Main:z from

Percentage Accurate: 91.6% → 99.4%

Time: 21.6s

Alternatives: 22

Speedup: 0.0×

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)) (sqrt x)) (- (sqrt (+ y 1)) (sqrt y)))
  (- (sqrt (+ z 1)) (sqrt z)))
 (- (sqrt (+ t 1)) (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), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.6% 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)) (sqrt x)) (- (sqrt (+ y 1)) (sqrt y)))
  (- (sqrt (+ z 1)) (sqrt z)))
 (- (sqrt (+ t 1)) (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), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(y, \mathsf{max}\left(x, z\right)\right)\\ t_2 := \mathsf{max}\left(\mathsf{min}\left(x, z\right), t\right)\\ t_3 := \mathsf{min}\left(\mathsf{min}\left(x, z\right), t\right)\\ t_4 := \sqrt{t\_3}\\ t_5 := \mathsf{max}\left(y, \mathsf{max}\left(x, z\right)\right)\\ t_6 := \mathsf{max}\left(t\_1, t\_2\right)\\ t_7 := \mathsf{min}\left(t\_5, t\_6\right)\\ t_8 := \mathsf{max}\left(t\_5, t\_6\right)\\ t_9 := \sqrt{t\_8}\\ t_10 := t\_8 - -1\\ t_11 := \mathsf{min}\left(t\_1, t\_2\right)\\ t_12 := \sqrt{1 + t\_11}\\ t_13 := \sqrt{t\_11}\\ t_14 := t\_13 + t\_12\\ t_15 := \left(t\_4 + \sqrt{1 + t\_3}\right) \cdot t\_14\\ \mathbf{if}\;t\_7 \leq 220000000:\\ \;\;\;\;\left(\sqrt{t\_3 - -1} - t\_4\right) - \left(\left(t\_13 - \sqrt{t\_11 - -1}\right) - \left(\left(\sqrt{t\_7 - -1} - \sqrt{t\_7}\right) - \frac{t\_8 - t\_10}{\sqrt{t\_10} + t\_9}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{1}{t\_7 \cdot \sqrt{\frac{1}{t\_7}}} + \left(\frac{1}{t\_14} + \left(\frac{t\_13}{t\_15} + \frac{t\_12}{t\_15}\right)\right)\right) + \left(\sqrt{t\_8 + 1} - t\_9\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin y (fmax x z)))
       (t_2 (fmax (fmin x z) t))
       (t_3 (fmin (fmin x z) t))
       (t_4 (sqrt t_3))
       (t_5 (fmax y (fmax x z)))
       (t_6 (fmax t_1 t_2))
       (t_7 (fmin t_5 t_6))
       (t_8 (fmax t_5 t_6))
       (t_9 (sqrt t_8))
       (t_10 (- t_8 -1))
       (t_11 (fmin t_1 t_2))
       (t_12 (sqrt (+ 1 t_11)))
       (t_13 (sqrt t_11))
       (t_14 (+ t_13 t_12))
       (t_15 (* (+ t_4 (sqrt (+ 1 t_3))) t_14)))
  (if (<= t_7 220000000)
    (-
     (- (sqrt (- t_3 -1)) t_4)
     (-
      (- t_13 (sqrt (- t_11 -1)))
      (-
       (- (sqrt (- t_7 -1)) (sqrt t_7))
       (/ (- t_8 t_10) (+ (sqrt t_10) t_9)))))
    (+
     (+
      (* 1/2 (/ 1 (* t_7 (sqrt (/ 1 t_7)))))
      (+ (/ 1 t_14) (+ (/ t_13 t_15) (/ t_12 t_15))))
     (- (sqrt (+ t_8 1)) t_9)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(y, fmax(x, z));
	double t_2 = fmax(fmin(x, z), t);
	double t_3 = fmin(fmin(x, z), t);
	double t_4 = sqrt(t_3);
	double t_5 = fmax(y, fmax(x, z));
	double t_6 = fmax(t_1, t_2);
	double t_7 = fmin(t_5, t_6);
	double t_8 = fmax(t_5, t_6);
	double t_9 = sqrt(t_8);
	double t_10 = t_8 - -1.0;
	double t_11 = fmin(t_1, t_2);
	double t_12 = sqrt((1.0 + t_11));
	double t_13 = sqrt(t_11);
	double t_14 = t_13 + t_12;
	double t_15 = (t_4 + sqrt((1.0 + t_3))) * t_14;
	double tmp;
	if (t_7 <= 220000000.0) {
		tmp = (sqrt((t_3 - -1.0)) - t_4) - ((t_13 - sqrt((t_11 - -1.0))) - ((sqrt((t_7 - -1.0)) - sqrt(t_7)) - ((t_8 - t_10) / (sqrt(t_10) + t_9))));
	} else {
		tmp = ((0.5 * (1.0 / (t_7 * sqrt((1.0 / t_7))))) + ((1.0 / t_14) + ((t_13 / t_15) + (t_12 / t_15)))) + (sqrt((t_8 + 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_15
    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(y, fmax(x, z))
    t_2 = fmax(fmin(x, z), t)
    t_3 = fmin(fmin(x, z), t)
    t_4 = sqrt(t_3)
    t_5 = fmax(y, fmax(x, z))
    t_6 = fmax(t_1, t_2)
    t_7 = fmin(t_5, t_6)
    t_8 = fmax(t_5, t_6)
    t_9 = sqrt(t_8)
    t_10 = t_8 - (-1.0d0)
    t_11 = fmin(t_1, t_2)
    t_12 = sqrt((1.0d0 + t_11))
    t_13 = sqrt(t_11)
    t_14 = t_13 + t_12
    t_15 = (t_4 + sqrt((1.0d0 + t_3))) * t_14
    if (t_7 <= 220000000.0d0) then
        tmp = (sqrt((t_3 - (-1.0d0))) - t_4) - ((t_13 - sqrt((t_11 - (-1.0d0)))) - ((sqrt((t_7 - (-1.0d0))) - sqrt(t_7)) - ((t_8 - t_10) / (sqrt(t_10) + t_9))))
    else
        tmp = ((0.5d0 * (1.0d0 / (t_7 * sqrt((1.0d0 / t_7))))) + ((1.0d0 / t_14) + ((t_13 / t_15) + (t_12 / t_15)))) + (sqrt((t_8 + 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 = fmin(y, fmax(x, z));
	double t_2 = fmax(fmin(x, z), t);
	double t_3 = fmin(fmin(x, z), t);
	double t_4 = Math.sqrt(t_3);
	double t_5 = fmax(y, fmax(x, z));
	double t_6 = fmax(t_1, t_2);
	double t_7 = fmin(t_5, t_6);
	double t_8 = fmax(t_5, t_6);
	double t_9 = Math.sqrt(t_8);
	double t_10 = t_8 - -1.0;
	double t_11 = fmin(t_1, t_2);
	double t_12 = Math.sqrt((1.0 + t_11));
	double t_13 = Math.sqrt(t_11);
	double t_14 = t_13 + t_12;
	double t_15 = (t_4 + Math.sqrt((1.0 + t_3))) * t_14;
	double tmp;
	if (t_7 <= 220000000.0) {
		tmp = (Math.sqrt((t_3 - -1.0)) - t_4) - ((t_13 - Math.sqrt((t_11 - -1.0))) - ((Math.sqrt((t_7 - -1.0)) - Math.sqrt(t_7)) - ((t_8 - t_10) / (Math.sqrt(t_10) + t_9))));
	} else {
		tmp = ((0.5 * (1.0 / (t_7 * Math.sqrt((1.0 / t_7))))) + ((1.0 / t_14) + ((t_13 / t_15) + (t_12 / t_15)))) + (Math.sqrt((t_8 + 1.0)) - t_9);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(y, fmax(x, z))
	t_2 = fmax(fmin(x, z), t)
	t_3 = fmin(fmin(x, z), t)
	t_4 = math.sqrt(t_3)
	t_5 = fmax(y, fmax(x, z))
	t_6 = fmax(t_1, t_2)
	t_7 = fmin(t_5, t_6)
	t_8 = fmax(t_5, t_6)
	t_9 = math.sqrt(t_8)
	t_10 = t_8 - -1.0
	t_11 = fmin(t_1, t_2)
	t_12 = math.sqrt((1.0 + t_11))
	t_13 = math.sqrt(t_11)
	t_14 = t_13 + t_12
	t_15 = (t_4 + math.sqrt((1.0 + t_3))) * t_14
	tmp = 0
	if t_7 <= 220000000.0:
		tmp = (math.sqrt((t_3 - -1.0)) - t_4) - ((t_13 - math.sqrt((t_11 - -1.0))) - ((math.sqrt((t_7 - -1.0)) - math.sqrt(t_7)) - ((t_8 - t_10) / (math.sqrt(t_10) + t_9))))
	else:
		tmp = ((0.5 * (1.0 / (t_7 * math.sqrt((1.0 / t_7))))) + ((1.0 / t_14) + ((t_13 / t_15) + (t_12 / t_15)))) + (math.sqrt((t_8 + 1.0)) - t_9)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(y, fmax(x, z))
	t_2 = fmax(fmin(x, z), t)
	t_3 = fmin(fmin(x, z), t)
	t_4 = sqrt(t_3)
	t_5 = fmax(y, fmax(x, z))
	t_6 = fmax(t_1, t_2)
	t_7 = fmin(t_5, t_6)
	t_8 = fmax(t_5, t_6)
	t_9 = sqrt(t_8)
	t_10 = Float64(t_8 - -1.0)
	t_11 = fmin(t_1, t_2)
	t_12 = sqrt(Float64(1.0 + t_11))
	t_13 = sqrt(t_11)
	t_14 = Float64(t_13 + t_12)
	t_15 = Float64(Float64(t_4 + sqrt(Float64(1.0 + t_3))) * t_14)
	tmp = 0.0
	if (t_7 <= 220000000.0)
		tmp = Float64(Float64(sqrt(Float64(t_3 - -1.0)) - t_4) - Float64(Float64(t_13 - sqrt(Float64(t_11 - -1.0))) - Float64(Float64(sqrt(Float64(t_7 - -1.0)) - sqrt(t_7)) - Float64(Float64(t_8 - t_10) / Float64(sqrt(t_10) + t_9)))));
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(1.0 / Float64(t_7 * sqrt(Float64(1.0 / t_7))))) + Float64(Float64(1.0 / t_14) + Float64(Float64(t_13 / t_15) + Float64(t_12 / t_15)))) + Float64(sqrt(Float64(t_8 + 1.0)) - t_9));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = min(y, max(x, z));
	t_2 = max(min(x, z), t);
	t_3 = min(min(x, z), t);
	t_4 = sqrt(t_3);
	t_5 = max(y, max(x, z));
	t_6 = max(t_1, t_2);
	t_7 = min(t_5, t_6);
	t_8 = max(t_5, t_6);
	t_9 = sqrt(t_8);
	t_10 = t_8 - -1.0;
	t_11 = min(t_1, t_2);
	t_12 = sqrt((1.0 + t_11));
	t_13 = sqrt(t_11);
	t_14 = t_13 + t_12;
	t_15 = (t_4 + sqrt((1.0 + t_3))) * t_14;
	tmp = 0.0;
	if (t_7 <= 220000000.0)
		tmp = (sqrt((t_3 - -1.0)) - t_4) - ((t_13 - sqrt((t_11 - -1.0))) - ((sqrt((t_7 - -1.0)) - sqrt(t_7)) - ((t_8 - t_10) / (sqrt(t_10) + t_9))));
	else
		tmp = ((0.5 * (1.0 / (t_7 * sqrt((1.0 / t_7))))) + ((1.0 / t_14) + ((t_13 / t_15) + (t_12 / t_15)))) + (sqrt((t_8 + 1.0)) - t_9);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Min[y, N[Max[x, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Min[x, z], $MachinePrecision], t], $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[y, N[Max[x, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[Max[t$95$1, t$95$2], $MachinePrecision]}, Block[{t$95$7 = N[Min[t$95$5, t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$5, t$95$6], $MachinePrecision]}, Block[{t$95$9 = N[Sqrt[t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[(t$95$8 - -1), $MachinePrecision]}, Block[{t$95$11 = N[Min[t$95$1, t$95$2], $MachinePrecision]}, Block[{t$95$12 = N[Sqrt[N[(1 + t$95$11), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$13 = N[Sqrt[t$95$11], $MachinePrecision]}, Block[{t$95$14 = N[(t$95$13 + t$95$12), $MachinePrecision]}, Block[{t$95$15 = N[(N[(t$95$4 + N[Sqrt[N[(1 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$14), $MachinePrecision]}, If[LessEqual[t$95$7, 220000000], N[(N[(N[Sqrt[N[(t$95$3 - -1), $MachinePrecision]], $MachinePrecision] - t$95$4), $MachinePrecision] - N[(N[(t$95$13 - N[Sqrt[N[(t$95$11 - -1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[N[(t$95$7 - -1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$7], $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$8 - t$95$10), $MachinePrecision] / N[(N[Sqrt[t$95$10], $MachinePrecision] + t$95$9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1/2 * N[(1 / N[(t$95$7 * N[Sqrt[N[(1 / t$95$7), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1 / t$95$14), $MachinePrecision] + N[(N[(t$95$13 / t$95$15), $MachinePrecision] + N[(t$95$12 / t$95$15), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$8 + 1), $MachinePrecision]], $MachinePrecision] - t$95$9), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if z < 2.2e8

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

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

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

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

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

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

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

      15. lower-unsound-+.f64 72.9%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 72.9%

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

   3. Applied rewrites 72.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. associate-+l- N/A

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

   5. Applied rewrites 91.9%

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

   if 2.2e8 < z

   1. Initial program 91.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 72.6%

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

      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 72.6%

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

   3. Applied rewrites 72.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

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

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

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

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

      6. lift-+.f64 N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

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

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

      15. lower-unsound-+.f64 72.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 72.7%

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

   5. Applied rewrites 72.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 92.4%

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

   8. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

   10. Applied rewrites 51.2%

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

Alternative 2: 98.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \sqrt{t\_2}\\ t_4 := \sqrt{t\_2 + 1} - t\_3\\ t_5 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_6 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_7 := \mathsf{min}\left(t\_6, t\right)\\ t_8 := \sqrt{t\_7}\\ t_9 := t\_7 - -1\\ t_10 := \mathsf{max}\left(t\_6, t\right)\\ t_11 := \mathsf{max}\left(t\_5, t\_10\right)\\ t_12 := \sqrt{t\_11}\\ t_13 := \sqrt{t\_11 + 1} - t\_12\\ t_14 := \mathsf{min}\left(t\_5, t\_10\right)\\ t_15 := \sqrt{t\_14}\\ \mathbf{if}\;\left(\left(\left(\sqrt{t\_7 + 1} - t\_8\right) + \left(\sqrt{t\_14 + 1} - t\_15\right)\right) + t\_4\right) + t\_13 \leq 0:\\ \;\;\;\;\left(\left(\frac{\frac{1}{2}}{t\_7 \cdot \sqrt{\frac{1}{t\_7}}} + \frac{\frac{1}{2}}{t\_14 \cdot \sqrt{\frac{1}{t\_14}}}\right) + t\_4\right) + t\_13\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t\_9 - t\_7}{\sqrt{t\_9} + t\_8} - \frac{-1}{\sqrt{t\_14 - -1} + t\_15}\right) - \left(\left(t\_3 - \sqrt{t\_2 - -1}\right) - \left(\sqrt{t\_11 - -1} - t\_12\right)\right)\\ \end{array} \]

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 (sqrt t_2))
       (t_4 (- (sqrt (+ t_2 1)) t_3))
       (t_5 (fmin (fmax x y) t_1))
       (t_6 (fmin (fmin x y) z))
       (t_7 (fmin t_6 t))
       (t_8 (sqrt t_7))
       (t_9 (- t_7 -1))
       (t_10 (fmax t_6 t))
       (t_11 (fmax t_5 t_10))
       (t_12 (sqrt t_11))
       (t_13 (- (sqrt (+ t_11 1)) t_12))
       (t_14 (fmin t_5 t_10))
       (t_15 (sqrt t_14)))
  (if (<=
       (+
        (+
         (+ (- (sqrt (+ t_7 1)) t_8) (- (sqrt (+ t_14 1)) t_15))
         t_4)
        t_13)
       0)
    (+
     (+
      (+
       (/ 1/2 (* t_7 (sqrt (/ 1 t_7))))
       (/ 1/2 (* t_14 (sqrt (/ 1 t_14)))))
      t_4)
     t_13)
    (-
     (-
      (/ (- t_9 t_7) (+ (sqrt t_9) t_8))
      (/ -1 (+ (sqrt (- t_14 -1)) t_15)))
     (- (- t_3 (sqrt (- t_2 -1))) (- (sqrt (- t_11 -1)) t_12))))))

