Main:z from

Percentage Accurate: 91.5% → 97.5%

Time: 17.7s

Alternatives: 19

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 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
  (- (sqrt (+ z 1.0)) (sqrt z)))
 (- (sqrt (+ t 1.0)) (sqrt t))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_3 := \mathsf{min}\left(t\_2, t\right)\\ t_4 := \mathsf{max}\left(t\_2, t\right)\\ t_5 := \mathsf{min}\left(t\_1, t\_4\right)\\ t_6 := \mathsf{max}\left(t\_1, t\_4\right)\\ t_7 := \sqrt{t\_6}\\ \mathbf{if}\;t\_3 \leq 42000000000:\\ \;\;\;\;\sqrt{t\_3 - -1} - \left(\left(\sqrt{t\_3} - \left(\sqrt{\mathsf{max}\left(x, y\right) - -1} - \sqrt{\mathsf{max}\left(x, y\right)}\right)\right) - \left(\sqrt{t\_5 - -1} - \left(\sqrt{t\_5} - \frac{1}{\sqrt{t\_6 - -1} + t\_7}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.5}{t\_3 \cdot \sqrt{\frac{1}{t\_3}}} + \frac{0.5}{\mathsf{max}\left(x, y\right) \cdot \sqrt{\frac{1}{\mathsf{max}\left(x, y\right)}}}\right) + \frac{0.5}{t\_5 \cdot \sqrt{\frac{1}{t\_5}}}\right) + \left(\sqrt{t\_6 + 1} - t\_7\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmin x y) z))
       (t_3 (fmin t_2 t))
       (t_4 (fmax t_2 t))
       (t_5 (fmin t_1 t_4))
       (t_6 (fmax t_1 t_4))
       (t_7 (sqrt t_6)))
  (if (<= t_3 42000000000.0)
    (-
     (sqrt (- t_3 -1.0))
     (-
      (- (sqrt t_3) (- (sqrt (- (fmax x y) -1.0)) (sqrt (fmax x y))))
      (-
       (sqrt (- t_5 -1.0))
       (- (sqrt t_5) (/ 1.0 (+ (sqrt (- t_6 -1.0)) t_7))))))
    (+
     (+
      (+
       (/ 0.5 (* t_3 (sqrt (/ 1.0 t_3))))
       (/ 0.5 (* (fmax x y) (sqrt (/ 1.0 (fmax x y))))))
      (/ 0.5 (* t_5 (sqrt (/ 1.0 t_5)))))
     (- (sqrt (+ t_6 1.0)) t_7)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmin(x, y), z);
	double t_3 = fmin(t_2, t);
	double t_4 = fmax(t_2, t);
	double t_5 = fmin(t_1, t_4);
	double t_6 = fmax(t_1, t_4);
	double t_7 = sqrt(t_6);
	double tmp;
	if (t_3 <= 42000000000.0) {
		tmp = sqrt((t_3 - -1.0)) - ((sqrt(t_3) - (sqrt((fmax(x, y) - -1.0)) - sqrt(fmax(x, y)))) - (sqrt((t_5 - -1.0)) - (sqrt(t_5) - (1.0 / (sqrt((t_6 - -1.0)) + t_7)))));
	} else {
		tmp = (((0.5 / (t_3 * sqrt((1.0 / t_3)))) + (0.5 / (fmax(x, y) * sqrt((1.0 / fmax(x, y)))))) + (0.5 / (t_5 * sqrt((1.0 / t_5))))) + (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_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 = fmax(fmin(x, y), z)
    t_2 = fmin(fmin(x, y), z)
    t_3 = fmin(t_2, t)
    t_4 = fmax(t_2, t)
    t_5 = fmin(t_1, t_4)
    t_6 = fmax(t_1, t_4)
    t_7 = sqrt(t_6)
    if (t_3 <= 42000000000.0d0) then
        tmp = sqrt((t_3 - (-1.0d0))) - ((sqrt(t_3) - (sqrt((fmax(x, y) - (-1.0d0))) - sqrt(fmax(x, y)))) - (sqrt((t_5 - (-1.0d0))) - (sqrt(t_5) - (1.0d0 / (sqrt((t_6 - (-1.0d0))) + t_7)))))
    else
        tmp = (((0.5d0 / (t_3 * sqrt((1.0d0 / t_3)))) + (0.5d0 / (fmax(x, y) * sqrt((1.0d0 / fmax(x, y)))))) + (0.5d0 / (t_5 * sqrt((1.0d0 / t_5))))) + (sqrt((t_6 + 1.0d0)) - t_7)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmin(x, y), z);
	double t_3 = fmin(t_2, t);
	double t_4 = fmax(t_2, t);
	double t_5 = fmin(t_1, t_4);
	double t_6 = fmax(t_1, t_4);
	double t_7 = Math.sqrt(t_6);
	double tmp;
	if (t_3 <= 42000000000.0) {
		tmp = Math.sqrt((t_3 - -1.0)) - ((Math.sqrt(t_3) - (Math.sqrt((fmax(x, y) - -1.0)) - Math.sqrt(fmax(x, y)))) - (Math.sqrt((t_5 - -1.0)) - (Math.sqrt(t_5) - (1.0 / (Math.sqrt((t_6 - -1.0)) + t_7)))));
	} else {
		tmp = (((0.5 / (t_3 * Math.sqrt((1.0 / t_3)))) + (0.5 / (fmax(x, y) * Math.sqrt((1.0 / fmax(x, y)))))) + (0.5 / (t_5 * Math.sqrt((1.0 / t_5))))) + (Math.sqrt((t_6 + 1.0)) - t_7);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmin(x, y), z)
	t_3 = fmin(t_2, t)
	t_4 = fmax(t_2, t)
	t_5 = fmin(t_1, t_4)
	t_6 = fmax(t_1, t_4)
	t_7 = math.sqrt(t_6)
	tmp = 0
	if t_3 <= 42000000000.0:
		tmp = math.sqrt((t_3 - -1.0)) - ((math.sqrt(t_3) - (math.sqrt((fmax(x, y) - -1.0)) - math.sqrt(fmax(x, y)))) - (math.sqrt((t_5 - -1.0)) - (math.sqrt(t_5) - (1.0 / (math.sqrt((t_6 - -1.0)) + t_7)))))
	else:
		tmp = (((0.5 / (t_3 * math.sqrt((1.0 / t_3)))) + (0.5 / (fmax(x, y) * math.sqrt((1.0 / fmax(x, y)))))) + (0.5 / (t_5 * math.sqrt((1.0 / t_5))))) + (math.sqrt((t_6 + 1.0)) - t_7)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmin(x, y), z)
	t_3 = fmin(t_2, t)
	t_4 = fmax(t_2, t)
	t_5 = fmin(t_1, t_4)
	t_6 = fmax(t_1, t_4)
	t_7 = sqrt(t_6)
	tmp = 0.0
	if (t_3 <= 42000000000.0)
		tmp = Float64(sqrt(Float64(t_3 - -1.0)) - Float64(Float64(sqrt(t_3) - Float64(sqrt(Float64(fmax(x, y) - -1.0)) - sqrt(fmax(x, y)))) - Float64(sqrt(Float64(t_5 - -1.0)) - Float64(sqrt(t_5) - Float64(1.0 / Float64(sqrt(Float64(t_6 - -1.0)) + t_7))))));
	else
		tmp = Float64(Float64(Float64(Float64(0.5 / Float64(t_3 * sqrt(Float64(1.0 / t_3)))) + Float64(0.5 / Float64(fmax(x, y) * sqrt(Float64(1.0 / fmax(x, y)))))) + Float64(0.5 / Float64(t_5 * sqrt(Float64(1.0 / t_5))))) + Float64(sqrt(Float64(t_6 + 1.0)) - t_7));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = min(min(x, y), z);
	t_3 = min(t_2, t);
	t_4 = max(t_2, t);
	t_5 = min(t_1, t_4);
	t_6 = max(t_1, t_4);
	t_7 = sqrt(t_6);
	tmp = 0.0;
	if (t_3 <= 42000000000.0)
		tmp = sqrt((t_3 - -1.0)) - ((sqrt(t_3) - (sqrt((max(x, y) - -1.0)) - sqrt(max(x, y)))) - (sqrt((t_5 - -1.0)) - (sqrt(t_5) - (1.0 / (sqrt((t_6 - -1.0)) + t_7)))));
	else
		tmp = (((0.5 / (t_3 * sqrt((1.0 / t_3)))) + (0.5 / (max(x, y) * sqrt((1.0 / max(x, y)))))) + (0.5 / (t_5 * sqrt((1.0 / t_5))))) + (sqrt((t_6 + 1.0)) - t_7);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$3 = N[Min[t$95$2, t], $MachinePrecision]}, Block[{t$95$4 = N[Max[t$95$2, t], $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$1, t$95$4], $MachinePrecision]}, Block[{t$95$6 = N[Max[t$95$1, t$95$4], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[t$95$6], $MachinePrecision]}, If[LessEqual[t$95$3, 42000000000.0], N[(N[Sqrt[N[(t$95$3 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Sqrt[t$95$3], $MachinePrecision] - N[(N[Sqrt[N[(N[Max[x, y], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[N[Max[x, y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[(t$95$5 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[t$95$5], $MachinePrecision] - N[(1.0 / N[(N[Sqrt[N[(t$95$6 - -1.0), $MachinePrecision]], $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 / N[(t$95$3 * N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(N[Max[x, y], $MachinePrecision] * N[Sqrt[N[(1.0 / N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(t$95$5 * N[Sqrt[N[(1.0 / t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$6 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$7), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2e10

