Main:z from

Percentage Accurate: 91.5% → 96.2%

Time: 15.2s

Alternatives: 17

Speedup: 0.4×

Specification

Math

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

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: 91.5% accurate, 1.0× speedup

Math

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

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: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(t\_1, t\right)\\ t_3 := \sqrt{t\_2 + 1} - \sqrt{t\_2}\\ t_4 := \mathsf{min}\left(t\_1, t\right)\\ t_5 := \sqrt{t\_4}\\ t_6 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_7 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_6\right)\\ t_8 := \sqrt{t\_7}\\ t_9 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_6\right)\\ \mathbf{if}\;\sqrt{t\_4 + 1} - t\_5 \leq 0.9999999999999994:\\ \;\;\;\;\left(\frac{1}{t\_5 + \sqrt{1 + t\_4}} + \left(\sqrt{t\_7 + 1} - t\_8\right)\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - t\_5\right) + \left(\sqrt{t\_9 + 1} - \sqrt{t\_9}\right)\right) + \frac{\left|-1 - t\_7\right| - \left|t\_7\right|}{\sqrt{t\_7 - -1} + t\_8}\right) + t\_3\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin (fmin x y) z))
       (t_2 (fmax t_1 t))
       (t_3 (- (sqrt (+ t_2 1.0)) (sqrt t_2)))
       (t_4 (fmin t_1 t))
       (t_5 (sqrt t_4))
       (t_6 (fmax (fmin x y) z))
       (t_7 (fmax (fmax x y) t_6))
       (t_8 (sqrt t_7))
       (t_9 (fmin (fmax x y) t_6)))
  (if (<= (- (sqrt (+ t_4 1.0)) t_5) 0.9999999999999994)
    (+
     (+ (/ 1.0 (+ t_5 (sqrt (+ 1.0 t_4)))) (- (sqrt (+ t_7 1.0)) t_8))
     t_3)
    (+
     (+
      (+ (- 1.0 t_5) (- (sqrt (+ t_9 1.0)) (sqrt t_9)))
      (/
       (- (fabs (- -1.0 t_7)) (fabs t_7))
       (+ (sqrt (- t_7 -1.0)) t_8)))
     t_3))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(t_1, t);
	double t_3 = sqrt((t_2 + 1.0)) - sqrt(t_2);
	double t_4 = fmin(t_1, t);
	double t_5 = sqrt(t_4);
	double t_6 = fmax(fmin(x, y), z);
	double t_7 = fmax(fmax(x, y), t_6);
	double t_8 = sqrt(t_7);
	double t_9 = fmin(fmax(x, y), t_6);
	double tmp;
	if ((sqrt((t_4 + 1.0)) - t_5) <= 0.9999999999999994) {
		tmp = ((1.0 / (t_5 + sqrt((1.0 + t_4)))) + (sqrt((t_7 + 1.0)) - t_8)) + t_3;
	} else {
		tmp = (((1.0 - t_5) + (sqrt((t_9 + 1.0)) - sqrt(t_9))) + ((fabs((-1.0 - t_7)) - fabs(t_7)) / (sqrt((t_7 - -1.0)) + t_8))) + t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmin(fmin(x, y), z)
    t_2 = fmax(t_1, t)
    t_3 = sqrt((t_2 + 1.0d0)) - sqrt(t_2)
    t_4 = fmin(t_1, t)
    t_5 = sqrt(t_4)
    t_6 = fmax(fmin(x, y), z)
    t_7 = fmax(fmax(x, y), t_6)
    t_8 = sqrt(t_7)
    t_9 = fmin(fmax(x, y), t_6)
    if ((sqrt((t_4 + 1.0d0)) - t_5) <= 0.9999999999999994d0) then
        tmp = ((1.0d0 / (t_5 + sqrt((1.0d0 + t_4)))) + (sqrt((t_7 + 1.0d0)) - t_8)) + t_3
    else
        tmp = (((1.0d0 - t_5) + (sqrt((t_9 + 1.0d0)) - sqrt(t_9))) + ((abs(((-1.0d0) - t_7)) - abs(t_7)) / (sqrt((t_7 - (-1.0d0))) + t_8))) + t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(t_1, t);
	double t_3 = Math.sqrt((t_2 + 1.0)) - Math.sqrt(t_2);
	double t_4 = fmin(t_1, t);
	double t_5 = Math.sqrt(t_4);
	double t_6 = fmax(fmin(x, y), z);
	double t_7 = fmax(fmax(x, y), t_6);
	double t_8 = Math.sqrt(t_7);
	double t_9 = fmin(fmax(x, y), t_6);
	double tmp;
	if ((Math.sqrt((t_4 + 1.0)) - t_5) <= 0.9999999999999994) {
		tmp = ((1.0 / (t_5 + Math.sqrt((1.0 + t_4)))) + (Math.sqrt((t_7 + 1.0)) - t_8)) + t_3;
	} else {
		tmp = (((1.0 - t_5) + (Math.sqrt((t_9 + 1.0)) - Math.sqrt(t_9))) + ((Math.abs((-1.0 - t_7)) - Math.abs(t_7)) / (Math.sqrt((t_7 - -1.0)) + t_8))) + t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = math.sqrt((t_2 + 1.0)) - math.sqrt(t_2)
	t_4 = fmin(t_1, t)
	t_5 = math.sqrt(t_4)
	t_6 = fmax(fmin(x, y), z)
	t_7 = fmax(fmax(x, y), t_6)
	t_8 = math.sqrt(t_7)
	t_9 = fmin(fmax(x, y), t_6)
	tmp = 0
	if (math.sqrt((t_4 + 1.0)) - t_5) <= 0.9999999999999994:
		tmp = ((1.0 / (t_5 + math.sqrt((1.0 + t_4)))) + (math.sqrt((t_7 + 1.0)) - t_8)) + t_3
	else:
		tmp = (((1.0 - t_5) + (math.sqrt((t_9 + 1.0)) - math.sqrt(t_9))) + ((math.fabs((-1.0 - t_7)) - math.fabs(t_7)) / (math.sqrt((t_7 - -1.0)) + t_8))) + t_3
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = Float64(sqrt(Float64(t_2 + 1.0)) - sqrt(t_2))
	t_4 = fmin(t_1, t)
	t_5 = sqrt(t_4)
	t_6 = fmax(fmin(x, y), z)
	t_7 = fmax(fmax(x, y), t_6)
	t_8 = sqrt(t_7)
	t_9 = fmin(fmax(x, y), t_6)
	tmp = 0.0
	if (Float64(sqrt(Float64(t_4 + 1.0)) - t_5) <= 0.9999999999999994)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_5 + sqrt(Float64(1.0 + t_4)))) + Float64(sqrt(Float64(t_7 + 1.0)) - t_8)) + t_3);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - t_5) + Float64(sqrt(Float64(t_9 + 1.0)) - sqrt(t_9))) + Float64(Float64(abs(Float64(-1.0 - t_7)) - abs(t_7)) / Float64(sqrt(Float64(t_7 - -1.0)) + t_8))) + t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = min(min(x, y), z);
	t_2 = max(t_1, t);
	t_3 = sqrt((t_2 + 1.0)) - sqrt(t_2);
	t_4 = min(t_1, t);
	t_5 = sqrt(t_4);
	t_6 = max(min(x, y), z);
	t_7 = max(max(x, y), t_6);
	t_8 = sqrt(t_7);
	t_9 = min(max(x, y), t_6);
	tmp = 0.0;
	if ((sqrt((t_4 + 1.0)) - t_5) <= 0.9999999999999994)
		tmp = ((1.0 / (t_5 + sqrt((1.0 + t_4)))) + (sqrt((t_7 + 1.0)) - t_8)) + t_3;
	else
		tmp = (((1.0 - t_5) + (sqrt((t_9 + 1.0)) - sqrt(t_9))) + ((abs((-1.0 - t_7)) - abs(t_7)) / (sqrt((t_7 - -1.0)) + t_8))) + t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[t$95$1, t], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Min[t$95$1, t], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[t$95$4], $MachinePrecision]}, Block[{t$95$6 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$7 = N[Max[N[Max[x, y], $MachinePrecision], t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[Sqrt[t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Min[N[Max[x, y], $MachinePrecision], t$95$6], $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(t$95$4 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$5), $MachinePrecision], 0.9999999999999994], N[(N[(N[(1.0 / N[(t$95$5 + N[Sqrt[N[(1.0 + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$7 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$8), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(N[(N[(1.0 - t$95$5), $MachinePrecision] + N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$9], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Abs[N[(-1.0 - t$95$7), $MachinePrecision]], $MachinePrecision] - N[Abs[t$95$7], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(t$95$7 - -1.0), $MachinePrecision]], $MachinePrecision] + t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.99999999999999944

   1. Initial program 91.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-+.f64 72.3%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 72.3%

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

   3. Applied rewrites 72.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. associate--l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 93.5%

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 52.7%

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

   8. Applied rewrites 52.7%

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

   if 0.99999999999999944 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 48.3%

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

   4. Applied rewrites 48.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. lower-unsound-+.f64 N/A

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

      8. lower-unsound-/.f64 N/A

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

   6. Applied rewrites 48.5%

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

Alternative 2: 96.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_3 := \mathsf{min}\left(t\_2, t\right)\\ t_4 := \mathsf{max}\left(t\_2, t\right)\\ t_5 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_6 := \mathsf{max}\left(t\_5, t\_4\right)\\ t_7 := \sqrt{t\_6}\\ t_8 := \mathsf{min}\left(t\_5, t\_4\right)\\ t_9 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_10 := \sqrt{t\_3}\\ t_11 := \sqrt{t\_9}\\ \mathbf{if}\;t\_8 \leq 1.36 \cdot 10^{+23}:\\ \;\;\;\;\left(\sqrt{t\_8 - -1} - \left(t\_10 - \sqrt{t\_3 - -1}\right)\right) - \left(\left(\sqrt{t\_8} - \left(\sqrt{t\_6 - -1} - t\_7\right)\right) - \left(\sqrt{t\_9 - -1} - t\_11\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t\_10 + \sqrt{1 + t\_3}} + \left(\sqrt{t\_9 + 1} - t\_11\right)\right) + \left(\sqrt{t\_6 + 1} - t\_7\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmin x y) z))
       (t_3 (fmin t_2 t))
       (t_4 (fmax t_2 t))
       (t_5 (fmin (fmax x y) t_1))
       (t_6 (fmax t_5 t_4))
       (t_7 (sqrt t_6))
       (t_8 (fmin t_5 t_4))
       (t_9 (fmax (fmax x y) t_1))
       (t_10 (sqrt t_3))
       (t_11 (sqrt t_9)))
  (if (<= t_8 1.36e+23)
    (-
     (- (sqrt (- t_8 -1.0)) (- t_10 (sqrt (- t_3 -1.0))))
     (-
      (- (sqrt t_8) (- (sqrt (- t_6 -1.0)) t_7))
      (- (sqrt (- t_9 -1.0)) t_11)))
    (+
     (+
      (/ 1.0 (+ t_10 (sqrt (+ 1.0 t_3))))
      (- (sqrt (+ t_9 1.0)) t_11))
     (- (sqrt (+ t_6 1.0)) t_7)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmin(x, y), z);
	double t_3 = fmin(t_2, t);
	double t_4 = fmax(t_2, t);
	double t_5 = fmin(fmax(x, y), t_1);
	double t_6 = fmax(t_5, t_4);
	double t_7 = sqrt(t_6);
	double t_8 = fmin(t_5, t_4);
	double t_9 = fmax(fmax(x, y), t_1);
	double t_10 = sqrt(t_3);
	double t_11 = sqrt(t_9);
	double tmp;
	if (t_8 <= 1.36e+23) {
		tmp = (sqrt((t_8 - -1.0)) - (t_10 - sqrt((t_3 - -1.0)))) - ((sqrt(t_8) - (sqrt((t_6 - -1.0)) - t_7)) - (sqrt((t_9 - -1.0)) - t_11));
	} else {
		tmp = ((1.0 / (t_10 + sqrt((1.0 + t_3)))) + (sqrt((t_9 + 1.0)) - t_11)) + (sqrt((t_6 + 1.0)) - t_7);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmin(fmin(x, y), z)
    t_3 = fmin(t_2, t)
    t_4 = fmax(t_2, t)
    t_5 = fmin(fmax(x, y), t_1)
    t_6 = fmax(t_5, t_4)
    t_7 = sqrt(t_6)
    t_8 = fmin(t_5, t_4)
    t_9 = fmax(fmax(x, y), t_1)
    t_10 = sqrt(t_3)
    t_11 = sqrt(t_9)
    if (t_8 <= 1.36d+23) then
        tmp = (sqrt((t_8 - (-1.0d0))) - (t_10 - sqrt((t_3 - (-1.0d0))))) - ((sqrt(t_8) - (sqrt((t_6 - (-1.0d0))) - t_7)) - (sqrt((t_9 - (-1.0d0))) - t_11))
    else
        tmp = ((1.0d0 / (t_10 + sqrt((1.0d0 + t_3)))) + (sqrt((t_9 + 1.0d0)) - t_11)) + (sqrt((t_6 + 1.0d0)) - t_7)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmin(x, y), z);
	double t_3 = fmin(t_2, t);
	double t_4 = fmax(t_2, t);
	double t_5 = fmin(fmax(x, y), t_1);
	double t_6 = fmax(t_5, t_4);
	double t_7 = Math.sqrt(t_6);
	double t_8 = fmin(t_5, t_4);
	double t_9 = fmax(fmax(x, y), t_1);
	double t_10 = Math.sqrt(t_3);
	double t_11 = Math.sqrt(t_9);
	double tmp;
	if (t_8 <= 1.36e+23) {
		tmp = (Math.sqrt((t_8 - -1.0)) - (t_10 - Math.sqrt((t_3 - -1.0)))) - ((Math.sqrt(t_8) - (Math.sqrt((t_6 - -1.0)) - t_7)) - (Math.sqrt((t_9 - -1.0)) - t_11));
	} else {
		tmp = ((1.0 / (t_10 + Math.sqrt((1.0 + t_3)))) + (Math.sqrt((t_9 + 1.0)) - t_11)) + (Math.sqrt((t_6 + 1.0)) - t_7);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmin(x, y), z)
	t_3 = fmin(t_2, t)
	t_4 = fmax(t_2, t)
	t_5 = fmin(fmax(x, y), t_1)
	t_6 = fmax(t_5, t_4)
	t_7 = math.sqrt(t_6)
	t_8 = fmin(t_5, t_4)
	t_9 = fmax(fmax(x, y), t_1)
	t_10 = math.sqrt(t_3)
	t_11 = math.sqrt(t_9)
	tmp = 0
	if t_8 <= 1.36e+23:
		tmp = (math.sqrt((t_8 - -1.0)) - (t_10 - math.sqrt((t_3 - -1.0)))) - ((math.sqrt(t_8) - (math.sqrt((t_6 - -1.0)) - t_7)) - (math.sqrt((t_9 - -1.0)) - t_11))
	else:
		tmp = ((1.0 / (t_10 + math.sqrt((1.0 + t_3)))) + (math.sqrt((t_9 + 1.0)) - t_11)) + (math.sqrt((t_6 + 1.0)) - t_7)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmin(x, y), z)
	t_3 = fmin(t_2, t)
	t_4 = fmax(t_2, t)
	t_5 = fmin(fmax(x, y), t_1)
	t_6 = fmax(t_5, t_4)
	t_7 = sqrt(t_6)
	t_8 = fmin(t_5, t_4)
	t_9 = fmax(fmax(x, y), t_1)
	t_10 = sqrt(t_3)
	t_11 = sqrt(t_9)
	tmp = 0.0
	if (t_8 <= 1.36e+23)
		tmp = Float64(Float64(sqrt(Float64(t_8 - -1.0)) - Float64(t_10 - sqrt(Float64(t_3 - -1.0)))) - Float64(Float64(sqrt(t_8) - Float64(sqrt(Float64(t_6 - -1.0)) - t_7)) - Float64(sqrt(Float64(t_9 - -1.0)) - t_11)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_10 + sqrt(Float64(1.0 + t_3)))) + Float64(sqrt(Float64(t_9 + 1.0)) - t_11)) + Float64(sqrt(Float64(t_6 + 1.0)) - t_7));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = min(min(x, y), z);
	t_3 = min(t_2, t);
	t_4 = max(t_2, t);
	t_5 = min(max(x, y), t_1);
	t_6 = max(t_5, t_4);
	t_7 = sqrt(t_6);
	t_8 = min(t_5, t_4);
	t_9 = max(max(x, y), t_1);
	t_10 = sqrt(t_3);
	t_11 = sqrt(t_9);
	tmp = 0.0;
	if (t_8 <= 1.36e+23)
		tmp = (sqrt((t_8 - -1.0)) - (t_10 - sqrt((t_3 - -1.0)))) - ((sqrt(t_8) - (sqrt((t_6 - -1.0)) - t_7)) - (sqrt((t_9 - -1.0)) - t_11));
	else
		tmp = ((1.0 / (t_10 + sqrt((1.0 + t_3)))) + (sqrt((t_9 + 1.0)) - t_11)) + (sqrt((t_6 + 1.0)) - t_7);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$3 = N[Min[t$95$2, t], $MachinePrecision]}, Block[{t$95$4 = N[Max[t$95$2, t], $MachinePrecision]}, Block[{t$95$5 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$6 = N[Max[t$95$5, t$95$4], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[Min[t$95$5, t$95$4], $MachinePrecision]}, Block[{t$95$9 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$3], $MachinePrecision]}, Block[{t$95$11 = N[Sqrt[t$95$9], $MachinePrecision]}, If[LessEqual[t$95$8, 1.36e+23], N[(N[(N[Sqrt[N[(t$95$8 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(t$95$10 - N[Sqrt[N[(t$95$3 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[t$95$8], $MachinePrecision] - N[(N[Sqrt[N[(t$95$6 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$7), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[(t$95$9 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$11), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(t$95$10 + N[Sqrt[N[(1.0 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$11), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$6 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$7), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.36e23