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 = sqrt(t_2);
	double t_4 = sqrt((t_2 + 1.0)) - t_3;
	double t_5 = fmin(fmax(x, y), t_1);
	double t_6 = fmin(fmin(x, y), z);
	double t_7 = fmin(t_6, t);
	double t_8 = sqrt(t_7);
	double t_9 = t_7 - -1.0;
	double t_10 = fmax(t_6, t);
	double t_11 = fmax(t_5, t_10);
	double t_12 = sqrt(t_11);
	double t_13 = sqrt((t_11 + 1.0)) - t_12;
	double t_14 = fmin(t_5, t_10);
	double t_15 = sqrt(t_14);
	double tmp;
	if (((((sqrt((t_7 + 1.0)) - t_8) + (sqrt((t_14 + 1.0)) - t_15)) + t_4) + t_13) <= 0.0) {
		tmp = (((0.5 / (t_7 * sqrt((1.0 / t_7)))) + (0.5 / (t_14 * sqrt((1.0 / t_14))))) + t_4) + t_13;
	} else {
		tmp = (((t_9 - t_7) / (sqrt(t_9) + t_8)) - (-1.0 / (sqrt((t_14 - -1.0)) + t_15))) - ((t_3 - sqrt((t_2 - -1.0))) - (sqrt((t_11 - -1.0)) - t_12));
	}
	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_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 = sqrt(t_2)
    t_4 = sqrt((t_2 + 1.0d0)) - t_3
    t_5 = fmin(fmax(x, y), t_1)
    t_6 = fmin(fmin(x, y), z)
    t_7 = fmin(t_6, t)
    t_8 = sqrt(t_7)
    t_9 = t_7 - (-1.0d0)
    t_10 = fmax(t_6, t)
    t_11 = fmax(t_5, t_10)
    t_12 = sqrt(t_11)
    t_13 = sqrt((t_11 + 1.0d0)) - t_12
    t_14 = fmin(t_5, t_10)
    t_15 = sqrt(t_14)
    if (((((sqrt((t_7 + 1.0d0)) - t_8) + (sqrt((t_14 + 1.0d0)) - t_15)) + t_4) + t_13) <= 0.0d0) then
        tmp = (((0.5d0 / (t_7 * sqrt((1.0d0 / t_7)))) + (0.5d0 / (t_14 * sqrt((1.0d0 / t_14))))) + t_4) + t_13
    else
        tmp = (((t_9 - t_7) / (sqrt(t_9) + t_8)) - ((-1.0d0) / (sqrt((t_14 - (-1.0d0))) + t_15))) - ((t_3 - sqrt((t_2 - (-1.0d0)))) - (sqrt((t_11 - (-1.0d0))) - t_12))
    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 = Math.sqrt(t_2);
	double t_4 = Math.sqrt((t_2 + 1.0)) - t_3;
	double t_5 = fmin(fmax(x, y), t_1);
	double t_6 = fmin(fmin(x, y), z);
	double t_7 = fmin(t_6, t);
	double t_8 = Math.sqrt(t_7);
	double t_9 = t_7 - -1.0;
	double t_10 = fmax(t_6, t);
	double t_11 = fmax(t_5, t_10);
	double t_12 = Math.sqrt(t_11);
	double t_13 = Math.sqrt((t_11 + 1.0)) - t_12;
	double t_14 = fmin(t_5, t_10);
	double t_15 = Math.sqrt(t_14);
	double tmp;
	if (((((Math.sqrt((t_7 + 1.0)) - t_8) + (Math.sqrt((t_14 + 1.0)) - t_15)) + t_4) + t_13) <= 0.0) {
		tmp = (((0.5 / (t_7 * Math.sqrt((1.0 / t_7)))) + (0.5 / (t_14 * Math.sqrt((1.0 / t_14))))) + t_4) + t_13;
	} else {
		tmp = (((t_9 - t_7) / (Math.sqrt(t_9) + t_8)) - (-1.0 / (Math.sqrt((t_14 - -1.0)) + t_15))) - ((t_3 - Math.sqrt((t_2 - -1.0))) - (Math.sqrt((t_11 - -1.0)) - t_12));
	}
	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 = math.sqrt(t_2)
	t_4 = math.sqrt((t_2 + 1.0)) - t_3
	t_5 = fmin(fmax(x, y), t_1)
	t_6 = fmin(fmin(x, y), z)
	t_7 = fmin(t_6, t)
	t_8 = math.sqrt(t_7)
	t_9 = t_7 - -1.0
	t_10 = fmax(t_6, t)
	t_11 = fmax(t_5, t_10)
	t_12 = math.sqrt(t_11)
	t_13 = math.sqrt((t_11 + 1.0)) - t_12
	t_14 = fmin(t_5, t_10)
	t_15 = math.sqrt(t_14)
	tmp = 0
	if ((((math.sqrt((t_7 + 1.0)) - t_8) + (math.sqrt((t_14 + 1.0)) - t_15)) + t_4) + t_13) <= 0.0:
		tmp = (((0.5 / (t_7 * math.sqrt((1.0 / t_7)))) + (0.5 / (t_14 * math.sqrt((1.0 / t_14))))) + t_4) + t_13
	else:
		tmp = (((t_9 - t_7) / (math.sqrt(t_9) + t_8)) - (-1.0 / (math.sqrt((t_14 - -1.0)) + t_15))) - ((t_3 - math.sqrt((t_2 - -1.0))) - (math.sqrt((t_11 - -1.0)) - t_12))
	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 = sqrt(t_2)
	t_4 = Float64(sqrt(Float64(t_2 + 1.0)) - t_3)
	t_5 = fmin(fmax(x, y), t_1)
	t_6 = fmin(fmin(x, y), z)
	t_7 = fmin(t_6, t)
	t_8 = sqrt(t_7)
	t_9 = Float64(t_7 - -1.0)
	t_10 = fmax(t_6, t)
	t_11 = fmax(t_5, t_10)
	t_12 = sqrt(t_11)
	t_13 = Float64(sqrt(Float64(t_11 + 1.0)) - t_12)
	t_14 = fmin(t_5, t_10)
	t_15 = sqrt(t_14)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(sqrt(Float64(t_7 + 1.0)) - t_8) + Float64(sqrt(Float64(t_14 + 1.0)) - t_15)) + t_4) + t_13) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(0.5 / Float64(t_7 * sqrt(Float64(1.0 / t_7)))) + Float64(0.5 / Float64(t_14 * sqrt(Float64(1.0 / t_14))))) + t_4) + t_13);
	else
		tmp = Float64(Float64(Float64(Float64(t_9 - t_7) / Float64(sqrt(t_9) + t_8)) - Float64(-1.0 / Float64(sqrt(Float64(t_14 - -1.0)) + t_15))) - Float64(Float64(t_3 - sqrt(Float64(t_2 - -1.0))) - Float64(sqrt(Float64(t_11 - -1.0)) - t_12)));
	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 = sqrt(t_2);
	t_4 = sqrt((t_2 + 1.0)) - t_3;
	t_5 = min(max(x, y), t_1);
	t_6 = min(min(x, y), z);
	t_7 = min(t_6, t);
	t_8 = sqrt(t_7);
	t_9 = t_7 - -1.0;
	t_10 = max(t_6, t);
	t_11 = max(t_5, t_10);
	t_12 = sqrt(t_11);
	t_13 = sqrt((t_11 + 1.0)) - t_12;
	t_14 = min(t_5, t_10);
	t_15 = sqrt(t_14);
	tmp = 0.0;
	if (((((sqrt((t_7 + 1.0)) - t_8) + (sqrt((t_14 + 1.0)) - t_15)) + t_4) + t_13) <= 0.0)
		tmp = (((0.5 / (t_7 * sqrt((1.0 / t_7)))) + (0.5 / (t_14 * sqrt((1.0 / t_14))))) + t_4) + t_13;
	else
		tmp = (((t_9 - t_7) / (sqrt(t_9) + t_8)) - (-1.0 / (sqrt((t_14 - -1.0)) + t_15))) - ((t_3 - sqrt((t_2 - -1.0))) - (sqrt((t_11 - -1.0)) - t_12));
	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[Sqrt[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t$95$2 + 1), $MachinePrecision]], $MachinePrecision] - t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$6 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$7 = N[Min[t$95$6, t], $MachinePrecision]}, Block[{t$95$8 = N[Sqrt[t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[(t$95$7 - -1), $MachinePrecision]}, Block[{t$95$10 = N[Max[t$95$6, t], $MachinePrecision]}, Block[{t$95$11 = N[Max[t$95$5, t$95$10], $MachinePrecision]}, Block[{t$95$12 = N[Sqrt[t$95$11], $MachinePrecision]}, Block[{t$95$13 = N[(N[Sqrt[N[(t$95$11 + 1), $MachinePrecision]], $MachinePrecision] - t$95$12), $MachinePrecision]}, Block[{t$95$14 = N[Min[t$95$5, t$95$10], $MachinePrecision]}, Block[{t$95$15 = N[Sqrt[t$95$14], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Sqrt[N[(t$95$7 + 1), $MachinePrecision]], $MachinePrecision] - t$95$8), $MachinePrecision] + N[(N[Sqrt[N[(t$95$14 + 1), $MachinePrecision]], $MachinePrecision] - t$95$15), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$13), $MachinePrecision], 0], N[(N[(N[(N[(1/2 / N[(t$95$7 * N[Sqrt[N[(1 / t$95$7), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1/2 / N[(t$95$14 * N[Sqrt[N[(1 / t$95$14), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$13), $MachinePrecision], N[(N[(N[(N[(t$95$9 - t$95$7), $MachinePrecision] / N[(N[Sqrt[t$95$9], $MachinePrecision] + t$95$8), $MachinePrecision]), $MachinePrecision] - N[(-1 / N[(N[Sqrt[N[(t$95$14 - -1), $MachinePrecision]], $MachinePrecision] + t$95$15), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$3 - N[Sqrt[N[(t$95$2 - -1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[(t$95$11 - -1), $MachinePrecision]], $MachinePrecision] - t$95$12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 0.0

   1. Initial program 91.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 47.9%

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 26.5%

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

   7. Applied rewrites 26.5%

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

   if 0.0 < (+.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 91.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 72.6%

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

      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 72.6%

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

   3. Applied rewrites 72.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

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

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

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

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

      6. lift-+.f64 N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

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

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

      15. lower-unsound-+.f64 72.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 72.7%

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

   5. Applied rewrites 72.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 92.4%

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

   9. Applied rewrites 93.9%

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

Alternative 3: 97.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_3 := \mathsf{min}\left(t\_2, t\right)\\ t_4 := \sqrt{t\_3}\\ t_5 := \mathsf{max}\left(t\_2, t\right)\\ t_6 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_5\right)\\ t_7 := \sqrt{t\_6 + 1} - \sqrt{t\_6}\\ t_8 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_5\right)\\ t_9 := \sqrt{t\_8}\\ t_10 := \sqrt{t\_1 + 1} - \sqrt{t\_1}\\ \mathbf{if}\;t\_8 \leq 32500000:\\ \;\;\;\;\left(\left(\left(1 + \sqrt{1 + t\_8}\right) - \left(t\_4 + t\_9\right)\right) + t\_10\right) + t\_7\\ \mathbf{elif}\;t\_8 \leq 210000000000000015434770284544:\\ \;\;\;\;\left(\left(\left(\sqrt{t\_3 + 1} - t\_4\right) + \frac{\frac{1}{2}}{t\_9}\right) + t\_10\right) + t\_7\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t\_4 + \sqrt{1 + t\_3}} + t\_10\right) + t\_7\\ \end{array} \]

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 (sqrt t_3))
       (t_5 (fmax t_2 t))
       (t_6 (fmax (fmax x y) t_5))
       (t_7 (- (sqrt (+ t_6 1)) (sqrt t_6)))
       (t_8 (fmin (fmax x y) t_5))
       (t_9 (sqrt t_8))
       (t_10 (- (sqrt (+ t_1 1)) (sqrt t_1))))
  (if (<= t_8 32500000)
    (+ (+ (- (+ 1 (sqrt (+ 1 t_8))) (+ t_4 t_9)) t_10) t_7)
    (if (<= t_8 210000000000000015434770284544)
      (+ (+ (+ (- (sqrt (+ t_3 1)) t_4) (/ 1/2 t_9)) t_10) t_7)
      (+ (+ (/ 1 (+ t_4 (sqrt (+ 1 t_3)))) t_10) 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 = sqrt(t_3);
	double t_5 = fmax(t_2, t);
	double t_6 = fmax(fmax(x, y), t_5);
	double t_7 = sqrt((t_6 + 1.0)) - sqrt(t_6);
	double t_8 = fmin(fmax(x, y), t_5);
	double t_9 = sqrt(t_8);
	double t_10 = sqrt((t_1 + 1.0)) - sqrt(t_1);
	double tmp;
	if (t_8 <= 32500000.0) {
		tmp = (((1.0 + sqrt((1.0 + t_8))) - (t_4 + t_9)) + t_10) + t_7;
	} else if (t_8 <= 2.1e+29) {
		tmp = (((sqrt((t_3 + 1.0)) - t_4) + (0.5 / t_9)) + t_10) + t_7;
	} else {
		tmp = ((1.0 / (t_4 + sqrt((1.0 + t_3)))) + t_10) + 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_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 = sqrt(t_3)
    t_5 = fmax(t_2, t)
    t_6 = fmax(fmax(x, y), t_5)
    t_7 = sqrt((t_6 + 1.0d0)) - sqrt(t_6)
    t_8 = fmin(fmax(x, y), t_5)
    t_9 = sqrt(t_8)
    t_10 = sqrt((t_1 + 1.0d0)) - sqrt(t_1)
    if (t_8 <= 32500000.0d0) then
        tmp = (((1.0d0 + sqrt((1.0d0 + t_8))) - (t_4 + t_9)) + t_10) + t_7
    else if (t_8 <= 2.1d+29) then
        tmp = (((sqrt((t_3 + 1.0d0)) - t_4) + (0.5d0 / t_9)) + t_10) + t_7
    else
        tmp = ((1.0d0 / (t_4 + sqrt((1.0d0 + t_3)))) + t_10) + 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 = Math.sqrt(t_3);
	double t_5 = fmax(t_2, t);
	double t_6 = fmax(fmax(x, y), t_5);
	double t_7 = Math.sqrt((t_6 + 1.0)) - Math.sqrt(t_6);
	double t_8 = fmin(fmax(x, y), t_5);
	double t_9 = Math.sqrt(t_8);
	double t_10 = Math.sqrt((t_1 + 1.0)) - Math.sqrt(t_1);
	double tmp;
	if (t_8 <= 32500000.0) {
		tmp = (((1.0 + Math.sqrt((1.0 + t_8))) - (t_4 + t_9)) + t_10) + t_7;
	} else if (t_8 <= 2.1e+29) {
		tmp = (((Math.sqrt((t_3 + 1.0)) - t_4) + (0.5 / t_9)) + t_10) + t_7;
	} else {
		tmp = ((1.0 / (t_4 + Math.sqrt((1.0 + t_3)))) + t_10) + 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 = math.sqrt(t_3)
	t_5 = fmax(t_2, t)
	t_6 = fmax(fmax(x, y), t_5)
	t_7 = math.sqrt((t_6 + 1.0)) - math.sqrt(t_6)
	t_8 = fmin(fmax(x, y), t_5)
	t_9 = math.sqrt(t_8)
	t_10 = math.sqrt((t_1 + 1.0)) - math.sqrt(t_1)
	tmp = 0
	if t_8 <= 32500000.0:
		tmp = (((1.0 + math.sqrt((1.0 + t_8))) - (t_4 + t_9)) + t_10) + t_7
	elif t_8 <= 2.1e+29:
		tmp = (((math.sqrt((t_3 + 1.0)) - t_4) + (0.5 / t_9)) + t_10) + t_7
	else:
		tmp = ((1.0 / (t_4 + math.sqrt((1.0 + t_3)))) + t_10) + 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 = sqrt(t_3)
	t_5 = fmax(t_2, t)
	t_6 = fmax(fmax(x, y), t_5)
	t_7 = Float64(sqrt(Float64(t_6 + 1.0)) - sqrt(t_6))
	t_8 = fmin(fmax(x, y), t_5)
	t_9 = sqrt(t_8)
	t_10 = Float64(sqrt(Float64(t_1 + 1.0)) - sqrt(t_1))
	tmp = 0.0
	if (t_8 <= 32500000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + sqrt(Float64(1.0 + t_8))) - Float64(t_4 + t_9)) + t_10) + t_7);
	elseif (t_8 <= 2.1e+29)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(t_3 + 1.0)) - t_4) + Float64(0.5 / t_9)) + t_10) + t_7);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_4 + sqrt(Float64(1.0 + t_3)))) + t_10) + 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 = sqrt(t_3);
	t_5 = max(t_2, t);
	t_6 = max(max(x, y), t_5);
	t_7 = sqrt((t_6 + 1.0)) - sqrt(t_6);
	t_8 = min(max(x, y), t_5);
	t_9 = sqrt(t_8);
	t_10 = sqrt((t_1 + 1.0)) - sqrt(t_1);
	tmp = 0.0;
	if (t_8 <= 32500000.0)
		tmp = (((1.0 + sqrt((1.0 + t_8))) - (t_4 + t_9)) + t_10) + t_7;
	elseif (t_8 <= 2.1e+29)
		tmp = (((sqrt((t_3 + 1.0)) - t_4) + (0.5 / t_9)) + t_10) + t_7;
	else
		tmp = ((1.0 / (t_4 + sqrt((1.0 + t_3)))) + t_10) + 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[Sqrt[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[Max[t$95$2, t], $MachinePrecision]}, Block[{t$95$6 = N[Max[N[Max[x, y], $MachinePrecision], t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[(N[Sqrt[N[(t$95$6 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$6], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[Min[N[Max[x, y], $MachinePrecision], t$95$5], $MachinePrecision]}, Block[{t$95$9 = N[Sqrt[t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[(N[Sqrt[N[(t$95$1 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$8, 32500000], N[(N[(N[(N[(1 + N[Sqrt[N[(1 + t$95$8), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$4 + t$95$9), $MachinePrecision]), $MachinePrecision] + t$95$10), $MachinePrecision] + t$95$7), $MachinePrecision], If[LessEqual[t$95$8, 210000000000000015434770284544], N[(N[(N[(N[(N[Sqrt[N[(t$95$3 + 1), $MachinePrecision]], $MachinePrecision] - t$95$4), $MachinePrecision] + N[(1/2 / t$95$9), $MachinePrecision]), $MachinePrecision] + t$95$10), $MachinePrecision] + t$95$7), $MachinePrecision], N[(N[(N[(1 / N[(t$95$4 + N[Sqrt[N[(1 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$10), $MachinePrecision] + t$95$7), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.25e7

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 36.6%

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

   4. Applied rewrites 36.6%

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

   if 3.25e7 < y < 2.1000000000000002e29

   1. Initial program 91.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 47.9%

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 47.9%

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

   7. Applied rewrites 47.9%

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

   if 2.1000000000000002e29 < y

   1. Initial program 91.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 72.6%

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

      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 72.6%

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

   3. Applied rewrites 72.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

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

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

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

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

      6. lift-+.f64 N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

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

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

      15. lower-unsound-+.f64 72.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 72.7%

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

   5. Applied rewrites 72.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 92.4%

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

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. 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%

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

   10. 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 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_4 := \mathsf{min}\left(t\_3, t\right)\\ t_5 := \mathsf{max}\left(t\_3, t\right)\\ t_6 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_7 := \mathsf{max}\left(t\_6, t\_5\right)\\ t_8 := \mathsf{min}\left(t\_2, t\_7\right)\\ t_9 := \sqrt{t\_8}\\ t_10 := \mathsf{min}\left(t\_6, t\_5\right)\\ t_11 := \mathsf{max}\left(t\_2, t\_7\right)\\ t_12 := t\_11 - -1\\ t_13 := \sqrt{t\_11}\\ \mathbf{if}\;t\_4 \leq 100000000:\\ \;\;\;\;\left(\sqrt{t\_4 - -1} - \sqrt{t\_4}\right) - \left(\left(\sqrt{t\_10} - \sqrt{t\_10 - -1}\right) - \left(\left(\sqrt{t\_8 - -1} - t\_9\right) - \frac{t\_11 - t\_12}{\sqrt{t\_12} + t\_13}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{1}{2}}{t\_4 \cdot \sqrt{\frac{1}{t\_4}}} + \frac{\frac{1}{2}}{t\_10 \cdot \sqrt{\frac{1}{t\_10}}}\right) + \left(\sqrt{t\_8 + 1} - t\_9\right)\right) + \left(\sqrt{t\_11 + 1} - t\_13\right)\\ \end{array} \]