   1. Initial program 91.5%

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

      1. lift--.f64 N/A

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

      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 72.6%

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

      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 72.6%

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

   3. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt 91.9%

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 54.4%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. lower--.f64 N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. associate-+l- N/A

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

      14. sub-negate N/A

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

      15. +-inverses N/A

          \[\leadsto \sqrt{x - -1} - \left(\left(\sqrt{x} - \left(\sqrt{y - -1} - \sqrt{y}\right)\right) - \left(\sqrt{z - -1} - \left(\sqrt{z} - \frac{\color{blue}{0} - -1}{\sqrt{t - -1} + \sqrt{t}}\right)\right)\right) \]

      16. metadata-eval N/A

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

   9. Applied rewrites 44.5%

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

   if 4.2e10 < x

   1. Initial program 91.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 47.4%

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in z around inf

      \[\leadsto \left(\left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\frac{1}{2}}{z \cdot \sqrt{\frac{1}{z}}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 26.2%

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

   7. Applied rewrites 26.2%

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

   8. Taylor expanded in y around inf

      \[\leadsto \left(\left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} + \color{blue}{\frac{\frac{1}{2}}{y \cdot \sqrt{\frac{1}{y}}}}\right) + \frac{0.5}{z \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 16.2%

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

   10. Applied rewrites 16.2%

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

Alternative 2: 97.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(t\_1, t\right)\\ t_3 := \sqrt{t\_2}\\ t_4 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_4\right)\\ t_6 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_4\right)\\ t_7 := \mathsf{max}\left(t\_1, t\right)\\ t_8 := \mathsf{max}\left(t\_6, t\_7\right)\\ t_9 := \mathsf{min}\left(t\_5, t\_8\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \mathsf{min}\left(t\_6, t\_7\right)\\ t_12 := \sqrt{t\_9 - -1}\\ t_13 := \mathsf{max}\left(t\_5, t\_8\right)\\ t_14 := \sqrt{t\_13 - -1}\\ t_15 := \sqrt{t\_13}\\ \mathbf{if}\;t\_2 \leq 42000000000:\\ \;\;\;\;\sqrt{t\_2 - -1} - \left(\left(t\_3 - \left(\sqrt{t\_11 - -1} - \sqrt{t\_11}\right)\right) - \left(t\_12 - \left(t\_10 - \frac{1}{t\_14 + t\_15}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{t\_3} - \left(\frac{-0.5}{t\_11 \cdot \sqrt{\frac{1}{t\_11}}} - \left(\left(t\_12 - t\_10\right) - \left(t\_15 - t\_14\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin (fmin x y) z))
       (t_2 (fmin t_1 t))
       (t_3 (sqrt t_2))
       (t_4 (fmax (fmin x y) z))
       (t_5 (fmax (fmax x y) t_4))
       (t_6 (fmin (fmax x y) t_4))
       (t_7 (fmax t_1 t))
       (t_8 (fmax t_6 t_7))
       (t_9 (fmin t_5 t_8))
       (t_10 (sqrt t_9))
       (t_11 (fmin t_6 t_7))
       (t_12 (sqrt (- t_9 -1.0)))
       (t_13 (fmax t_5 t_8))
       (t_14 (sqrt (- t_13 -1.0)))
       (t_15 (sqrt t_13)))
  (if (<= t_2 42000000000.0)
    (-
     (sqrt (- t_2 -1.0))
     (-
      (- t_3 (- (sqrt (- t_11 -1.0)) (sqrt t_11)))
      (- t_12 (- t_10 (/ 1.0 (+ t_14 t_15))))))
    (-
     (/ 0.5 t_3)
     (-
      (/ -0.5 (* t_11 (sqrt (/ 1.0 t_11))))
      (- (- t_12 t_10) (- t_15 t_14)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmin(t_1, t);
	double t_3 = sqrt(t_2);
	double t_4 = fmax(fmin(x, y), z);
	double t_5 = fmax(fmax(x, y), t_4);
	double t_6 = fmin(fmax(x, y), t_4);
	double t_7 = fmax(t_1, t);
	double t_8 = fmax(t_6, t_7);
	double t_9 = fmin(t_5, t_8);
	double t_10 = sqrt(t_9);
	double t_11 = fmin(t_6, t_7);
	double t_12 = sqrt((t_9 - -1.0));
	double t_13 = fmax(t_5, t_8);
	double t_14 = sqrt((t_13 - -1.0));
	double t_15 = sqrt(t_13);
	double tmp;
	if (t_2 <= 42000000000.0) {
		tmp = sqrt((t_2 - -1.0)) - ((t_3 - (sqrt((t_11 - -1.0)) - sqrt(t_11))) - (t_12 - (t_10 - (1.0 / (t_14 + t_15)))));
	} else {
		tmp = (0.5 / t_3) - ((-0.5 / (t_11 * sqrt((1.0 / t_11)))) - ((t_12 - t_10) - (t_15 - t_14)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_14
    real(8) :: t_15
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmin(fmin(x, y), z)
    t_2 = fmin(t_1, t)
    t_3 = sqrt(t_2)
    t_4 = fmax(fmin(x, y), z)
    t_5 = fmax(fmax(x, y), t_4)