   1. Initial program 91.5%

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

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

      8. lower--.f64 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 53.7%

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

   if 1.36e23 < y

   1. Initial program 91.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-+.f64 72.3%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 72.3%

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

   3. Applied rewrites 72.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. associate--l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 93.5%

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 52.7%

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

   8. Applied rewrites 52.7%

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

Alternative 3: 96.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(t\_1, t\right)\\ t_3 := \mathsf{min}\left(t\_1, t\right)\\ t_4 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ \left(\left(\frac{1 + \left(t\_3 - t\_3\right)}{\sqrt{t\_3 - -1} + \sqrt{t\_3}} + \left(\sqrt{\mathsf{max}\left(x, y\right) + 1} - \sqrt{\mathsf{max}\left(x, y\right)}\right)\right) + \left(\sqrt{t\_4 + 1} - \sqrt{t\_4}\right)\right) + \left(\sqrt{t\_2 + 1} - \sqrt{t\_2}\right) \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin (fmin x y) z))
       (t_2 (fmax t_1 t))
       (t_3 (fmin t_1 t))
       (t_4 (fmax (fmin x y) z)))
  (+
   (+
    (+
     (/ (+ 1.0 (- t_3 t_3)) (+ (sqrt (- t_3 -1.0)) (sqrt t_3)))
     (- (sqrt (+ (fmax x y) 1.0)) (sqrt (fmax x y))))
    (- (sqrt (+ t_4 1.0)) (sqrt t_4)))
   (- (sqrt (+ t_2 1.0)) (sqrt t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(t_1, t);
	double t_3 = fmin(t_1, t);
	double t_4 = fmax(fmin(x, y), z);
	return ((((1.0 + (t_3 - t_3)) / (sqrt((t_3 - -1.0)) + sqrt(t_3))) + (sqrt((fmax(x, y) + 1.0)) - sqrt(fmax(x, y)))) + (sqrt((t_4 + 1.0)) - sqrt(t_4))) + (sqrt((t_2 + 1.0)) - sqrt(t_2));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    t_1 = fmin(fmin(x, y), z)
    t_2 = fmax(t_1, t)
    t_3 = fmin(t_1, t)
    t_4 = fmax(fmin(x, y), z)
    code = ((((1.0d0 + (t_3 - t_3)) / (sqrt((t_3 - (-1.0d0))) + sqrt(t_3))) + (sqrt((fmax(x, y) + 1.0d0)) - sqrt(fmax(x, y)))) + (sqrt((t_4 + 1.0d0)) - sqrt(t_4))) + (sqrt((t_2 + 1.0d0)) - sqrt(t_2))
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(t_1, t);
	double t_3 = fmin(t_1, t);
	double t_4 = fmax(fmin(x, y), z);
	return ((((1.0 + (t_3 - t_3)) / (Math.sqrt((t_3 - -1.0)) + Math.sqrt(t_3))) + (Math.sqrt((fmax(x, y) + 1.0)) - Math.sqrt(fmax(x, y)))) + (Math.sqrt((t_4 + 1.0)) - Math.sqrt(t_4))) + (Math.sqrt((t_2 + 1.0)) - Math.sqrt(t_2));
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = fmin(t_1, t)
	t_4 = fmax(fmin(x, y), z)
	return ((((1.0 + (t_3 - t_3)) / (math.sqrt((t_3 - -1.0)) + math.sqrt(t_3))) + (math.sqrt((fmax(x, y) + 1.0)) - math.sqrt(fmax(x, y)))) + (math.sqrt((t_4 + 1.0)) - math.sqrt(t_4))) + (math.sqrt((t_2 + 1.0)) - math.sqrt(t_2))

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = fmin(t_1, t)
	t_4 = fmax(fmin(x, y), z)
	return Float64(Float64(Float64(Float64(Float64(1.0 + Float64(t_3 - t_3)) / Float64(sqrt(Float64(t_3 - -1.0)) + sqrt(t_3))) + Float64(sqrt(Float64(fmax(x, y) + 1.0)) - sqrt(fmax(x, y)))) + Float64(sqrt(Float64(t_4 + 1.0)) - sqrt(t_4))) + Float64(sqrt(Float64(t_2 + 1.0)) - sqrt(t_2)))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = min(min(x, y), z);
	t_2 = max(t_1, t);
	t_3 = min(t_1, t);
	t_4 = max(min(x, y), z);
	tmp = ((((1.0 + (t_3 - t_3)) / (sqrt((t_3 - -1.0)) + sqrt(t_3))) + (sqrt((max(x, y) + 1.0)) - sqrt(max(x, y)))) + (sqrt((t_4 + 1.0)) - sqrt(t_4))) + (sqrt((t_2 + 1.0)) - sqrt(t_2));
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[t$95$1, t], $MachinePrecision]}, Block[{t$95$3 = N[Min[t$95$1, t], $MachinePrecision]}, Block[{t$95$4 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, N[(N[(N[(N[(N[(1.0 + N[(t$95$3 - t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(t$95$3 - -1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(N[Max[x, y], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[N[Max[x, y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$4 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 91.5%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound--.f64 N/A

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

   5. lower-unsound-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-+.f64 N/A

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

   11. add-flip N/A

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

   12. lower--.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-unsound-*.f64 N/A

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

   15. lower-unsound-+.f64 72.3%

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

   16. lift-+.f64 N/A

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

   17. add-flip N/A

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

   18. lower--.f64 N/A

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

   19. metadata-eval 72.3%

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

3. Applied rewrites 72.3%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. metadata-eval N/A

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

   4. add-flip N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. associate--l+ N/A

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

   11. lower-+.f64 N/A

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

   12. lower--.f64 93.5%

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

5. Applied rewrites 93.5%

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

Alternative 4: 95.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(t\_1, t\right)\\ t_3 := \sqrt{t\_2}\\ t_4 := \mathsf{max}\left(t\_1, t\right)\\ t_5 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_6 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_5\right)\\ t_7 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_5\right)\\ t_8 := \mathsf{min}\left(t\_6, t\_4\right)\\ t_9 := \mathsf{max}\left(t\_6, t\_4\right)\\ t_10 := \mathsf{min}\left(t\_7, t\_9\right)\\ t_11 := \sqrt{t\_10}\\ t_12 := \mathsf{max}\left(t\_7, t\_9\right)\\ t_13 := \sqrt{t\_12 + 1} - \sqrt{t\_12}\\ \mathbf{if}\;t\_8 \leq 1.36 \cdot 10^{+23}:\\ \;\;\;\;\left(\left(\left(\left(\sqrt{t\_10 - -1} - t\_11\right) + \sqrt{t\_2 - -1}\right) - \left(t\_3 - \sqrt{t\_8 - -1}\right)\right) - \sqrt{t\_8}\right) + t\_13\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t\_3 + \sqrt{1 + t\_2}} + \left(\sqrt{t\_10 + 1} - t\_11\right)\right) + t\_13\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin (fmin x y) z))
       (t_2 (fmin t_1 t))
       (t_3 (sqrt t_2))
       (t_4 (fmax t_1 t))
       (t_5 (fmax (fmin x y) z))
       (t_6 (fmin (fmax x y) t_5))
       (t_7 (fmax (fmax x y) t_5))
       (t_8 (fmin t_6 t_4))
       (t_9 (fmax t_6 t_4))
       (t_10 (fmin t_7 t_9))
       (t_11 (sqrt t_10))
       (t_12 (fmax t_7 t_9))
       (t_13 (- (sqrt (+ t_12 1.0)) (sqrt t_12))))
  (if (<= t_8 1.36e+23)
    (+
     (-
      (-
       (+ (- (sqrt (- t_10 -1.0)) t_11) (sqrt (- t_2 -1.0)))
       (- t_3 (sqrt (- t_8 -1.0))))
      (sqrt t_8))
     t_13)
    (+
     (+
      (/ 1.0 (+ t_3 (sqrt (+ 1.0 t_2))))
      (- (sqrt (+ t_10 1.0)) t_11))
     t_13))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmin(t_1, t);
	double t_3 = sqrt(t_2);
	double t_4 = fmax(t_1, t);
	double t_5 = fmax(fmin(x, y), z);
	double t_6 = fmin(fmax(x, y), t_5);
	double t_7 = fmax(fmax(x, y), t_5);
	double t_8 = fmin(t_6, t_4);
	double t_9 = fmax(t_6, t_4);
	double t_10 = fmin(t_7, t_9);
	double t_11 = sqrt(t_10);
	double t_12 = fmax(t_7, t_9);
	double t_13 = sqrt((t_12 + 1.0)) - sqrt(t_12);
	double tmp;
	if (t_8 <= 1.36e+23) {
		tmp = ((((sqrt((t_10 - -1.0)) - t_11) + sqrt((t_2 - -1.0))) - (t_3 - sqrt((t_8 - -1.0)))) - sqrt(t_8)) + t_13;
	} else {
		tmp = ((1.0 / (t_3 + sqrt((1.0 + t_2)))) + (sqrt((t_10 + 1.0)) - t_11)) + t_13;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmin(fmin(x, y), z)
    t_2 = fmin(t_1, t)
    t_3 = sqrt(t_2)
    t_4 = fmax(t_1, t)
    t_5 = fmax(fmin(x, y), z)
    t_6 = fmin(fmax(x, y), t_5)
    t_7 = fmax(fmax(x, y), t_5)
    t_8 = fmin(t_6, t_4)
    t_9 = fmax(t_6, t_4)
    t_10 = fmin(t_7, t_9)
    t_11 = sqrt(t_10)
    t_12 = fmax(t_7, t_9)
    t_13 = sqrt((t_12 + 1.0d0)) - sqrt(t_12)
    if (t_8 <= 1.36d+23) then
        tmp = ((((sqrt((t_10 - (-1.0d0))) - t_11) + sqrt((t_2 - (-1.0d0)))) - (t_3 - sqrt((t_8 - (-1.0d0))))) - sqrt(t_8)) + t_13
    else
        tmp = ((1.0d0 / (t_3 + sqrt((1.0d0 + t_2)))) + (sqrt((t_10 + 1.0d0)) - t_11)) + t_13
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmin(t_1, t);
	double t_3 = Math.sqrt(t_2);
	double t_4 = fmax(t_1, t);
	double t_5 = fmax(fmin(x, y), z);
	double t_6 = fmin(fmax(x, y), t_5);
	double t_7 = fmax(fmax(x, y), t_5);
	double t_8 = fmin(t_6, t_4);
	double t_9 = fmax(t_6, t_4);
	double t_10 = fmin(t_7, t_9);
	double t_11 = Math.sqrt(t_10);
	double t_12 = fmax(t_7, t_9);
	double t_13 = Math.sqrt((t_12 + 1.0)) - Math.sqrt(t_12);
	double tmp;
	if (t_8 <= 1.36e+23) {
		tmp = ((((Math.sqrt((t_10 - -1.0)) - t_11) + Math.sqrt((t_2 - -1.0))) - (t_3 - Math.sqrt((t_8 - -1.0)))) - Math.sqrt(t_8)) + t_13;
	} else {
		tmp = ((1.0 / (t_3 + Math.sqrt((1.0 + t_2)))) + (Math.sqrt((t_10 + 1.0)) - t_11)) + t_13;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmin(t_1, t)
	t_3 = math.sqrt(t_2)
	t_4 = fmax(t_1, t)
	t_5 = fmax(fmin(x, y), z)
	t_6 = fmin(fmax(x, y), t_5)
	t_7 = fmax(fmax(x, y), t_5)
	t_8 = fmin(t_6, t_4)
	t_9 = fmax(t_6, t_4)
	t_10 = fmin(t_7, t_9)
	t_11 = math.sqrt(t_10)
	t_12 = fmax(t_7, t_9)
	t_13 = math.sqrt((t_12 + 1.0)) - math.sqrt(t_12)
	tmp = 0
	if t_8 <= 1.36e+23:
		tmp = ((((math.sqrt((t_10 - -1.0)) - t_11) + math.sqrt((t_2 - -1.0))) - (t_3 - math.sqrt((t_8 - -1.0)))) - math.sqrt(t_8)) + t_13
	else:
		tmp = ((1.0 / (t_3 + math.sqrt((1.0 + t_2)))) + (math.sqrt((t_10 + 1.0)) - t_11)) + t_13
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmin(t_1, t)
	t_3 = sqrt(t_2)
	t_4 = fmax(t_1, t)
	t_5 = fmax(fmin(x, y), z)
	t_6 = fmin(fmax(x, y), t_5)
	t_7 = fmax(fmax(x, y), t_5)
	t_8 = fmin(t_6, t_4)
	t_9 = fmax(t_6, t_4)
	t_10 = fmin(t_7, t_9)
	t_11 = sqrt(t_10)
	t_12 = fmax(t_7, t_9)
	t_13 = Float64(sqrt(Float64(t_12 + 1.0)) - sqrt(t_12))
	tmp = 0.0
	if (t_8 <= 1.36e+23)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(t_10 - -1.0)) - t_11) + sqrt(Float64(t_2 - -1.0))) - Float64(t_3 - sqrt(Float64(t_8 - -1.0)))) - sqrt(t_8)) + t_13);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_3 + sqrt(Float64(1.0 + t_2)))) + Float64(sqrt(Float64(t_10 + 1.0)) - t_11)) + t_13);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = min(min(x, y), z);
	t_2 = min(t_1, t);
	t_3 = sqrt(t_2);
	t_4 = max(t_1, t);
	t_5 = max(min(x, y), z);
	t_6 = min(max(x, y), t_5);
	t_7 = max(max(x, y), t_5);
	t_8 = min(t_6, t_4);
	t_9 = max(t_6, t_4);
	t_10 = min(t_7, t_9);
	t_11 = sqrt(t_10);
	t_12 = max(t_7, t_9);
	t_13 = sqrt((t_12 + 1.0)) - sqrt(t_12);
	tmp = 0.0;
	if (t_8 <= 1.36e+23)
		tmp = ((((sqrt((t_10 - -1.0)) - t_11) + sqrt((t_2 - -1.0))) - (t_3 - sqrt((t_8 - -1.0)))) - sqrt(t_8)) + t_13;
	else
		tmp = ((1.0 / (t_3 + sqrt((1.0 + t_2)))) + (sqrt((t_10 + 1.0)) - t_11)) + t_13;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[t$95$1, t], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Max[t$95$1, t], $MachinePrecision]}, Block[{t$95$5 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$6 = N[Min[N[Max[x, y], $MachinePrecision], t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Max[N[Max[x, y], $MachinePrecision], t$95$5], $MachinePrecision]}, Block[{t$95$8 = N[Min[t$95$6, t$95$4], $MachinePrecision]}, Block[{t$95$9 = N[Max[t$95$6, t$95$4], $MachinePrecision]}, Block[{t$95$10 = N[Min[t$95$7, t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Sqrt[t$95$10], $MachinePrecision]}, Block[{t$95$12 = N[Max[t$95$7, t$95$9], $MachinePrecision]}, Block[{t$95$13 = N[(N[Sqrt[N[(t$95$12 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$12], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$8, 1.36e+23], N[(N[(N[(N[(N[(N[Sqrt[N[(t$95$10 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$11), $MachinePrecision] + N[Sqrt[N[(t$95$2 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$3 - N[Sqrt[N[(t$95$8 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[t$95$8], $MachinePrecision]), $MachinePrecision] + t$95$13), $MachinePrecision], N[(N[(N[(1.0 / N[(t$95$3 + N[Sqrt[N[(1.0 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$10 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$11), $MachinePrecision]), $MachinePrecision] + t$95$13), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.36e23