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 (fmin x y) z))
       (t_4 (fmin t_3 t))
       (t_5 (fmax t_3 t))
       (t_6 (fmin (fmax x y) t_1))
       (t_7 (fmax t_6 t_5))
       (t_8 (fmin t_2 t_7))
       (t_9 (sqrt t_8))
       (t_10 (fmin t_6 t_5))
       (t_11 (fmax t_2 t_7))
       (t_12 (- t_11 -1))
       (t_13 (sqrt t_11)))
  (if (<= t_4 100000000)
    (-
     (- (sqrt (- t_4 -1)) (sqrt t_4))
     (-
      (- (sqrt t_10) (sqrt (- t_10 -1)))
      (-
       (- (sqrt (- t_8 -1)) t_9)
       (/ (- t_11 t_12) (+ (sqrt t_12) t_13)))))
    (+
     (+
      (+
       (/ 1/2 (* t_4 (sqrt (/ 1 t_4))))
       (/ 1/2 (* t_10 (sqrt (/ 1 t_10)))))
      (- (sqrt (+ t_8 1)) t_9))
     (- (sqrt (+ t_11 1)) 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(fmin(x, y), z);
	double t_4 = fmin(t_3, t);
	double t_5 = fmax(t_3, t);
	double t_6 = fmin(fmax(x, y), t_1);
	double t_7 = fmax(t_6, t_5);
	double t_8 = fmin(t_2, t_7);
	double t_9 = sqrt(t_8);
	double t_10 = fmin(t_6, t_5);
	double t_11 = fmax(t_2, t_7);
	double t_12 = t_11 - -1.0;
	double t_13 = sqrt(t_11);
	double tmp;
	if (t_4 <= 100000000.0) {
		tmp = (sqrt((t_4 - -1.0)) - sqrt(t_4)) - ((sqrt(t_10) - sqrt((t_10 - -1.0))) - ((sqrt((t_8 - -1.0)) - t_9) - ((t_11 - t_12) / (sqrt(t_12) + t_13))));
	} else {
		tmp = (((0.5 / (t_4 * sqrt((1.0 / t_4)))) + (0.5 / (t_10 * sqrt((1.0 / t_10))))) + (sqrt((t_8 + 1.0)) - t_9)) + (sqrt((t_11 + 1.0)) - 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 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmin(x, y), z)
    t_4 = fmin(t_3, t)
    t_5 = fmax(t_3, t)
    t_6 = fmin(fmax(x, y), t_1)
    t_7 = fmax(t_6, t_5)
    t_8 = fmin(t_2, t_7)
    t_9 = sqrt(t_8)
    t_10 = fmin(t_6, t_5)
    t_11 = fmax(t_2, t_7)
    t_12 = t_11 - (-1.0d0)
    t_13 = sqrt(t_11)
    if (t_4 <= 100000000.0d0) then
        tmp = (sqrt((t_4 - (-1.0d0))) - sqrt(t_4)) - ((sqrt(t_10) - sqrt((t_10 - (-1.0d0)))) - ((sqrt((t_8 - (-1.0d0))) - t_9) - ((t_11 - t_12) / (sqrt(t_12) + t_13))))
    else
        tmp = (((0.5d0 / (t_4 * sqrt((1.0d0 / t_4)))) + (0.5d0 / (t_10 * sqrt((1.0d0 / t_10))))) + (sqrt((t_8 + 1.0d0)) - t_9)) + (sqrt((t_11 + 1.0d0)) - 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(fmin(x, y), z);
	double t_4 = fmin(t_3, t);
	double t_5 = fmax(t_3, t);
	double t_6 = fmin(fmax(x, y), t_1);
	double t_7 = fmax(t_6, t_5);
	double t_8 = fmin(t_2, t_7);
	double t_9 = Math.sqrt(t_8);
	double t_10 = fmin(t_6, t_5);
	double t_11 = fmax(t_2, t_7);
	double t_12 = t_11 - -1.0;
	double t_13 = Math.sqrt(t_11);
	double tmp;
	if (t_4 <= 100000000.0) {
		tmp = (Math.sqrt((t_4 - -1.0)) - Math.sqrt(t_4)) - ((Math.sqrt(t_10) - Math.sqrt((t_10 - -1.0))) - ((Math.sqrt((t_8 - -1.0)) - t_9) - ((t_11 - t_12) / (Math.sqrt(t_12) + t_13))));
	} else {
		tmp = (((0.5 / (t_4 * Math.sqrt((1.0 / t_4)))) + (0.5 / (t_10 * Math.sqrt((1.0 / t_10))))) + (Math.sqrt((t_8 + 1.0)) - t_9)) + (Math.sqrt((t_11 + 1.0)) - 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(fmin(x, y), z)
	t_4 = fmin(t_3, t)
	t_5 = fmax(t_3, t)
	t_6 = fmin(fmax(x, y), t_1)
	t_7 = fmax(t_6, t_5)
	t_8 = fmin(t_2, t_7)
	t_9 = math.sqrt(t_8)
	t_10 = fmin(t_6, t_5)
	t_11 = fmax(t_2, t_7)
	t_12 = t_11 - -1.0
	t_13 = math.sqrt(t_11)
	tmp = 0
	if t_4 <= 100000000.0:
		tmp = (math.sqrt((t_4 - -1.0)) - math.sqrt(t_4)) - ((math.sqrt(t_10) - math.sqrt((t_10 - -1.0))) - ((math.sqrt((t_8 - -1.0)) - t_9) - ((t_11 - t_12) / (math.sqrt(t_12) + t_13))))
	else:
		tmp = (((0.5 / (t_4 * math.sqrt((1.0 / t_4)))) + (0.5 / (t_10 * math.sqrt((1.0 / t_10))))) + (math.sqrt((t_8 + 1.0)) - t_9)) + (math.sqrt((t_11 + 1.0)) - 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(fmin(x, y), z)
	t_4 = fmin(t_3, t)
	t_5 = fmax(t_3, t)
	t_6 = fmin(fmax(x, y), t_1)
	t_7 = fmax(t_6, t_5)
	t_8 = fmin(t_2, t_7)
	t_9 = sqrt(t_8)
	t_10 = fmin(t_6, t_5)
	t_11 = fmax(t_2, t_7)
	t_12 = Float64(t_11 - -1.0)
	t_13 = sqrt(t_11)
	tmp = 0.0
	if (t_4 <= 100000000.0)
		tmp = Float64(Float64(sqrt(Float64(t_4 - -1.0)) - sqrt(t_4)) - Float64(Float64(sqrt(t_10) - sqrt(Float64(t_10 - -1.0))) - Float64(Float64(sqrt(Float64(t_8 - -1.0)) - t_9) - Float64(Float64(t_11 - t_12) / Float64(sqrt(t_12) + t_13)))));
	else
		tmp = Float64(Float64(Float64(Float64(0.5 / Float64(t_4 * sqrt(Float64(1.0 / t_4)))) + Float64(0.5 / Float64(t_10 * sqrt(Float64(1.0 / t_10))))) + Float64(sqrt(Float64(t_8 + 1.0)) - t_9)) + Float64(sqrt(Float64(t_11 + 1.0)) - 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(min(x, y), z);
	t_4 = min(t_3, t);
	t_5 = max(t_3, t);
	t_6 = min(max(x, y), t_1);
	t_7 = max(t_6, t_5);
	t_8 = min(t_2, t_7);
	t_9 = sqrt(t_8);
	t_10 = min(t_6, t_5);
	t_11 = max(t_2, t_7);
	t_12 = t_11 - -1.0;
	t_13 = sqrt(t_11);
	tmp = 0.0;
	if (t_4 <= 100000000.0)
		tmp = (sqrt((t_4 - -1.0)) - sqrt(t_4)) - ((sqrt(t_10) - sqrt((t_10 - -1.0))) - ((sqrt((t_8 - -1.0)) - t_9) - ((t_11 - t_12) / (sqrt(t_12) + t_13))));
	else
		tmp = (((0.5 / (t_4 * sqrt((1.0 / t_4)))) + (0.5 / (t_10 * sqrt((1.0 / t_10))))) + (sqrt((t_8 + 1.0)) - t_9)) + (sqrt((t_11 + 1.0)) - 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[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$4 = N[Min[t$95$3, t], $MachinePrecision]}, Block[{t$95$5 = N[Max[t$95$3, t], $MachinePrecision]}, Block[{t$95$6 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$6, t$95$5], $MachinePrecision]}, Block[{t$95$8 = N[Min[t$95$2, t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Sqrt[t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Min[t$95$6, t$95$5], $MachinePrecision]}, Block[{t$95$11 = N[Max[t$95$2, t$95$7], $MachinePrecision]}, Block[{t$95$12 = N[(t$95$11 - -1), $MachinePrecision]}, Block[{t$95$13 = N[Sqrt[t$95$11], $MachinePrecision]}, If[LessEqual[t$95$4, 100000000], N[(N[(N[Sqrt[N[(t$95$4 - -1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[t$95$10], $MachinePrecision] - N[Sqrt[N[(t$95$10 - -1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[N[(t$95$8 - -1), $MachinePrecision]], $MachinePrecision] - t$95$9), $MachinePrecision] - N[(N[(t$95$11 - t$95$12), $MachinePrecision] / N[(N[Sqrt[t$95$12], $MachinePrecision] + t$95$13), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1/2 / N[(t$95$4 * N[Sqrt[N[(1 / t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1/2 / N[(t$95$10 * N[Sqrt[N[(1 / t$95$10), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$8 + 1), $MachinePrecision]], $MachinePrecision] - t$95$9), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$11 + 1), $MachinePrecision]], $MachinePrecision] - t$95$13), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e8

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

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

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

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

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

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

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

      15. lower-unsound-+.f64 72.9%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 72.9%

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

   3. Applied rewrites 72.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. associate-+l- N/A

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

   5. Applied rewrites 91.9%

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

   if 1e8 < x

   1. Initial program 91.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 47.9%

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 26.5%

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

   7. Applied rewrites 26.5%

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

Alternative 5: 96.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \sqrt{t\_1}\\ t_3 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_4 := \mathsf{min}\left(t\_3, t\right)\\ t_5 := \mathsf{max}\left(t\_3, t\right)\\ t_6 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_5\right)\\ t_7 := \sqrt{t\_6}\\ t_8 := t\_6 - -1\\ t_9 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_5\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \sqrt{t\_4}\\ \mathbf{if}\;\sqrt{t\_9 + 1} - t\_10 \leq \frac{3022314549036573}{151115727451828646838272}:\\ \;\;\;\;\left(\frac{1}{t\_11 + \sqrt{1 + t\_4}} + \left(\sqrt{t\_1 + 1} - t\_2\right)\right) + \left(\sqrt{t\_6 + 1} - t\_7\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t\_4 - -1} - t\_11\right) - \left(\left(t\_10 - \sqrt{t\_9 - -1}\right) - \left(\left(\sqrt{t\_1 - -1} - t\_2\right) - \frac{t\_6 - t\_8}{\sqrt{t\_8} + t\_7}\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmax (fmin x y) z))
       (t_2 (sqrt t_1))
       (t_3 (fmin (fmin x y) z))
       (t_4 (fmin t_3 t))
       (t_5 (fmax t_3 t))
       (t_6 (fmax (fmax x y) t_5))
       (t_7 (sqrt t_6))
       (t_8 (- t_6 -1))
       (t_9 (fmin (fmax x y) t_5))
       (t_10 (sqrt t_9))
       (t_11 (sqrt t_4)))
  (if (<=
       (- (sqrt (+ t_9 1)) t_10)
       3022314549036573/151115727451828646838272)
    (+
     (+ (/ 1 (+ t_11 (sqrt (+ 1 t_4)))) (- (sqrt (+ t_1 1)) t_2))
     (- (sqrt (+ t_6 1)) t_7))
    (-
     (- (sqrt (- t_4 -1)) t_11)
     (-
      (- t_10 (sqrt (- t_9 -1)))
      (-
       (- (sqrt (- t_1 -1)) t_2)
       (/ (- t_6 t_8) (+ (sqrt t_8) t_7))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = sqrt(t_1);
	double t_3 = fmin(fmin(x, y), z);
	double t_4 = fmin(t_3, t);
	double t_5 = fmax(t_3, t);
	double t_6 = fmax(fmax(x, y), t_5);
	double t_7 = sqrt(t_6);
	double t_8 = t_6 - -1.0;
	double t_9 = fmin(fmax(x, y), t_5);
	double t_10 = sqrt(t_9);
	double t_11 = sqrt(t_4);
	double tmp;
	if ((sqrt((t_9 + 1.0)) - t_10) <= 2e-8) {
		tmp = ((1.0 / (t_11 + sqrt((1.0 + t_4)))) + (sqrt((t_1 + 1.0)) - t_2)) + (sqrt((t_6 + 1.0)) - t_7);
	} else {
		tmp = (sqrt((t_4 - -1.0)) - t_11) - ((t_10 - sqrt((t_9 - -1.0))) - ((sqrt((t_1 - -1.0)) - t_2) - ((t_6 - t_8) / (sqrt(t_8) + 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 = sqrt(t_1)
    t_3 = fmin(fmin(x, y), z)
    t_4 = fmin(t_3, t)
    t_5 = fmax(t_3, t)
    t_6 = fmax(fmax(x, y), t_5)
    t_7 = sqrt(t_6)
    t_8 = t_6 - (-1.0d0)
    t_9 = fmin(fmax(x, y), t_5)
    t_10 = sqrt(t_9)
    t_11 = sqrt(t_4)
    if ((sqrt((t_9 + 1.0d0)) - t_10) <= 2d-8) then
        tmp = ((1.0d0 / (t_11 + sqrt((1.0d0 + t_4)))) + (sqrt((t_1 + 1.0d0)) - t_2)) + (sqrt((t_6 + 1.0d0)) - t_7)
    else
        tmp = (sqrt((t_4 - (-1.0d0))) - t_11) - ((t_10 - sqrt((t_9 - (-1.0d0)))) - ((sqrt((t_1 - (-1.0d0))) - t_2) - ((t_6 - t_8) / (sqrt(t_8) + 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 = Math.sqrt(t_1);
	double t_3 = fmin(fmin(x, y), z);
	double t_4 = fmin(t_3, t);
	double t_5 = fmax(t_3, t);
	double t_6 = fmax(fmax(x, y), t_5);
	double t_7 = Math.sqrt(t_6);
	double t_8 = t_6 - -1.0;
	double t_9 = fmin(fmax(x, y), t_5);
	double t_10 = Math.sqrt(t_9);
	double t_11 = Math.sqrt(t_4);
	double tmp;
	if ((Math.sqrt((t_9 + 1.0)) - t_10) <= 2e-8) {
		tmp = ((1.0 / (t_11 + Math.sqrt((1.0 + t_4)))) + (Math.sqrt((t_1 + 1.0)) - t_2)) + (Math.sqrt((t_6 + 1.0)) - t_7);
	} else {
		tmp = (Math.sqrt((t_4 - -1.0)) - t_11) - ((t_10 - Math.sqrt((t_9 - -1.0))) - ((Math.sqrt((t_1 - -1.0)) - t_2) - ((t_6 - t_8) / (Math.sqrt(t_8) + t_7))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = math.sqrt(t_1)
	t_3 = fmin(fmin(x, y), z)
	t_4 = fmin(t_3, t)
	t_5 = fmax(t_3, t)
	t_6 = fmax(fmax(x, y), t_5)
	t_7 = math.sqrt(t_6)
	t_8 = t_6 - -1.0
	t_9 = fmin(fmax(x, y), t_5)
	t_10 = math.sqrt(t_9)
	t_11 = math.sqrt(t_4)
	tmp = 0
	if (math.sqrt((t_9 + 1.0)) - t_10) <= 2e-8:
		tmp = ((1.0 / (t_11 + math.sqrt((1.0 + t_4)))) + (math.sqrt((t_1 + 1.0)) - t_2)) + (math.sqrt((t_6 + 1.0)) - t_7)
	else:
		tmp = (math.sqrt((t_4 - -1.0)) - t_11) - ((t_10 - math.sqrt((t_9 - -1.0))) - ((math.sqrt((t_1 - -1.0)) - t_2) - ((t_6 - t_8) / (math.sqrt(t_8) + t_7))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = sqrt(t_1)
	t_3 = fmin(fmin(x, y), z)
	t_4 = fmin(t_3, t)
	t_5 = fmax(t_3, t)
	t_6 = fmax(fmax(x, y), t_5)
	t_7 = sqrt(t_6)
	t_8 = Float64(t_6 - -1.0)
	t_9 = fmin(fmax(x, y), t_5)
	t_10 = sqrt(t_9)
	t_11 = sqrt(t_4)
	tmp = 0.0
	if (Float64(sqrt(Float64(t_9 + 1.0)) - t_10) <= 2e-8)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_11 + sqrt(Float64(1.0 + t_4)))) + Float64(sqrt(Float64(t_1 + 1.0)) - t_2)) + Float64(sqrt(Float64(t_6 + 1.0)) - t_7));
	else
		tmp = Float64(Float64(sqrt(Float64(t_4 - -1.0)) - t_11) - Float64(Float64(t_10 - sqrt(Float64(t_9 - -1.0))) - Float64(Float64(sqrt(Float64(t_1 - -1.0)) - t_2) - Float64(Float64(t_6 - t_8) / Float64(sqrt(t_8) + t_7)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = sqrt(t_1);
	t_3 = min(min(x, y), z);
	t_4 = min(t_3, t);
	t_5 = max(t_3, t);
	t_6 = max(max(x, y), t_5);
	t_7 = sqrt(t_6);
	t_8 = t_6 - -1.0;
	t_9 = min(max(x, y), t_5);
	t_10 = sqrt(t_9);
	t_11 = sqrt(t_4);
	tmp = 0.0;
	if ((sqrt((t_9 + 1.0)) - t_10) <= 2e-8)
		tmp = ((1.0 / (t_11 + sqrt((1.0 + t_4)))) + (sqrt((t_1 + 1.0)) - t_2)) + (sqrt((t_6 + 1.0)) - t_7);
	else
		tmp = (sqrt((t_4 - -1.0)) - t_11) - ((t_10 - sqrt((t_9 - -1.0))) - ((sqrt((t_1 - -1.0)) - t_2) - ((t_6 - t_8) / (sqrt(t_8) + 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[Sqrt[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$4 = N[Min[t$95$3, t], $MachinePrecision]}, Block[{t$95$5 = N[Max[t$95$3, t], $MachinePrecision]}, Block[{t$95$6 = N[Max[N[Max[x, y], $MachinePrecision], t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[(t$95$6 - -1), $MachinePrecision]}, Block[{t$95$9 = N[Min[N[Max[x, y], $MachinePrecision], t$95$5], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Sqrt[t$95$4], $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(t$95$9 + 1), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision], 3022314549036573/151115727451828646838272], N[(N[(N[(1 / N[(t$95$11 + N[Sqrt[N[(1 + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$1 + 1), $MachinePrecision]], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$6 + 1), $MachinePrecision]], $MachinePrecision] - t$95$7), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(t$95$4 - -1), $MachinePrecision]], $MachinePrecision] - t$95$11), $MachinePrecision] - N[(N[(t$95$10 - N[Sqrt[N[(t$95$9 - -1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[N[(t$95$1 - -1), $MachinePrecision]], $MachinePrecision] - t$95$2), $MachinePrecision] - N[(N[(t$95$6 - t$95$8), $MachinePrecision] / N[(N[Sqrt[t$95$8], $MachinePrecision] + t$95$7), $MachinePrecision]), $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)) < 2e-8

   1. Initial program 91.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 72.6%

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

      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 72.6%

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

   3. Applied rewrites 72.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

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

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

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

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

      6. lift-+.f64 N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

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

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

      15. lower-unsound-+.f64 72.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 72.7%

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

   5. Applied rewrites 72.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 92.4%

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

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. 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%

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

   10. 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 2e-8 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

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

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

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

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

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

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

      15. lower-unsound-+.f64 72.9%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 72.9%