    t_6 = fmin(fmax(x, y), t_4)
    t_7 = fmax(t_1, t)
    t_8 = fmax(t_6, t_7)
    t_9 = fmin(t_5, t_8)
    t_10 = sqrt(t_9)
    t_11 = fmin(t_6, t_7)
    t_12 = sqrt((t_9 - (-1.0d0)))
    t_13 = fmax(t_5, t_8)
    t_14 = sqrt((t_13 - (-1.0d0)))
    t_15 = sqrt(t_13)
    if (t_2 <= 42000000000.0d0) then
        tmp = sqrt((t_2 - (-1.0d0))) - ((t_3 - (sqrt((t_11 - (-1.0d0))) - sqrt(t_11))) - (t_12 - (t_10 - (1.0d0 / (t_14 + t_15)))))
    else
        tmp = (0.5d0 / t_3) - (((-0.5d0) / (t_11 * sqrt((1.0d0 / t_11)))) - ((t_12 - t_10) - (t_15 - t_14)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmin(t_1, t);
	double t_3 = Math.sqrt(t_2);
	double t_4 = fmax(fmin(x, y), z);
	double t_5 = fmax(fmax(x, y), t_4);
	double t_6 = fmin(fmax(x, y), t_4);
	double t_7 = fmax(t_1, t);
	double t_8 = fmax(t_6, t_7);
	double t_9 = fmin(t_5, t_8);
	double t_10 = Math.sqrt(t_9);
	double t_11 = fmin(t_6, t_7);
	double t_12 = Math.sqrt((t_9 - -1.0));
	double t_13 = fmax(t_5, t_8);
	double t_14 = Math.sqrt((t_13 - -1.0));
	double t_15 = Math.sqrt(t_13);
	double tmp;
	if (t_2 <= 42000000000.0) {
		tmp = Math.sqrt((t_2 - -1.0)) - ((t_3 - (Math.sqrt((t_11 - -1.0)) - Math.sqrt(t_11))) - (t_12 - (t_10 - (1.0 / (t_14 + t_15)))));
	} else {
		tmp = (0.5 / t_3) - ((-0.5 / (t_11 * Math.sqrt((1.0 / t_11)))) - ((t_12 - t_10) - (t_15 - t_14)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmin(t_1, t)
	t_3 = math.sqrt(t_2)
	t_4 = fmax(fmin(x, y), z)
	t_5 = fmax(fmax(x, y), t_4)
	t_6 = fmin(fmax(x, y), t_4)
	t_7 = fmax(t_1, t)
	t_8 = fmax(t_6, t_7)
	t_9 = fmin(t_5, t_8)
	t_10 = math.sqrt(t_9)
	t_11 = fmin(t_6, t_7)
	t_12 = math.sqrt((t_9 - -1.0))
	t_13 = fmax(t_5, t_8)
	t_14 = math.sqrt((t_13 - -1.0))
	t_15 = math.sqrt(t_13)
	tmp = 0
	if t_2 <= 42000000000.0:
		tmp = math.sqrt((t_2 - -1.0)) - ((t_3 - (math.sqrt((t_11 - -1.0)) - math.sqrt(t_11))) - (t_12 - (t_10 - (1.0 / (t_14 + t_15)))))
	else:
		tmp = (0.5 / t_3) - ((-0.5 / (t_11 * math.sqrt((1.0 / t_11)))) - ((t_12 - t_10) - (t_15 - t_14)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmin(t_1, t)
	t_3 = sqrt(t_2)
	t_4 = fmax(fmin(x, y), z)
	t_5 = fmax(fmax(x, y), t_4)
	t_6 = fmin(fmax(x, y), t_4)
	t_7 = fmax(t_1, t)
	t_8 = fmax(t_6, t_7)
	t_9 = fmin(t_5, t_8)
	t_10 = sqrt(t_9)
	t_11 = fmin(t_6, t_7)
	t_12 = sqrt(Float64(t_9 - -1.0))
	t_13 = fmax(t_5, t_8)
	t_14 = sqrt(Float64(t_13 - -1.0))
	t_15 = sqrt(t_13)
	tmp = 0.0
	if (t_2 <= 42000000000.0)
		tmp = Float64(sqrt(Float64(t_2 - -1.0)) - Float64(Float64(t_3 - Float64(sqrt(Float64(t_11 - -1.0)) - sqrt(t_11))) - Float64(t_12 - Float64(t_10 - Float64(1.0 / Float64(t_14 + t_15))))));
	else
		tmp = Float64(Float64(0.5 / t_3) - Float64(Float64(-0.5 / Float64(t_11 * sqrt(Float64(1.0 / t_11)))) - Float64(Float64(t_12 - t_10) - Float64(t_15 - t_14))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = min(min(x, y), z);
	t_2 = min(t_1, t);
	t_3 = sqrt(t_2);
	t_4 = max(min(x, y), z);
	t_5 = max(max(x, y), t_4);
	t_6 = min(max(x, y), t_4);
	t_7 = max(t_1, t);
	t_8 = max(t_6, t_7);
	t_9 = min(t_5, t_8);
	t_10 = sqrt(t_9);
	t_11 = min(t_6, t_7);
	t_12 = sqrt((t_9 - -1.0));
	t_13 = max(t_5, t_8);
	t_14 = sqrt((t_13 - -1.0));
	t_15 = sqrt(t_13);
	tmp = 0.0;
	if (t_2 <= 42000000000.0)
		tmp = sqrt((t_2 - -1.0)) - ((t_3 - (sqrt((t_11 - -1.0)) - sqrt(t_11))) - (t_12 - (t_10 - (1.0 / (t_14 + t_15)))));
	else
		tmp = (0.5 / t_3) - ((-0.5 / (t_11 * sqrt((1.0 / t_11)))) - ((t_12 - t_10) - (t_15 - t_14)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[t$95$1, t], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$5 = N[Max[N[Max[x, y], $MachinePrecision], t$95$4], $MachinePrecision]}, Block[{t$95$6 = N[Min[N[Max[x, y], $MachinePrecision], t$95$4], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$1, t], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$6, t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$5, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Min[t$95$6, t$95$7], $MachinePrecision]}, Block[{t$95$12 = N[Sqrt[N[(t$95$9 - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$13 = N[Max[t$95$5, t$95$8], $MachinePrecision]}, Block[{t$95$14 = N[Sqrt[N[(t$95$13 - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$15 = N[Sqrt[t$95$13], $MachinePrecision]}, If[LessEqual[t$95$2, 42000000000.0], N[(N[Sqrt[N[(t$95$2 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(t$95$3 - N[(N[Sqrt[N[(t$95$11 - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$11], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$12 - N[(t$95$10 - N[(1.0 / N[(t$95$14 + t$95$15), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / t$95$3), $MachinePrecision] - N[(N[(-0.5 / N[(t$95$11 * N[Sqrt[N[(1.0 / t$95$11), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$12 - t$95$10), $MachinePrecision] - N[(t$95$15 - t$95$14), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2e10