   1. Initial program 91.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-+l- N/A

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

      6. associate-+r- N/A

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

      7. lift--.f64 N/A

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

      8. associate--r- N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 37.8%

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

   if 1.36e23 < y

   1. Initial program 91.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-+.f64 72.3%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 72.3%

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

   3. Applied rewrites 72.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. associate--l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 93.5%

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 52.7%

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

   8. Applied rewrites 52.7%

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

Alternative 5: 95.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(t\_1, t\right)\\ t_3 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_4 := \sqrt{t\_3 + 1} - \sqrt{t\_3}\\ t_5 := \mathsf{min}\left(t\_1, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \sqrt{t\_2 + 1} - \sqrt{t\_2}\\ \mathbf{if}\;\mathsf{max}\left(x, y\right) \leq 1.05 \cdot 10^{+21}:\\ \;\;\;\;\left(\left(\left(\sqrt{t\_5 + 1} - t\_6\right) + \left(\sqrt{\mathsf{max}\left(x, y\right) + 1} - \sqrt{\mathsf{max}\left(x, y\right)}\right)\right) + t\_4\right) + t\_7\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t\_6 + \sqrt{1 + t\_5}} + t\_4\right) + t\_7\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin (fmin x y) z))
       (t_2 (fmax t_1 t))
       (t_3 (fmax (fmin x y) z))
       (t_4 (- (sqrt (+ t_3 1.0)) (sqrt t_3)))
       (t_5 (fmin t_1 t))
       (t_6 (sqrt t_5))
       (t_7 (- (sqrt (+ t_2 1.0)) (sqrt t_2))))
  (if (<= (fmax x y) 1.05e+21)
    (+
     (+
      (+
       (- (sqrt (+ t_5 1.0)) t_6)
       (- (sqrt (+ (fmax x y) 1.0)) (sqrt (fmax x y))))
      t_4)
     t_7)
    (+ (+ (/ 1.0 (+ t_6 (sqrt (+ 1.0 t_5)))) t_4) t_7))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(t_1, t);
	double t_3 = fmax(fmin(x, y), z);
	double t_4 = sqrt((t_3 + 1.0)) - sqrt(t_3);
	double t_5 = fmin(t_1, t);
	double t_6 = sqrt(t_5);
	double t_7 = sqrt((t_2 + 1.0)) - sqrt(t_2);
	double tmp;
	if (fmax(x, y) <= 1.05e+21) {
		tmp = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((fmax(x, y) + 1.0)) - sqrt(fmax(x, y)))) + t_4) + t_7;
	} else {
		tmp = ((1.0 / (t_6 + sqrt((1.0 + t_5)))) + t_4) + t_7;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = fmin(fmin(x, y), z)
    t_2 = fmax(t_1, t)
    t_3 = fmax(fmin(x, y), z)
    t_4 = sqrt((t_3 + 1.0d0)) - sqrt(t_3)
    t_5 = fmin(t_1, t)
    t_6 = sqrt(t_5)
    t_7 = sqrt((t_2 + 1.0d0)) - sqrt(t_2)
    if (fmax(x, y) <= 1.05d+21) then
        tmp = (((sqrt((t_5 + 1.0d0)) - t_6) + (sqrt((fmax(x, y) + 1.0d0)) - sqrt(fmax(x, y)))) + t_4) + t_7
    else
        tmp = ((1.0d0 / (t_6 + sqrt((1.0d0 + t_5)))) + t_4) + t_7
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(t_1, t);
	double t_3 = fmax(fmin(x, y), z);
	double t_4 = Math.sqrt((t_3 + 1.0)) - Math.sqrt(t_3);
	double t_5 = fmin(t_1, t);
	double t_6 = Math.sqrt(t_5);
	double t_7 = Math.sqrt((t_2 + 1.0)) - Math.sqrt(t_2);
	double tmp;
	if (fmax(x, y) <= 1.05e+21) {
		tmp = (((Math.sqrt((t_5 + 1.0)) - t_6) + (Math.sqrt((fmax(x, y) + 1.0)) - Math.sqrt(fmax(x, y)))) + t_4) + t_7;
	} else {
		tmp = ((1.0 / (t_6 + Math.sqrt((1.0 + t_5)))) + t_4) + t_7;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = fmax(fmin(x, y), z)
	t_4 = math.sqrt((t_3 + 1.0)) - math.sqrt(t_3)
	t_5 = fmin(t_1, t)
	t_6 = math.sqrt(t_5)
	t_7 = math.sqrt((t_2 + 1.0)) - math.sqrt(t_2)
	tmp = 0
	if fmax(x, y) <= 1.05e+21:
		tmp = (((math.sqrt((t_5 + 1.0)) - t_6) + (math.sqrt((fmax(x, y) + 1.0)) - math.sqrt(fmax(x, y)))) + t_4) + t_7
	else:
		tmp = ((1.0 / (t_6 + math.sqrt((1.0 + t_5)))) + t_4) + t_7
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = fmax(fmin(x, y), z)
	t_4 = Float64(sqrt(Float64(t_3 + 1.0)) - sqrt(t_3))
	t_5 = fmin(t_1, t)
	t_6 = sqrt(t_5)
	t_7 = Float64(sqrt(Float64(t_2 + 1.0)) - sqrt(t_2))
	tmp = 0.0
	if (fmax(x, y) <= 1.05e+21)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + Float64(sqrt(Float64(fmax(x, y) + 1.0)) - sqrt(fmax(x, y)))) + t_4) + t_7);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_6 + sqrt(Float64(1.0 + t_5)))) + t_4) + t_7);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = min(min(x, y), z);
	t_2 = max(t_1, t);
	t_3 = max(min(x, y), z);
	t_4 = sqrt((t_3 + 1.0)) - sqrt(t_3);
	t_5 = min(t_1, t);
	t_6 = sqrt(t_5);
	t_7 = sqrt((t_2 + 1.0)) - sqrt(t_2);
	tmp = 0.0;
	if (max(x, y) <= 1.05e+21)
		tmp = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((max(x, y) + 1.0)) - sqrt(max(x, y)))) + t_4) + t_7;
	else
		tmp = ((1.0 / (t_6 + sqrt((1.0 + t_5)))) + t_4) + t_7;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[t$95$1, t], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t$95$3 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$1, t], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[(N[Sqrt[N[(t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[x, y], $MachinePrecision], 1.05e+21], N[(N[(N[(N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + N[(N[Sqrt[N[(N[Max[x, y], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[N[Max[x, y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$7), $MachinePrecision], N[(N[(N[(1.0 / N[(t$95$6 + N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$7), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e21

   1. Initial program 91.5%

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

   if 1.05e21 < y

   1. Initial program 91.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-+.f64 72.3%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 72.3%

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

   3. Applied rewrites 72.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. associate--l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 93.5%

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 52.7%

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

   8. Applied rewrites 52.7%

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

Alternative 6: 95.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{max}\left(t\_4, t\right)\\ t_8 := \mathsf{max}\left(t\_3, t\_7\right)\\ t_9 := \mathsf{min}\left(t\_2, t\_8\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \mathsf{min}\left(t\_3, t\_7\right)\\ t_12 := \mathsf{max}\left(t\_2, t\_8\right)\\ t_13 := \sqrt{t\_12 + 1} - \sqrt{t\_12}\\ t_14 := \sqrt{t\_11}\\ t_15 := \sqrt{t\_9 + 1} - t\_10\\ t_16 := \left(\left(\left(\sqrt{t\_5 + 1} - t\_6\right) + \left(\sqrt{t\_11 + 1} - t\_14\right)\right) + t\_15\right) + t\_13\\ t_17 := \sqrt{1 + t\_5}\\ \mathbf{if}\;t\_16 \leq 1:\\ \;\;\;\;\left(\frac{1}{t\_6 + t\_17} + t\_15\right) + t\_13\\ \mathbf{elif}\;t\_16 \leq 2.9999996:\\ \;\;\;\;\left(\sqrt{t\_11 - -1} - \left(t\_6 - \sqrt{t\_5 - -1}\right)\right) - \left(t\_14 - \left(\sqrt{t\_9 - -1} - t\_10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \left(t\_17 + \sqrt{1 + t\_11}\right)\right) - \left(t\_6 + \left(t\_14 + t\_10\right)\right)\right) + t\_13\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmax (fmin x y) z))
       (t_2 (fmax (fmax x y) t_1))
       (t_3 (fmin (fmax x y) t_1))
       (t_4 (fmin (fmin x y) z))
       (t_5 (fmin t_4 t))
       (t_6 (sqrt t_5))
       (t_7 (fmax t_4 t))
       (t_8 (fmax t_3 t_7))
       (t_9 (fmin t_2 t_8))
       (t_10 (sqrt t_9))
       (t_11 (fmin t_3 t_7))
       (t_12 (fmax t_2 t_8))
       (t_13 (- (sqrt (+ t_12 1.0)) (sqrt t_12)))
       (t_14 (sqrt t_11))
       (t_15 (- (sqrt (+ t_9 1.0)) t_10))
       (t_16
        (+
         (+
          (+ (- (sqrt (+ t_5 1.0)) t_6) (- (sqrt (+ t_11 1.0)) t_14))
          t_15)
         t_13))
       (t_17 (sqrt (+ 1.0 t_5))))
  (if (<= t_16 1.0)
    (+ (+ (/ 1.0 (+ t_6 t_17)) t_15) t_13)
    (if (<= t_16 2.9999996)
      (-
       (- (sqrt (- t_11 -1.0)) (- t_6 (sqrt (- t_5 -1.0))))
       (- t_14 (- (sqrt (- t_9 -1.0)) t_10)))
      (+
       (- (+ 1.0 (+ t_17 (sqrt (+ 1.0 t_11)))) (+ t_6 (+ t_14 t_10)))
       t_13)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = fmax(t_2, t_8);
	double t_13 = sqrt((t_12 + 1.0)) - sqrt(t_12);
	double t_14 = sqrt(t_11);
	double t_15 = sqrt((t_9 + 1.0)) - t_10;
	double t_16 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_14)) + t_15) + t_13;
	double t_17 = sqrt((1.0 + t_5));
	double tmp;
	if (t_16 <= 1.0) {
		tmp = ((1.0 / (t_6 + t_17)) + t_15) + t_13;
	} else if (t_16 <= 2.9999996) {
		tmp = (sqrt((t_11 - -1.0)) - (t_6 - sqrt((t_5 - -1.0)))) - (t_14 - (sqrt((t_9 - -1.0)) - t_10));
	} else {
		tmp = ((1.0 + (t_17 + sqrt((1.0 + t_11)))) - (t_6 + (t_14 + t_10))) + t_13;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_14
    real(8) :: t_15
    real(8) :: t_16
    real(8) :: t_17
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmin(t_4, t)
    t_6 = sqrt(t_5)
    t_7 = fmax(t_4, t)
    t_8 = fmax(t_3, t_7)
    t_9 = fmin(t_2, t_8)
    t_10 = sqrt(t_9)
    t_11 = fmin(t_3, t_7)
    t_12 = fmax(t_2, t_8)
    t_13 = sqrt((t_12 + 1.0d0)) - sqrt(t_12)
    t_14 = sqrt(t_11)
    t_15 = sqrt((t_9 + 1.0d0)) - t_10
    t_16 = (((sqrt((t_5 + 1.0d0)) - t_6) + (sqrt((t_11 + 1.0d0)) - t_14)) + t_15) + t_13
    t_17 = sqrt((1.0d0 + t_5))
    if (t_16 <= 1.0d0) then
        tmp = ((1.0d0 / (t_6 + t_17)) + t_15) + t_13
    else if (t_16 <= 2.9999996d0) then
        tmp = (sqrt((t_11 - (-1.0d0))) - (t_6 - sqrt((t_5 - (-1.0d0))))) - (t_14 - (sqrt((t_9 - (-1.0d0))) - t_10))
    else
        tmp = ((1.0d0 + (t_17 + sqrt((1.0d0 + t_11)))) - (t_6 + (t_14 + t_10))) + t_13
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = Math.sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = Math.sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = fmax(t_2, t_8);
	double t_13 = Math.sqrt((t_12 + 1.0)) - Math.sqrt(t_12);
	double t_14 = Math.sqrt(t_11);
	double t_15 = Math.sqrt((t_9 + 1.0)) - t_10;
	double t_16 = (((Math.sqrt((t_5 + 1.0)) - t_6) + (Math.sqrt((t_11 + 1.0)) - t_14)) + t_15) + t_13;
	double t_17 = Math.sqrt((1.0 + t_5));
	double tmp;
	if (t_16 <= 1.0) {
		tmp = ((1.0 / (t_6 + t_17)) + t_15) + t_13;
	} else if (t_16 <= 2.9999996) {
		tmp = (Math.sqrt((t_11 - -1.0)) - (t_6 - Math.sqrt((t_5 - -1.0)))) - (t_14 - (Math.sqrt((t_9 - -1.0)) - t_10));
	} else {
		tmp = ((1.0 + (t_17 + Math.sqrt((1.0 + t_11)))) - (t_6 + (t_14 + t_10))) + t_13;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = math.sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = math.sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = fmax(t_2, t_8)
	t_13 = math.sqrt((t_12 + 1.0)) - math.sqrt(t_12)
	t_14 = math.sqrt(t_11)
	t_15 = math.sqrt((t_9 + 1.0)) - t_10
	t_16 = (((math.sqrt((t_5 + 1.0)) - t_6) + (math.sqrt((t_11 + 1.0)) - t_14)) + t_15) + t_13
	t_17 = math.sqrt((1.0 + t_5))
	tmp = 0
	if t_16 <= 1.0:
		tmp = ((1.0 / (t_6 + t_17)) + t_15) + t_13
	elif t_16 <= 2.9999996:
		tmp = (math.sqrt((t_11 - -1.0)) - (t_6 - math.sqrt((t_5 - -1.0)))) - (t_14 - (math.sqrt((t_9 - -1.0)) - t_10))
	else:
		tmp = ((1.0 + (t_17 + math.sqrt((1.0 + t_11)))) - (t_6 + (t_14 + t_10))) + t_13
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = fmax(t_2, t_8)
	t_13 = Float64(sqrt(Float64(t_12 + 1.0)) - sqrt(t_12))
	t_14 = sqrt(t_11)
	t_15 = Float64(sqrt(Float64(t_9 + 1.0)) - t_10)
	t_16 = Float64(Float64(Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + Float64(sqrt(Float64(t_11 + 1.0)) - t_14)) + t_15) + t_13)
	t_17 = sqrt(Float64(1.0 + t_5))
	tmp = 0.0
	if (t_16 <= 1.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_6 + t_17)) + t_15) + t_13);
	elseif (t_16 <= 2.9999996)
		tmp = Float64(Float64(sqrt(Float64(t_11 - -1.0)) - Float64(t_6 - sqrt(Float64(t_5 - -1.0)))) - Float64(t_14 - Float64(sqrt(Float64(t_9 - -1.0)) - t_10)));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(t_17 + sqrt(Float64(1.0 + t_11)))) - Float64(t_6 + Float64(t_14 + t_10))) + t_13);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = max(max(x, y), t_1);
	t_3 = min(max(x, y), t_1);
	t_4 = min(min(x, y), z);
	t_5 = min(t_4, t);
	t_6 = sqrt(t_5);
	t_7 = max(t_4, t);
	t_8 = max(t_3, t_7);
	t_9 = min(t_2, t_8);
	t_10 = sqrt(t_9);
	t_11 = min(t_3, t_7);
	t_12 = max(t_2, t_8);
	t_13 = sqrt((t_12 + 1.0)) - sqrt(t_12);
	t_14 = sqrt(t_11);
	t_15 = sqrt((t_9 + 1.0)) - t_10;
	t_16 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_14)) + t_15) + t_13;
	t_17 = sqrt((1.0 + t_5));
	tmp = 0.0;
	if (t_16 <= 1.0)
		tmp = ((1.0 / (t_6 + t_17)) + t_15) + t_13;
	elseif (t_16 <= 2.9999996)
		tmp = (sqrt((t_11 - -1.0)) - (t_6 - sqrt((t_5 - -1.0)))) - (t_14 - (sqrt((t_9 - -1.0)) - t_10));
	else
		tmp = ((1.0 + (t_17 + sqrt((1.0 + t_11)))) - (t_6 + (t_14 + t_10))) + t_13;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$4, t], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Min[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$12 = N[Max[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$13 = N[(N[Sqrt[N[(t$95$12 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$12], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$14 = N[Sqrt[t$95$11], $MachinePrecision]}, Block[{t$95$15 = N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]}, Block[{t$95$16 = N[(N[(N[(N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + N[(N[Sqrt[N[(t$95$11 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$14), $MachinePrecision]), $MachinePrecision] + t$95$15), $MachinePrecision] + t$95$13), $MachinePrecision]}, Block[{t$95$17 = N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$16, 1.0], N[(N[(N[(1.0 / N[(t$95$6 + t$95$17), $MachinePrecision]), $MachinePrecision] + t$95$15), $MachinePrecision] + t$95$13), $MachinePrecision], If[LessEqual[t$95$16, 2.9999996], N[(N[(N[Sqrt[N[(t$95$11 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(t$95$6 - N[Sqrt[N[(t$95$5 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$14 - N[(N[Sqrt[N[(t$95$9 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(t$95$17 + N[Sqrt[N[(1.0 + t$95$11), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$6 + N[(t$95$14 + t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$13), $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 91.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-+.f64 72.3%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 72.3%