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

   3. Applied rewrites 72.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. associate-+l- N/A

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

   5. Applied rewrites 91.9%

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

Alternative 6: 96.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_2\right)\\ t_4 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_2\right)\\ t_5 := \sqrt{t\_3 + 1} - \sqrt{t\_3}\\ t_6 := \mathsf{min}\left(t\_1, t\right)\\ t_7 := \sqrt{t\_6}\\ t_8 := \mathsf{max}\left(t\_1, t\right)\\ t_9 := \mathsf{min}\left(t\_4, t\_8\right)\\ t_10 := t\_7 + \sqrt{t\_9}\\ t_11 := \mathsf{max}\left(t\_4, t\_8\right)\\ t_12 := \sqrt{t\_11 + 1} - \sqrt{t\_11}\\ t_13 := \sqrt{1 + t\_6}\\ \mathbf{if}\;t\_9 \leq \frac{287769207549869}{147573952589676412928}:\\ \;\;\;\;\left(\left(\left(2 + \frac{1}{2} \cdot t\_9\right) - t\_10\right) + t\_5\right) + t\_12\\ \mathbf{elif}\;t\_9 \leq 560000000000000:\\ \;\;\;\;\left(\left(t\_13 + \sqrt{1 + t\_9}\right) - t\_10\right) + t\_12\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t\_7 + t\_13} + t\_5\right) + t\_12\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin (fmin x y) z))
       (t_2 (fmax (fmin x y) z))
       (t_3 (fmax (fmax x y) t_2))
       (t_4 (fmin (fmax x y) t_2))
       (t_5 (- (sqrt (+ t_3 1)) (sqrt t_3)))
       (t_6 (fmin t_1 t))
       (t_7 (sqrt t_6))
       (t_8 (fmax t_1 t))
       (t_9 (fmin t_4 t_8))
       (t_10 (+ t_7 (sqrt t_9)))
       (t_11 (fmax t_4 t_8))
       (t_12 (- (sqrt (+ t_11 1)) (sqrt t_11)))
       (t_13 (sqrt (+ 1 t_6))))
  (if (<= t_9 287769207549869/147573952589676412928)
    (+ (+ (- (+ 2 (* 1/2 t_9)) t_10) t_5) t_12)
    (if (<= t_9 560000000000000)
      (+ (- (+ t_13 (sqrt (+ 1 t_9))) t_10) t_12)
      (+ (+ (/ 1 (+ t_7 t_13)) t_5) t_12)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(fmin(x, y), z);
	double t_3 = fmax(fmax(x, y), t_2);
	double t_4 = fmin(fmax(x, y), t_2);
	double t_5 = sqrt((t_3 + 1.0)) - sqrt(t_3);
	double t_6 = fmin(t_1, t);
	double t_7 = sqrt(t_6);
	double t_8 = fmax(t_1, t);
	double t_9 = fmin(t_4, t_8);
	double t_10 = t_7 + sqrt(t_9);
	double t_11 = fmax(t_4, t_8);
	double t_12 = sqrt((t_11 + 1.0)) - sqrt(t_11);
	double t_13 = sqrt((1.0 + t_6));
	double tmp;
	if (t_9 <= 1.95e-6) {
		tmp = (((2.0 + (0.5 * t_9)) - t_10) + t_5) + t_12;
	} else if (t_9 <= 5.6e+14) {
		tmp = ((t_13 + sqrt((1.0 + t_9))) - t_10) + t_12;
	} else {
		tmp = ((1.0 / (t_7 + t_13)) + t_5) + t_12;
	}
	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 = fmax(fmin(x, y), z)
    t_3 = fmax(fmax(x, y), t_2)
    t_4 = fmin(fmax(x, y), t_2)
    t_5 = sqrt((t_3 + 1.0d0)) - sqrt(t_3)
    t_6 = fmin(t_1, t)
    t_7 = sqrt(t_6)
    t_8 = fmax(t_1, t)
    t_9 = fmin(t_4, t_8)
    t_10 = t_7 + sqrt(t_9)
    t_11 = fmax(t_4, t_8)
    t_12 = sqrt((t_11 + 1.0d0)) - sqrt(t_11)
    t_13 = sqrt((1.0d0 + t_6))
    if (t_9 <= 1.95d-6) then
        tmp = (((2.0d0 + (0.5d0 * t_9)) - t_10) + t_5) + t_12
    else if (t_9 <= 5.6d+14) then
        tmp = ((t_13 + sqrt((1.0d0 + t_9))) - t_10) + t_12
    else
        tmp = ((1.0d0 / (t_7 + t_13)) + t_5) + t_12
    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(fmin(x, y), z);
	double t_3 = fmax(fmax(x, y), t_2);
	double t_4 = fmin(fmax(x, y), t_2);
	double t_5 = Math.sqrt((t_3 + 1.0)) - Math.sqrt(t_3);
	double t_6 = fmin(t_1, t);
	double t_7 = Math.sqrt(t_6);
	double t_8 = fmax(t_1, t);
	double t_9 = fmin(t_4, t_8);
	double t_10 = t_7 + Math.sqrt(t_9);
	double t_11 = fmax(t_4, t_8);
	double t_12 = Math.sqrt((t_11 + 1.0)) - Math.sqrt(t_11);
	double t_13 = Math.sqrt((1.0 + t_6));
	double tmp;
	if (t_9 <= 1.95e-6) {
		tmp = (((2.0 + (0.5 * t_9)) - t_10) + t_5) + t_12;
	} else if (t_9 <= 5.6e+14) {
		tmp = ((t_13 + Math.sqrt((1.0 + t_9))) - t_10) + t_12;
	} else {
		tmp = ((1.0 / (t_7 + t_13)) + t_5) + t_12;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(fmin(x, y), z)
	t_3 = fmax(fmax(x, y), t_2)
	t_4 = fmin(fmax(x, y), t_2)
	t_5 = math.sqrt((t_3 + 1.0)) - math.sqrt(t_3)
	t_6 = fmin(t_1, t)
	t_7 = math.sqrt(t_6)
	t_8 = fmax(t_1, t)
	t_9 = fmin(t_4, t_8)
	t_10 = t_7 + math.sqrt(t_9)
	t_11 = fmax(t_4, t_8)
	t_12 = math.sqrt((t_11 + 1.0)) - math.sqrt(t_11)
	t_13 = math.sqrt((1.0 + t_6))
	tmp = 0
	if t_9 <= 1.95e-6:
		tmp = (((2.0 + (0.5 * t_9)) - t_10) + t_5) + t_12
	elif t_9 <= 5.6e+14:
		tmp = ((t_13 + math.sqrt((1.0 + t_9))) - t_10) + t_12
	else:
		tmp = ((1.0 / (t_7 + t_13)) + t_5) + t_12
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(fmin(x, y), z)
	t_3 = fmax(fmax(x, y), t_2)
	t_4 = fmin(fmax(x, y), t_2)
	t_5 = Float64(sqrt(Float64(t_3 + 1.0)) - sqrt(t_3))
	t_6 = fmin(t_1, t)
	t_7 = sqrt(t_6)
	t_8 = fmax(t_1, t)
	t_9 = fmin(t_4, t_8)
	t_10 = Float64(t_7 + sqrt(t_9))
	t_11 = fmax(t_4, t_8)
	t_12 = Float64(sqrt(Float64(t_11 + 1.0)) - sqrt(t_11))
	t_13 = sqrt(Float64(1.0 + t_6))
	tmp = 0.0
	if (t_9 <= 1.95e-6)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(0.5 * t_9)) - t_10) + t_5) + t_12);
	elseif (t_9 <= 5.6e+14)
		tmp = Float64(Float64(Float64(t_13 + sqrt(Float64(1.0 + t_9))) - t_10) + t_12);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_7 + t_13)) + t_5) + t_12);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = min(min(x, y), z);
	t_2 = max(min(x, y), z);
	t_3 = max(max(x, y), t_2);
	t_4 = min(max(x, y), t_2);
	t_5 = sqrt((t_3 + 1.0)) - sqrt(t_3);
	t_6 = min(t_1, t);
	t_7 = sqrt(t_6);
	t_8 = max(t_1, t);
	t_9 = min(t_4, t_8);
	t_10 = t_7 + sqrt(t_9);
	t_11 = max(t_4, t_8);
	t_12 = sqrt((t_11 + 1.0)) - sqrt(t_11);
	t_13 = sqrt((1.0 + t_6));
	tmp = 0.0;
	if (t_9 <= 1.95e-6)
		tmp = (((2.0 + (0.5 * t_9)) - t_10) + t_5) + t_12;
	elseif (t_9 <= 5.6e+14)
		tmp = ((t_13 + sqrt((1.0 + t_9))) - t_10) + t_12;
	else
		tmp = ((1.0 / (t_7 + t_13)) + t_5) + t_12;
	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[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Max[x, y], $MachinePrecision], t$95$2], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(t$95$3 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Min[t$95$1, t], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$1, t], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$4, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[(t$95$7 + N[Sqrt[t$95$9], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$11 = N[Max[t$95$4, t$95$8], $MachinePrecision]}, Block[{t$95$12 = N[(N[Sqrt[N[(t$95$11 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$11], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$13 = N[Sqrt[N[(1 + t$95$6), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$9, 287769207549869/147573952589676412928], N[(N[(N[(N[(2 + N[(1/2 * t$95$9), $MachinePrecision]), $MachinePrecision] - t$95$10), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$12), $MachinePrecision], If[LessEqual[t$95$9, 560000000000000], N[(N[(N[(t$95$13 + N[Sqrt[N[(1 + t$95$9), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$10), $MachinePrecision] + t$95$12), $MachinePrecision], N[(N[(N[(1 / N[(t$95$7 + t$95$13), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$12), $MachinePrecision]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.95e-6

   1. Initial program 91.6%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 35.2%

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

   4. Applied rewrites 35.2%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 26.1%

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

   7. Applied rewrites 26.1%

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

   if 1.95e-6 < y < 5.6e14

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 36.6%

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 29.2%

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

   7. Applied rewrites 29.2%

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

   if 5.6e14 < y

   1. Initial program 91.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 72.6%

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

      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 72.6%

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

   3. Applied rewrites 72.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

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

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

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

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

      6. lift-+.f64 N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

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

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

      15. lower-unsound-+.f64 72.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 72.7%

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

   5. Applied rewrites 72.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 92.4%

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

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. 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%

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

   10. 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 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 96.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(t\_1, t\right)\\ t_3 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_4 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_3\right)\\ t_5 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_3\right)\\ t_6 := \mathsf{min}\left(t\_5, t\_2\right)\\ t_7 := \mathsf{max}\left(t\_5, t\_2\right)\\ t_8 := \sqrt{t\_7 + 1} - \sqrt{t\_7}\\ t_9 := \mathsf{min}\left(t\_1, t\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := t\_10 + \sqrt{t\_6}\\ t_12 := \sqrt{t\_4 + 1} - \sqrt{t\_4}\\ t_13 := \sqrt{1 + t\_9}\\ \mathbf{if}\;t\_6 \leq \frac{287769207549869}{147573952589676412928}:\\ \;\;\;\;\left(\left(\left(2 + \frac{1}{2} \cdot t\_6\right) - t\_11\right) + t\_12\right) + t\_8\\ \mathbf{elif}\;t\_6 \leq 560000000000000:\\ \;\;\;\;\left(t\_13 + \left(\sqrt{1 + t\_6} + \frac{1}{2} \cdot \frac{1}{t\_7 \cdot \sqrt{\frac{1}{t\_7}}}\right)\right) - t\_11\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t\_10 + t\_13} + t\_12\right) + t\_8\\ \end{array} \]

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 (fmax (fmax x y) t_3))
       (t_5 (fmin (fmax x y) t_3))
       (t_6 (fmin t_5 t_2))
       (t_7 (fmax t_5 t_2))
       (t_8 (- (sqrt (+ t_7 1)) (sqrt t_7)))
       (t_9 (fmin t_1 t))
       (t_10 (sqrt t_9))
       (t_11 (+ t_10 (sqrt t_6)))
       (t_12 (- (sqrt (+ t_4 1)) (sqrt t_4)))
       (t_13 (sqrt (+ 1 t_9))))
  (if (<= t_6 287769207549869/147573952589676412928)
    (+ (+ (- (+ 2 (* 1/2 t_6)) t_11) t_12) t_8)
    (if (<= t_6 560000000000000)
      (-
       (+
        t_13
        (+ (sqrt (+ 1 t_6)) (* 1/2 (/ 1 (* t_7 (sqrt (/ 1 t_7)))))))
       t_11)
      (+ (+ (/ 1 (+ t_10 t_13)) t_12) t_8)))))

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 = fmax(fmax(x, y), t_3);
	double t_5 = fmin(fmax(x, y), t_3);
	double t_6 = fmin(t_5, t_2);
	double t_7 = fmax(t_5, t_2);
	double t_8 = sqrt((t_7 + 1.0)) - sqrt(t_7);
	double t_9 = fmin(t_1, t);
	double t_10 = sqrt(t_9);
	double t_11 = t_10 + sqrt(t_6);
	double t_12 = sqrt((t_4 + 1.0)) - sqrt(t_4);
	double t_13 = sqrt((1.0 + t_9));
	double tmp;
	if (t_6 <= 1.95e-6) {
		tmp = (((2.0 + (0.5 * t_6)) - t_11) + t_12) + t_8;
	} else if (t_6 <= 5.6e+14) {
		tmp = (t_13 + (sqrt((1.0 + t_6)) + (0.5 * (1.0 / (t_7 * sqrt((1.0 / t_7))))))) - t_11;
	} else {
		tmp = ((1.0 / (t_10 + t_13)) + t_12) + t_8;
	}
	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 = fmax(t_1, t)
    t_3 = fmax(fmin(x, y), z)
    t_4 = fmax(fmax(x, y), t_3)
    t_5 = fmin(fmax(x, y), t_3)
    t_6 = fmin(t_5, t_2)
    t_7 = fmax(t_5, t_2)
    t_8 = sqrt((t_7 + 1.0d0)) - sqrt(t_7)
    t_9 = fmin(t_1, t)
    t_10 = sqrt(t_9)
    t_11 = t_10 + sqrt(t_6)
    t_12 = sqrt((t_4 + 1.0d0)) - sqrt(t_4)
    t_13 = sqrt((1.0d0 + t_9))
    if (t_6 <= 1.95d-6) then
        tmp = (((2.0d0 + (0.5d0 * t_6)) - t_11) + t_12) + t_8
    else if (t_6 <= 5.6d+14) then
        tmp = (t_13 + (sqrt((1.0d0 + t_6)) + (0.5d0 * (1.0d0 / (t_7 * sqrt((1.0d0 / t_7))))))) - t_11
    else
        tmp = ((1.0d0 / (t_10 + t_13)) + t_12) + t_8
    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 = fmax(fmax(x, y), t_3);
	double t_5 = fmin(fmax(x, y), t_3);
	double t_6 = fmin(t_5, t_2);
	double t_7 = fmax(t_5, t_2);
	double t_8 = Math.sqrt((t_7 + 1.0)) - Math.sqrt(t_7);
	double t_9 = fmin(t_1, t);
	double t_10 = Math.sqrt(t_9);
	double t_11 = t_10 + Math.sqrt(t_6);
	double t_12 = Math.sqrt((t_4 + 1.0)) - Math.sqrt(t_4);
	double t_13 = Math.sqrt((1.0 + t_9));
	double tmp;
	if (t_6 <= 1.95e-6) {
		tmp = (((2.0 + (0.5 * t_6)) - t_11) + t_12) + t_8;
	} else if (t_6 <= 5.6e+14) {
		tmp = (t_13 + (Math.sqrt((1.0 + t_6)) + (0.5 * (1.0 / (t_7 * Math.sqrt((1.0 / t_7))))))) - t_11;
	} else {
		tmp = ((1.0 / (t_10 + t_13)) + t_12) + t_8;
	}
	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 = fmax(fmax(x, y), t_3)
	t_5 = fmin(fmax(x, y), t_3)
	t_6 = fmin(t_5, t_2)
	t_7 = fmax(t_5, t_2)
	t_8 = math.sqrt((t_7 + 1.0)) - math.sqrt(t_7)
	t_9 = fmin(t_1, t)
	t_10 = math.sqrt(t_9)
	t_11 = t_10 + math.sqrt(t_6)
	t_12 = math.sqrt((t_4 + 1.0)) - math.sqrt(t_4)
	t_13 = math.sqrt((1.0 + t_9))
	tmp = 0
	if t_6 <= 1.95e-6:
		tmp = (((2.0 + (0.5 * t_6)) - t_11) + t_12) + t_8
	elif t_6 <= 5.6e+14:
		tmp = (t_13 + (math.sqrt((1.0 + t_6)) + (0.5 * (1.0 / (t_7 * math.sqrt((1.0 / t_7))))))) - t_11
	else:
		tmp = ((1.0 / (t_10 + t_13)) + t_12) + t_8
	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 = fmax(fmax(x, y), t_3)
	t_5 = fmin(fmax(x, y), t_3)
	t_6 = fmin(t_5, t_2)
	t_7 = fmax(t_5, t_2)
	t_8 = Float64(sqrt(Float64(t_7 + 1.0)) - sqrt(t_7))
	t_9 = fmin(t_1, t)
	t_10 = sqrt(t_9)
	t_11 = Float64(t_10 + sqrt(t_6))
	t_12 = Float64(sqrt(Float64(t_4 + 1.0)) - sqrt(t_4))
	t_13 = sqrt(Float64(1.0 + t_9))
	tmp = 0.0
	if (t_6 <= 1.95e-6)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(0.5 * t_6)) - t_11) + t_12) + t_8);
	elseif (t_6 <= 5.6e+14)
		tmp = Float64(Float64(t_13 + Float64(sqrt(Float64(1.0 + t_6)) + Float64(0.5 * Float64(1.0 / Float64(t_7 * sqrt(Float64(1.0 / t_7))))))) - t_11);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_10 + t_13)) + t_12) + t_8);
	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 = max(max(x, y), t_3);
	t_5 = min(max(x, y), t_3);
	t_6 = min(t_5, t_2);
	t_7 = max(t_5, t_2);
	t_8 = sqrt((t_7 + 1.0)) - sqrt(t_7);
	t_9 = min(t_1, t);
	t_10 = sqrt(t_9);
	t_11 = t_10 + sqrt(t_6);
	t_12 = sqrt((t_4 + 1.0)) - sqrt(t_4);
	t_13 = sqrt((1.0 + t_9));
	tmp = 0.0;
	if (t_6 <= 1.95e-6)
		tmp = (((2.0 + (0.5 * t_6)) - t_11) + t_12) + t_8;
	elseif (t_6 <= 5.6e+14)
		tmp = (t_13 + (sqrt((1.0 + t_6)) + (0.5 * (1.0 / (t_7 * sqrt((1.0 / t_7))))))) - t_11;
	else
		tmp = ((1.0 / (t_10 + t_13)) + t_12) + t_8;
	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[Max[N[Max[x, y], $MachinePrecision], t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[Min[N[Max[x, y], $MachinePrecision], t$95$3], $MachinePrecision]}, Block[{t$95$6 = N[Min[t$95$5, t$95$2], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$5, t$95$2], $MachinePrecision]}, Block[{t$95$8 = N[(N[Sqrt[N[(t$95$7 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$7], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$1, t], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[(t$95$10 + N[Sqrt[t$95$6], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$12 = N[(N[Sqrt[N[(t$95$4 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$13 = N[Sqrt[N[(1 + t$95$9), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$6, 287769207549869/147573952589676412928], N[(N[(N[(N[(2 + N[(1/2 * t$95$6), $MachinePrecision]), $MachinePrecision] - t$95$11), $MachinePrecision] + t$95$12), $MachinePrecision] + t$95$8), $MachinePrecision], If[LessEqual[t$95$6, 560000000000000], N[(N[(t$95$13 + N[(N[Sqrt[N[(1 + t$95$6), $MachinePrecision]], $MachinePrecision] + N[(1/2 * N[(1 / N[(t$95$7 * N[Sqrt[N[(1 / t$95$7), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$11), $MachinePrecision], N[(N[(N[(1 / N[(t$95$10 + t$95$13), $MachinePrecision]), $MachinePrecision] + t$95$12), $MachinePrecision] + t$95$8), $MachinePrecision]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.95e-6

   1. Initial program 91.6%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 35.2%

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

   4. Applied rewrites 35.2%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 26.1%

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

   7. Applied rewrites 26.1%

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

   if 1.95e-6 < y < 5.6e14

   1. Initial program 91.6%

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

   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 11.7%

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

   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 13.3%

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   10. Applied rewrites 12.3%

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

   if 5.6e14 < y

   1. Initial program 91.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 72.6%

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

      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 72.6%

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

   3. Applied rewrites 72.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

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

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

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

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

      6. lift-+.f64 N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

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

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

      15. lower-unsound-+.f64 72.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 72.7%

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

   5. Applied rewrites 72.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 92.4%

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

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. 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%