   1. Initial program 91.5%

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

      1. lift--.f64 N/A

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

      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 72.6%

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

      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 72.6%

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

   3. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt 91.9%

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 54.4%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. lower--.f64 N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. associate-+l- N/A

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

      14. sub-negate N/A

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

      15. +-inverses N/A

          \[\leadsto \sqrt{x - -1} - \left(\left(\sqrt{x} - \left(\sqrt{y - -1} - \sqrt{y}\right)\right) - \left(\sqrt{z - -1} - \left(\sqrt{z} - \frac{\color{blue}{0} - -1}{\sqrt{t - -1} + \sqrt{t}}\right)\right)\right) \]

      16. metadata-eval N/A

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

   9. Applied rewrites 44.5%

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

   if 4.2e10 < x

   1. Initial program 91.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 47.4%

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

   4. Applied rewrites 47.4%

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

   6. Applied rewrites 47.4%

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

   7. Taylor expanded in y around inf

      \[\leadsto \frac{0.5}{\sqrt{x}} - \left(\color{blue}{\frac{\frac{-1}{2}}{y \cdot \sqrt{\frac{1}{y}}}} - \left(\left(\sqrt{z - -1} - \sqrt{z}\right) - \left(\sqrt{t} - \sqrt{t - -1}\right)\right)\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 26.3%

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

   9. Applied rewrites 26.3%

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

Alternative 3: 96.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{max}\left(t\_4, t\right)\\ t_8 := \mathsf{max}\left(t\_3, t\_7\right)\\ t_9 := \sqrt{t\_8}\\ t_10 := \mathsf{min}\left(t\_3, t\_7\right)\\ t_11 := \sqrt{t\_8 + 1} - t\_9\\ t_12 := \sqrt{t\_10}\\ t_13 := \sqrt{t\_10 + 1} - t\_12\\ t_14 := \left(\sqrt{t\_5 + 1} - t\_6\right) + t\_13\\ \mathbf{if}\;t\_14 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.5}{t\_6} - \left(\left(t\_9 + t\_12\right) - \left(\sqrt{1 + t\_8} + \sqrt{1 + t\_10}\right)\right)\\ \mathbf{elif}\;t\_14 \leq 1.002:\\ \;\;\;\;\left(\left(\sqrt{1 + t\_5} + 0.5 \cdot \frac{1}{t\_10 \cdot \sqrt{\frac{1}{t\_10}}}\right) - t\_6\right) + t\_11\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - t\_6\right) + t\_13\right) + \left(\sqrt{t\_2 + 1} - \sqrt{t\_2}\right)\right) + t\_11\\ \end{array} \]