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

   3. Applied rewrites 72.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. associate--l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 93.5%

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 52.7%

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

   8. Applied rewrites 52.7%

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

   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))) < 2.9999996000000002

   1. Initial program 91.5%

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

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

      8. lower--.f64 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 53.7%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 31.1%

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

   6. Applied rewrites 31.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lift--.f64 N/A

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

      9. add-flip N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. metadata-eval N/A

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

      14. add-flip N/A

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

      15. lift-+.f64 N/A

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

      16. lower--.f64 31.7%

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

      17. lift-+.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 31.7%

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

   8. Applied rewrites 31.7%

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

   if 2.9999996000000002 < (+.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 91.5%

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

   2. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 50.5%

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 23.1%

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

   7. Applied rewrites 23.1%

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

Alternative 7: 95.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{max}\left(t\_4, t\right)\\ t_8 := \mathsf{max}\left(t\_3, t\_7\right)\\ t_9 := \mathsf{min}\left(t\_2, t\_8\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \sqrt{t\_9 + 1} - t\_10\\ t_12 := \mathsf{min}\left(t\_3, t\_7\right)\\ t_13 := \mathsf{max}\left(t\_2, t\_8\right)\\ t_14 := \sqrt{t\_13 + 1} - \sqrt{t\_13}\\ t_15 := \sqrt{t\_12}\\ t_16 := \sqrt{t\_12 + 1} - t\_15\\ t_17 := \left(\left(\left(\sqrt{t\_5 + 1} - t\_6\right) + t\_16\right) + t\_11\right) + t\_14\\ \mathbf{if}\;t\_17 \leq 1:\\ \;\;\;\;\left(\frac{1}{t\_6 + \sqrt{1 + t\_5}} + t\_11\right) + t\_14\\ \mathbf{elif}\;t\_17 \leq 2.999999999995:\\ \;\;\;\;\left(\sqrt{t\_12 - -1} - \left(t\_6 - \sqrt{t\_5 - -1}\right)\right) - \left(t\_15 - \left(\sqrt{t\_9 - -1} - t\_10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - t\_6\right) + t\_16\right) + \left(1 - t\_10\right)\right) + t\_14\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmax (fmin x y) z))
       (t_2 (fmax (fmax x y) t_1))
       (t_3 (fmin (fmax x y) t_1))
       (t_4 (fmin (fmin x y) z))
       (t_5 (fmin t_4 t))
       (t_6 (sqrt t_5))
       (t_7 (fmax t_4 t))
       (t_8 (fmax t_3 t_7))
       (t_9 (fmin t_2 t_8))
       (t_10 (sqrt t_9))
       (t_11 (- (sqrt (+ t_9 1.0)) t_10))
       (t_12 (fmin t_3 t_7))
       (t_13 (fmax t_2 t_8))
       (t_14 (- (sqrt (+ t_13 1.0)) (sqrt t_13)))
       (t_15 (sqrt t_12))
       (t_16 (- (sqrt (+ t_12 1.0)) t_15))
       (t_17 (+ (+ (+ (- (sqrt (+ t_5 1.0)) t_6) t_16) t_11) t_14)))
  (if (<= t_17 1.0)
    (+ (+ (/ 1.0 (+ t_6 (sqrt (+ 1.0 t_5)))) t_11) t_14)
    (if (<= t_17 2.999999999995)
      (-
       (- (sqrt (- t_12 -1.0)) (- t_6 (sqrt (- t_5 -1.0))))
       (- t_15 (- (sqrt (- t_9 -1.0)) t_10)))
      (+ (+ (+ (- 1.0 t_6) t_16) (- 1.0 t_10)) t_14)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = sqrt(t_9);
	double t_11 = sqrt((t_9 + 1.0)) - t_10;
	double t_12 = fmin(t_3, t_7);
	double t_13 = fmax(t_2, t_8);
	double t_14 = sqrt((t_13 + 1.0)) - sqrt(t_13);
	double t_15 = sqrt(t_12);
	double t_16 = sqrt((t_12 + 1.0)) - t_15;
	double t_17 = (((sqrt((t_5 + 1.0)) - t_6) + t_16) + t_11) + t_14;
	double tmp;
	if (t_17 <= 1.0) {
		tmp = ((1.0 / (t_6 + sqrt((1.0 + t_5)))) + t_11) + t_14;
	} else if (t_17 <= 2.999999999995) {
		tmp = (sqrt((t_12 - -1.0)) - (t_6 - sqrt((t_5 - -1.0)))) - (t_15 - (sqrt((t_9 - -1.0)) - t_10));
	} else {
		tmp = (((1.0 - t_6) + t_16) + (1.0 - t_10)) + t_14;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_14
    real(8) :: t_15
    real(8) :: t_16
    real(8) :: t_17
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmin(t_4, t)
    t_6 = sqrt(t_5)
    t_7 = fmax(t_4, t)
    t_8 = fmax(t_3, t_7)
    t_9 = fmin(t_2, t_8)
    t_10 = sqrt(t_9)
    t_11 = sqrt((t_9 + 1.0d0)) - t_10
    t_12 = fmin(t_3, t_7)
    t_13 = fmax(t_2, t_8)
    t_14 = sqrt((t_13 + 1.0d0)) - sqrt(t_13)
    t_15 = sqrt(t_12)
    t_16 = sqrt((t_12 + 1.0d0)) - t_15
    t_17 = (((sqrt((t_5 + 1.0d0)) - t_6) + t_16) + t_11) + t_14
    if (t_17 <= 1.0d0) then
        tmp = ((1.0d0 / (t_6 + sqrt((1.0d0 + t_5)))) + t_11) + t_14
    else if (t_17 <= 2.999999999995d0) then
        tmp = (sqrt((t_12 - (-1.0d0))) - (t_6 - sqrt((t_5 - (-1.0d0))))) - (t_15 - (sqrt((t_9 - (-1.0d0))) - t_10))
    else
        tmp = (((1.0d0 - t_6) + t_16) + (1.0d0 - t_10)) + t_14
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = Math.sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = Math.sqrt(t_9);
	double t_11 = Math.sqrt((t_9 + 1.0)) - t_10;
	double t_12 = fmin(t_3, t_7);
	double t_13 = fmax(t_2, t_8);
	double t_14 = Math.sqrt((t_13 + 1.0)) - Math.sqrt(t_13);
	double t_15 = Math.sqrt(t_12);
	double t_16 = Math.sqrt((t_12 + 1.0)) - t_15;
	double t_17 = (((Math.sqrt((t_5 + 1.0)) - t_6) + t_16) + t_11) + t_14;
	double tmp;
	if (t_17 <= 1.0) {
		tmp = ((1.0 / (t_6 + Math.sqrt((1.0 + t_5)))) + t_11) + t_14;
	} else if (t_17 <= 2.999999999995) {
		tmp = (Math.sqrt((t_12 - -1.0)) - (t_6 - Math.sqrt((t_5 - -1.0)))) - (t_15 - (Math.sqrt((t_9 - -1.0)) - t_10));
	} else {
		tmp = (((1.0 - t_6) + t_16) + (1.0 - t_10)) + t_14;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = math.sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = math.sqrt(t_9)
	t_11 = math.sqrt((t_9 + 1.0)) - t_10
	t_12 = fmin(t_3, t_7)
	t_13 = fmax(t_2, t_8)
	t_14 = math.sqrt((t_13 + 1.0)) - math.sqrt(t_13)
	t_15 = math.sqrt(t_12)
	t_16 = math.sqrt((t_12 + 1.0)) - t_15
	t_17 = (((math.sqrt((t_5 + 1.0)) - t_6) + t_16) + t_11) + t_14
	tmp = 0
	if t_17 <= 1.0:
		tmp = ((1.0 / (t_6 + math.sqrt((1.0 + t_5)))) + t_11) + t_14
	elif t_17 <= 2.999999999995:
		tmp = (math.sqrt((t_12 - -1.0)) - (t_6 - math.sqrt((t_5 - -1.0)))) - (t_15 - (math.sqrt((t_9 - -1.0)) - t_10))
	else:
		tmp = (((1.0 - t_6) + t_16) + (1.0 - t_10)) + t_14
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = sqrt(t_9)
	t_11 = Float64(sqrt(Float64(t_9 + 1.0)) - t_10)
	t_12 = fmin(t_3, t_7)
	t_13 = fmax(t_2, t_8)
	t_14 = Float64(sqrt(Float64(t_13 + 1.0)) - sqrt(t_13))
	t_15 = sqrt(t_12)
	t_16 = Float64(sqrt(Float64(t_12 + 1.0)) - t_15)
	t_17 = Float64(Float64(Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + t_16) + t_11) + t_14)
	tmp = 0.0
	if (t_17 <= 1.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_6 + sqrt(Float64(1.0 + t_5)))) + t_11) + t_14);
	elseif (t_17 <= 2.999999999995)
		tmp = Float64(Float64(sqrt(Float64(t_12 - -1.0)) - Float64(t_6 - sqrt(Float64(t_5 - -1.0)))) - Float64(t_15 - Float64(sqrt(Float64(t_9 - -1.0)) - t_10)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - t_6) + t_16) + Float64(1.0 - t_10)) + t_14);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = max(max(x, y), t_1);
	t_3 = min(max(x, y), t_1);
	t_4 = min(min(x, y), z);
	t_5 = min(t_4, t);
	t_6 = sqrt(t_5);
	t_7 = max(t_4, t);
	t_8 = max(t_3, t_7);
	t_9 = min(t_2, t_8);
	t_10 = sqrt(t_9);
	t_11 = sqrt((t_9 + 1.0)) - t_10;
	t_12 = min(t_3, t_7);
	t_13 = max(t_2, t_8);
	t_14 = sqrt((t_13 + 1.0)) - sqrt(t_13);
	t_15 = sqrt(t_12);
	t_16 = sqrt((t_12 + 1.0)) - t_15;
	t_17 = (((sqrt((t_5 + 1.0)) - t_6) + t_16) + t_11) + t_14;
	tmp = 0.0;
	if (t_17 <= 1.0)
		tmp = ((1.0 / (t_6 + sqrt((1.0 + t_5)))) + t_11) + t_14;
	elseif (t_17 <= 2.999999999995)
		tmp = (sqrt((t_12 - -1.0)) - (t_6 - sqrt((t_5 - -1.0)))) - (t_15 - (sqrt((t_9 - -1.0)) - t_10));
	else
		tmp = (((1.0 - t_6) + t_16) + (1.0 - t_10)) + t_14;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$4, t], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]}, Block[{t$95$12 = N[Min[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$13 = N[Max[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$14 = N[(N[Sqrt[N[(t$95$13 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$13], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$15 = N[Sqrt[t$95$12], $MachinePrecision]}, Block[{t$95$16 = N[(N[Sqrt[N[(t$95$12 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$15), $MachinePrecision]}, Block[{t$95$17 = N[(N[(N[(N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + t$95$16), $MachinePrecision] + t$95$11), $MachinePrecision] + t$95$14), $MachinePrecision]}, If[LessEqual[t$95$17, 1.0], N[(N[(N[(1.0 / N[(t$95$6 + N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$11), $MachinePrecision] + t$95$14), $MachinePrecision], If[LessEqual[t$95$17, 2.999999999995], N[(N[(N[Sqrt[N[(t$95$12 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(t$95$6 - N[Sqrt[N[(t$95$5 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$15 - N[(N[Sqrt[N[(t$95$9 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 - t$95$6), $MachinePrecision] + t$95$16), $MachinePrecision] + N[(1.0 - t$95$10), $MachinePrecision]), $MachinePrecision] + t$95$14), $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 91.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-+.f64 72.3%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 72.3%

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

   3. Applied rewrites 72.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. associate--l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 93.5%

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 52.7%

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

   8. Applied rewrites 52.7%

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

   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))) < 2.999999999995

   1. Initial program 91.5%

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

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

      8. lower--.f64 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 53.7%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 31.1%

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

   6. Applied rewrites 31.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lift--.f64 N/A

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

      9. add-flip N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. metadata-eval N/A

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

      14. add-flip N/A

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

      15. lift-+.f64 N/A

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

      16. lower--.f64 31.7%

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

      17. lift-+.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 31.7%

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

   8. Applied rewrites 31.7%

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

   if 2.999999999995 < (+.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 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 48.3%