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

   10. 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 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 96.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \sqrt{t\_1}\\ t_3 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_4 := \mathsf{min}\left(t\_3, t\right)\\ t_5 := \mathsf{max}\left(t\_3, t\right)\\ t_6 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_5\right)\\ t_7 := \sqrt{t\_6}\\ t_8 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_5\right)\\ t_9 := \sqrt{t\_8}\\ t_10 := \sqrt{t\_4}\\ \mathbf{if}\;\sqrt{t\_8 + 1} - t\_9 \leq \frac{3022314549036573}{151115727451828646838272}:\\ \;\;\;\;\left(\frac{1}{t\_10 + \sqrt{1 + t\_4}} + \left(\sqrt{t\_1 + 1} - t\_2\right)\right) + \left(\sqrt{t\_6 + 1} - t\_7\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\_10\right) + \left(\sqrt{t\_4 - -1} - \left(\left(\left(t\_9 - \left(\sqrt{t\_1 - -1} - t\_2\right)\right) - \sqrt{t\_8 - -1}\right) - \left(\sqrt{t\_6 - -1} - t\_7\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmax (fmin x y) z))
       (t_2 (sqrt t_1))
       (t_3 (fmin (fmin x y) z))
       (t_4 (fmin t_3 t))
       (t_5 (fmax t_3 t))
       (t_6 (fmax (fmax x y) t_5))
       (t_7 (sqrt t_6))
       (t_8 (fmin (fmax x y) t_5))
       (t_9 (sqrt t_8))
       (t_10 (sqrt t_4)))
  (if (<=
       (- (sqrt (+ t_8 1)) t_9)
       3022314549036573/151115727451828646838272)
    (+
     (+ (/ 1 (+ t_10 (sqrt (+ 1 t_4)))) (- (sqrt (+ t_1 1)) t_2))
     (- (sqrt (+ t_6 1)) t_7))
    (+
     (- t_10)
     (-
      (sqrt (- t_4 -1))
      (-
       (- (- t_9 (- (sqrt (- t_1 -1)) t_2)) (sqrt (- t_8 -1)))
       (- (sqrt (- t_6 -1)) t_7)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = sqrt(t_1);
	double t_3 = fmin(fmin(x, y), z);
	double t_4 = fmin(t_3, t);
	double t_5 = fmax(t_3, t);
	double t_6 = fmax(fmax(x, y), t_5);
	double t_7 = sqrt(t_6);
	double t_8 = fmin(fmax(x, y), t_5);
	double t_9 = sqrt(t_8);
	double t_10 = sqrt(t_4);
	double tmp;
	if ((sqrt((t_8 + 1.0)) - t_9) <= 2e-8) {
		tmp = ((1.0 / (t_10 + sqrt((1.0 + t_4)))) + (sqrt((t_1 + 1.0)) - t_2)) + (sqrt((t_6 + 1.0)) - t_7);
	} else {
		tmp = -t_10 + (sqrt((t_4 - -1.0)) - (((t_9 - (sqrt((t_1 - -1.0)) - t_2)) - sqrt((t_8 - -1.0))) - (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_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 = sqrt(t_1)
    t_3 = fmin(fmin(x, y), z)
    t_4 = fmin(t_3, t)
    t_5 = fmax(t_3, t)
    t_6 = fmax(fmax(x, y), t_5)
    t_7 = sqrt(t_6)
    t_8 = fmin(fmax(x, y), t_5)
    t_9 = sqrt(t_8)
    t_10 = sqrt(t_4)
    if ((sqrt((t_8 + 1.0d0)) - t_9) <= 2d-8) then
        tmp = ((1.0d0 / (t_10 + sqrt((1.0d0 + t_4)))) + (sqrt((t_1 + 1.0d0)) - t_2)) + (sqrt((t_6 + 1.0d0)) - t_7)
    else
        tmp = -t_10 + (sqrt((t_4 - (-1.0d0))) - (((t_9 - (sqrt((t_1 - (-1.0d0))) - t_2)) - sqrt((t_8 - (-1.0d0)))) - (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 = Math.sqrt(t_1);
	double t_3 = fmin(fmin(x, y), z);
	double t_4 = fmin(t_3, t);
	double t_5 = fmax(t_3, t);
	double t_6 = fmax(fmax(x, y), t_5);
	double t_7 = Math.sqrt(t_6);
	double t_8 = fmin(fmax(x, y), t_5);
	double t_9 = Math.sqrt(t_8);
	double t_10 = Math.sqrt(t_4);
	double tmp;
	if ((Math.sqrt((t_8 + 1.0)) - t_9) <= 2e-8) {
		tmp = ((1.0 / (t_10 + Math.sqrt((1.0 + t_4)))) + (Math.sqrt((t_1 + 1.0)) - t_2)) + (Math.sqrt((t_6 + 1.0)) - t_7);
	} else {
		tmp = -t_10 + (Math.sqrt((t_4 - -1.0)) - (((t_9 - (Math.sqrt((t_1 - -1.0)) - t_2)) - Math.sqrt((t_8 - -1.0))) - (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 = math.sqrt(t_1)
	t_3 = fmin(fmin(x, y), z)
	t_4 = fmin(t_3, t)
	t_5 = fmax(t_3, t)
	t_6 = fmax(fmax(x, y), t_5)
	t_7 = math.sqrt(t_6)
	t_8 = fmin(fmax(x, y), t_5)
	t_9 = math.sqrt(t_8)
	t_10 = math.sqrt(t_4)
	tmp = 0
	if (math.sqrt((t_8 + 1.0)) - t_9) <= 2e-8:
		tmp = ((1.0 / (t_10 + math.sqrt((1.0 + t_4)))) + (math.sqrt((t_1 + 1.0)) - t_2)) + (math.sqrt((t_6 + 1.0)) - t_7)
	else:
		tmp = -t_10 + (math.sqrt((t_4 - -1.0)) - (((t_9 - (math.sqrt((t_1 - -1.0)) - t_2)) - math.sqrt((t_8 - -1.0))) - (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 = sqrt(t_1)
	t_3 = fmin(fmin(x, y), z)
	t_4 = fmin(t_3, t)
	t_5 = fmax(t_3, t)
	t_6 = fmax(fmax(x, y), t_5)
	t_7 = sqrt(t_6)
	t_8 = fmin(fmax(x, y), t_5)
	t_9 = sqrt(t_8)
	t_10 = sqrt(t_4)
	tmp = 0.0
	if (Float64(sqrt(Float64(t_8 + 1.0)) - t_9) <= 2e-8)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_10 + sqrt(Float64(1.0 + t_4)))) + Float64(sqrt(Float64(t_1 + 1.0)) - t_2)) + Float64(sqrt(Float64(t_6 + 1.0)) - t_7));
	else
		tmp = Float64(Float64(-t_10) + Float64(sqrt(Float64(t_4 - -1.0)) - Float64(Float64(Float64(t_9 - Float64(sqrt(Float64(t_1 - -1.0)) - t_2)) - sqrt(Float64(t_8 - -1.0))) - 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 = sqrt(t_1);
	t_3 = min(min(x, y), z);
	t_4 = min(t_3, t);
	t_5 = max(t_3, t);
	t_6 = max(max(x, y), t_5);
	t_7 = sqrt(t_6);
	t_8 = min(max(x, y), t_5);
	t_9 = sqrt(t_8);
	t_10 = sqrt(t_4);
	tmp = 0.0;
	if ((sqrt((t_8 + 1.0)) - t_9) <= 2e-8)
		tmp = ((1.0 / (t_10 + sqrt((1.0 + t_4)))) + (sqrt((t_1 + 1.0)) - t_2)) + (sqrt((t_6 + 1.0)) - t_7);
	else
		tmp = -t_10 + (sqrt((t_4 - -1.0)) - (((t_9 - (sqrt((t_1 - -1.0)) - t_2)) - sqrt((t_8 - -1.0))) - (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[Sqrt[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$4 = N[Min[t$95$3, t], $MachinePrecision]}, Block[{t$95$5 = N[Max[t$95$3, t], $MachinePrecision]}, Block[{t$95$6 = N[Max[N[Max[x, y], $MachinePrecision], t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[Min[N[Max[x, y], $MachinePrecision], t$95$5], $MachinePrecision]}, Block[{t$95$9 = N[Sqrt[t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$4], $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(t$95$8 + 1), $MachinePrecision]], $MachinePrecision] - t$95$9), $MachinePrecision], 3022314549036573/151115727451828646838272], N[(N[(N[(1 / N[(t$95$10 + N[Sqrt[N[(1 + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$1 + 1), $MachinePrecision]], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$6 + 1), $MachinePrecision]], $MachinePrecision] - t$95$7), $MachinePrecision]), $MachinePrecision], N[((-t$95$10) + N[(N[Sqrt[N[(t$95$4 - -1), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(t$95$9 - N[(N[Sqrt[N[(t$95$1 - -1), $MachinePrecision]], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(t$95$8 - -1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[(t$95$6 - -1), $MachinePrecision]], $MachinePrecision] - t$95$7), $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)) < 2e-8

   1. Initial program 91.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 72.6%

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

      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 72.6%

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

   3. Applied rewrites 72.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

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

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

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

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

      6. lift-+.f64 N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

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

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

      15. lower-unsound-+.f64 72.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 72.7%

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

   5. Applied rewrites 72.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 92.4%

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

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. 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%

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

   10. 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 2e-8 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 91.6%

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

   3. Applied rewrites 43.8%

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

Alternative 9: 96.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_3 := \mathsf{max}\left(t\_2, t\right)\\ t_4 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_3\right)\\ t_5 := \sqrt{t\_4 + 1} - \sqrt{t\_4}\\ t_6 := \sqrt{t\_1 + 1} - \sqrt{t\_1}\\ t_7 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_3\right)\\ t_8 := \mathsf{min}\left(t\_2, t\right)\\ t_9 := \sqrt{t\_8}\\ \mathbf{if}\;t\_7 \leq 560000000000000:\\ \;\;\;\;\left(\left(\left(\sqrt{t\_8 + 1} - t\_9\right) + \left(\sqrt{t\_7 + 1} - \sqrt{t\_7}\right)\right) + t\_6\right) + t\_5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t\_9 + \sqrt{1 + t\_8}} + t\_6\right) + t\_5\\ \end{array} \]

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 (fmax t_2 t))
       (t_4 (fmax (fmax x y) t_3))
       (t_5 (- (sqrt (+ t_4 1)) (sqrt t_4)))
       (t_6 (- (sqrt (+ t_1 1)) (sqrt t_1)))
       (t_7 (fmin (fmax x y) t_3))
       (t_8 (fmin t_2 t))
       (t_9 (sqrt t_8)))
  (if (<= t_7 560000000000000)
    (+
     (+
      (+ (- (sqrt (+ t_8 1)) t_9) (- (sqrt (+ t_7 1)) (sqrt t_7)))
      t_6)
     t_5)
    (+ (+ (/ 1 (+ t_9 (sqrt (+ 1 t_8)))) t_6) t_5))))

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 = fmax(t_2, t);
	double t_4 = fmax(fmax(x, y), t_3);
	double t_5 = sqrt((t_4 + 1.0)) - sqrt(t_4);
	double t_6 = sqrt((t_1 + 1.0)) - sqrt(t_1);
	double t_7 = fmin(fmax(x, y), t_3);
	double t_8 = fmin(t_2, t);
	double t_9 = sqrt(t_8);
	double tmp;
	if (t_7 <= 5.6e+14) {
		tmp = (((sqrt((t_8 + 1.0)) - t_9) + (sqrt((t_7 + 1.0)) - sqrt(t_7))) + t_6) + t_5;
	} else {
		tmp = ((1.0 / (t_9 + sqrt((1.0 + t_8)))) + t_6) + t_5;
	}
	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(fmin(x, y), z)
    t_2 = fmin(fmin(x, y), z)
    t_3 = fmax(t_2, t)
    t_4 = fmax(fmax(x, y), t_3)
    t_5 = sqrt((t_4 + 1.0d0)) - sqrt(t_4)
    t_6 = sqrt((t_1 + 1.0d0)) - sqrt(t_1)
    t_7 = fmin(fmax(x, y), t_3)
    t_8 = fmin(t_2, t)
    t_9 = sqrt(t_8)
    if (t_7 <= 5.6d+14) then
        tmp = (((sqrt((t_8 + 1.0d0)) - t_9) + (sqrt((t_7 + 1.0d0)) - sqrt(t_7))) + t_6) + t_5
    else
        tmp = ((1.0d0 / (t_9 + sqrt((1.0d0 + t_8)))) + t_6) + t_5
    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 = fmax(t_2, t);
	double t_4 = fmax(fmax(x, y), t_3);
	double t_5 = Math.sqrt((t_4 + 1.0)) - Math.sqrt(t_4);
	double t_6 = Math.sqrt((t_1 + 1.0)) - Math.sqrt(t_1);
	double t_7 = fmin(fmax(x, y), t_3);
	double t_8 = fmin(t_2, t);
	double t_9 = Math.sqrt(t_8);
	double tmp;
	if (t_7 <= 5.6e+14) {
		tmp = (((Math.sqrt((t_8 + 1.0)) - t_9) + (Math.sqrt((t_7 + 1.0)) - Math.sqrt(t_7))) + t_6) + t_5;
	} else {
		tmp = ((1.0 / (t_9 + Math.sqrt((1.0 + t_8)))) + t_6) + t_5;
	}
	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 = fmax(t_2, t)
	t_4 = fmax(fmax(x, y), t_3)
	t_5 = math.sqrt((t_4 + 1.0)) - math.sqrt(t_4)
	t_6 = math.sqrt((t_1 + 1.0)) - math.sqrt(t_1)
	t_7 = fmin(fmax(x, y), t_3)
	t_8 = fmin(t_2, t)
	t_9 = math.sqrt(t_8)
	tmp = 0
	if t_7 <= 5.6e+14:
		tmp = (((math.sqrt((t_8 + 1.0)) - t_9) + (math.sqrt((t_7 + 1.0)) - math.sqrt(t_7))) + t_6) + t_5
	else:
		tmp = ((1.0 / (t_9 + math.sqrt((1.0 + t_8)))) + t_6) + t_5
	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 = fmax(t_2, t)
	t_4 = fmax(fmax(x, y), t_3)
	t_5 = Float64(sqrt(Float64(t_4 + 1.0)) - sqrt(t_4))
	t_6 = Float64(sqrt(Float64(t_1 + 1.0)) - sqrt(t_1))
	t_7 = fmin(fmax(x, y), t_3)
	t_8 = fmin(t_2, t)
	t_9 = sqrt(t_8)
	tmp = 0.0
	if (t_7 <= 5.6e+14)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(t_8 + 1.0)) - t_9) + Float64(sqrt(Float64(t_7 + 1.0)) - sqrt(t_7))) + t_6) + t_5);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_9 + sqrt(Float64(1.0 + t_8)))) + t_6) + t_5);
	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 = max(t_2, t);
	t_4 = max(max(x, y), t_3);
	t_5 = sqrt((t_4 + 1.0)) - sqrt(t_4);
	t_6 = sqrt((t_1 + 1.0)) - sqrt(t_1);
	t_7 = min(max(x, y), t_3);
	t_8 = min(t_2, t);
	t_9 = sqrt(t_8);
	tmp = 0.0;
	if (t_7 <= 5.6e+14)
		tmp = (((sqrt((t_8 + 1.0)) - t_9) + (sqrt((t_7 + 1.0)) - sqrt(t_7))) + t_6) + t_5;
	else
		tmp = ((1.0 / (t_9 + sqrt((1.0 + t_8)))) + t_6) + t_5;
	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[Max[t$95$2, t], $MachinePrecision]}, Block[{t$95$4 = N[Max[N[Max[x, y], $MachinePrecision], t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(t$95$4 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(t$95$1 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Min[N[Max[x, y], $MachinePrecision], t$95$3], $MachinePrecision]}, Block[{t$95$8 = N[Min[t$95$2, t], $MachinePrecision]}, Block[{t$95$9 = N[Sqrt[t$95$8], $MachinePrecision]}, If[LessEqual[t$95$7, 560000000000000], N[(N[(N[(N[(N[Sqrt[N[(t$95$8 + 1), $MachinePrecision]], $MachinePrecision] - t$95$9), $MachinePrecision] + N[(N[Sqrt[N[(t$95$7 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$7], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision] + t$95$5), $MachinePrecision], N[(N[(N[(1 / N[(t$95$9 + N[Sqrt[N[(1 + t$95$8), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision] + t$95$5), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.6e14

   1. Initial program 91.6%

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

   if 5.6e14 < y

   1. Initial program 91.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 72.6%

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

      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 72.6%

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

   3. Applied rewrites 72.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

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

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

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

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

      6. lift-+.f64 N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

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

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

      15. lower-unsound-+.f64 72.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 72.7%

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

   5. Applied rewrites 72.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 92.4%

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

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. 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%

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

   10. 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 10: 96.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \sqrt{t\_2 + 1} - \sqrt{t\_2}\\ t_5 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_6 := \mathsf{min}\left(t\_5, t\right)\\ t_7 := \sqrt{t\_6}\\ t_8 := \mathsf{max}\left(t\_5, t\right)\\ t_9 := \mathsf{max}\left(t\_3, t\_8\right)\\ t_10 := \sqrt{t\_9 + 1} - \sqrt{t\_9}\\ t_11 := \mathsf{min}\left(t\_3, t\_8\right)\\ t_12 := \sqrt{t\_11}\\ \mathbf{if}\;\left(\left(\left(\sqrt{t\_6 + 1} - t\_7\right) + \left(\sqrt{t\_11 + 1} - t\_12\right)\right) + t\_4\right) + t\_10 \leq \frac{4503599717442489}{4503599627370496}:\\ \;\;\;\;\left(\frac{1}{t\_7 + \sqrt{1 + t\_6}} + t\_4\right) + t\_10\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 + \sqrt{1 + t\_11}\right) - \left(t\_7 + t\_12\right)\right) + t\_4\right) + t\_10\\ \end{array} \]

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 (- (sqrt (+ t_2 1)) (sqrt t_2)))
       (t_5 (fmin (fmin x y) z))
       (t_6 (fmin t_5 t))
       (t_7 (sqrt t_6))
       (t_8 (fmax t_5 t))
       (t_9 (fmax t_3 t_8))
       (t_10 (- (sqrt (+ t_9 1)) (sqrt t_9)))
       (t_11 (fmin t_3 t_8))
       (t_12 (sqrt t_11)))
  (if (<=
       (+
        (+
         (+ (- (sqrt (+ t_6 1)) t_7) (- (sqrt (+ t_11 1)) t_12))
         t_4)
        t_10)
       4503599717442489/4503599627370496)
    (+ (+ (/ 1 (+ t_7 (sqrt (+ 1 t_6)))) t_4) t_10)
    (+ (+ (- (+ 1 (sqrt (+ 1 t_11))) (+ t_7 t_12)) t_4) t_10))))

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 = sqrt((t_2 + 1.0)) - sqrt(t_2);
	double t_5 = fmin(fmin(x, y), z);
	double t_6 = fmin(t_5, t);
	double t_7 = sqrt(t_6);
	double t_8 = fmax(t_5, t);
	double t_9 = fmax(t_3, t_8);
	double t_10 = sqrt((t_9 + 1.0)) - sqrt(t_9);
	double t_11 = fmin(t_3, t_8);
	double t_12 = sqrt(t_11);
	double tmp;
	if (((((sqrt((t_6 + 1.0)) - t_7) + (sqrt((t_11 + 1.0)) - t_12)) + t_4) + t_10) <= 1.00000002) {
		tmp = ((1.0 / (t_7 + sqrt((1.0 + t_6)))) + t_4) + t_10;
	} else {
		tmp = (((1.0 + sqrt((1.0 + t_11))) - (t_7 + t_12)) + t_4) + 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 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    t_4 = sqrt((t_2 + 1.0d0)) - sqrt(t_2)
    t_5 = fmin(fmin(x, y), z)
    t_6 = fmin(t_5, t)
    t_7 = sqrt(t_6)
    t_8 = fmax(t_5, t)
    t_9 = fmax(t_3, t_8)
    t_10 = sqrt((t_9 + 1.0d0)) - sqrt(t_9)
    t_11 = fmin(t_3, t_8)
    t_12 = sqrt(t_11)
    if (((((sqrt((t_6 + 1.0d0)) - t_7) + (sqrt((t_11 + 1.0d0)) - t_12)) + t_4) + t_10) <= 1.00000002d0) then
        tmp = ((1.0d0 / (t_7 + sqrt((1.0d0 + t_6)))) + t_4) + t_10
    else
        tmp = (((1.0d0 + sqrt((1.0d0 + t_11))) - (t_7 + t_12)) + t_4) + 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 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = Math.sqrt((t_2 + 1.0)) - Math.sqrt(t_2);
	double t_5 = fmin(fmin(x, y), z);
	double t_6 = fmin(t_5, t);
	double t_7 = Math.sqrt(t_6);
	double t_8 = fmax(t_5, t);
	double t_9 = fmax(t_3, t_8);
	double t_10 = Math.sqrt((t_9 + 1.0)) - Math.sqrt(t_9);
	double t_11 = fmin(t_3, t_8);
	double t_12 = Math.sqrt(t_11);
	double tmp;
	if (((((Math.sqrt((t_6 + 1.0)) - t_7) + (Math.sqrt((t_11 + 1.0)) - t_12)) + t_4) + t_10) <= 1.00000002) {
		tmp = ((1.0 / (t_7 + Math.sqrt((1.0 + t_6)))) + t_4) + t_10;
	} else {
		tmp = (((1.0 + Math.sqrt((1.0 + t_11))) - (t_7 + t_12)) + t_4) + t_10;
	}
	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 = math.sqrt((t_2 + 1.0)) - math.sqrt(t_2)
	t_5 = fmin(fmin(x, y), z)
	t_6 = fmin(t_5, t)
	t_7 = math.sqrt(t_6)
	t_8 = fmax(t_5, t)
	t_9 = fmax(t_3, t_8)
	t_10 = math.sqrt((t_9 + 1.0)) - math.sqrt(t_9)
	t_11 = fmin(t_3, t_8)
	t_12 = math.sqrt(t_11)
	tmp = 0
	if ((((math.sqrt((t_6 + 1.0)) - t_7) + (math.sqrt((t_11 + 1.0)) - t_12)) + t_4) + t_10) <= 1.00000002:
		tmp = ((1.0 / (t_7 + math.sqrt((1.0 + t_6)))) + t_4) + t_10
	else:
		tmp = (((1.0 + math.sqrt((1.0 + t_11))) - (t_7 + t_12)) + t_4) + t_10
	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 = Float64(sqrt(Float64(t_2 + 1.0)) - sqrt(t_2))
	t_5 = fmin(fmin(x, y), z)
	t_6 = fmin(t_5, t)
	t_7 = sqrt(t_6)
	t_8 = fmax(t_5, t)
	t_9 = fmax(t_3, t_8)
	t_10 = Float64(sqrt(Float64(t_9 + 1.0)) - sqrt(t_9))
	t_11 = fmin(t_3, t_8)
	t_12 = sqrt(t_11)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(sqrt(Float64(t_6 + 1.0)) - t_7) + Float64(sqrt(Float64(t_11 + 1.0)) - t_12)) + t_4) + t_10) <= 1.00000002)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_7 + sqrt(Float64(1.0 + t_6)))) + t_4) + t_10);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + sqrt(Float64(1.0 + t_11))) - Float64(t_7 + t_12)) + t_4) + t_10);
	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 = sqrt((t_2 + 1.0)) - sqrt(t_2);
	t_5 = min(min(x, y), z);
	t_6 = min(t_5, t);
	t_7 = sqrt(t_6);
	t_8 = max(t_5, t);
	t_9 = max(t_3, t_8);
	t_10 = sqrt((t_9 + 1.0)) - sqrt(t_9);
	t_11 = min(t_3, t_8);
	t_12 = sqrt(t_11);
	tmp = 0.0;
	if (((((sqrt((t_6 + 1.0)) - t_7) + (sqrt((t_11 + 1.0)) - t_12)) + t_4) + t_10) <= 1.00000002)
		tmp = ((1.0 / (t_7 + sqrt((1.0 + t_6)))) + t_4) + t_10;
	else
		tmp = (((1.0 + sqrt((1.0 + t_11))) - (t_7 + t_12)) + t_4) + 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[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[(N[Sqrt[N[(t$95$2 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$6 = N[Min[t$95$5, t], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$5, t], $MachinePrecision]}, Block[{t$95$9 = N[Max[t$95$3, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[(N[Sqrt[N[(t$95$9 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$9], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$11 = N[Min[t$95$3, t$95$8], $MachinePrecision]}, Block[{t$95$12 = N[Sqrt[t$95$11], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Sqrt[N[(t$95$6 + 1), $MachinePrecision]], $MachinePrecision] - t$95$7), $MachinePrecision] + N[(N[Sqrt[N[(t$95$11 + 1), $MachinePrecision]], $MachinePrecision] - t$95$12), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$10), $MachinePrecision], 4503599717442489/4503599627370496], N[(N[(N[(1 / N[(t$95$7 + N[Sqrt[N[(1 + t$95$6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$10), $MachinePrecision], N[(N[(N[(N[(1 + N[Sqrt[N[(1 + t$95$11), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$7 + t$95$12), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$10), $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))) < 1.0000000200000001