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 (sqrt t_8))
       (t_10 (fmin t_3 t_7))
       (t_11 (- (sqrt (+ t_8 1.0)) t_9))
       (t_12 (sqrt t_10))
       (t_13 (- (sqrt (+ t_10 1.0)) t_12))
       (t_14 (+ (- (sqrt (+ t_5 1.0)) t_6) t_13)))
  (if (<= t_14 4e-6)
    (-
     (/ 0.5 t_6)
     (- (+ t_9 t_12) (+ (sqrt (+ 1.0 t_8)) (sqrt (+ 1.0 t_10)))))
    (if (<= t_14 1.002)
      (+
       (-
        (+
         (sqrt (+ 1.0 t_5))
         (* 0.5 (/ 1.0 (* t_10 (sqrt (/ 1.0 t_10))))))
        t_6)
       t_11)
      (+
       (+ (+ (- 1.0 t_6) t_13) (- (sqrt (+ t_2 1.0)) (sqrt t_2)))
       t_11)))))

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 = sqrt(t_8);
	double t_10 = fmin(t_3, t_7);
	double t_11 = sqrt((t_8 + 1.0)) - t_9;
	double t_12 = sqrt(t_10);
	double t_13 = sqrt((t_10 + 1.0)) - t_12;
	double t_14 = (sqrt((t_5 + 1.0)) - t_6) + t_13;
	double tmp;
	if (t_14 <= 4e-6) {
		tmp = (0.5 / t_6) - ((t_9 + t_12) - (sqrt((1.0 + t_8)) + sqrt((1.0 + t_10))));
	} else if (t_14 <= 1.002) {
		tmp = ((sqrt((1.0 + t_5)) + (0.5 * (1.0 / (t_10 * sqrt((1.0 / t_10)))))) - t_6) + t_11;
	} else {
		tmp = (((1.0 - t_6) + t_13) + (sqrt((t_2 + 1.0)) - sqrt(t_2))) + t_11;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_14
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmin(t_4, t)
    t_6 = sqrt(t_5)
    t_7 = fmax(t_4, t)
    t_8 = fmax(t_3, t_7)
    t_9 = sqrt(t_8)
    t_10 = fmin(t_3, t_7)
    t_11 = sqrt((t_8 + 1.0d0)) - t_9
    t_12 = sqrt(t_10)
    t_13 = sqrt((t_10 + 1.0d0)) - t_12
    t_14 = (sqrt((t_5 + 1.0d0)) - t_6) + t_13
    if (t_14 <= 4d-6) then
        tmp = (0.5d0 / t_6) - ((t_9 + t_12) - (sqrt((1.0d0 + t_8)) + sqrt((1.0d0 + t_10))))
    else if (t_14 <= 1.002d0) then
        tmp = ((sqrt((1.0d0 + t_5)) + (0.5d0 * (1.0d0 / (t_10 * sqrt((1.0d0 / t_10)))))) - t_6) + t_11
    else
        tmp = (((1.0d0 - t_6) + t_13) + (sqrt((t_2 + 1.0d0)) - sqrt(t_2))) + t_11
    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 = Math.sqrt(t_8);
	double t_10 = fmin(t_3, t_7);
	double t_11 = Math.sqrt((t_8 + 1.0)) - t_9;
	double t_12 = Math.sqrt(t_10);
	double t_13 = Math.sqrt((t_10 + 1.0)) - t_12;
	double t_14 = (Math.sqrt((t_5 + 1.0)) - t_6) + t_13;
	double tmp;
	if (t_14 <= 4e-6) {
		tmp = (0.5 / t_6) - ((t_9 + t_12) - (Math.sqrt((1.0 + t_8)) + Math.sqrt((1.0 + t_10))));
	} else if (t_14 <= 1.002) {
		tmp = ((Math.sqrt((1.0 + t_5)) + (0.5 * (1.0 / (t_10 * Math.sqrt((1.0 / t_10)))))) - t_6) + t_11;
	} else {
		tmp = (((1.0 - t_6) + t_13) + (Math.sqrt((t_2 + 1.0)) - Math.sqrt(t_2))) + t_11;
	}
	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 = math.sqrt(t_8)
	t_10 = fmin(t_3, t_7)
	t_11 = math.sqrt((t_8 + 1.0)) - t_9
	t_12 = math.sqrt(t_10)
	t_13 = math.sqrt((t_10 + 1.0)) - t_12
	t_14 = (math.sqrt((t_5 + 1.0)) - t_6) + t_13
	tmp = 0
	if t_14 <= 4e-6:
		tmp = (0.5 / t_6) - ((t_9 + t_12) - (math.sqrt((1.0 + t_8)) + math.sqrt((1.0 + t_10))))
	elif t_14 <= 1.002:
		tmp = ((math.sqrt((1.0 + t_5)) + (0.5 * (1.0 / (t_10 * math.sqrt((1.0 / t_10)))))) - t_6) + t_11
	else:
		tmp = (((1.0 - t_6) + t_13) + (math.sqrt((t_2 + 1.0)) - math.sqrt(t_2))) + t_11
	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 = sqrt(t_8)
	t_10 = fmin(t_3, t_7)
	t_11 = Float64(sqrt(Float64(t_8 + 1.0)) - t_9)
	t_12 = sqrt(t_10)
	t_13 = Float64(sqrt(Float64(t_10 + 1.0)) - t_12)
	t_14 = Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + t_13)
	tmp = 0.0
	if (t_14 <= 4e-6)
		tmp = Float64(Float64(0.5 / t_6) - Float64(Float64(t_9 + t_12) - Float64(sqrt(Float64(1.0 + t_8)) + sqrt(Float64(1.0 + t_10)))));
	elseif (t_14 <= 1.002)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t_5)) + Float64(0.5 * Float64(1.0 / Float64(t_10 * sqrt(Float64(1.0 / t_10)))))) - t_6) + t_11);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - t_6) + t_13) + Float64(sqrt(Float64(t_2 + 1.0)) - sqrt(t_2))) + t_11);
	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 = sqrt(t_8);
	t_10 = min(t_3, t_7);
	t_11 = sqrt((t_8 + 1.0)) - t_9;
	t_12 = sqrt(t_10);
	t_13 = sqrt((t_10 + 1.0)) - t_12;
	t_14 = (sqrt((t_5 + 1.0)) - t_6) + t_13;
	tmp = 0.0;
	if (t_14 <= 4e-6)
		tmp = (0.5 / t_6) - ((t_9 + t_12) - (sqrt((1.0 + t_8)) + sqrt((1.0 + t_10))));
	elseif (t_14 <= 1.002)
		tmp = ((sqrt((1.0 + t_5)) + (0.5 * (1.0 / (t_10 * sqrt((1.0 / t_10)))))) - t_6) + t_11;
	else
		tmp = (((1.0 - t_6) + t_13) + (sqrt((t_2 + 1.0)) - sqrt(t_2))) + t_11;
	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[Sqrt[t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Min[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$11 = N[(N[Sqrt[N[(t$95$8 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$9), $MachinePrecision]}, Block[{t$95$12 = N[Sqrt[t$95$10], $MachinePrecision]}, Block[{t$95$13 = N[(N[Sqrt[N[(t$95$10 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$12), $MachinePrecision]}, Block[{t$95$14 = N[(N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + t$95$13), $MachinePrecision]}, If[LessEqual[t$95$14, 4e-6], N[(N[(0.5 / t$95$6), $MachinePrecision] - N[(N[(t$95$9 + t$95$12), $MachinePrecision] - N[(N[Sqrt[N[(1.0 + t$95$8), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + t$95$10), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$14, 1.002], N[(N[(N[(N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[(t$95$10 * N[Sqrt[N[(1.0 / t$95$10), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision] + t$95$11), $MachinePrecision], N[(N[(N[(N[(1.0 - t$95$6), $MachinePrecision] + t$95$13), $MachinePrecision] + N[(N[Sqrt[N[(t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$11), $MachinePrecision]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.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))) < 3.9999999999999998e-6