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 24.2%

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

   7. Applied rewrites 24.2%

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

Alternative 8: 95.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{\mathsf{max}\left(x, y\right) + 1} - \sqrt{\mathsf{max}\left(x, y\right)}\\ t_2 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_3 := \mathsf{max}\left(t\_2, t\right)\\ t_4 := \mathsf{min}\left(t\_2, t\right)\\ t_5 := \sqrt{t\_4}\\ t_6 := \sqrt{t\_3 + 1} - \sqrt{t\_3}\\ t_7 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_8 := \sqrt{t\_7 + 1} - \sqrt{t\_7}\\ \mathbf{if}\;\left(\left(\left(\sqrt{t\_4 + 1} - t\_5\right) + t\_1\right) + t\_8\right) + t\_6 \leq 1:\\ \;\;\;\;\left(\frac{1}{t\_5 + \sqrt{1 + t\_4}} + t\_8\right) + t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - t\_5\right) + t\_1\right) + t\_8\right) + t\_6\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (- (sqrt (+ (fmax x y) 1.0)) (sqrt (fmax x y))))
       (t_2 (fmin (fmin x y) z))
       (t_3 (fmax t_2 t))
       (t_4 (fmin t_2 t))
       (t_5 (sqrt t_4))
       (t_6 (- (sqrt (+ t_3 1.0)) (sqrt t_3)))
       (t_7 (fmax (fmin x y) z))
       (t_8 (- (sqrt (+ t_7 1.0)) (sqrt t_7))))
  (if (<= (+ (+ (+ (- (sqrt (+ t_4 1.0)) t_5) t_1) t_8) t_6) 1.0)
    (+ (+ (/ 1.0 (+ t_5 (sqrt (+ 1.0 t_4)))) t_8) t_6)
    (+ (+ (+ (- 1.0 t_5) t_1) t_8) t_6))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((fmax(x, y) + 1.0)) - sqrt(fmax(x, y));
	double t_2 = fmin(fmin(x, y), z);
	double t_3 = fmax(t_2, t);
	double t_4 = fmin(t_2, t);
	double t_5 = sqrt(t_4);
	double t_6 = sqrt((t_3 + 1.0)) - sqrt(t_3);
	double t_7 = fmax(fmin(x, y), z);
	double t_8 = sqrt((t_7 + 1.0)) - sqrt(t_7);
	double tmp;
	if (((((sqrt((t_4 + 1.0)) - t_5) + t_1) + t_8) + t_6) <= 1.0) {
		tmp = ((1.0 / (t_5 + sqrt((1.0 + t_4)))) + t_8) + t_6;
	} else {
		tmp = (((1.0 - t_5) + t_1) + t_8) + t_6;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: tmp
    t_1 = sqrt((fmax(x, y) + 1.0d0)) - sqrt(fmax(x, y))
    t_2 = fmin(fmin(x, y), z)
    t_3 = fmax(t_2, t)
    t_4 = fmin(t_2, t)
    t_5 = sqrt(t_4)
    t_6 = sqrt((t_3 + 1.0d0)) - sqrt(t_3)
    t_7 = fmax(fmin(x, y), z)
    t_8 = sqrt((t_7 + 1.0d0)) - sqrt(t_7)
    if (((((sqrt((t_4 + 1.0d0)) - t_5) + t_1) + t_8) + t_6) <= 1.0d0) then
        tmp = ((1.0d0 / (t_5 + sqrt((1.0d0 + t_4)))) + t_8) + t_6
    else
        tmp = (((1.0d0 - t_5) + t_1) + t_8) + t_6
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((fmax(x, y) + 1.0)) - Math.sqrt(fmax(x, y));
	double t_2 = fmin(fmin(x, y), z);
	double t_3 = fmax(t_2, t);
	double t_4 = fmin(t_2, t);
	double t_5 = Math.sqrt(t_4);
	double t_6 = Math.sqrt((t_3 + 1.0)) - Math.sqrt(t_3);
	double t_7 = fmax(fmin(x, y), z);
	double t_8 = Math.sqrt((t_7 + 1.0)) - Math.sqrt(t_7);
	double tmp;
	if (((((Math.sqrt((t_4 + 1.0)) - t_5) + t_1) + t_8) + t_6) <= 1.0) {
		tmp = ((1.0 / (t_5 + Math.sqrt((1.0 + t_4)))) + t_8) + t_6;
	} else {
		tmp = (((1.0 - t_5) + t_1) + t_8) + t_6;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((fmax(x, y) + 1.0)) - math.sqrt(fmax(x, y))
	t_2 = fmin(fmin(x, y), z)
	t_3 = fmax(t_2, t)
	t_4 = fmin(t_2, t)
	t_5 = math.sqrt(t_4)
	t_6 = math.sqrt((t_3 + 1.0)) - math.sqrt(t_3)
	t_7 = fmax(fmin(x, y), z)
	t_8 = math.sqrt((t_7 + 1.0)) - math.sqrt(t_7)
	tmp = 0
	if ((((math.sqrt((t_4 + 1.0)) - t_5) + t_1) + t_8) + t_6) <= 1.0:
		tmp = ((1.0 / (t_5 + math.sqrt((1.0 + t_4)))) + t_8) + t_6
	else:
		tmp = (((1.0 - t_5) + t_1) + t_8) + t_6
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(fmax(x, y) + 1.0)) - sqrt(fmax(x, y)))
	t_2 = fmin(fmin(x, y), z)
	t_3 = fmax(t_2, t)
	t_4 = fmin(t_2, t)
	t_5 = sqrt(t_4)
	t_6 = Float64(sqrt(Float64(t_3 + 1.0)) - sqrt(t_3))
	t_7 = fmax(fmin(x, y), z)
	t_8 = Float64(sqrt(Float64(t_7 + 1.0)) - sqrt(t_7))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(sqrt(Float64(t_4 + 1.0)) - t_5) + t_1) + t_8) + t_6) <= 1.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_5 + sqrt(Float64(1.0 + t_4)))) + t_8) + t_6);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - t_5) + t_1) + t_8) + t_6);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((max(x, y) + 1.0)) - sqrt(max(x, y));
	t_2 = min(min(x, y), z);
	t_3 = max(t_2, t);
	t_4 = min(t_2, t);
	t_5 = sqrt(t_4);
	t_6 = sqrt((t_3 + 1.0)) - sqrt(t_3);
	t_7 = max(min(x, y), z);
	t_8 = sqrt((t_7 + 1.0)) - sqrt(t_7);
	tmp = 0.0;
	if (((((sqrt((t_4 + 1.0)) - t_5) + t_1) + t_8) + t_6) <= 1.0)
		tmp = ((1.0 / (t_5 + sqrt((1.0 + t_4)))) + t_8) + t_6;
	else
		tmp = (((1.0 - t_5) + t_1) + t_8) + t_6;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(N[Max[x, y], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[N[Max[x, y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$3 = N[Max[t$95$2, t], $MachinePrecision]}, Block[{t$95$4 = N[Min[t$95$2, t], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[t$95$4], $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(t$95$3 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$8 = N[(N[Sqrt[N[(t$95$7 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$7], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Sqrt[N[(t$95$4 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$5), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$8), $MachinePrecision] + t$95$6), $MachinePrecision], 1.0], N[(N[(N[(1.0 / N[(t$95$5 + N[Sqrt[N[(1.0 + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$8), $MachinePrecision] + t$95$6), $MachinePrecision], N[(N[(N[(N[(1.0 - t$95$5), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$8), $MachinePrecision] + t$95$6), $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))) < 1

   1. Initial program 91.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-+.f64 72.3%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 72.3%

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

   3. Applied rewrites 72.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. associate--l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 93.5%

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 52.7%

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

   8. Applied rewrites 52.7%

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

   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)))

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 48.3%

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

   4. Applied rewrites 48.3%

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

Alternative 9: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{max}\left(t\_4, t\right)\\ t_8 := \mathsf{max}\left(t\_3, t\_7\right)\\ t_9 := \mathsf{min}\left(t\_2, t\_8\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \mathsf{min}\left(t\_3, t\_7\right)\\ t_12 := \mathsf{max}\left(t\_2, t\_8\right)\\ t_13 := \sqrt{t\_12 + 1} - \sqrt{t\_12}\\ t_14 := \sqrt{t\_11}\\ t_15 := \sqrt{t\_11 + 1} - t\_14\\ t_16 := \left(\left(\left(\sqrt{t\_5 + 1} - t\_6\right) + t\_15\right) + \left(\sqrt{t\_9 + 1} - t\_10\right)\right) + t\_13\\ \mathbf{if}\;t\_16 \leq 1:\\ \;\;\;\;\left(\left(\sqrt{1 + t\_5} - t\_6\right) + \frac{0.5}{t\_9 \cdot \sqrt{\frac{1}{t\_9}}}\right) + t\_13\\ \mathbf{elif}\;t\_16 \leq 2.999999999995:\\ \;\;\;\;\left(\sqrt{t\_11 - -1} - \left(t\_6 - \sqrt{t\_5 - -1}\right)\right) - \left(t\_14 - \left(\sqrt{t\_9 - -1} - t\_10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - t\_6\right) + t\_15\right) + \left(1 - t\_10\right)\right) + t\_13\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmax (fmin x y) z))
       (t_2 (fmax (fmax x y) t_1))
       (t_3 (fmin (fmax x y) t_1))
       (t_4 (fmin (fmin x y) z))
       (t_5 (fmin t_4 t))
       (t_6 (sqrt t_5))
       (t_7 (fmax t_4 t))
       (t_8 (fmax t_3 t_7))
       (t_9 (fmin t_2 t_8))
       (t_10 (sqrt t_9))
       (t_11 (fmin t_3 t_7))
       (t_12 (fmax t_2 t_8))
       (t_13 (- (sqrt (+ t_12 1.0)) (sqrt t_12)))
       (t_14 (sqrt t_11))
       (t_15 (- (sqrt (+ t_11 1.0)) t_14))
       (t_16
        (+
         (+
          (+ (- (sqrt (+ t_5 1.0)) t_6) t_15)
          (- (sqrt (+ t_9 1.0)) t_10))
         t_13)))
  (if (<= t_16 1.0)
    (+
     (+ (- (sqrt (+ 1.0 t_5)) t_6) (/ 0.5 (* t_9 (sqrt (/ 1.0 t_9)))))
     t_13)
    (if (<= t_16 2.999999999995)
      (-
       (- (sqrt (- t_11 -1.0)) (- t_6 (sqrt (- t_5 -1.0))))
       (- t_14 (- (sqrt (- t_9 -1.0)) t_10)))
      (+ (+ (+ (- 1.0 t_6) t_15) (- 1.0 t_10)) t_13)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = fmax(t_2, t_8);
	double t_13 = sqrt((t_12 + 1.0)) - sqrt(t_12);
	double t_14 = sqrt(t_11);
	double t_15 = sqrt((t_11 + 1.0)) - t_14;
	double t_16 = (((sqrt((t_5 + 1.0)) - t_6) + t_15) + (sqrt((t_9 + 1.0)) - t_10)) + t_13;
	double tmp;
	if (t_16 <= 1.0) {
		tmp = ((sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_9 * sqrt((1.0 / t_9))))) + t_13;
	} else if (t_16 <= 2.999999999995) {
		tmp = (sqrt((t_11 - -1.0)) - (t_6 - sqrt((t_5 - -1.0)))) - (t_14 - (sqrt((t_9 - -1.0)) - t_10));
	} else {
		tmp = (((1.0 - t_6) + t_15) + (1.0 - t_10)) + t_13;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_14
    real(8) :: t_15
    real(8) :: t_16
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmin(t_4, t)
    t_6 = sqrt(t_5)
    t_7 = fmax(t_4, t)
    t_8 = fmax(t_3, t_7)
    t_9 = fmin(t_2, t_8)
    t_10 = sqrt(t_9)
    t_11 = fmin(t_3, t_7)
    t_12 = fmax(t_2, t_8)
    t_13 = sqrt((t_12 + 1.0d0)) - sqrt(t_12)
    t_14 = sqrt(t_11)
    t_15 = sqrt((t_11 + 1.0d0)) - t_14
    t_16 = (((sqrt((t_5 + 1.0d0)) - t_6) + t_15) + (sqrt((t_9 + 1.0d0)) - t_10)) + t_13
    if (t_16 <= 1.0d0) then
        tmp = ((sqrt((1.0d0 + t_5)) - t_6) + (0.5d0 / (t_9 * sqrt((1.0d0 / t_9))))) + t_13
    else if (t_16 <= 2.999999999995d0) then
        tmp = (sqrt((t_11 - (-1.0d0))) - (t_6 - sqrt((t_5 - (-1.0d0))))) - (t_14 - (sqrt((t_9 - (-1.0d0))) - t_10))
    else
        tmp = (((1.0d0 - t_6) + t_15) + (1.0d0 - t_10)) + t_13
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = Math.sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = Math.sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = fmax(t_2, t_8);
	double t_13 = Math.sqrt((t_12 + 1.0)) - Math.sqrt(t_12);
	double t_14 = Math.sqrt(t_11);
	double t_15 = Math.sqrt((t_11 + 1.0)) - t_14;
	double t_16 = (((Math.sqrt((t_5 + 1.0)) - t_6) + t_15) + (Math.sqrt((t_9 + 1.0)) - t_10)) + t_13;
	double tmp;
	if (t_16 <= 1.0) {
		tmp = ((Math.sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_9 * Math.sqrt((1.0 / t_9))))) + t_13;
	} else if (t_16 <= 2.999999999995) {
		tmp = (Math.sqrt((t_11 - -1.0)) - (t_6 - Math.sqrt((t_5 - -1.0)))) - (t_14 - (Math.sqrt((t_9 - -1.0)) - t_10));
	} else {
		tmp = (((1.0 - t_6) + t_15) + (1.0 - t_10)) + t_13;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = math.sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = math.sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = fmax(t_2, t_8)
	t_13 = math.sqrt((t_12 + 1.0)) - math.sqrt(t_12)
	t_14 = math.sqrt(t_11)
	t_15 = math.sqrt((t_11 + 1.0)) - t_14
	t_16 = (((math.sqrt((t_5 + 1.0)) - t_6) + t_15) + (math.sqrt((t_9 + 1.0)) - t_10)) + t_13
	tmp = 0
	if t_16 <= 1.0:
		tmp = ((math.sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_9 * math.sqrt((1.0 / t_9))))) + t_13
	elif t_16 <= 2.999999999995:
		tmp = (math.sqrt((t_11 - -1.0)) - (t_6 - math.sqrt((t_5 - -1.0)))) - (t_14 - (math.sqrt((t_9 - -1.0)) - t_10))
	else:
		tmp = (((1.0 - t_6) + t_15) + (1.0 - t_10)) + t_13
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = fmax(t_2, t_8)
	t_13 = Float64(sqrt(Float64(t_12 + 1.0)) - sqrt(t_12))
	t_14 = sqrt(t_11)
	t_15 = Float64(sqrt(Float64(t_11 + 1.0)) - t_14)
	t_16 = Float64(Float64(Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + t_15) + Float64(sqrt(Float64(t_9 + 1.0)) - t_10)) + t_13)
	tmp = 0.0
	if (t_16 <= 1.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t_5)) - t_6) + Float64(0.5 / Float64(t_9 * sqrt(Float64(1.0 / t_9))))) + t_13);
	elseif (t_16 <= 2.999999999995)
		tmp = Float64(Float64(sqrt(Float64(t_11 - -1.0)) - Float64(t_6 - sqrt(Float64(t_5 - -1.0)))) - Float64(t_14 - Float64(sqrt(Float64(t_9 - -1.0)) - t_10)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - t_6) + t_15) + Float64(1.0 - t_10)) + t_13);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = max(max(x, y), t_1);
	t_3 = min(max(x, y), t_1);
	t_4 = min(min(x, y), z);
	t_5 = min(t_4, t);
	t_6 = sqrt(t_5);
	t_7 = max(t_4, t);
	t_8 = max(t_3, t_7);
	t_9 = min(t_2, t_8);
	t_10 = sqrt(t_9);
	t_11 = min(t_3, t_7);
	t_12 = max(t_2, t_8);
	t_13 = sqrt((t_12 + 1.0)) - sqrt(t_12);
	t_14 = sqrt(t_11);
	t_15 = sqrt((t_11 + 1.0)) - t_14;
	t_16 = (((sqrt((t_5 + 1.0)) - t_6) + t_15) + (sqrt((t_9 + 1.0)) - t_10)) + t_13;
	tmp = 0.0;
	if (t_16 <= 1.0)
		tmp = ((sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_9 * sqrt((1.0 / t_9))))) + t_13;
	elseif (t_16 <= 2.999999999995)
		tmp = (sqrt((t_11 - -1.0)) - (t_6 - sqrt((t_5 - -1.0)))) - (t_14 - (sqrt((t_9 - -1.0)) - t_10));
	else
		tmp = (((1.0 - t_6) + t_15) + (1.0 - t_10)) + t_13;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$4, t], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Min[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$12 = N[Max[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$13 = N[(N[Sqrt[N[(t$95$12 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$12], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$14 = N[Sqrt[t$95$11], $MachinePrecision]}, Block[{t$95$15 = N[(N[Sqrt[N[(t$95$11 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$14), $MachinePrecision]}, Block[{t$95$16 = N[(N[(N[(N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + t$95$15), $MachinePrecision] + N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision] + t$95$13), $MachinePrecision]}, If[LessEqual[t$95$16, 1.0], N[(N[(N[(N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + N[(0.5 / N[(t$95$9 * N[Sqrt[N[(1.0 / t$95$9), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$13), $MachinePrecision], If[LessEqual[t$95$16, 2.999999999995], N[(N[(N[Sqrt[N[(t$95$11 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(t$95$6 - N[Sqrt[N[(t$95$5 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$14 - N[(N[Sqrt[N[(t$95$9 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 - t$95$6), $MachinePrecision] + t$95$15), $MachinePrecision] + N[(1.0 - t$95$10), $MachinePrecision]), $MachinePrecision] + t$95$13), $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 91.5%

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

   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 48.5%

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

   4. Applied rewrites 48.5%

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

   5. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right)} + \frac{0.5}{z \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 27.8%

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

   7. Applied rewrites 27.8%

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

   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))) < 2.999999999995

   1. Initial program 91.5%

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

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

      8. lower--.f64 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 53.7%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 31.1%

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

   6. Applied rewrites 31.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lift--.f64 N/A

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

      9. add-flip N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. metadata-eval N/A

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

      14. add-flip N/A

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

      15. lift-+.f64 N/A

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

      16. lower--.f64 31.7%

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

      17. lift-+.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 31.7%

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

   8. Applied rewrites 31.7%

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

   if 2.999999999995 < (+.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 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 48.3%