   1. Initial program 91.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 72.6%

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

      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 72.6%

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

   3. Applied rewrites 72.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

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

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

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

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

      6. lift-+.f64 N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

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

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

      15. lower-unsound-+.f64 72.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 72.7%

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

   5. Applied rewrites 72.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 92.4%

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

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. 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%

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

   10. 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 1.0000000200000001 < (+.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 91.6%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 36.6%

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

   4. Applied rewrites 36.6%

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

Alternative 11: 91.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_2\right)\\ t_4 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_2\right)\\ t_5 := \sqrt{t\_3 + 1} - \sqrt{t\_3}\\ t_6 := \mathsf{min}\left(t\_1, t\right)\\ t_7 := \sqrt{t\_6}\\ t_8 := \mathsf{max}\left(t\_1, t\right)\\ t_9 := \mathsf{min}\left(t\_4, t\_8\right)\\ t_10 := t\_7 + \sqrt{t\_9}\\ t_11 := \mathsf{max}\left(t\_4, t\_8\right)\\ t_12 := \sqrt{t\_11 + 1} - \sqrt{t\_11}\\ t_13 := \sqrt{1 + t\_6}\\ \mathbf{if}\;t\_9 \leq \frac{287769207549869}{147573952589676412928}:\\ \;\;\;\;\left(\left(\left(2 + \frac{1}{2} \cdot t\_9\right) - t\_10\right) + t\_5\right) + t\_12\\ \mathbf{elif}\;t\_9 \leq 560000000000000:\\ \;\;\;\;\left(\left(t\_13 + \sqrt{1 + t\_9}\right) - t\_10\right) + t\_12\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_13 - t\_7\right) + t\_5\right) + t\_12\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin (fmin x y) z))
       (t_2 (fmax (fmin x y) z))
       (t_3 (fmax (fmax x y) t_2))
       (t_4 (fmin (fmax x y) t_2))
       (t_5 (- (sqrt (+ t_3 1)) (sqrt t_3)))
       (t_6 (fmin t_1 t))
       (t_7 (sqrt t_6))
       (t_8 (fmax t_1 t))
       (t_9 (fmin t_4 t_8))
       (t_10 (+ t_7 (sqrt t_9)))
       (t_11 (fmax t_4 t_8))
       (t_12 (- (sqrt (+ t_11 1)) (sqrt t_11)))
       (t_13 (sqrt (+ 1 t_6))))
  (if (<= t_9 287769207549869/147573952589676412928)
    (+ (+ (- (+ 2 (* 1/2 t_9)) t_10) t_5) t_12)
    (if (<= t_9 560000000000000)
      (+ (- (+ t_13 (sqrt (+ 1 t_9))) t_10) t_12)
      (+ (+ (- t_13 t_7) t_5) t_12)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(fmin(x, y), z);
	double t_3 = fmax(fmax(x, y), t_2);
	double t_4 = fmin(fmax(x, y), t_2);
	double t_5 = sqrt((t_3 + 1.0)) - sqrt(t_3);
	double t_6 = fmin(t_1, t);
	double t_7 = sqrt(t_6);
	double t_8 = fmax(t_1, t);
	double t_9 = fmin(t_4, t_8);
	double t_10 = t_7 + sqrt(t_9);
	double t_11 = fmax(t_4, t_8);
	double t_12 = sqrt((t_11 + 1.0)) - sqrt(t_11);
	double t_13 = sqrt((1.0 + t_6));
	double tmp;
	if (t_9 <= 1.95e-6) {
		tmp = (((2.0 + (0.5 * t_9)) - t_10) + t_5) + t_12;
	} else if (t_9 <= 5.6e+14) {
		tmp = ((t_13 + sqrt((1.0 + t_9))) - t_10) + t_12;
	} else {
		tmp = ((t_13 - t_7) + t_5) + t_12;
	}
	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 = fmax(fmin(x, y), z)
    t_3 = fmax(fmax(x, y), t_2)
    t_4 = fmin(fmax(x, y), t_2)
    t_5 = sqrt((t_3 + 1.0d0)) - sqrt(t_3)
    t_6 = fmin(t_1, t)
    t_7 = sqrt(t_6)
    t_8 = fmax(t_1, t)
    t_9 = fmin(t_4, t_8)
    t_10 = t_7 + sqrt(t_9)
    t_11 = fmax(t_4, t_8)
    t_12 = sqrt((t_11 + 1.0d0)) - sqrt(t_11)
    t_13 = sqrt((1.0d0 + t_6))
    if (t_9 <= 1.95d-6) then
        tmp = (((2.0d0 + (0.5d0 * t_9)) - t_10) + t_5) + t_12
    else if (t_9 <= 5.6d+14) then
        tmp = ((t_13 + sqrt((1.0d0 + t_9))) - t_10) + t_12
    else
        tmp = ((t_13 - t_7) + t_5) + t_12
    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(fmin(x, y), z);
	double t_3 = fmax(fmax(x, y), t_2);
	double t_4 = fmin(fmax(x, y), t_2);
	double t_5 = Math.sqrt((t_3 + 1.0)) - Math.sqrt(t_3);
	double t_6 = fmin(t_1, t);
	double t_7 = Math.sqrt(t_6);
	double t_8 = fmax(t_1, t);
	double t_9 = fmin(t_4, t_8);
	double t_10 = t_7 + Math.sqrt(t_9);
	double t_11 = fmax(t_4, t_8);
	double t_12 = Math.sqrt((t_11 + 1.0)) - Math.sqrt(t_11);
	double t_13 = Math.sqrt((1.0 + t_6));
	double tmp;
	if (t_9 <= 1.95e-6) {
		tmp = (((2.0 + (0.5 * t_9)) - t_10) + t_5) + t_12;
	} else if (t_9 <= 5.6e+14) {
		tmp = ((t_13 + Math.sqrt((1.0 + t_9))) - t_10) + t_12;
	} else {
		tmp = ((t_13 - t_7) + t_5) + t_12;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(fmin(x, y), z)
	t_3 = fmax(fmax(x, y), t_2)
	t_4 = fmin(fmax(x, y), t_2)
	t_5 = math.sqrt((t_3 + 1.0)) - math.sqrt(t_3)
	t_6 = fmin(t_1, t)
	t_7 = math.sqrt(t_6)
	t_8 = fmax(t_1, t)
	t_9 = fmin(t_4, t_8)
	t_10 = t_7 + math.sqrt(t_9)
	t_11 = fmax(t_4, t_8)
	t_12 = math.sqrt((t_11 + 1.0)) - math.sqrt(t_11)
	t_13 = math.sqrt((1.0 + t_6))
	tmp = 0
	if t_9 <= 1.95e-6:
		tmp = (((2.0 + (0.5 * t_9)) - t_10) + t_5) + t_12
	elif t_9 <= 5.6e+14:
		tmp = ((t_13 + math.sqrt((1.0 + t_9))) - t_10) + t_12
	else:
		tmp = ((t_13 - t_7) + t_5) + t_12
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(fmin(x, y), z)
	t_3 = fmax(fmax(x, y), t_2)
	t_4 = fmin(fmax(x, y), t_2)
	t_5 = Float64(sqrt(Float64(t_3 + 1.0)) - sqrt(t_3))
	t_6 = fmin(t_1, t)
	t_7 = sqrt(t_6)
	t_8 = fmax(t_1, t)
	t_9 = fmin(t_4, t_8)
	t_10 = Float64(t_7 + sqrt(t_9))
	t_11 = fmax(t_4, t_8)
	t_12 = Float64(sqrt(Float64(t_11 + 1.0)) - sqrt(t_11))
	t_13 = sqrt(Float64(1.0 + t_6))
	tmp = 0.0
	if (t_9 <= 1.95e-6)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(0.5 * t_9)) - t_10) + t_5) + t_12);
	elseif (t_9 <= 5.6e+14)
		tmp = Float64(Float64(Float64(t_13 + sqrt(Float64(1.0 + t_9))) - t_10) + t_12);
	else
		tmp = Float64(Float64(Float64(t_13 - t_7) + t_5) + t_12);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = min(min(x, y), z);
	t_2 = max(min(x, y), z);
	t_3 = max(max(x, y), t_2);
	t_4 = min(max(x, y), t_2);
	t_5 = sqrt((t_3 + 1.0)) - sqrt(t_3);
	t_6 = min(t_1, t);
	t_7 = sqrt(t_6);
	t_8 = max(t_1, t);
	t_9 = min(t_4, t_8);
	t_10 = t_7 + sqrt(t_9);
	t_11 = max(t_4, t_8);
	t_12 = sqrt((t_11 + 1.0)) - sqrt(t_11);
	t_13 = sqrt((1.0 + t_6));
	tmp = 0.0;
	if (t_9 <= 1.95e-6)
		tmp = (((2.0 + (0.5 * t_9)) - t_10) + t_5) + t_12;
	elseif (t_9 <= 5.6e+14)
		tmp = ((t_13 + sqrt((1.0 + t_9))) - t_10) + t_12;
	else
		tmp = ((t_13 - t_7) + t_5) + t_12;
	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[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Max[x, y], $MachinePrecision], t$95$2], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(t$95$3 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Min[t$95$1, t], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$1, t], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$4, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[(t$95$7 + N[Sqrt[t$95$9], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$11 = N[Max[t$95$4, t$95$8], $MachinePrecision]}, Block[{t$95$12 = N[(N[Sqrt[N[(t$95$11 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$11], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$13 = N[Sqrt[N[(1 + t$95$6), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$9, 287769207549869/147573952589676412928], N[(N[(N[(N[(2 + N[(1/2 * t$95$9), $MachinePrecision]), $MachinePrecision] - t$95$10), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$12), $MachinePrecision], If[LessEqual[t$95$9, 560000000000000], N[(N[(N[(t$95$13 + N[Sqrt[N[(1 + t$95$9), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$10), $MachinePrecision] + t$95$12), $MachinePrecision], N[(N[(N[(t$95$13 - t$95$7), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$12), $MachinePrecision]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.95e-6

   1. Initial program 91.6%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 35.2%

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

   4. Applied rewrites 35.2%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 26.1%

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

   7. Applied rewrites 26.1%

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

   if 1.95e-6 < y < 5.6e14

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 36.6%

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 29.2%

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

   7. Applied rewrites 29.2%

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

   if 5.6e14 < y

   1. Initial program 91.6%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 50.8%

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

   4. Applied rewrites 50.8%

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

Alternative 12: 91.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(t\_1, t\right)\\ t_3 := \mathsf{max}\left(t\_1, t\right)\\ t_4 := \sqrt{t\_2}\\ t_5 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_6 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_5\right)\\ t_7 := \sqrt{t\_6 + 1} - \sqrt{t\_6}\\ t_8 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_5\right)\\ t_9 := \mathsf{max}\left(t\_8, t\_3\right)\\ t_10 := \sqrt{t\_9 + 1} - \sqrt{t\_9}\\ t_11 := \mathsf{min}\left(t\_8, t\_3\right)\\ t_12 := t\_4 + \sqrt{t\_11}\\ t_13 := \sqrt{1 + t\_2}\\ \mathbf{if}\;t\_11 \leq \frac{1590140912926291}{649037107316853453566312041152512}:\\ \;\;\;\;\left(\left(2 - t\_12\right) + t\_7\right) + t\_10\\ \mathbf{elif}\;t\_11 \leq 560000000000000:\\ \;\;\;\;\left(\left(t\_13 + \sqrt{1 + t\_11}\right) - t\_12\right) + t\_10\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_13 - t\_4\right) + t\_7\right) + t\_10\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin (fmin x y) z))
       (t_2 (fmin t_1 t))
       (t_3 (fmax t_1 t))
       (t_4 (sqrt t_2))
       (t_5 (fmax (fmin x y) z))
       (t_6 (fmax (fmax x y) t_5))
       (t_7 (- (sqrt (+ t_6 1)) (sqrt t_6)))
       (t_8 (fmin (fmax x y) t_5))
       (t_9 (fmax t_8 t_3))
       (t_10 (- (sqrt (+ t_9 1)) (sqrt t_9)))
       (t_11 (fmin t_8 t_3))
       (t_12 (+ t_4 (sqrt t_11)))
       (t_13 (sqrt (+ 1 t_2))))
  (if (<= t_11 1590140912926291/649037107316853453566312041152512)
    (+ (+ (- 2 t_12) t_7) t_10)
    (if (<= t_11 560000000000000)
      (+ (- (+ t_13 (sqrt (+ 1 t_11))) t_12) t_10)
      (+ (+ (- t_13 t_4) t_7) t_10)))))

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 = fmax(t_1, t);
	double t_4 = sqrt(t_2);
	double t_5 = fmax(fmin(x, y), z);
	double t_6 = fmax(fmax(x, y), t_5);
	double t_7 = sqrt((t_6 + 1.0)) - sqrt(t_6);
	double t_8 = fmin(fmax(x, y), t_5);
	double t_9 = fmax(t_8, t_3);
	double t_10 = sqrt((t_9 + 1.0)) - sqrt(t_9);
	double t_11 = fmin(t_8, t_3);
	double t_12 = t_4 + sqrt(t_11);
	double t_13 = sqrt((1.0 + t_2));
	double tmp;
	if (t_11 <= 2.45e-18) {
		tmp = ((2.0 - t_12) + t_7) + t_10;
	} else if (t_11 <= 5.6e+14) {
		tmp = ((t_13 + sqrt((1.0 + t_11))) - t_12) + t_10;
	} else {
		tmp = ((t_13 - t_4) + t_7) + 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_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 = fmax(t_1, t)
    t_4 = sqrt(t_2)
    t_5 = fmax(fmin(x, y), z)
    t_6 = fmax(fmax(x, y), t_5)
    t_7 = sqrt((t_6 + 1.0d0)) - sqrt(t_6)
    t_8 = fmin(fmax(x, y), t_5)
    t_9 = fmax(t_8, t_3)
    t_10 = sqrt((t_9 + 1.0d0)) - sqrt(t_9)
    t_11 = fmin(t_8, t_3)
    t_12 = t_4 + sqrt(t_11)
    t_13 = sqrt((1.0d0 + t_2))
    if (t_11 <= 2.45d-18) then
        tmp = ((2.0d0 - t_12) + t_7) + t_10
    else if (t_11 <= 5.6d+14) then
        tmp = ((t_13 + sqrt((1.0d0 + t_11))) - t_12) + t_10
    else
        tmp = ((t_13 - t_4) + t_7) + t_10
    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 = fmax(t_1, t);
	double t_4 = Math.sqrt(t_2);
	double t_5 = fmax(fmin(x, y), z);
	double t_6 = fmax(fmax(x, y), t_5);
	double t_7 = Math.sqrt((t_6 + 1.0)) - Math.sqrt(t_6);
	double t_8 = fmin(fmax(x, y), t_5);
	double t_9 = fmax(t_8, t_3);
	double t_10 = Math.sqrt((t_9 + 1.0)) - Math.sqrt(t_9);
	double t_11 = fmin(t_8, t_3);
	double t_12 = t_4 + Math.sqrt(t_11);
	double t_13 = Math.sqrt((1.0 + t_2));
	double tmp;
	if (t_11 <= 2.45e-18) {
		tmp = ((2.0 - t_12) + t_7) + t_10;
	} else if (t_11 <= 5.6e+14) {
		tmp = ((t_13 + Math.sqrt((1.0 + t_11))) - t_12) + t_10;
	} else {
		tmp = ((t_13 - t_4) + t_7) + t_10;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmin(t_1, t)
	t_3 = fmax(t_1, t)
	t_4 = math.sqrt(t_2)
	t_5 = fmax(fmin(x, y), z)
	t_6 = fmax(fmax(x, y), t_5)
	t_7 = math.sqrt((t_6 + 1.0)) - math.sqrt(t_6)
	t_8 = fmin(fmax(x, y), t_5)
	t_9 = fmax(t_8, t_3)
	t_10 = math.sqrt((t_9 + 1.0)) - math.sqrt(t_9)
	t_11 = fmin(t_8, t_3)
	t_12 = t_4 + math.sqrt(t_11)
	t_13 = math.sqrt((1.0 + t_2))
	tmp = 0
	if t_11 <= 2.45e-18:
		tmp = ((2.0 - t_12) + t_7) + t_10
	elif t_11 <= 5.6e+14:
		tmp = ((t_13 + math.sqrt((1.0 + t_11))) - t_12) + t_10
	else:
		tmp = ((t_13 - t_4) + t_7) + t_10
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmin(t_1, t)
	t_3 = fmax(t_1, t)
	t_4 = sqrt(t_2)
	t_5 = fmax(fmin(x, y), z)
	t_6 = fmax(fmax(x, y), t_5)
	t_7 = Float64(sqrt(Float64(t_6 + 1.0)) - sqrt(t_6))
	t_8 = fmin(fmax(x, y), t_5)
	t_9 = fmax(t_8, t_3)
	t_10 = Float64(sqrt(Float64(t_9 + 1.0)) - sqrt(t_9))
	t_11 = fmin(t_8, t_3)
	t_12 = Float64(t_4 + sqrt(t_11))
	t_13 = sqrt(Float64(1.0 + t_2))
	tmp = 0.0
	if (t_11 <= 2.45e-18)
		tmp = Float64(Float64(Float64(2.0 - t_12) + t_7) + t_10);
	elseif (t_11 <= 5.6e+14)
		tmp = Float64(Float64(Float64(t_13 + sqrt(Float64(1.0 + t_11))) - t_12) + t_10);
	else
		tmp = Float64(Float64(Float64(t_13 - t_4) + t_7) + t_10);
	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 = max(t_1, t);
	t_4 = sqrt(t_2);
	t_5 = max(min(x, y), z);
	t_6 = max(max(x, y), t_5);
	t_7 = sqrt((t_6 + 1.0)) - sqrt(t_6);
	t_8 = min(max(x, y), t_5);
	t_9 = max(t_8, t_3);
	t_10 = sqrt((t_9 + 1.0)) - sqrt(t_9);
	t_11 = min(t_8, t_3);
	t_12 = t_4 + sqrt(t_11);
	t_13 = sqrt((1.0 + t_2));
	tmp = 0.0;
	if (t_11 <= 2.45e-18)
		tmp = ((2.0 - t_12) + t_7) + t_10;
	elseif (t_11 <= 5.6e+14)
		tmp = ((t_13 + sqrt((1.0 + t_11))) - t_12) + t_10;
	else
		tmp = ((t_13 - t_4) + t_7) + t_10;
	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[Max[t$95$1, t], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$2], $MachinePrecision]}, Block[{t$95$5 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$6 = N[Max[N[Max[x, y], $MachinePrecision], t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[(N[Sqrt[N[(t$95$6 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$6], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[Min[N[Max[x, y], $MachinePrecision], t$95$5], $MachinePrecision]}, Block[{t$95$9 = N[Max[t$95$8, t$95$3], $MachinePrecision]}, Block[{t$95$10 = N[(N[Sqrt[N[(t$95$9 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$9], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$11 = N[Min[t$95$8, t$95$3], $MachinePrecision]}, Block[{t$95$12 = N[(t$95$4 + N[Sqrt[t$95$11], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$13 = N[Sqrt[N[(1 + t$95$2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$11, 1590140912926291/649037107316853453566312041152512], N[(N[(N[(2 - t$95$12), $MachinePrecision] + t$95$7), $MachinePrecision] + t$95$10), $MachinePrecision], If[LessEqual[t$95$11, 560000000000000], N[(N[(N[(t$95$13 + N[Sqrt[N[(1 + t$95$11), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$12), $MachinePrecision] + t$95$10), $MachinePrecision], N[(N[(N[(t$95$13 - t$95$4), $MachinePrecision] + t$95$7), $MachinePrecision] + t$95$10), $MachinePrecision]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.4500000000000001e-18

