqrt((1.0 + x)) + math.sqrt((1.0 + y))) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = math.sqrt((x - -1.0)) + -math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5e+15)
		tmp = Float64(Float64(sqrt(Float64(1.0 + x)) + sqrt(Float64(1.0 + y))) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(sqrt(Float64(x - -1.0)) + Float64(-sqrt(x)));
	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 (y <= 5e+15)
		tmp = (sqrt((1.0 + x)) + sqrt((1.0 + y))) - (sqrt(x) + sqrt(y));
	else
		tmp = sqrt((x - -1.0)) + -sqrt(x);
	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[y, 5e+15], N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] + (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5e15

   1. Initial program 91.1%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 32.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 60.7%

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

   7. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      9. lower-sqrt.f64 46.3

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

   9. Applied rewrites 46.3%

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

   if 5e15 < y

   1. Initial program 91.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-+r- N/A

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

      7. associate-+l- N/A

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

   3. Applied rewrites 90.0%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 35.9

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

   6. Applied rewrites 35.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \sqrt{x - -1} + -1 \cdot \color{blue}{\sqrt{x}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{x - -1} + -1 \cdot \sqrt{x} \]

      2. lower-sqrt.f64 35.9

         \[\leadsto \sqrt{x - -1} + -1 \cdot \sqrt{x} \]

   9. Applied rewrites 35.9%

      \[\leadsto \sqrt{x - -1} + -1 \cdot \color{blue}{\sqrt{x}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{x - -1} + -1 \cdot \sqrt{x} \]

       2. mul-1-neg N/A

          \[\leadsto \sqrt{x - -1} + \left(\mathsf{neg}\left(\sqrt{x}\right)\right) \]

       3. lift-neg.f64 35.9

          \[\leadsto \sqrt{x - -1} + \left(-\sqrt{x}\right) \]

   11. Applied rewrites 35.9%

       \[\leadsto \sqrt{x - -1} + \color{blue}{\left(-\sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 63.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\sqrt{y} - \sqrt{1 + y}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (-
  (- (- (sqrt (- t -1.0)) (sqrt t)) (- (sqrt x) 1.0))
  (- (sqrt y) (sqrt (+ 1.0 y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return ((sqrt((t - -1.0)) - sqrt(t)) - (sqrt(x) - 1.0)) - (sqrt(y) - sqrt((1.0 + y)));
}

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 = ((sqrt((t - (-1.0d0))) - sqrt(t)) - (sqrt(x) - 1.0d0)) - (sqrt(y) - sqrt((1.0d0 + y)))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return ((Math.sqrt((t - -1.0)) - Math.sqrt(t)) - (Math.sqrt(x) - 1.0)) - (Math.sqrt(y) - Math.sqrt((1.0 + y)));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return ((math.sqrt((t - -1.0)) - math.sqrt(t)) - (math.sqrt(x) - 1.0)) - (math.sqrt(y) - math.sqrt((1.0 + y)))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(Float64(sqrt(Float64(t - -1.0)) - sqrt(t)) - Float64(sqrt(x) - 1.0)) - Float64(sqrt(y) - sqrt(Float64(1.0 + y))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = ((sqrt((t - -1.0)) - sqrt(t)) - (sqrt(x) - 1.0)) - (sqrt(y) - sqrt((1.0 + y)));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[(N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.1%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

3. Applied rewrites 91.1%

   \[\leadsto \color{blue}{\left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \sqrt{x - -1}\right)\right) - \left(\left(\sqrt{y} - \left(\sqrt{z - -1} - \sqrt{z}\right)\right) - \sqrt{y - -1}\right)} \]

4. Taylor expanded in x around 0

   \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \color{blue}{1}\right)\right) - \left(\left(\sqrt{y} - \left(\sqrt{z - -1} - \sqrt{z}\right)\right) - \sqrt{y - -1}\right) \]
5. Step-by-step derivation