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 24.2%

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

   7. Applied rewrites 24.2%

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

Alternative 10: 91.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{max}\left(t\_4, t\right)\\ t_8 := \mathsf{max}\left(t\_3, t\_7\right)\\ t_9 := \mathsf{min}\left(t\_2, t\_8\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \mathsf{min}\left(t\_3, t\_7\right)\\ t_12 := \mathsf{max}\left(t\_2, t\_8\right)\\ t_13 := \sqrt{t\_12 + 1} - \sqrt{t\_12}\\ t_14 := \sqrt{t\_11}\\ t_15 := \sqrt{t\_11 + 1} - t\_14\\ t_16 := \left(\left(\left(\sqrt{t\_5 + 1} - t\_6\right) + t\_15\right) + \left(\sqrt{t\_9 + 1} - t\_10\right)\right) + t\_13\\ \mathbf{if}\;t\_16 \leq 1:\\ \;\;\;\;\left(\sqrt{1 + t\_5} + 0.5 \cdot \frac{1}{t\_12 \cdot \sqrt{\frac{1}{t\_12}}}\right) - t\_6\\ \mathbf{elif}\;t\_16 \leq 2.999999999995:\\ \;\;\;\;\left(\sqrt{t\_11 - -1} - \left(t\_6 - \sqrt{t\_5 - -1}\right)\right) - \left(t\_14 - \left(\sqrt{t\_9 - -1} - t\_10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - t\_6\right) + t\_15\right) + \left(1 - t\_10\right)\right) + t\_13\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmax (fmin x y) z))
       (t_2 (fmax (fmax x y) t_1))
       (t_3 (fmin (fmax x y) t_1))
       (t_4 (fmin (fmin x y) z))
       (t_5 (fmin t_4 t))
       (t_6 (sqrt t_5))
       (t_7 (fmax t_4 t))
       (t_8 (fmax t_3 t_7))
       (t_9 (fmin t_2 t_8))
       (t_10 (sqrt t_9))
       (t_11 (fmin t_3 t_7))
       (t_12 (fmax t_2 t_8))
       (t_13 (- (sqrt (+ t_12 1.0)) (sqrt t_12)))
       (t_14 (sqrt t_11))
       (t_15 (- (sqrt (+ t_11 1.0)) t_14))
       (t_16
        (+
         (+
          (+ (- (sqrt (+ t_5 1.0)) t_6) t_15)
          (- (sqrt (+ t_9 1.0)) t_10))
         t_13)))
  (if (<= t_16 1.0)
    (-
     (+
      (sqrt (+ 1.0 t_5))
      (* 0.5 (/ 1.0 (* t_12 (sqrt (/ 1.0 t_12))))))
     t_6)
    (if (<= t_16 2.999999999995)
      (-
       (- (sqrt (- t_11 -1.0)) (- t_6 (sqrt (- t_5 -1.0))))
       (- t_14 (- (sqrt (- t_9 -1.0)) t_10)))
      (+ (+ (+ (- 1.0 t_6) t_15) (- 1.0 t_10)) t_13)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = fmax(t_2, t_8);
	double t_13 = sqrt((t_12 + 1.0)) - sqrt(t_12);
	double t_14 = sqrt(t_11);
	double t_15 = sqrt((t_11 + 1.0)) - t_14;
	double t_16 = (((sqrt((t_5 + 1.0)) - t_6) + t_15) + (sqrt((t_9 + 1.0)) - t_10)) + t_13;
	double tmp;
	if (t_16 <= 1.0) {
		tmp = (sqrt((1.0 + t_5)) + (0.5 * (1.0 / (t_12 * sqrt((1.0 / t_12)))))) - t_6;
	} else if (t_16 <= 2.999999999995) {
		tmp = (sqrt((t_11 - -1.0)) - (t_6 - sqrt((t_5 - -1.0)))) - (t_14 - (sqrt((t_9 - -1.0)) - t_10));
	} else {
		tmp = (((1.0 - t_6) + t_15) + (1.0 - t_10)) + t_13;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_14
    real(8) :: t_15
    real(8) :: t_16
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmin(t_4, t)
    t_6 = sqrt(t_5)
    t_7 = fmax(t_4, t)
    t_8 = fmax(t_3, t_7)
    t_9 = fmin(t_2, t_8)
    t_10 = sqrt(t_9)
    t_11 = fmin(t_3, t_7)
    t_12 = fmax(t_2, t_8)
    t_13 = sqrt((t_12 + 1.0d0)) - sqrt(t_12)
    t_14 = sqrt(t_11)
    t_15 = sqrt((t_11 + 1.0d0)) - t_14
    t_16 = (((sqrt((t_5 + 1.0d0)) - t_6) + t_15) + (sqrt((t_9 + 1.0d0)) - t_10)) + t_13
    if (t_16 <= 1.0d0) then
        tmp = (sqrt((1.0d0 + t_5)) + (0.5d0 * (1.0d0 / (t_12 * sqrt((1.0d0 / t_12)))))) - t_6
    else if (t_16 <= 2.999999999995d0) then
        tmp = (sqrt((t_11 - (-1.0d0))) - (t_6 - sqrt((t_5 - (-1.0d0))))) - (t_14 - (sqrt((t_9 - (-1.0d0))) - t_10))
    else
        tmp = (((1.0d0 - t_6) + t_15) + (1.0d0 - t_10)) + t_13
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = Math.sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = Math.sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = fmax(t_2, t_8);
	double t_13 = Math.sqrt((t_12 + 1.0)) - Math.sqrt(t_12);
	double t_14 = Math.sqrt(t_11);
	double t_15 = Math.sqrt((t_11 + 1.0)) - t_14;
	double t_16 = (((Math.sqrt((t_5 + 1.0)) - t_6) + t_15) + (Math.sqrt((t_9 + 1.0)) - t_10)) + t_13;
	double tmp;
	if (t_16 <= 1.0) {
		tmp = (Math.sqrt((1.0 + t_5)) + (0.5 * (1.0 / (t_12 * Math.sqrt((1.0 / t_12)))))) - t_6;
	} else if (t_16 <= 2.999999999995) {
		tmp = (Math.sqrt((t_11 - -1.0)) - (t_6 - Math.sqrt((t_5 - -1.0)))) - (t_14 - (Math.sqrt((t_9 - -1.0)) - t_10));
	} else {
		tmp = (((1.0 - t_6) + t_15) + (1.0 - t_10)) + t_13;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = math.sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = math.sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = fmax(t_2, t_8)
	t_13 = math.sqrt((t_12 + 1.0)) - math.sqrt(t_12)
	t_14 = math.sqrt(t_11)
	t_15 = math.sqrt((t_11 + 1.0)) - t_14
	t_16 = (((math.sqrt((t_5 + 1.0)) - t_6) + t_15) + (math.sqrt((t_9 + 1.0)) - t_10)) + t_13
	tmp = 0
	if t_16 <= 1.0:
		tmp = (math.sqrt((1.0 + t_5)) + (0.5 * (1.0 / (t_12 * math.sqrt((1.0 / t_12)))))) - t_6
	elif t_16 <= 2.999999999995:
		tmp = (math.sqrt((t_11 - -1.0)) - (t_6 - math.sqrt((t_5 - -1.0)))) - (t_14 - (math.sqrt((t_9 - -1.0)) - t_10))
	else:
		tmp = (((1.0 - t_6) + t_15) + (1.0 - t_10)) + t_13
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = fmax(t_2, t_8)
	t_13 = Float64(sqrt(Float64(t_12 + 1.0)) - sqrt(t_12))
	t_14 = sqrt(t_11)
	t_15 = Float64(sqrt(Float64(t_11 + 1.0)) - t_14)
	t_16 = Float64(Float64(Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + t_15) + Float64(sqrt(Float64(t_9 + 1.0)) - t_10)) + t_13)
	tmp = 0.0
	if (t_16 <= 1.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t_5)) + Float64(0.5 * Float64(1.0 / Float64(t_12 * sqrt(Float64(1.0 / t_12)))))) - t_6);
	elseif (t_16 <= 2.999999999995)
		tmp = Float64(Float64(sqrt(Float64(t_11 - -1.0)) - Float64(t_6 - sqrt(Float64(t_5 - -1.0)))) - Float64(t_14 - Float64(sqrt(Float64(t_9 - -1.0)) - t_10)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - t_6) + t_15) + Float64(1.0 - t_10)) + t_13);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = max(max(x, y), t_1);
	t_3 = min(max(x, y), t_1);
	t_4 = min(min(x, y), z);
	t_5 = min(t_4, t);
	t_6 = sqrt(t_5);
	t_7 = max(t_4, t);
	t_8 = max(t_3, t_7);
	t_9 = min(t_2, t_8);
	t_10 = sqrt(t_9);
	t_11 = min(t_3, t_7);
	t_12 = max(t_2, t_8);
	t_13 = sqrt((t_12 + 1.0)) - sqrt(t_12);
	t_14 = sqrt(t_11);
	t_15 = sqrt((t_11 + 1.0)) - t_14;
	t_16 = (((sqrt((t_5 + 1.0)) - t_6) + t_15) + (sqrt((t_9 + 1.0)) - t_10)) + t_13;
	tmp = 0.0;
	if (t_16 <= 1.0)
		tmp = (sqrt((1.0 + t_5)) + (0.5 * (1.0 / (t_12 * sqrt((1.0 / t_12)))))) - t_6;
	elseif (t_16 <= 2.999999999995)
		tmp = (sqrt((t_11 - -1.0)) - (t_6 - sqrt((t_5 - -1.0)))) - (t_14 - (sqrt((t_9 - -1.0)) - t_10));
	else
		tmp = (((1.0 - t_6) + t_15) + (1.0 - t_10)) + t_13;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$4, t], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Min[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$12 = N[Max[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$13 = N[(N[Sqrt[N[(t$95$12 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$12], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$14 = N[Sqrt[t$95$11], $MachinePrecision]}, Block[{t$95$15 = N[(N[Sqrt[N[(t$95$11 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$14), $MachinePrecision]}, Block[{t$95$16 = N[(N[(N[(N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + t$95$15), $MachinePrecision] + N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision] + t$95$13), $MachinePrecision]}, If[LessEqual[t$95$16, 1.0], N[(N[(N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[(t$95$12 * N[Sqrt[N[(1.0 / t$95$12), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision], If[LessEqual[t$95$16, 2.999999999995], N[(N[(N[Sqrt[N[(t$95$11 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(t$95$6 - N[Sqrt[N[(t$95$5 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$14 - N[(N[Sqrt[N[(t$95$9 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 - t$95$6), $MachinePrecision] + t$95$15), $MachinePrecision] + N[(1.0 - t$95$10), $MachinePrecision]), $MachinePrecision] + t$95$13), $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 91.5%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 12.1%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 13.8%

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

   7. Applied rewrites 13.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \sqrt{1 + t} - \sqrt{t} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

      3. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

      4. lower-sqrt.f64 15.5%

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

   10. Applied rewrites 15.5%

       \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

   11. Taylor expanded in t around inf

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-sqrt.f64 13.3%

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

   13. Applied rewrites 13.3%

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

   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))) < 2.999999999995

   1. Initial program 91.5%

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

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

      8. lower--.f64 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 53.7%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 31.1%

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

   6. Applied rewrites 31.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lift--.f64 N/A

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

      9. add-flip N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. metadata-eval N/A

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

      14. add-flip N/A

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

      15. lift-+.f64 N/A

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

      16. lower--.f64 31.7%

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

      17. lift-+.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 31.7%

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

   8. Applied rewrites 31.7%

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

   if 2.999999999995 < (+.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 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 48.3%

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 24.2%

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

   7. Applied rewrites 24.2%

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

Alternative 11: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(y, \mathsf{max}\left(x, z\right)\right)\\ t_2 := \mathsf{min}\left(y, \mathsf{max}\left(x, z\right)\right)\\ t_3 := \mathsf{min}\left(\mathsf{min}\left(x, z\right), t\right)\\ t_4 := \sqrt{t\_3}\\ t_5 := \mathsf{max}\left(\mathsf{min}\left(x, z\right), t\right)\\ t_6 := \mathsf{max}\left(t\_2, t\_5\right)\\ t_7 := \mathsf{min}\left(t\_1, t\_6\right)\\ t_8 := \mathsf{min}\left(t\_2, t\_5\right)\\ t_9 := \mathsf{max}\left(t\_1, t\_6\right)\\ \mathbf{if}\;t\_8 \leq 1.75 \cdot 10^{+30}:\\ \;\;\;\;\left(\sqrt{t\_8 - -1} - \left(t\_4 - \sqrt{t\_3 - -1}\right)\right) - \left(\sqrt{t\_8} - \left(\sqrt{t\_7 - -1} - \sqrt{t\_7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t\_3} + 0.5 \cdot \frac{1}{t\_9 \cdot \sqrt{\frac{1}{t\_9}}}\right) - t\_4\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmax y (fmax x z)))
       (t_2 (fmin y (fmax x z)))
       (t_3 (fmin (fmin x z) t))
       (t_4 (sqrt t_3))
       (t_5 (fmax (fmin x z) t))
       (t_6 (fmax t_2 t_5))
       (t_7 (fmin t_1 t_6))
       (t_8 (fmin t_2 t_5))
       (t_9 (fmax t_1 t_6)))
  (if (<= t_8 1.75e+30)
    (-
     (- (sqrt (- t_8 -1.0)) (- t_4 (sqrt (- t_3 -1.0))))
     (- (sqrt t_8) (- (sqrt (- t_7 -1.0)) (sqrt t_7))))
    (-
     (+ (sqrt (+ 1.0 t_3)) (* 0.5 (/ 1.0 (* t_9 (sqrt (/ 1.0 t_9))))))
     t_4))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(y, fmax(x, z));
	double t_2 = fmin(y, fmax(x, z));
	double t_3 = fmin(fmin(x, z), t);
	double t_4 = sqrt(t_3);
	double t_5 = fmax(fmin(x, z), t);
	double t_6 = fmax(t_2, t_5);
	double t_7 = fmin(t_1, t_6);
	double t_8 = fmin(t_2, t_5);
	double t_9 = fmax(t_1, t_6);
	double tmp;
	if (t_8 <= 1.75e+30) {
		tmp = (sqrt((t_8 - -1.0)) - (t_4 - sqrt((t_3 - -1.0)))) - (sqrt(t_8) - (sqrt((t_7 - -1.0)) - sqrt(t_7)));
	} else {
		tmp = (sqrt((1.0 + t_3)) + (0.5 * (1.0 / (t_9 * sqrt((1.0 / t_9)))))) - t_4;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(y, fmax(x, z))
    t_2 = fmin(y, fmax(x, z))
    t_3 = fmin(fmin(x, z), t)
    t_4 = sqrt(t_3)
    t_5 = fmax(fmin(x, z), t)
    t_6 = fmax(t_2, t_5)
    t_7 = fmin(t_1, t_6)
    t_8 = fmin(t_2, t_5)
    t_9 = fmax(t_1, t_6)
    if (t_8 <= 1.75d+30) then
        tmp = (sqrt((t_8 - (-1.0d0))) - (t_4 - sqrt((t_3 - (-1.0d0))))) - (sqrt(t_8) - (sqrt((t_7 - (-1.0d0))) - sqrt(t_7)))
    else
        tmp = (sqrt((1.0d0 + t_3)) + (0.5d0 * (1.0d0 / (t_9 * sqrt((1.0d0 / t_9)))))) - t_4
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(y, fmax(x, z));
	double t_2 = fmin(y, fmax(x, z));
	double t_3 = fmin(fmin(x, z), t);
	double t_4 = Math.sqrt(t_3);
	double t_5 = fmax(fmin(x, z), t);
	double t_6 = fmax(t_2, t_5);
	double t_7 = fmin(t_1, t_6);
	double t_8 = fmin(t_2, t_5);
	double t_9 = fmax(t_1, t_6);
	double tmp;
	if (t_8 <= 1.75e+30) {
		tmp = (Math.sqrt((t_8 - -1.0)) - (t_4 - Math.sqrt((t_3 - -1.0)))) - (Math.sqrt(t_8) - (Math.sqrt((t_7 - -1.0)) - Math.sqrt(t_7)));
	} else {
		tmp = (Math.sqrt((1.0 + t_3)) + (0.5 * (1.0 / (t_9 * Math.sqrt((1.0 / t_9)))))) - t_4;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(y, fmax(x, z))
	t_2 = fmin(y, fmax(x, z))
	t_3 = fmin(fmin(x, z), t)
	t_4 = math.sqrt(t_3)
	t_5 = fmax(fmin(x, z), t)
	t_6 = fmax(t_2, t_5)
	t_7 = fmin(t_1, t_6)
	t_8 = fmin(t_2, t_5)
	t_9 = fmax(t_1, t_6)
	tmp = 0
	if t_8 <= 1.75e+30:
		tmp = (math.sqrt((t_8 - -1.0)) - (t_4 - math.sqrt((t_3 - -1.0)))) - (math.sqrt(t_8) - (math.sqrt((t_7 - -1.0)) - math.sqrt(t_7)))
	else:
		tmp = (math.sqrt((1.0 + t_3)) + (0.5 * (1.0 / (t_9 * math.sqrt((1.0 / t_9)))))) - t_4
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(y, fmax(x, z))
	t_2 = fmin(y, fmax(x, z))
	t_3 = fmin(fmin(x, z), t)
	t_4 = sqrt(t_3)
	t_5 = fmax(fmin(x, z), t)
	t_6 = fmax(t_2, t_5)
	t_7 = fmin(t_1, t_6)
	t_8 = fmin(t_2, t_5)
	t_9 = fmax(t_1, t_6)
	tmp = 0.0
	if (t_8 <= 1.75e+30)
		tmp = Float64(Float64(sqrt(Float64(t_8 - -1.0)) - Float64(t_4 - sqrt(Float64(t_3 - -1.0)))) - Float64(sqrt(t_8) - Float64(sqrt(Float64(t_7 - -1.0)) - sqrt(t_7))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t_3)) + Float64(0.5 * Float64(1.0 / Float64(t_9 * sqrt(Float64(1.0 / t_9)))))) - t_4);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(y, max(x, z));
	t_2 = min(y, max(x, z));
	t_3 = min(min(x, z), t);
	t_4 = sqrt(t_3);
	t_5 = max(min(x, z), t);
	t_6 = max(t_2, t_5);
	t_7 = min(t_1, t_6);
	t_8 = min(t_2, t_5);
	t_9 = max(t_1, t_6);
	tmp = 0.0;
	if (t_8 <= 1.75e+30)
		tmp = (sqrt((t_8 - -1.0)) - (t_4 - sqrt((t_3 - -1.0)))) - (sqrt(t_8) - (sqrt((t_7 - -1.0)) - sqrt(t_7)));
	else
		tmp = (sqrt((1.0 + t_3)) + (0.5 * (1.0 / (t_9 * sqrt((1.0 / t_9)))))) - t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[y, N[Max[x, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[y, N[Max[x, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Min[x, z], $MachinePrecision], t], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[Max[N[Min[x, z], $MachinePrecision], t], $MachinePrecision]}, Block[{t$95$6 = N[Max[t$95$2, t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Min[t$95$1, t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[Min[t$95$2, t$95$5], $MachinePrecision]}, Block[{t$95$9 = N[Max[t$95$1, t$95$6], $MachinePrecision]}, If[LessEqual[t$95$8, 1.75e+30], N[(N[(N[Sqrt[N[(t$95$8 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(t$95$4 - N[Sqrt[N[(t$95$3 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t$95$8], $MachinePrecision] - N[(N[Sqrt[N[(t$95$7 - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$7], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t$95$3), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[(t$95$9 * N[Sqrt[N[(1.0 / t$95$9), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.7500000000000001e30