   1. Initial program 91.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 47.4%

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

   4. Applied rewrites 47.4%

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

   6. Applied rewrites 47.4%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 15.6%

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

   9. Applied rewrites 15.6%

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

   if 3.9999999999999998e-6 < (+.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))) < 1.002

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in z around 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 29.7%

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

   7. Applied rewrites 29.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 21.7%

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

   10. Applied rewrites 21.7%

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

   11. Taylor expanded in y around inf

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-sqrt.f64 26.1%

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

   13. Applied rewrites 26.1%

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

   if 1.002 < (+.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)))

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 48.8%

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

   4. Applied rewrites 48.8%

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

Alternative 4: 96.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(y, \mathsf{max}\left(x, z\right)\right)\\ t_2 := \mathsf{min}\left(t\_1, t\right)\\ t_3 := \mathsf{max}\left(t\_1, t\right)\\ t_4 := \sqrt{t\_3}\\ t_5 := \sqrt{\mathsf{min}\left(x, z\right)}\\ t_6 := \left(\sqrt{\mathsf{min}\left(x, z\right) + 1} - t\_5\right) + \left(\sqrt{t\_2 + 1} - \sqrt{t\_2}\right)\\ t_7 := \mathsf{max}\left(y, \mathsf{max}\left(x, z\right)\right)\\ t_8 := \sqrt{t\_7}\\ \mathbf{if}\;t\_6 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.5}{t\_5} - \left(\frac{-0.5}{t\_2 \cdot \sqrt{\frac{1}{t\_2}}} - \left(\left(\sqrt{t\_7 - -1} - t\_8\right) - \left(t\_4 - \sqrt{t\_3 - -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_6 + \left(\sqrt{t\_7 + 1} - t\_8\right)\right) + \left(\sqrt{t\_3 + 1} - t\_4\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin y (fmax x z)))
       (t_2 (fmin t_1 t))
       (t_3 (fmax t_1 t))
       (t_4 (sqrt t_3))
       (t_5 (sqrt (fmin x z)))
       (t_6
        (+
         (- (sqrt (+ (fmin x z) 1.0)) t_5)
         (- (sqrt (+ t_2 1.0)) (sqrt t_2))))
       (t_7 (fmax y (fmax x z)))
       (t_8 (sqrt t_7)))
  (if (<= t_6 4e-6)
    (-
     (/ 0.5 t_5)
     (-
      (/ -0.5 (* t_2 (sqrt (/ 1.0 t_2))))
      (- (- (sqrt (- t_7 -1.0)) t_8) (- t_4 (sqrt (- t_3 -1.0))))))
    (+
     (+ t_6 (- (sqrt (+ t_7 1.0)) t_8))
     (- (sqrt (+ t_3 1.0)) t_4)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(y, fmax(x, z));
	double t_2 = fmin(t_1, t);
	double t_3 = fmax(t_1, t);
	double t_4 = sqrt(t_3);
	double t_5 = sqrt(fmin(x, z));
	double t_6 = (sqrt((fmin(x, z) + 1.0)) - t_5) + (sqrt((t_2 + 1.0)) - sqrt(t_2));
	double t_7 = fmax(y, fmax(x, z));
	double t_8 = sqrt(t_7);
	double tmp;
	if (t_6 <= 4e-6) {
		tmp = (0.5 / t_5) - ((-0.5 / (t_2 * sqrt((1.0 / t_2)))) - ((sqrt((t_7 - -1.0)) - t_8) - (t_4 - sqrt((t_3 - -1.0)))));
	} else {
		tmp = (t_6 + (sqrt((t_7 + 1.0)) - t_8)) + (sqrt((t_3 + 1.0)) - t_4);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: tmp
    t_1 = fmin(y, fmax(x, z))
    t_2 = fmin(t_1, t)
    t_3 = fmax(t_1, t)
    t_4 = sqrt(t_3)
    t_5 = sqrt(fmin(x, z))
    t_6 = (sqrt((fmin(x, z) + 1.0d0)) - t_5) + (sqrt((t_2 + 1.0d0)) - sqrt(t_2))
    t_7 = fmax(y, fmax(x, z))
    t_8 = sqrt(t_7)
    if (t_6 <= 4d-6) then
        tmp = (0.5d0 / t_5) - (((-0.5d0) / (t_2 * sqrt((1.0d0 / t_2)))) - ((sqrt((t_7 - (-1.0d0))) - t_8) - (t_4 - sqrt((t_3 - (-1.0d0))))))
    else
        tmp = (t_6 + (sqrt((t_7 + 1.0d0)) - t_8)) + (sqrt((t_3 + 1.0d0)) - 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(y, fmax(x, z));
	double t_2 = fmin(t_1, t);
	double t_3 = fmax(t_1, t);