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 36.6%

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 24.2%

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

   if 2.4500000000000001e-18 < y < 5.6e14

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 36.6%

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 29.2%

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

   7. Applied rewrites 29.2%

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

   if 5.6e14 < y

   1. Initial program 91.6%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 50.8%

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

   4. Applied rewrites 50.8%

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

Alternative 13: 91.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(t\_1, t\right)\\ t_3 := \mathsf{max}\left(t\_1, t\right)\\ t_4 := \sqrt{t\_2}\\ t_5 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_6 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_5\right)\\ t_7 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_5\right)\\ t_8 := \mathsf{max}\left(t\_7, t\_3\right)\\ t_9 := \sqrt{t\_8 + 1} - \sqrt{t\_8}\\ t_10 := \mathsf{min}\left(t\_7, t\_3\right)\\ t_11 := t\_4 + \sqrt{t\_10}\\ t_12 := \sqrt{1 + t\_2}\\ \mathbf{if}\;t\_10 \leq \frac{1590140912926291}{649037107316853453566312041152512}:\\ \;\;\;\;\left(\left(2 - t\_11\right) + \left(\sqrt{t\_6 + 1} - \sqrt{t\_6}\right)\right) + t\_9\\ \mathbf{elif}\;t\_10 \leq 560000000000000:\\ \;\;\;\;\left(\left(t\_12 + \sqrt{1 + t\_10}\right) - t\_11\right) + t\_9\\ \mathbf{else}:\\ \;\;\;\;\left(t\_12 + \frac{1}{2} \cdot \frac{1}{t\_8 \cdot \sqrt{\frac{1}{t\_8}}}\right) - t\_4\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin (fmin x y) z))
       (t_2 (fmin t_1 t))
       (t_3 (fmax t_1 t))
       (t_4 (sqrt t_2))
       (t_5 (fmax (fmin x y) z))
       (t_6 (fmax (fmax x y) t_5))
       (t_7 (fmin (fmax x y) t_5))
       (t_8 (fmax t_7 t_3))
       (t_9 (- (sqrt (+ t_8 1)) (sqrt t_8)))
       (t_10 (fmin t_7 t_3))
       (t_11 (+ t_4 (sqrt t_10)))
       (t_12 (sqrt (+ 1 t_2))))
  (if (<= t_10 1590140912926291/649037107316853453566312041152512)
    (+ (+ (- 2 t_11) (- (sqrt (+ t_6 1)) (sqrt t_6))) t_9)
    (if (<= t_10 560000000000000)
      (+ (- (+ t_12 (sqrt (+ 1 t_10))) t_11) t_9)
      (- (+ t_12 (* 1/2 (/ 1 (* t_8 (sqrt (/ 1 t_8)))))) t_4)))))

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 = fmax(t_1, t);
	double t_4 = sqrt(t_2);
	double t_5 = fmax(fmin(x, y), z);
	double t_6 = fmax(fmax(x, y), t_5);
	double t_7 = fmin(fmax(x, y), t_5);
	double t_8 = fmax(t_7, t_3);
	double t_9 = sqrt((t_8 + 1.0)) - sqrt(t_8);
	double t_10 = fmin(t_7, t_3);
	double t_11 = t_4 + sqrt(t_10);
	double t_12 = sqrt((1.0 + t_2));
	double tmp;
	if (t_10 <= 2.45e-18) {
		tmp = ((2.0 - t_11) + (sqrt((t_6 + 1.0)) - sqrt(t_6))) + t_9;
	} else if (t_10 <= 5.6e+14) {
		tmp = ((t_12 + sqrt((1.0 + t_10))) - t_11) + t_9;
	} else {
		tmp = (t_12 + (0.5 * (1.0 / (t_8 * sqrt((1.0 / t_8)))))) - 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_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 = fmin(fmin(x, y), z)
    t_2 = fmin(t_1, t)
    t_3 = fmax(t_1, t)
    t_4 = sqrt(t_2)
    t_5 = fmax(fmin(x, y), z)
    t_6 = fmax(fmax(x, y), t_5)
    t_7 = fmin(fmax(x, y), t_5)
    t_8 = fmax(t_7, t_3)
    t_9 = sqrt((t_8 + 1.0d0)) - sqrt(t_8)
    t_10 = fmin(t_7, t_3)
    t_11 = t_4 + sqrt(t_10)
    t_12 = sqrt((1.0d0 + t_2))
    if (t_10 <= 2.45d-18) then
        tmp = ((2.0d0 - t_11) + (sqrt((t_6 + 1.0d0)) - sqrt(t_6))) + t_9
    else if (t_10 <= 5.6d+14) then
        tmp = ((t_12 + sqrt((1.0d0 + t_10))) - t_11) + t_9
    else
        tmp = (t_12 + (0.5d0 * (1.0d0 / (t_8 * sqrt((1.0d0 / t_8)))))) - t_4
    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 = fmax(t_1, t);
	double t_4 = Math.sqrt(t_2);
	double t_5 = fmax(fmin(x, y), z);
	double t_6 = fmax(fmax(x, y), t_5);
	double t_7 = fmin(fmax(x, y), t_5);
	double t_8 = fmax(t_7, t_3);
	double t_9 = Math.sqrt((t_8 + 1.0)) - Math.sqrt(t_8);
	double t_10 = fmin(t_7, t_3);
	double t_11 = t_4 + Math.sqrt(t_10);
	double t_12 = Math.sqrt((1.0 + t_2));
	double tmp;
	if (t_10 <= 2.45e-18) {
		tmp = ((2.0 - t_11) + (Math.sqrt((t_6 + 1.0)) - Math.sqrt(t_6))) + t_9;
	} else if (t_10 <= 5.6e+14) {
		tmp = ((t_12 + Math.sqrt((1.0 + t_10))) - t_11) + t_9;
	} else {
		tmp = (t_12 + (0.5 * (1.0 / (t_8 * Math.sqrt((1.0 / t_8)))))) - t_4;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmin(t_1, t)
	t_3 = fmax(t_1, t)
	t_4 = math.sqrt(t_2)
	t_5 = fmax(fmin(x, y), z)
	t_6 = fmax(fmax(x, y), t_5)
	t_7 = fmin(fmax(x, y), t_5)
	t_8 = fmax(t_7, t_3)
	t_9 = math.sqrt((t_8 + 1.0)) - math.sqrt(t_8)
	t_10 = fmin(t_7, t_3)
	t_11 = t_4 + math.sqrt(t_10)
	t_12 = math.sqrt((1.0 + t_2))
	tmp = 0
	if t_10 <= 2.45e-18:
		tmp = ((2.0 - t_11) + (math.sqrt((t_6 + 1.0)) - math.sqrt(t_6))) + t_9
	elif t_10 <= 5.6e+14:
		tmp = ((t_12 + math.sqrt((1.0 + t_10))) - t_11) + t_9
	else:
		tmp = (t_12 + (0.5 * (1.0 / (t_8 * math.sqrt((1.0 / t_8)))))) - t_4
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmin(t_1, t)
	t_3 = fmax(t_1, t)
	t_4 = sqrt(t_2)
	t_5 = fmax(fmin(x, y), z)
	t_6 = fmax(fmax(x, y), t_5)
	t_7 = fmin(fmax(x, y), t_5)
	t_8 = fmax(t_7, t_3)
	t_9 = Float64(sqrt(Float64(t_8 + 1.0)) - sqrt(t_8))
	t_10 = fmin(t_7, t_3)
	t_11 = Float64(t_4 + sqrt(t_10))
	t_12 = sqrt(Float64(1.0 + t_2))
	tmp = 0.0
	if (t_10 <= 2.45e-18)
		tmp = Float64(Float64(Float64(2.0 - t_11) + Float64(sqrt(Float64(t_6 + 1.0)) - sqrt(t_6))) + t_9);
	elseif (t_10 <= 5.6e+14)
		tmp = Float64(Float64(Float64(t_12 + sqrt(Float64(1.0 + t_10))) - t_11) + t_9);
	else
		tmp = Float64(Float64(t_12 + Float64(0.5 * Float64(1.0 / Float64(t_8 * sqrt(Float64(1.0 / t_8)))))) - t_4);
	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 = max(t_1, t);
	t_4 = sqrt(t_2);
	t_5 = max(min(x, y), z);
	t_6 = max(max(x, y), t_5);
	t_7 = min(max(x, y), t_5);
	t_8 = max(t_7, t_3);
	t_9 = sqrt((t_8 + 1.0)) - sqrt(t_8);
	t_10 = min(t_7, t_3);
	t_11 = t_4 + sqrt(t_10);
	t_12 = sqrt((1.0 + t_2));
	tmp = 0.0;
	if (t_10 <= 2.45e-18)
		tmp = ((2.0 - t_11) + (sqrt((t_6 + 1.0)) - sqrt(t_6))) + t_9;
	elseif (t_10 <= 5.6e+14)
		tmp = ((t_12 + sqrt((1.0 + t_10))) - t_11) + t_9;
	else
		tmp = (t_12 + (0.5 * (1.0 / (t_8 * sqrt((1.0 / t_8)))))) - t_4;
	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[Max[t$95$1, t], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$2], $MachinePrecision]}, Block[{t$95$5 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$6 = N[Max[N[Max[x, y], $MachinePrecision], t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Min[N[Max[x, y], $MachinePrecision], t$95$5], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$7, t$95$3], $MachinePrecision]}, Block[{t$95$9 = N[(N[Sqrt[N[(t$95$8 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$8], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[Min[t$95$7, t$95$3], $MachinePrecision]}, Block[{t$95$11 = N[(t$95$4 + N[Sqrt[t$95$10], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$12 = N[Sqrt[N[(1 + t$95$2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$10, 1590140912926291/649037107316853453566312041152512], N[(N[(N[(2 - t$95$11), $MachinePrecision] + N[(N[Sqrt[N[(t$95$6 + 1), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$6], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$9), $MachinePrecision], If[LessEqual[t$95$10, 560000000000000], N[(N[(N[(t$95$12 + N[Sqrt[N[(1 + t$95$10), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$11), $MachinePrecision] + t$95$9), $MachinePrecision], N[(N[(t$95$12 + N[(1/2 * N[(1 / N[(t$95$8 * N[Sqrt[N[(1 / t$95$8), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.4500000000000001e-18

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 36.6%

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 24.2%

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

   if 2.4500000000000001e-18 < y < 5.6e14

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 36.6%

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 29.2%

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

   7. Applied rewrites 29.2%

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

   if 5.6e14 < y

   1. Initial program 91.6%

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

   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 11.7%

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

   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 13.3%

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

   7. Applied rewrites 13.3%

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

   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 13.4%

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

   13. Applied rewrites 13.4%

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

Alternative 14: 85.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{max}\left(t\_4, t\right)\\ t_8 := \mathsf{max}\left(t\_3, t\_7\right)\\ t_9 := \mathsf{min}\left(t\_2, t\_8\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \mathsf{min}\left(t\_3, t\_7\right)\\ t_12 := \mathsf{max}\left(t\_2, t\_8\right)\\ t_13 := \sqrt{t\_12 + 1} - \sqrt{t\_12}\\ t_14 := \sqrt{1 + t\_11}\\ t_15 := \sqrt{t\_11}\\ t_16 := \left(\left(\left(\sqrt{t\_5 + 1} - t\_6\right) + \left(\sqrt{t\_11 + 1} - t\_15\right)\right) + \left(\sqrt{t\_9 + 1} - t\_10\right)\right) + t\_13\\ t_17 := \sqrt{1 + t\_5}\\ \mathbf{if}\;t\_16 \leq \frac{4503599717442489}{4503599627370496}:\\ \;\;\;\;\left(t\_17 + \frac{1}{2} \cdot \frac{1}{t\_12 \cdot \sqrt{\frac{1}{t\_12}}}\right) - t\_6\\ \mathbf{elif}\;t\_16 \leq 2:\\ \;\;\;\;\left(\left(t\_17 + t\_14\right) - \left(t\_6 + t\_15\right)\right) + t\_13\\ \mathbf{else}:\\ \;\;\;\;\left(t\_17 + \left(t\_14 + \sqrt{1 + t\_9}\right)\right) - \left(t\_6 + \left(t\_15 + t\_10\right)\right)\\ \end{array} \]

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_12 1)) (sqrt t_12)))
       (t_14 (sqrt (+ 1 t_11)))
       (t_15 (sqrt t_11))
       (t_16
        (+
         (+
          (+ (- (sqrt (+ t_5 1)) t_6) (- (sqrt (+ t_11 1)) t_15))
          (- (sqrt (+ t_9 1)) t_10))
         t_13))
       (t_17 (sqrt (+ 1 t_5))))
  (if (<= t_16 4503599717442489/4503599627370496)
    (- (+ t_17 (* 1/2 (/ 1 (* t_12 (sqrt (/ 1 t_12)))))) t_6)
    (if (<= t_16 2)
      (+ (- (+ t_17 t_14) (+ t_6 t_15)) t_13)
      (- (+ t_17 (+ t_14 (sqrt (+ 1 t_9)))) (+ t_6 (+ t_15 t_10)))))))

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((1.0 + t_11));
	double t_15 = sqrt(t_11);
	double t_16 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_15)) + (sqrt((t_9 + 1.0)) - t_10)) + t_13;
	double t_17 = sqrt((1.0 + t_5));
	double tmp;
	if (t_16 <= 1.00000002) {
		tmp = (t_17 + (0.5 * (1.0 / (t_12 * sqrt((1.0 / t_12)))))) - t_6;
	} else if (t_16 <= 2.0) {
		tmp = ((t_17 + t_14) - (t_6 + t_15)) + t_13;
	} else {
		tmp = (t_17 + (t_14 + sqrt((1.0 + t_9)))) - (t_6 + (t_15 + 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_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((1.0d0 + t_11))
    t_15 = sqrt(t_11)
    t_16 = (((sqrt((t_5 + 1.0d0)) - t_6) + (sqrt((t_11 + 1.0d0)) - t_15)) + (sqrt((t_9 + 1.0d0)) - t_10)) + t_13
    t_17 = sqrt((1.0d0 + t_5))
    if (t_16 <= 1.00000002d0) then
        tmp = (t_17 + (0.5d0 * (1.0d0 / (t_12 * sqrt((1.0d0 / t_12)))))) - t_6
    else if (t_16 <= 2.0d0) then
        tmp = ((t_17 + t_14) - (t_6 + t_15)) + t_13
    else
        tmp = (t_17 + (t_14 + sqrt((1.0d0 + t_9)))) - (t_6 + (t_15 + 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 = 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((1.0 + t_11));
	double t_15 = Math.sqrt(t_11);
	double t_16 = (((Math.sqrt((t_5 + 1.0)) - t_6) + (Math.sqrt((t_11 + 1.0)) - t_15)) + (Math.sqrt((t_9 + 1.0)) - t_10)) + t_13;
	double t_17 = Math.sqrt((1.0 + t_5));
	double tmp;
	if (t_16 <= 1.00000002) {
		tmp = (t_17 + (0.5 * (1.0 / (t_12 * Math.sqrt((1.0 / t_12)))))) - t_6;
	} else if (t_16 <= 2.0) {
		tmp = ((t_17 + t_14) - (t_6 + t_15)) + t_13;
	} else {
		tmp = (t_17 + (t_14 + Math.sqrt((1.0 + t_9)))) - (t_6 + (t_15 + t_10));
	}
	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((1.0 + t_11))
	t_15 = math.sqrt(t_11)
	t_16 = (((math.sqrt((t_5 + 1.0)) - t_6) + (math.sqrt((t_11 + 1.0)) - t_15)) + (math.sqrt((t_9 + 1.0)) - t_10)) + t_13
	t_17 = math.sqrt((1.0 + t_5))
	tmp = 0
	if t_16 <= 1.00000002:
		tmp = (t_17 + (0.5 * (1.0 / (t_12 * math.sqrt((1.0 / t_12)))))) - t_6
	elif t_16 <= 2.0:
		tmp = ((t_17 + t_14) - (t_6 + t_15)) + t_13
	else:
		tmp = (t_17 + (t_14 + math.sqrt((1.0 + t_9)))) - (t_6 + (t_15 + t_10))
	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(Float64(1.0 + t_11))
	t_15 = sqrt(t_11)
	t_16 = Float64(Float64(Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + Float64(sqrt(Float64(t_11 + 1.0)) - t_15)) + Float64(sqrt(Float64(t_9 + 1.0)) - t_10)) + t_13)
	t_17 = sqrt(Float64(1.0 + t_5))
	tmp = 0.0
	if (t_16 <= 1.00000002)
		tmp = Float64(Float64(t_17 + Float64(0.5 * Float64(1.0 / Float64(t_12 * sqrt(Float64(1.0 / t_12)))))) - t_6);
	elseif (t_16 <= 2.0)
		tmp = Float64(Float64(Float64(t_17 + t_14) - Float64(t_6 + t_15)) + t_13);
	else
		tmp = Float64(Float64(t_17 + Float64(t_14 + sqrt(Float64(1.0 + t_9)))) - Float64(t_6 + Float64(t_15 + t_10)));
	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((1.0 + t_11));
	t_15 = sqrt(t_11);
	t_16 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_15)) + (sqrt((t_9 + 1.0)) - t_10)) + t_13;
	t_17 = sqrt((1.0 + t_5));
	tmp = 0.0;
	if (t_16 <= 1.00000002)
		tmp = (t_17 + (0.5 * (1.0 / (t_12 * sqrt((1.0 / t_12)))))) - t_6;
	elseif (t_16 <= 2.0)
		tmp = ((t_17 + t_14) - (t_6 + t_15)) + t_13;
	else
		tmp = (t_17 + (t_14 + sqrt((1.0 + t_9)))) - (t_6 + (t_15 + 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[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), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$12], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$14 = N[Sqrt[N[(1 + t$95$11), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$15 = N[Sqrt[t$95$11], $MachinePrecision]}, Block[{t$95$16 = N[(N[(N[(N[(N[Sqrt[N[(t$95$5 + 1), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + N[(N[Sqrt[N[(t$95$11 + 1), $MachinePrecision]], $MachinePrecision] - t$95$15), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$9 + 1), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision] + t$95$13), $MachinePrecision]}, Block[{t$95$17 = N[Sqrt[N[(1 + t$95$5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$16, 4503599717442489/4503599627370496], N[(N[(t$95$17 + N[(1/2 * N[(1 / N[(t$95$12 * N[Sqrt[N[(1 / t$95$12), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision], If[LessEqual[t$95$16, 2], N[(N[(N[(t$95$17 + t$95$14), $MachinePrecision] - N[(t$95$6 + t$95$15), $MachinePrecision]), $MachinePrecision] + t$95$13), $MachinePrecision], N[(N[(t$95$17 + N[(t$95$14 + N[Sqrt[N[(1 + t$95$9), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$6 + N[(t$95$15 + t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.0000000200000001

   1. Initial program 91.6%

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

   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 11.7%

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

   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 13.3%

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

   7. Applied rewrites 13.3%

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

   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 13.4%

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

   13. Applied rewrites 13.4%

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

   if 1.0000000200000001 < (+.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

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 36.6%

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 29.2%

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

   7. Applied rewrites 29.2%

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

   if 2 < (+.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 91.6%

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

   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 11.7%

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

   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 13.3%

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

   7. Applied rewrites 13.3%

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

   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

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-+.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-+.f64 N/A

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

   13. Applied rewrites 11.8%

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

Alternative 15: 81.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{max}\left(t\_4, t\right)\\ t_8 := \mathsf{max}\left(t\_3, t\_7\right)\\ t_9 := \mathsf{min}\left(t\_2, t\_8\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \mathsf{min}\left(t\_3, t\_7\right)\\ t_12 := \mathsf{max}\left(t\_2, t\_8\right)\\ t_13 := \sqrt{t\_12}\\ t_14 := \sqrt{t\_11}\\ t_15 := \left(\left(\left(\sqrt{t\_5 + 1} - t\_6\right) + \left(\sqrt{t\_11 + 1} - t\_14\right)\right) + \left(\sqrt{t\_9 + 1} - t\_10\right)\right) + \left(\sqrt{t\_12 + 1} - t\_13\right)\\ t_16 := \sqrt{1 + t\_5}\\ \mathbf{if}\;t\_15 \leq \frac{3}{2}:\\ \;\;\;\;\left(t\_16 + \frac{1}{2} \cdot \frac{1}{t\_12 \cdot \sqrt{\frac{1}{t\_12}}}\right) - t\_6\\ \mathbf{elif}\;t\_15 \leq 2:\\ \;\;\;\;\left(\sqrt{t\_5 - -1} + \sqrt{t\_11 - -1}\right) + \left(\sqrt{t\_12 - -1} - \left(\left(t\_14 + t\_6\right) + t\_13\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_16 + \left(\sqrt{1 + t\_11} + \sqrt{1 + t\_9}\right)\right) - \left(t\_6 + \left(t\_14 + t\_10\right)\right)\\ \end{array} \]