   1. Initial program 91.5%

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

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

      8. lower--.f64 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 53.7%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 31.1%

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

   6. Applied rewrites 31.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lift--.f64 N/A

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

      9. add-flip N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. metadata-eval N/A

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

      14. add-flip N/A

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

      15. lift-+.f64 N/A

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

      16. lower--.f64 31.7%

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

      17. lift-+.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 31.7%

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

   8. Applied rewrites 31.7%

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

   if 1.7500000000000001e30 < y

   1. Initial program 91.5%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 12.1%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 13.8%

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

   7. Applied rewrites 13.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \sqrt{1 + t} - \sqrt{t} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

      3. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

      4. lower-sqrt.f64 15.5%

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

   10. Applied rewrites 15.5%

       \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

   11. Taylor expanded in t around inf

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-sqrt.f64 13.3%

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

   13. Applied rewrites 13.3%

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

Alternative 12: 81.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \mathsf{max}\left(t\_4, t\right)\\ t_7 := \mathsf{min}\left(t\_2, t\_6\right)\\ t_8 := \mathsf{max}\left(t\_2, t\_6\right)\\ t_9 := \mathsf{max}\left(t\_3, t\_8\right)\\ t_10 := \mathsf{min}\left(t\_3, t\_8\right)\\ t_11 := \sqrt{t\_7}\\ t_12 := \sqrt{t\_5}\\ \mathbf{if}\;\sqrt{t\_7 + 1} - t\_11 \leq 0.1:\\ \;\;\;\;\left(\sqrt{1 + t\_5} + 0.5 \cdot \frac{1}{t\_9 \cdot \sqrt{\frac{1}{t\_9}}}\right) - t\_12\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t\_7 - -1} - \left(t\_12 - 1\right)\right) - \left(\left(t\_11 + \sqrt{t\_10}\right) - \sqrt{1 + t\_10}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmax x y) t_1))
       (t_3 (fmax (fmax x y) t_1))
       (t_4 (fmin (fmin x y) z))
       (t_5 (fmin t_4 t))
       (t_6 (fmax t_4 t))
       (t_7 (fmin t_2 t_6))
       (t_8 (fmax t_2 t_6))
       (t_9 (fmax t_3 t_8))
       (t_10 (fmin t_3 t_8))
       (t_11 (sqrt t_7))
       (t_12 (sqrt t_5)))
  (if (<= (- (sqrt (+ t_7 1.0)) t_11) 0.1)
    (-
     (+ (sqrt (+ 1.0 t_5)) (* 0.5 (/ 1.0 (* t_9 (sqrt (/ 1.0 t_9))))))
     t_12)
    (-
     (- (sqrt (- t_7 -1.0)) (- t_12 1.0))
     (- (+ t_11 (sqrt t_10)) (sqrt (+ 1.0 t_10)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = fmax(t_4, t);
	double t_7 = fmin(t_2, t_6);
	double t_8 = fmax(t_2, t_6);
	double t_9 = fmax(t_3, t_8);
	double t_10 = fmin(t_3, t_8);
	double t_11 = sqrt(t_7);
	double t_12 = sqrt(t_5);
	double tmp;
	if ((sqrt((t_7 + 1.0)) - t_11) <= 0.1) {
		tmp = (sqrt((1.0 + t_5)) + (0.5 * (1.0 / (t_9 * sqrt((1.0 / t_9)))))) - t_12;
	} else {
		tmp = (sqrt((t_7 - -1.0)) - (t_12 - 1.0)) - ((t_11 + sqrt(t_10)) - sqrt((1.0 + t_10)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmin(fmax(x, y), t_1)
    t_3 = fmax(fmax(x, y), t_1)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmin(t_4, t)
    t_6 = fmax(t_4, t)
    t_7 = fmin(t_2, t_6)
    t_8 = fmax(t_2, t_6)
    t_9 = fmax(t_3, t_8)
    t_10 = fmin(t_3, t_8)
    t_11 = sqrt(t_7)
    t_12 = sqrt(t_5)
    if ((sqrt((t_7 + 1.0d0)) - t_11) <= 0.1d0) then
        tmp = (sqrt((1.0d0 + t_5)) + (0.5d0 * (1.0d0 / (t_9 * sqrt((1.0d0 / t_9)))))) - t_12
    else
        tmp = (sqrt((t_7 - (-1.0d0))) - (t_12 - 1.0d0)) - ((t_11 + sqrt(t_10)) - sqrt((1.0d0 + t_10)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = fmax(t_4, t);
	double t_7 = fmin(t_2, t_6);
	double t_8 = fmax(t_2, t_6);
	double t_9 = fmax(t_3, t_8);
	double t_10 = fmin(t_3, t_8);
	double t_11 = Math.sqrt(t_7);
	double t_12 = Math.sqrt(t_5);
	double tmp;
	if ((Math.sqrt((t_7 + 1.0)) - t_11) <= 0.1) {
		tmp = (Math.sqrt((1.0 + t_5)) + (0.5 * (1.0 / (t_9 * Math.sqrt((1.0 / t_9)))))) - t_12;
	} else {
		tmp = (Math.sqrt((t_7 - -1.0)) - (t_12 - 1.0)) - ((t_11 + Math.sqrt(t_10)) - Math.sqrt((1.0 + t_10)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = fmax(t_4, t)
	t_7 = fmin(t_2, t_6)
	t_8 = fmax(t_2, t_6)
	t_9 = fmax(t_3, t_8)
	t_10 = fmin(t_3, t_8)
	t_11 = math.sqrt(t_7)
	t_12 = math.sqrt(t_5)
	tmp = 0
	if (math.sqrt((t_7 + 1.0)) - t_11) <= 0.1:
		tmp = (math.sqrt((1.0 + t_5)) + (0.5 * (1.0 / (t_9 * math.sqrt((1.0 / t_9)))))) - t_12
	else:
		tmp = (math.sqrt((t_7 - -1.0)) - (t_12 - 1.0)) - ((t_11 + math.sqrt(t_10)) - math.sqrt((1.0 + t_10)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = fmax(t_4, t)
	t_7 = fmin(t_2, t_6)
	t_8 = fmax(t_2, t_6)
	t_9 = fmax(t_3, t_8)
	t_10 = fmin(t_3, t_8)
	t_11 = sqrt(t_7)
	t_12 = sqrt(t_5)
	tmp = 0.0
	if (Float64(sqrt(Float64(t_7 + 1.0)) - t_11) <= 0.1)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t_5)) + Float64(0.5 * Float64(1.0 / Float64(t_9 * sqrt(Float64(1.0 / t_9)))))) - t_12);
	else
		tmp = Float64(Float64(sqrt(Float64(t_7 - -1.0)) - Float64(t_12 - 1.0)) - Float64(Float64(t_11 + sqrt(t_10)) - sqrt(Float64(1.0 + t_10))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = min(max(x, y), t_1);
	t_3 = max(max(x, y), t_1);
	t_4 = min(min(x, y), z);
	t_5 = min(t_4, t);
	t_6 = max(t_4, t);
	t_7 = min(t_2, t_6);
	t_8 = max(t_2, t_6);
	t_9 = max(t_3, t_8);
	t_10 = min(t_3, t_8);
	t_11 = sqrt(t_7);
	t_12 = sqrt(t_5);
	tmp = 0.0;
	if ((sqrt((t_7 + 1.0)) - t_11) <= 0.1)
		tmp = (sqrt((1.0 + t_5)) + (0.5 * (1.0 / (t_9 * sqrt((1.0 / t_9)))))) - t_12;
	else
		tmp = (sqrt((t_7 - -1.0)) - (t_12 - 1.0)) - ((t_11 + sqrt(t_10)) - sqrt((1.0 + t_10)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$4, t], $MachinePrecision]}, Block[{t$95$6 = N[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$7 = N[Min[t$95$2, t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$2, t$95$6], $MachinePrecision]}, Block[{t$95$9 = N[Max[t$95$3, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Min[t$95$3, t$95$8], $MachinePrecision]}, Block[{t$95$11 = N[Sqrt[t$95$7], $MachinePrecision]}, Block[{t$95$12 = N[Sqrt[t$95$5], $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(t$95$7 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$11), $MachinePrecision], 0.1], N[(N[(N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[(t$95$9 * N[Sqrt[N[(1.0 / t$95$9), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$12), $MachinePrecision], N[(N[(N[Sqrt[N[(t$95$7 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(t$95$12 - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$11 + N[Sqrt[t$95$10], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(1.0 + t$95$10), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 0.10000000000000001

   1. Initial program 91.5%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 12.1%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 13.8%

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

   7. Applied rewrites 13.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \sqrt{1 + t} - \sqrt{t} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

      3. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

      4. lower-sqrt.f64 15.5%

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

   10. Applied rewrites 15.5%

       \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

   11. Taylor expanded in t around inf

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-sqrt.f64 13.3%

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

   13. Applied rewrites 13.3%

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

   if 0.10000000000000001 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 91.5%

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

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

      8. lower--.f64 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 53.7%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 31.1%

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

   6. Applied rewrites 31.1%

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

   7. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 16.7%

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

   9. Applied rewrites 16.7%

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

Alternative 13: 81.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{max}\left(t\_4, t\right)\\ t_8 := \mathsf{max}\left(t\_3, t\_7\right)\\ t_9 := \mathsf{min}\left(t\_2, t\_8\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \mathsf{min}\left(t\_3, t\_7\right)\\ t_12 := \mathsf{max}\left(t\_2, t\_8\right)\\ t_13 := \sqrt{t\_11}\\ t_14 := \sqrt{1 + t\_5}\\ \mathbf{if}\;\left(\left(\left(\sqrt{t\_5 + 1} - t\_6\right) + \left(\sqrt{t\_11 + 1} - t\_13\right)\right) + \left(\sqrt{t\_9 + 1} - t\_10\right)\right) + \left(\sqrt{t\_12 + 1} - \sqrt{t\_12}\right) \leq 1.5:\\ \;\;\;\;\left(t\_14 + 0.5 \cdot \frac{1}{t\_12 \cdot \sqrt{\frac{1}{t\_12}}}\right) - t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + t\_14\right) - t\_6\right) - \left(\left(t\_13 + t\_10\right) - \sqrt{1 + t\_9}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmax (fmin x y) z))
       (t_2 (fmax (fmax x y) t_1))
       (t_3 (fmin (fmax x y) t_1))
       (t_4 (fmin (fmin x y) z))
       (t_5 (fmin t_4 t))
       (t_6 (sqrt t_5))
       (t_7 (fmax t_4 t))
       (t_8 (fmax t_3 t_7))
       (t_9 (fmin t_2 t_8))
       (t_10 (sqrt t_9))
       (t_11 (fmin t_3 t_7))
       (t_12 (fmax t_2 t_8))
       (t_13 (sqrt t_11))
       (t_14 (sqrt (+ 1.0 t_5))))
  (if (<=
       (+
        (+
         (+ (- (sqrt (+ t_5 1.0)) t_6) (- (sqrt (+ t_11 1.0)) t_13))
         (- (sqrt (+ t_9 1.0)) t_10))
        (- (sqrt (+ t_12 1.0)) (sqrt t_12)))
       1.5)
    (- (+ t_14 (* 0.5 (/ 1.0 (* t_12 (sqrt (/ 1.0 t_12)))))) t_6)
    (- (- (+ 1.0 t_14) t_6) (- (+ t_13 t_10) (sqrt (+ 1.0 t_9)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = fmax(t_2, t_8);
	double t_13 = sqrt(t_11);
	double t_14 = sqrt((1.0 + t_5));
	double tmp;
	if (((((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_13)) + (sqrt((t_9 + 1.0)) - t_10)) + (sqrt((t_12 + 1.0)) - sqrt(t_12))) <= 1.5) {
		tmp = (t_14 + (0.5 * (1.0 / (t_12 * sqrt((1.0 / t_12)))))) - t_6;
	} else {
		tmp = ((1.0 + t_14) - t_6) - ((t_13 + t_10) - sqrt((1.0 + t_9)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_14
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmin(t_4, t)
    t_6 = sqrt(t_5)
    t_7 = fmax(t_4, t)
    t_8 = fmax(t_3, t_7)
    t_9 = fmin(t_2, t_8)
    t_10 = sqrt(t_9)
    t_11 = fmin(t_3, t_7)
    t_12 = fmax(t_2, t_8)
    t_13 = sqrt(t_11)
    t_14 = sqrt((1.0d0 + t_5))
    if (((((sqrt((t_5 + 1.0d0)) - t_6) + (sqrt((t_11 + 1.0d0)) - t_13)) + (sqrt((t_9 + 1.0d0)) - t_10)) + (sqrt((t_12 + 1.0d0)) - sqrt(t_12))) <= 1.5d0) then
        tmp = (t_14 + (0.5d0 * (1.0d0 / (t_12 * sqrt((1.0d0 / t_12)))))) - t_6
    else
        tmp = ((1.0d0 + t_14) - t_6) - ((t_13 + t_10) - sqrt((1.0d0 + t_9)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = Math.sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = Math.sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = fmax(t_2, t_8);
	double t_13 = Math.sqrt(t_11);
	double t_14 = Math.sqrt((1.0 + t_5));
	double tmp;
	if (((((Math.sqrt((t_5 + 1.0)) - t_6) + (Math.sqrt((t_11 + 1.0)) - t_13)) + (Math.sqrt((t_9 + 1.0)) - t_10)) + (Math.sqrt((t_12 + 1.0)) - Math.sqrt(t_12))) <= 1.5) {
		tmp = (t_14 + (0.5 * (1.0 / (t_12 * Math.sqrt((1.0 / t_12)))))) - t_6;
	} else {
		tmp = ((1.0 + t_14) - t_6) - ((t_13 + t_10) - Math.sqrt((1.0 + t_9)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = math.sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = math.sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = fmax(t_2, t_8)
	t_13 = math.sqrt(t_11)
	t_14 = math.sqrt((1.0 + t_5))
	tmp = 0
	if ((((math.sqrt((t_5 + 1.0)) - t_6) + (math.sqrt((t_11 + 1.0)) - t_13)) + (math.sqrt((t_9 + 1.0)) - t_10)) + (math.sqrt((t_12 + 1.0)) - math.sqrt(t_12))) <= 1.5:
		tmp = (t_14 + (0.5 * (1.0 / (t_12 * math.sqrt((1.0 / t_12)))))) - t_6
	else:
		tmp = ((1.0 + t_14) - t_6) - ((t_13 + t_10) - math.sqrt((1.0 + t_9)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = fmax(t_2, t_8)
	t_13 = sqrt(t_11)
	t_14 = sqrt(Float64(1.0 + t_5))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + Float64(sqrt(Float64(t_11 + 1.0)) - t_13)) + Float64(sqrt(Float64(t_9 + 1.0)) - t_10)) + Float64(sqrt(Float64(t_12 + 1.0)) - sqrt(t_12))) <= 1.5)
		tmp = Float64(Float64(t_14 + Float64(0.5 * Float64(1.0 / Float64(t_12 * sqrt(Float64(1.0 / t_12)))))) - t_6);
	else
		tmp = Float64(Float64(Float64(1.0 + t_14) - t_6) - Float64(Float64(t_13 + t_10) - sqrt(Float64(1.0 + t_9))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = max(max(x, y), t_1);
	t_3 = min(max(x, y), t_1);
	t_4 = min(min(x, y), z);
	t_5 = min(t_4, t);
	t_6 = sqrt(t_5);
	t_7 = max(t_4, t);
	t_8 = max(t_3, t_7);
	t_9 = min(t_2, t_8);
	t_10 = sqrt(t_9);
	t_11 = min(t_3, t_7);
	t_12 = max(t_2, t_8);
	t_13 = sqrt(t_11);
	t_14 = sqrt((1.0 + t_5));
	tmp = 0.0;
	if (((((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_13)) + (sqrt((t_9 + 1.0)) - t_10)) + (sqrt((t_12 + 1.0)) - sqrt(t_12))) <= 1.5)
		tmp = (t_14 + (0.5 * (1.0 / (t_12 * sqrt((1.0 / t_12)))))) - t_6;
	else
		tmp = ((1.0 + t_14) - t_6) - ((t_13 + t_10) - sqrt((1.0 + t_9)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$4, t], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Min[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$12 = N[Max[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$13 = N[Sqrt[t$95$11], $MachinePrecision]}, Block[{t$95$14 = N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + N[(N[Sqrt[N[(t$95$11 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$13), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$12 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$12], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], N[(N[(t$95$14 + N[(0.5 * N[(1.0 / N[(t$95$12 * N[Sqrt[N[(1.0 / t$95$12), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision], N[(N[(N[(1.0 + t$95$14), $MachinePrecision] - t$95$6), $MachinePrecision] - N[(N[(t$95$13 + t$95$10), $MachinePrecision] - N[Sqrt[N[(1.0 + t$95$9), $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))) < 1.5