	double t_4 = Math.sqrt(t_3);
	double t_5 = Math.sqrt(fmin(x, z));
	double t_6 = (Math.sqrt((fmin(x, z) + 1.0)) - t_5) + (Math.sqrt((t_2 + 1.0)) - Math.sqrt(t_2));
	double t_7 = fmax(y, fmax(x, z));
	double t_8 = Math.sqrt(t_7);
	double tmp;
	if (t_6 <= 4e-6) {
		tmp = (0.5 / t_5) - ((-0.5 / (t_2 * Math.sqrt((1.0 / t_2)))) - ((Math.sqrt((t_7 - -1.0)) - t_8) - (t_4 - Math.sqrt((t_3 - -1.0)))));
	} else {
		tmp = (t_6 + (Math.sqrt((t_7 + 1.0)) - t_8)) + (Math.sqrt((t_3 + 1.0)) - t_4);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(y, fmax(x, z))
	t_2 = fmin(t_1, t)
	t_3 = fmax(t_1, t)
	t_4 = math.sqrt(t_3)
	t_5 = math.sqrt(fmin(x, z))
	t_6 = (math.sqrt((fmin(x, z) + 1.0)) - t_5) + (math.sqrt((t_2 + 1.0)) - math.sqrt(t_2))
	t_7 = fmax(y, fmax(x, z))
	t_8 = math.sqrt(t_7)
	tmp = 0
	if t_6 <= 4e-6:
		tmp = (0.5 / t_5) - ((-0.5 / (t_2 * math.sqrt((1.0 / t_2)))) - ((math.sqrt((t_7 - -1.0)) - t_8) - (t_4 - math.sqrt((t_3 - -1.0)))))
	else:
		tmp = (t_6 + (math.sqrt((t_7 + 1.0)) - t_8)) + (math.sqrt((t_3 + 1.0)) - t_4)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(y, fmax(x, z))
	t_2 = fmin(t_1, t)
	t_3 = fmax(t_1, t)
	t_4 = sqrt(t_3)
	t_5 = sqrt(fmin(x, z))
	t_6 = Float64(Float64(sqrt(Float64(fmin(x, z) + 1.0)) - t_5) + Float64(sqrt(Float64(t_2 + 1.0)) - sqrt(t_2)))
	t_7 = fmax(y, fmax(x, z))
	t_8 = sqrt(t_7)
	tmp = 0.0
	if (t_6 <= 4e-6)
		tmp = Float64(Float64(0.5 / t_5) - Float64(Float64(-0.5 / Float64(t_2 * sqrt(Float64(1.0 / t_2)))) - Float64(Float64(sqrt(Float64(t_7 - -1.0)) - t_8) - Float64(t_4 - sqrt(Float64(t_3 - -1.0))))));
	else
		tmp = Float64(Float64(t_6 + Float64(sqrt(Float64(t_7 + 1.0)) - t_8)) + Float64(sqrt(Float64(t_3 + 1.0)) - t_4));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = min(y, max(x, z));
	t_2 = min(t_1, t);
	t_3 = max(t_1, t);
	t_4 = sqrt(t_3);
	t_5 = sqrt(min(x, z));
	t_6 = (sqrt((min(x, z) + 1.0)) - t_5) + (sqrt((t_2 + 1.0)) - sqrt(t_2));
	t_7 = max(y, max(x, z));
	t_8 = sqrt(t_7);
	tmp = 0.0;
	if (t_6 <= 4e-6)
		tmp = (0.5 / t_5) - ((-0.5 / (t_2 * sqrt((1.0 / t_2)))) - ((sqrt((t_7 - -1.0)) - t_8) - (t_4 - sqrt((t_3 - -1.0)))));
	else
		tmp = (t_6 + (sqrt((t_7 + 1.0)) - t_8)) + (sqrt((t_3 + 1.0)) - t_4);
	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[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$3], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[Min[x, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[Sqrt[N[(N[Min[x, z], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$5), $MachinePrecision] + N[(N[Sqrt[N[(t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Max[y, N[Max[x, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$8 = N[Sqrt[t$95$7], $MachinePrecision]}, If[LessEqual[t$95$6, 4e-6], N[(N[(0.5 / t$95$5), $MachinePrecision] - N[(N[(-0.5 / N[(t$95$2 * N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[N[(t$95$7 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$8), $MachinePrecision] - N[(t$95$4 - N[Sqrt[N[(t$95$3 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$6 + N[(N[Sqrt[N[(t$95$7 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$8), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$3 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.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))) < 3.9999999999999998e-6

   1. Initial program 91.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 47.4%

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

   4. Applied rewrites 47.4%

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

   6. Applied rewrites 47.4%

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

   7. Taylor expanded in y around inf

      \[\leadsto \frac{0.5}{\sqrt{x}} - \left(\color{blue}{\frac{\frac{-1}{2}}{y \cdot \sqrt{\frac{1}{y}}}} - \left(\left(\sqrt{z - -1} - \sqrt{z}\right) - \left(\sqrt{t} - \sqrt{t - -1}\right)\right)\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 26.3%

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

   9. Applied rewrites 26.3%

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

   if 3.9999999999999998e-6 < (+.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)))