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_12))
       (t_14 (sqrt t_11))
       (t_15
        (+
         (+
          (+ (- (sqrt (+ t_5 1)) t_6) (- (sqrt (+ t_11 1)) t_14))
          (- (sqrt (+ t_9 1)) t_10))
         (- (sqrt (+ t_12 1)) t_13)))
       (t_16 (sqrt (+ 1 t_5))))
  (if (<= t_15 3/2)
    (- (+ t_16 (* 1/2 (/ 1 (* t_12 (sqrt (/ 1 t_12)))))) t_6)
    (if (<= t_15 2)
      (+
       (+ (sqrt (- t_5 -1)) (sqrt (- t_11 -1)))
       (- (sqrt (- t_12 -1)) (+ (+ t_14 t_6) t_13)))
      (-
       (+ t_16 (+ (sqrt (+ 1 t_11)) (sqrt (+ 1 t_9))))
       (+ t_6 (+ t_14 t_10)))))))

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);
	double t_14 = sqrt(t_11);
	double t_15 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_14)) + (sqrt((t_9 + 1.0)) - t_10)) + (sqrt((t_12 + 1.0)) - t_13);
	double t_16 = sqrt((1.0 + t_5));
	double tmp;
	if (t_15 <= 1.5) {
		tmp = (t_16 + (0.5 * (1.0 / (t_12 * sqrt((1.0 / t_12)))))) - t_6;
	} else if (t_15 <= 2.0) {
		tmp = (sqrt((t_5 - -1.0)) + sqrt((t_11 - -1.0))) + (sqrt((t_12 - -1.0)) - ((t_14 + t_6) + t_13));
	} else {
		tmp = (t_16 + (sqrt((1.0 + t_11)) + sqrt((1.0 + t_9)))) - (t_6 + (t_14 + 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_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)
    t_14 = sqrt(t_11)
    t_15 = (((sqrt((t_5 + 1.0d0)) - t_6) + (sqrt((t_11 + 1.0d0)) - t_14)) + (sqrt((t_9 + 1.0d0)) - t_10)) + (sqrt((t_12 + 1.0d0)) - t_13)
    t_16 = sqrt((1.0d0 + t_5))
    if (t_15 <= 1.5d0) then
        tmp = (t_16 + (0.5d0 * (1.0d0 / (t_12 * sqrt((1.0d0 / t_12)))))) - t_6
    else if (t_15 <= 2.0d0) then
        tmp = (sqrt((t_5 - (-1.0d0))) + sqrt((t_11 - (-1.0d0)))) + (sqrt((t_12 - (-1.0d0))) - ((t_14 + t_6) + t_13))
    else
        tmp = (t_16 + (sqrt((1.0d0 + t_11)) + sqrt((1.0d0 + t_9)))) - (t_6 + (t_14 + 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 = 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);
	double t_14 = Math.sqrt(t_11);
	double t_15 = (((Math.sqrt((t_5 + 1.0)) - t_6) + (Math.sqrt((t_11 + 1.0)) - t_14)) + (Math.sqrt((t_9 + 1.0)) - t_10)) + (Math.sqrt((t_12 + 1.0)) - t_13);
	double t_16 = Math.sqrt((1.0 + t_5));
	double tmp;
	if (t_15 <= 1.5) {
		tmp = (t_16 + (0.5 * (1.0 / (t_12 * Math.sqrt((1.0 / t_12)))))) - t_6;
	} else if (t_15 <= 2.0) {
		tmp = (Math.sqrt((t_5 - -1.0)) + Math.sqrt((t_11 - -1.0))) + (Math.sqrt((t_12 - -1.0)) - ((t_14 + t_6) + t_13));
	} else {
		tmp = (t_16 + (Math.sqrt((1.0 + t_11)) + Math.sqrt((1.0 + t_9)))) - (t_6 + (t_14 + t_10));
	}
	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)
	t_14 = math.sqrt(t_11)
	t_15 = (((math.sqrt((t_5 + 1.0)) - t_6) + (math.sqrt((t_11 + 1.0)) - t_14)) + (math.sqrt((t_9 + 1.0)) - t_10)) + (math.sqrt((t_12 + 1.0)) - t_13)
	t_16 = math.sqrt((1.0 + t_5))
	tmp = 0
	if t_15 <= 1.5:
		tmp = (t_16 + (0.5 * (1.0 / (t_12 * math.sqrt((1.0 / t_12)))))) - t_6
	elif t_15 <= 2.0:
		tmp = (math.sqrt((t_5 - -1.0)) + math.sqrt((t_11 - -1.0))) + (math.sqrt((t_12 - -1.0)) - ((t_14 + t_6) + t_13))
	else:
		tmp = (t_16 + (math.sqrt((1.0 + t_11)) + math.sqrt((1.0 + t_9)))) - (t_6 + (t_14 + t_10))
	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_12)
	t_14 = sqrt(t_11)
	t_15 = Float64(Float64(Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + Float64(sqrt(Float64(t_11 + 1.0)) - t_14)) + Float64(sqrt(Float64(t_9 + 1.0)) - t_10)) + Float64(sqrt(Float64(t_12 + 1.0)) - t_13))
	t_16 = sqrt(Float64(1.0 + t_5))
	tmp = 0.0
	if (t_15 <= 1.5)
		tmp = Float64(Float64(t_16 + Float64(0.5 * Float64(1.0 / Float64(t_12 * sqrt(Float64(1.0 / t_12)))))) - t_6);
	elseif (t_15 <= 2.0)
		tmp = Float64(Float64(sqrt(Float64(t_5 - -1.0)) + sqrt(Float64(t_11 - -1.0))) + Float64(sqrt(Float64(t_12 - -1.0)) - Float64(Float64(t_14 + t_6) + t_13)));
	else
		tmp = Float64(Float64(t_16 + Float64(sqrt(Float64(1.0 + t_11)) + sqrt(Float64(1.0 + t_9)))) - Float64(t_6 + Float64(t_14 + t_10)));
	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);
	t_14 = sqrt(t_11);
	t_15 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_14)) + (sqrt((t_9 + 1.0)) - t_10)) + (sqrt((t_12 + 1.0)) - t_13);
	t_16 = sqrt((1.0 + t_5));
	tmp = 0.0;
	if (t_15 <= 1.5)
		tmp = (t_16 + (0.5 * (1.0 / (t_12 * sqrt((1.0 / t_12)))))) - t_6;
	elseif (t_15 <= 2.0)
		tmp = (sqrt((t_5 - -1.0)) + sqrt((t_11 - -1.0))) + (sqrt((t_12 - -1.0)) - ((t_14 + t_6) + t_13));
	else
		tmp = (t_16 + (sqrt((1.0 + t_11)) + sqrt((1.0 + t_9)))) - (t_6 + (t_14 + 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[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$12], $MachinePrecision]}, Block[{t$95$14 = N[Sqrt[t$95$11], $MachinePrecision]}, Block[{t$95$15 = N[(N[(N[(N[(N[Sqrt[N[(t$95$5 + 1), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + N[(N[Sqrt[N[(t$95$11 + 1), $MachinePrecision]], $MachinePrecision] - t$95$14), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$9 + 1), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$12 + 1), $MachinePrecision]], $MachinePrecision] - t$95$13), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$16 = N[Sqrt[N[(1 + t$95$5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$15, 3/2], N[(N[(t$95$16 + N[(1/2 * N[(1 / N[(t$95$12 * N[Sqrt[N[(1 / t$95$12), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision], If[LessEqual[t$95$15, 2], N[(N[(N[Sqrt[N[(t$95$5 - -1), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(t$95$11 - -1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$12 - -1), $MachinePrecision]], $MachinePrecision] - N[(N[(t$95$14 + t$95$6), $MachinePrecision] + t$95$13), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$16 + N[(N[Sqrt[N[(1 + t$95$11), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1 + t$95$9), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$6 + N[(t$95$14 + t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.5

   1. Initial program 91.6%

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

   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 11.7%

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

   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 13.3%

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

   7. Applied rewrites 13.3%

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

   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 13.4%

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

   13. Applied rewrites 13.4%

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

   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))) < 2

   1. Initial program 91.6%

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

   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 11.7%

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

      1. lift--.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. lift-+.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. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. add-flip N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 18.1%

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

   6. Applied rewrites 18.1%

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

   if 2 < (+.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 91.6%

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

   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 11.7%

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

   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 13.3%

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

   7. Applied rewrites 13.3%

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

   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

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-+.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-+.f64 N/A

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

   13. Applied rewrites 11.8%

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

Alternative 16: 76.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\right)\\ t_2 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_3 := \mathsf{max}\left(t\_2, t\right)\\ t_4 := \mathsf{max}\left(t\_1, t\_3\right)\\ t_5 := \mathsf{min}\left(t\_1, t\_3\right)\\ t_6 := \mathsf{min}\left(t\_2, t\right)\\ t_7 := \sqrt{t\_6}\\ \mathbf{if}\;t\_5 \leq \frac{5188146770730811}{288230376151711744}:\\ \;\;\;\;\left(\sqrt{t\_6 - -1} + \sqrt{t\_5 - -1}\right) + \left(\sqrt{t\_4 - -1} - \left(\left(\sqrt{t\_5} + t\_7\right) + \sqrt{t\_4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t\_6} + \frac{1}{2} \cdot \frac{1}{t\_4 \cdot \sqrt{\frac{1}{t\_4}}}\right) - t\_7\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin (fmax x y) (fmax (fmin x y) z)))
       (t_2 (fmin (fmin x y) z))
       (t_3 (fmax t_2 t))
       (t_4 (fmax t_1 t_3))
       (t_5 (fmin t_1 t_3))
       (t_6 (fmin t_2 t))
       (t_7 (sqrt t_6)))
  (if (<= t_5 5188146770730811/288230376151711744)
    (+
     (+ (sqrt (- t_6 -1)) (sqrt (- t_5 -1)))
     (- (sqrt (- t_4 -1)) (+ (+ (sqrt t_5) t_7) (sqrt t_4))))
    (-
     (+ (sqrt (+ 1 t_6)) (* 1/2 (/ 1 (* t_4 (sqrt (/ 1 t_4))))))
     t_7))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmax(x, y), fmax(fmin(x, y), z));
	double t_2 = fmin(fmin(x, y), z);
	double t_3 = fmax(t_2, t);
	double t_4 = fmax(t_1, t_3);
	double t_5 = fmin(t_1, t_3);
	double t_6 = fmin(t_2, t);
	double t_7 = sqrt(t_6);
	double tmp;
	if (t_5 <= 0.018) {
		tmp = (sqrt((t_6 - -1.0)) + sqrt((t_5 - -1.0))) + (sqrt((t_4 - -1.0)) - ((sqrt(t_5) + t_7) + sqrt(t_4)));
	} else {
		tmp = (sqrt((1.0 + t_6)) + (0.5 * (1.0 / (t_4 * sqrt((1.0 / 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(fmax(x, y), fmax(fmin(x, y), z))
    t_2 = fmin(fmin(x, y), z)
    t_3 = fmax(t_2, t)
    t_4 = fmax(t_1, t_3)
    t_5 = fmin(t_1, t_3)
    t_6 = fmin(t_2, t)
    t_7 = sqrt(t_6)
    if (t_5 <= 0.018d0) then
        tmp = (sqrt((t_6 - (-1.0d0))) + sqrt((t_5 - (-1.0d0)))) + (sqrt((t_4 - (-1.0d0))) - ((sqrt(t_5) + t_7) + sqrt(t_4)))
    else
        tmp = (sqrt((1.0d0 + t_6)) + (0.5d0 * (1.0d0 / (t_4 * sqrt((1.0d0 / 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(fmax(x, y), fmax(fmin(x, y), z));
	double t_2 = fmin(fmin(x, y), z);
	double t_3 = fmax(t_2, t);
	double t_4 = fmax(t_1, t_3);
	double t_5 = fmin(t_1, t_3);
	double t_6 = fmin(t_2, t);
	double t_7 = Math.sqrt(t_6);
	double tmp;
	if (t_5 <= 0.018) {
		tmp = (Math.sqrt((t_6 - -1.0)) + Math.sqrt((t_5 - -1.0))) + (Math.sqrt((t_4 - -1.0)) - ((Math.sqrt(t_5) + t_7) + Math.sqrt(t_4)));
	} else {
		tmp = (Math.sqrt((1.0 + t_6)) + (0.5 * (1.0 / (t_4 * Math.sqrt((1.0 / t_4)))))) - t_7;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmax(x, y), fmax(fmin(x, y), z))
	t_2 = fmin(fmin(x, y), z)
	t_3 = fmax(t_2, t)
	t_4 = fmax(t_1, t_3)
	t_5 = fmin(t_1, t_3)
	t_6 = fmin(t_2, t)
	t_7 = math.sqrt(t_6)
	tmp = 0
	if t_5 <= 0.018:
		tmp = (math.sqrt((t_6 - -1.0)) + math.sqrt((t_5 - -1.0))) + (math.sqrt((t_4 - -1.0)) - ((math.sqrt(t_5) + t_7) + math.sqrt(t_4)))
	else:
		tmp = (math.sqrt((1.0 + t_6)) + (0.5 * (1.0 / (t_4 * math.sqrt((1.0 / t_4)))))) - t_7
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(fmax(x, y), fmax(fmin(x, y), z))
	t_2 = fmin(fmin(x, y), z)
	t_3 = fmax(t_2, t)
	t_4 = fmax(t_1, t_3)
	t_5 = fmin(t_1, t_3)
	t_6 = fmin(t_2, t)
	t_7 = sqrt(t_6)
	tmp = 0.0
	if (t_5 <= 0.018)
		tmp = Float64(Float64(sqrt(Float64(t_6 - -1.0)) + sqrt(Float64(t_5 - -1.0))) + Float64(sqrt(Float64(t_4 - -1.0)) - Float64(Float64(sqrt(t_5) + t_7) + sqrt(t_4))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t_6)) + Float64(0.5 * Float64(1.0 / Float64(t_4 * sqrt(Float64(1.0 / t_4)))))) - t_7);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = min(max(x, y), max(min(x, y), z));
	t_2 = min(min(x, y), z);
	t_3 = max(t_2, t);
	t_4 = max(t_1, t_3);
	t_5 = min(t_1, t_3);
	t_6 = min(t_2, t);
	t_7 = sqrt(t_6);
	tmp = 0.0;
	if (t_5 <= 0.018)
		tmp = (sqrt((t_6 - -1.0)) + sqrt((t_5 - -1.0))) + (sqrt((t_4 - -1.0)) - ((sqrt(t_5) + t_7) + sqrt(t_4)));
	else
		tmp = (sqrt((1.0 + t_6)) + (0.5 * (1.0 / (t_4 * sqrt((1.0 / t_4)))))) - t_7;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Min[N[Max[x, y], $MachinePrecision], N[Max[N[Min[x, y], $MachinePrecision], z], $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[Max[t$95$1, t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$1, t$95$3], $MachinePrecision]}, Block[{t$95$6 = N[Min[t$95$2, t], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[t$95$6], $MachinePrecision]}, If[LessEqual[t$95$5, 5188146770730811/288230376151711744], N[(N[(N[Sqrt[N[(t$95$6 - -1), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(t$95$5 - -1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$4 - -1), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Sqrt[t$95$5], $MachinePrecision] + t$95$7), $MachinePrecision] + N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1 + t$95$6), $MachinePrecision]], $MachinePrecision] + N[(1/2 * N[(1 / N[(t$95$4 * N[Sqrt[N[(1 / t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$7), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.017999999999999999

   1. Initial program 91.6%

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

   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 11.7%

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

      1. lift--.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. lift-+.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. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. add-flip N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 18.1%

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

   6. Applied rewrites 18.1%

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

   if 0.017999999999999999 < y

   1. Initial program 91.6%

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

   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 11.7%

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

   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 13.3%

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

   7. Applied rewrites 13.3%

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

   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 13.4%

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

   13. Applied rewrites 13.4%

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

Alternative 17: 49.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right), \mathsf{max}\left(\mathsf{max}\left(x, y\right), \mathsf{max}\left(t\_1, t\right)\right)\right)\\ t_3 := \mathsf{min}\left(t\_1, t\right)\\ \left(\sqrt{1 + t\_3} + \frac{1}{2} \cdot \frac{1}{t\_2 \cdot \sqrt{\frac{1}{t\_2}}}\right) - \sqrt{t\_3} \end{array} \]

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 t_3)) (* 1/2 (/ 1 (* t_2 (sqrt (/ 1 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 + t$95$3), $MachinePrecision]], $MachinePrecision] + N[(1/2 * N[(1 / N[(t$95$2 * N[Sqrt[N[(1 / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 91.6%

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

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 11.7%

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

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 13.3%

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

7. Applied rewrites 13.3%

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

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 13.4%

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

13. Applied rewrites 13.4%

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

Alternative 18: 35.2% accurate, 0.1× 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))
   (- (sqrt (- t_2 -1)) (+ (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), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(t$95$2 - -1), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[t$95$2], $MachinePrecision] + N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 91.6%

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

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 11.7%

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

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 13.3%

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

7. Applied rewrites 13.3%

   \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{t} + \sqrt{x}\right)} \]
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 21.5%

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

   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 21.5%

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

9. Applied rewrites 21.5%

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

Alternative 19: 25.6% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x y z t)
  :precision binary64
  (- (+ 1 (+ (sqrt (+ 1 x)) (* 1/2 t))) (+ (sqrt t) (sqrt x))))

C

double code(double x, double y, double z, double t) {
	return (1.0 + (sqrt((1.0 + x)) + (0.5 * t))) - (sqrt(t) + sqrt(x));
}

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 + x)) + (0.5d0 * t))) - (sqrt(t) + sqrt(x))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (1.0 + (Math.sqrt((1.0 + x)) + (0.5 * t))) - (Math.sqrt(t) + Math.sqrt(x));
}

Python

def code(x, y, z, t):
	return (1.0 + (math.sqrt((1.0 + x)) + (0.5 * t))) - (math.sqrt(t) + math.sqrt(x))

Julia

function code(x, y, z, t)
	return Float64(Float64(1.0 + Float64(sqrt(Float64(1.0 + x)) + Float64(0.5 * t))) - Float64(sqrt(t) + sqrt(x)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (1.0 + (sqrt((1.0 + x)) + (0.5 * t))) - (sqrt(t) + sqrt(x));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(1 + N[(N[Sqrt[N[(1 + x), $MachinePrecision]], $MachinePrecision] + N[(1/2 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

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

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 11.7%

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

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 13.3%

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

7. Applied rewrites 13.3%

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

8. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-*.f64 13.3%

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

10. Applied rewrites 13.3%

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

Alternative 20: 15.5% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x y z t)
  :precision binary64
  (- (+ 1 (+ (sqrt (+ 1 t)) (* 1/2 x))) (+ (sqrt t) (sqrt x))))

C

double code(double x, double y, double z, double t) {
	return (1.0 + (sqrt((1.0 + t)) + (0.5 * x))) - (sqrt(t) + sqrt(x));
}

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)) + (0.5d0 * x))) - (sqrt(t) + sqrt(x))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (1.0 + (Math.sqrt((1.0 + t)) + (0.5 * x))) - (Math.sqrt(t) + Math.sqrt(x));
}

Python

def code(x, y, z, t):
	return (1.0 + (math.sqrt((1.0 + t)) + (0.5 * x))) - (math.sqrt(t) + math.sqrt(x))

Julia

function code(x, y, z, t)
	return Float64(Float64(1.0 + Float64(sqrt(Float64(1.0 + t)) + Float64(0.5 * x))) - Float64(sqrt(t) + sqrt(x)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (1.0 + (sqrt((1.0 + t)) + (0.5 * x))) - (sqrt(t) + sqrt(x));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(1 + N[(N[Sqrt[N[(1 + t), $MachinePrecision]], $MachinePrecision] + N[(1/2 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

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

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 11.7%

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

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 13.3%

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

7. Applied rewrites 13.3%

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

8. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-*.f64 13.2%

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

10. Applied rewrites 13.2%

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

Alternative 21: 13.3% accurate, 0.5× 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 (sqrt (+ 1 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 + N[Sqrt[N[(1 + 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 91.6%

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

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 11.7%

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

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 13.3%

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

7. Applied rewrites 13.3%

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

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 11.5%

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

10. Applied rewrites 11.5%

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

Alternative 22: 13.2% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (x y z t)
  :precision binary64
  (- (sqrt (+ 1 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 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

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

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 11.7%

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

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 13.3%

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

7. Applied rewrites 13.3%

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

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 2025271 -o generate:evaluate
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64
  (+ (+ (+ (- (sqrt (+ x 1)) (sqrt x)) (- (sqrt (+ y 1)) (sqrt y))) (- (sqrt (+ z 1)) (sqrt z))) (- (sqrt (+ t 1)) (sqrt t))))