   1. Initial program 91.5%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 12.1%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 13.8%

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

   7. Applied rewrites 13.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \sqrt{1 + t} - \sqrt{t} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

      3. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

      4. lower-sqrt.f64 15.5%

         \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

   10. Applied rewrites 15.5%

       \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

   11. Taylor expanded in t around inf

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-sqrt.f64 13.3%

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

   13. Applied rewrites 13.3%

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

   if 1.5 < (+.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 91.5%

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

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

      8. lower--.f64 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 53.7%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 31.1%

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

   6. Applied rewrites 31.1%

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

   7. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 21.2%

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

   9. Applied rewrites 21.2%

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

Alternative 14: 50.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right), \mathsf{max}\left(\mathsf{max}\left(x, y\right), \mathsf{max}\left(t\_1, t\right)\right)\right)\\ t_3 := \mathsf{min}\left(t\_1, t\right)\\ \left(\sqrt{1 + t\_3} + 0.5 \cdot \frac{1}{t\_2 \cdot \sqrt{\frac{1}{t\_2}}}\right) - \sqrt{t\_3} \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin (fmin x y) z))
       (t_2 (fmax (fmax (fmin x y) z) (fmax (fmax x y) (fmax t_1 t))))
       (t_3 (fmin t_1 t)))
  (-
   (+ (sqrt (+ 1.0 t_3)) (* 0.5 (/ 1.0 (* t_2 (sqrt (/ 1.0 t_2))))))
   (sqrt t_3))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(fmax(fmin(x, y), z), fmax(fmax(x, y), fmax(t_1, t)));
	double t_3 = fmin(t_1, t);
	return (sqrt((1.0 + t_3)) + (0.5 * (1.0 / (t_2 * sqrt((1.0 / t_2)))))) - sqrt(t_3);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_1 = fmin(fmin(x, y), z)
    t_2 = fmax(fmax(fmin(x, y), z), fmax(fmax(x, y), fmax(t_1, t)))
    t_3 = fmin(t_1, t)
    code = (sqrt((1.0d0 + t_3)) + (0.5d0 * (1.0d0 / (t_2 * sqrt((1.0d0 / t_2)))))) - sqrt(t_3)
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(fmax(fmin(x, y), z), fmax(fmax(x, y), fmax(t_1, t)));
	double t_3 = fmin(t_1, t);
	return (Math.sqrt((1.0 + t_3)) + (0.5 * (1.0 / (t_2 * Math.sqrt((1.0 / t_2)))))) - Math.sqrt(t_3);
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(fmax(fmin(x, y), z), fmax(fmax(x, y), fmax(t_1, t)))
	t_3 = fmin(t_1, t)
	return (math.sqrt((1.0 + t_3)) + (0.5 * (1.0 / (t_2 * math.sqrt((1.0 / t_2)))))) - math.sqrt(t_3)

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(fmax(fmin(x, y), z), fmax(fmax(x, y), fmax(t_1, t)))
	t_3 = fmin(t_1, t)
	return Float64(Float64(sqrt(Float64(1.0 + t_3)) + Float64(0.5 * Float64(1.0 / Float64(t_2 * sqrt(Float64(1.0 / t_2)))))) - sqrt(t_3))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = min(min(x, y), z);
	t_2 = max(max(min(x, y), z), max(max(x, y), max(t_1, t)));
	t_3 = min(t_1, t);
	tmp = (sqrt((1.0 + t_3)) + (0.5 * (1.0 / (t_2 * sqrt((1.0 / t_2)))))) - sqrt(t_3);
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision], N[Max[N[Max[x, y], $MachinePrecision], N[Max[t$95$1, t], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Min[t$95$1, t], $MachinePrecision]}, N[(N[(N[Sqrt[N[(1.0 + t$95$3), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[(t$95$2 * N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 91.5%

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

2. Taylor expanded in z around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-+.f64 N/A

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

4. Applied rewrites 12.1%

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

5. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 13.8%

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

7. Applied rewrites 13.8%

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

8. Taylor expanded in x around inf

   \[\leadsto \sqrt{1 + t} - \sqrt{t} \]
9. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

   3. lower-+.f64 N/A

      \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

   4. lower-sqrt.f64 15.5%

      \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

10. Applied rewrites 15.5%

    \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

11. Taylor expanded in t around inf

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

    1. lower--.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + x} + \frac{1}{2} \cdot \frac{1}{t \cdot \sqrt{\frac{1}{t}}}\right) - \sqrt{x} \]

    4. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + x} + \frac{1}{2} \cdot \frac{1}{t \cdot \sqrt{\frac{1}{t}}}\right) - \sqrt{x} \]

    5. lower-*.f64 N/A

       \[\leadsto \left(\sqrt{1 + x} + \frac{1}{2} \cdot \frac{1}{t \cdot \sqrt{\frac{1}{t}}}\right) - \sqrt{x} \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\sqrt{1 + x} + \frac{1}{2} \cdot \frac{1}{t \cdot \sqrt{\frac{1}{t}}}\right) - \sqrt{x} \]

    7. lower-*.f64 N/A

       \[\leadsto \left(\sqrt{1 + x} + \frac{1}{2} \cdot \frac{1}{t \cdot \sqrt{\frac{1}{t}}}\right) - \sqrt{x} \]

    8. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + x} + \frac{1}{2} \cdot \frac{1}{t \cdot \sqrt{\frac{1}{t}}}\right) - \sqrt{x} \]

    9. lower-/.f64 N/A

       \[\leadsto \left(\sqrt{1 + x} + \frac{1}{2} \cdot \frac{1}{t \cdot \sqrt{\frac{1}{t}}}\right) - \sqrt{x} \]

    10. lower-sqrt.f64 13.3%

        \[\leadsto \left(\sqrt{1 + x} + 0.5 \cdot \frac{1}{t \cdot \sqrt{\frac{1}{t}}}\right) - \sqrt{x} \]

13. Applied rewrites 13.3%

    \[\leadsto \left(\sqrt{1 + x} + 0.5 \cdot \frac{1}{t \cdot \sqrt{\frac{1}{t}}}\right) - \sqrt{x} \]
14. Add Preprocessing

Alternative 15: 35.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(t\_1, t\right)\\ t_3 := \mathsf{min}\left(t\_1, t\right)\\ \sqrt{t\_3 - -1} + \left(\sqrt{t\_2 - -1} - \left(\sqrt{t\_2} + \sqrt{t\_3}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin (fmin x y) z))
       (t_2 (fmax t_1 t))
       (t_3 (fmin t_1 t)))
  (+
   (sqrt (- t_3 -1.0))
   (- (sqrt (- t_2 -1.0)) (+ (sqrt t_2) (sqrt t_3))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(t_1, t);
	double t_3 = fmin(t_1, t);
	return sqrt((t_3 - -1.0)) + (sqrt((t_2 - -1.0)) - (sqrt(t_2) + sqrt(t_3)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_1 = fmin(fmin(x, y), z)
    t_2 = fmax(t_1, t)
    t_3 = fmin(t_1, t)
    code = sqrt((t_3 - (-1.0d0))) + (sqrt((t_2 - (-1.0d0))) - (sqrt(t_2) + sqrt(t_3)))
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(t_1, t);
	double t_3 = fmin(t_1, t);
	return Math.sqrt((t_3 - -1.0)) + (Math.sqrt((t_2 - -1.0)) - (Math.sqrt(t_2) + Math.sqrt(t_3)));
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = fmin(t_1, t)
	return math.sqrt((t_3 - -1.0)) + (math.sqrt((t_2 - -1.0)) - (math.sqrt(t_2) + math.sqrt(t_3)))

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = fmin(t_1, t)
	return Float64(sqrt(Float64(t_3 - -1.0)) + Float64(sqrt(Float64(t_2 - -1.0)) - Float64(sqrt(t_2) + sqrt(t_3))))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = min(min(x, y), z);
	t_2 = max(t_1, t);
	t_3 = min(t_1, t);
	tmp = sqrt((t_3 - -1.0)) + (sqrt((t_2 - -1.0)) - (sqrt(t_2) + sqrt(t_3)));
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[t$95$1, t], $MachinePrecision]}, Block[{t$95$3 = N[Min[t$95$1, t], $MachinePrecision]}, N[(N[Sqrt[N[(t$95$3 - -1.0), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(t$95$2 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[t$95$2], $MachinePrecision] + N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 91.5%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

2. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\color{blue}{\sqrt{t}} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{\color{blue}{t}} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   9. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   10. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

4. Applied rewrites 12.1%

   \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

5. Taylor expanded in y around inf

   \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{t} + \sqrt{x}\right)} \]
6. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \color{blue}{\sqrt{x}}\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{\color{blue}{x}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   6. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   9. lower-sqrt.f64 13.8%

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

7. Applied rewrites 13.8%

   \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{t} + \sqrt{x}\right)} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \color{blue}{\sqrt{x}}\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{\color{blue}{x}}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \sqrt{\color{blue}{x}}\right) \]

   4. associate--l+ N/A

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + t} - \color{blue}{\left(\sqrt{t} + \sqrt{x}\right)}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + t} - \color{blue}{\left(\sqrt{t} + \sqrt{x}\right)}\right) \]

   6. lift-+.f64 N/A

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + t} - \left(\sqrt{\color{blue}{t}} + \sqrt{x}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + t} - \left(\sqrt{\color{blue}{t}} + \sqrt{x}\right)\right) \]

   8. add-flip N/A

      \[\leadsto \sqrt{x - \left(\mathsf{neg}\left(1\right)\right)} + \left(\sqrt{1 + t} - \left(\sqrt{\color{blue}{t}} + \sqrt{x}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \sqrt{x - -1} + \left(\sqrt{1 + t} - \left(\sqrt{t} + \sqrt{x}\right)\right) \]

   10. lift--.f64 N/A

       \[\leadsto \sqrt{x - -1} + \left(\sqrt{1 + t} - \left(\sqrt{\color{blue}{t}} + \sqrt{x}\right)\right) \]

   11. lower--.f64 21.7%

       \[\leadsto \sqrt{x - -1} + \left(\sqrt{1 + t} - \left(\sqrt{t} + \color{blue}{\sqrt{x}}\right)\right) \]

   12. lift-+.f64 N/A

       \[\leadsto \sqrt{x - -1} + \left(\sqrt{1 + t} - \left(\sqrt{t} + \sqrt{x}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \sqrt{x - -1} + \left(\sqrt{t + 1} - \left(\sqrt{t} + \sqrt{x}\right)\right) \]

   14. add-flip N/A

       \[\leadsto \sqrt{x - -1} + \left(\sqrt{t - \left(\mathsf{neg}\left(1\right)\right)} - \left(\sqrt{t} + \sqrt{x}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \sqrt{x - -1} + \left(\sqrt{t - -1} - \left(\sqrt{t} + \sqrt{x}\right)\right) \]

   16. lower--.f64 21.7%

       \[\leadsto \sqrt{x - -1} + \left(\sqrt{t - -1} - \left(\sqrt{t} + \sqrt{x}\right)\right) \]

9. Applied rewrites 21.7%

   \[\leadsto \sqrt{x - -1} + \left(\sqrt{t - -1} - \color{blue}{\left(\sqrt{t} + \sqrt{x}\right)}\right) \]
10. Add Preprocessing

Alternative 16: 25.9% accurate, 1.9× speedup

Math

\[\left(1 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \sqrt{\mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)}\right) \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (- (+ 1.0 (sqrt (+ 1.0 t))) (+ (sqrt t) (sqrt (fmin (fmin x y) z)))))

C

double code(double x, double y, double z, double t) {
	return (1.0 + sqrt((1.0 + t))) - (sqrt(t) + sqrt(fmin(fmin(x, y), z)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (1.0d0 + sqrt((1.0d0 + t))) - (sqrt(t) + sqrt(fmin(fmin(x, y), z)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (1.0 + Math.sqrt((1.0 + t))) - (Math.sqrt(t) + Math.sqrt(fmin(fmin(x, y), z)));
}

Python

def code(x, y, z, t):
	return (1.0 + math.sqrt((1.0 + t))) - (math.sqrt(t) + math.sqrt(fmin(fmin(x, y), z)))

Julia

function code(x, y, z, t)
	return Float64(Float64(1.0 + sqrt(Float64(1.0 + t))) - Float64(sqrt(t) + sqrt(fmin(fmin(x, y), z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (1.0 + sqrt((1.0 + t))) - (sqrt(t) + sqrt(min(min(x, y), z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(1.0 + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

2. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\color{blue}{\sqrt{t}} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{\color{blue}{t}} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   9. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   10. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

4. Applied rewrites 12.1%

   \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

5. Taylor expanded in y around inf

   \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{t} + \sqrt{x}\right)} \]
6. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \color{blue}{\sqrt{x}}\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{\color{blue}{x}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   6. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   9. lower-sqrt.f64 13.8%

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

7. Applied rewrites 13.8%

   \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{t} + \sqrt{x}\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \left(1 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \sqrt{\color{blue}{x}}\right) \]
9. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(1 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \left(1 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   3. lower-+.f64 12.0%

      \[\leadsto \left(1 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

10. Applied rewrites 12.0%

    \[\leadsto \left(1 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \sqrt{\color{blue}{x}}\right) \]
11. Add Preprocessing

Alternative 17: 15.5% accurate, 4.5× speedup

Math

\[\sqrt{1 + t} - \sqrt{t} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (- (sqrt (+ 1.0 t)) (sqrt t)))

C

double code(double x, double y, double z, double t) {
	return sqrt((1.0 + t)) - 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((1.0d0 + t)) - sqrt(t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.sqrt((1.0 + t)) - Math.sqrt(t);
}

Python

def code(x, y, z, t):
	return math.sqrt((1.0 + t)) - math.sqrt(t)

Julia

function code(x, y, z, t)
	return Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = sqrt((1.0 + t)) - sqrt(t);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

2. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\color{blue}{\sqrt{t}} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{\color{blue}{t}} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   9. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   10. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

4. Applied rewrites 12.1%

   \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

5. Taylor expanded in y around inf

   \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{t} + \sqrt{x}\right)} \]
6. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \color{blue}{\sqrt{x}}\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{\color{blue}{x}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   6. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

   9. lower-sqrt.f64 13.8%

      \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right) \]

7. Applied rewrites 13.8%

   \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{t} + \sqrt{x}\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto \sqrt{1 + t} - \sqrt{t} \]
9. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

   3. lower-+.f64 N/A

      \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

   4. lower-sqrt.f64 15.5%

      \[\leadsto \sqrt{1 + t} - \sqrt{t} \]

10. Applied rewrites 15.5%

    \[\leadsto \sqrt{1 + t} - \sqrt{t} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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))))