6. Applied rewrites 89.9%

   \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - \color{blue}{1}\right)\right) - \left(\left(\sqrt{y} - \left(\sqrt{z - -1} - \sqrt{z}\right)\right) - \sqrt{y - -1}\right) \]

7. Taylor expanded in z around inf

   \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \color{blue}{\left(\sqrt{y} - \sqrt{1 + y}\right)} \]
8. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\sqrt{y} - \color{blue}{\sqrt{1 + y}}\right) \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\sqrt{y} - \sqrt{\color{blue}{1 + y}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\sqrt{y} - \sqrt{1 + y}\right) \]

   4. lower-+.f64 63.9

      \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \left(\sqrt{y} - \sqrt{1 + y}\right) \]

9. Applied rewrites 63.9%

   \[\leadsto \left(\left(\sqrt{t - -1} - \sqrt{t}\right) - \left(\sqrt{x} - 1\right)\right) - \color{blue}{\left(\sqrt{y} - \sqrt{1 + y}\right)} \]
10. Add Preprocessing

Alternative 21: 35.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{x - -1} + \left(-\sqrt{x}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (+ (sqrt (- x -1.0)) (- (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt((x - -1.0)) + -sqrt(x);
}

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 = sqrt((x - (-1.0d0))) + -sqrt(x)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.sqrt((x - -1.0)) + -Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt((x - -1.0)) + -math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(sqrt(Float64(x - -1.0)) + Float64(-sqrt(x)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt((x - -1.0)) + -sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] + (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 91.1%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   5. lift--.f64 N/A

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   6. associate-+r- N/A

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   7. associate-+l- N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

3. Applied rewrites 90.0%

   \[\leadsto \color{blue}{\sqrt{x - -1} + \left(\left(\left(-\sqrt{x}\right) + \sqrt{y - -1}\right) - \left(\left(\sqrt{y} - \left(\sqrt{t - -1} - \sqrt{t}\right)\right) - \left(\sqrt{z - -1} - \sqrt{z}\right)\right)\right)} \]

4. Taylor expanded in x around inf

   \[\leadsto \sqrt{x - -1} + \color{blue}{-1 \cdot \left(x \cdot \sqrt{\frac{1}{x}}\right)} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sqrt{x - -1} + -1 \cdot \color{blue}{\left(x \cdot \sqrt{\frac{1}{x}}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{x - -1} + -1 \cdot \left(x \cdot \color{blue}{\sqrt{\frac{1}{x}}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{x - -1} + -1 \cdot \left(x \cdot \sqrt{\frac{1}{x}}\right) \]

   4. lower-/.f64 35.9

      \[\leadsto \sqrt{x - -1} + -1 \cdot \left(x \cdot \sqrt{\frac{1}{x}}\right) \]

6. Applied rewrites 35.9%

   \[\leadsto \sqrt{x - -1} + \color{blue}{-1 \cdot \left(x \cdot \sqrt{\frac{1}{x}}\right)} \]

7. Taylor expanded in x around 0

   \[\leadsto \sqrt{x - -1} + -1 \cdot \color{blue}{\sqrt{x}} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sqrt{x - -1} + -1 \cdot \sqrt{x} \]

   2. lower-sqrt.f64 35.9

      \[\leadsto \sqrt{x - -1} + -1 \cdot \sqrt{x} \]

9. Applied rewrites 35.9%

   \[\leadsto \sqrt{x - -1} + -1 \cdot \color{blue}{\sqrt{x}} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \sqrt{x - -1} + -1 \cdot \sqrt{x} \]

    2. mul-1-neg N/A

       \[\leadsto \sqrt{x - -1} + \left(\mathsf{neg}\left(\sqrt{x}\right)\right) \]

    3. lift-neg.f64 35.9

       \[\leadsto \sqrt{x - -1} + \left(-\sqrt{x}\right) \]

11. Applied rewrites 35.9%

    \[\leadsto \sqrt{x - -1} + \color{blue}{\left(-\sqrt{x}\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64
  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))