   1. Initial program 91.5%

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

Alternative 5: 95.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(t\_1, t\right)\\ t_3 := \mathsf{min}\left(t\_1, t\right)\\ t_4 := \sqrt{t\_3}\\ t_5 := \sqrt{t\_3 + 1} - t\_4\\ t_6 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_7 := \mathsf{max}\left(t\_6, t\_2\right)\\ t_8 := \sqrt{t\_7}\\ t_9 := \sqrt{\mathsf{max}\left(x, y\right)}\\ t_10 := \mathsf{min}\left(t\_6, t\_2\right)\\ \mathbf{if}\;t\_5 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.5}{t\_4} - \left(\left(t\_8 + t\_9\right) - \left(\sqrt{1 + t\_7} + \sqrt{1 + \mathsf{max}\left(x, y\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_5 + \left(\sqrt{\mathsf{max}\left(x, y\right) + 1} - t\_9\right)\right) + \left(\sqrt{t\_10 + 1} - \sqrt{t\_10}\right)\right) + \left(\sqrt{t\_7 + 1} - t\_8\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin (fmin x y) z))
       (t_2 (fmax t_1 t))
       (t_3 (fmin t_1 t))
       (t_4 (sqrt t_3))
       (t_5 (- (sqrt (+ t_3 1.0)) t_4))
       (t_6 (fmax (fmin x y) z))
       (t_7 (fmax t_6 t_2))
       (t_8 (sqrt t_7))
       (t_9 (sqrt (fmax x y)))
       (t_10 (fmin t_6 t_2)))
  (if (<= t_5 4e-6)
    (-
     (/ 0.5 t_4)
     (- (+ t_8 t_9) (+ (sqrt (+ 1.0 t_7)) (sqrt (+ 1.0 (fmax x y))))))
    (+
     (+
      (+ t_5 (- (sqrt (+ (fmax x y) 1.0)) t_9))
      (- (sqrt (+ t_10 1.0)) (sqrt t_10)))
     (- (sqrt (+ t_7 1.0)) 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 = fmin(t_1, t);
	double t_4 = sqrt(t_3);
	double t_5 = sqrt((t_3 + 1.0)) - t_4;
	double t_6 = fmax(fmin(x, y), z);
	double t_7 = fmax(t_6, t_2);
	double t_8 = sqrt(t_7);
	double t_9 = sqrt(fmax(x, y));
	double t_10 = fmin(t_6, t_2);
	double tmp;
	if (t_5 <= 4e-6) {
		tmp = (0.5 / t_4) - ((t_8 + t_9) - (sqrt((1.0 + t_7)) + sqrt((1.0 + fmax(x, y)))));
	} else {
		tmp = ((t_5 + (sqrt((fmax(x, y) + 1.0)) - t_9)) + (sqrt((t_10 + 1.0)) - sqrt(t_10))) + (sqrt((t_7 + 1.0)) - 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_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 = fmin(t_1, t)
    t_4 = sqrt(t_3)
    t_5 = sqrt((t_3 + 1.0d0)) - t_4
    t_6 = fmax(fmin(x, y), z)
    t_7 = fmax(t_6, t_2)
    t_8 = sqrt(t_7)
    t_9 = sqrt(fmax(x, y))
    t_10 = fmin(t_6, t_2)
    if (t_5 <= 4d-6) then
        tmp = (0.5d0 / t_4) - ((t_8 + t_9) - (sqrt((1.0d0 + t_7)) + sqrt((1.0d0 + fmax(x, y)))))
    else
        tmp = ((t_5 + (sqrt((fmax(x, y) + 1.0d0)) - t_9)) + (sqrt((t_10 + 1.0d0)) - sqrt(t_10))) + (sqrt((t_7 + 1.0d0)) - 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 = fmin(t_1, t);
	double t_4 = Math.sqrt(t_3);
	double t_5 = Math.sqrt((t_3 + 1.0)) - t_4;
	double t_6 = fmax(fmin(x, y), z);
	double t_7 = fmax(t_6, t_2);
	double t_8 = Math.sqrt(t_7);
	double t_9 = Math.sqrt(fmax(x, y));
	double t_10 = fmin(t_6, t_2);
	double tmp;
	if (t_5 <= 4e-6) {
		tmp = (0.5 / t_4) - ((t_8 + t_9) - (Math.sqrt((1.0 + t_7)) + Math.sqrt((1.0 + fmax(x, y)))));
	} else {
		tmp = ((t_5 + (Math.sqrt((fmax(x, y) + 1.0)) - t_9)) + (Math.sqrt((t_10 + 1.0)) - Math.sqrt(t_10))) + (Math.sqrt((t_7 + 1.0)) - 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 = fmin(t_1, t)
	t_4 = math.sqrt(t_3)
	t_5 = math.sqrt((t_3 + 1.0)) - t_4
	t_6 = fmax(fmin(x, y), z)
	t_7 = fmax(t_6, t_2)
	t_8 = math.sqrt(t_7)
	t_9 = math.sqrt(fmax(x, y))
	t_10 = fmin(t_6, t_2)
	tmp = 0
	if t_5 <= 4e-6:
		tmp = (0.5 / t_4) - ((t_8 + t_9) - (math.sqrt((1.0 + t_7)) + math.sqrt((1.0 + fmax(x, y)))))
	else:
		tmp = ((t_5 + (math.sqrt((fmax(x, y) + 1.0)) - t_9)) + (math.sqrt((t_10 + 1.0)) - math.sqrt(t_10))) + (math.sqrt((t_7 + 1.0)) - 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 = fmin(t_1, t)
	t_4 = sqrt(t_3)
	t_5 = Float64(sqrt(Float64(t_3 + 1.0)) - t_4)
	t_6 = fmax(fmin(x, y), z)
	t_7 = fmax(t_6, t_2)
	t_8 = sqrt(t_7)
	t_9 = sqrt(fmax(x, y))
	t_10 = fmin(t_6, t_2)
	tmp = 0.0
	if (t_5 <= 4e-6)
		tmp = Float64(Float64(0.5 / t_4) - Float64(Float64(t_8 + t_9) - Float64(sqrt(Float64(1.0 + t_7)) + sqrt(Float64(1.0 + fmax(x, y))))));
	else
		tmp = Float64(Float64(Float64(t_5 + Float64(sqrt(Float64(fmax(x, y) + 1.0)) - t_9)) + Float64(sqrt(Float64(t_10 + 1.0)) - sqrt(t_10))) + Float64(sqrt(Float64(t_7 + 1.0)) - 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 = min(t_1, t);
	t_4 = sqrt(t_3);
	t_5 = sqrt((t_3 + 1.0)) - t_4;
	t_6 = max(min(x, y), z);
	t_7 = max(t_6, t_2);
	t_8 = sqrt(t_7);
	t_9 = sqrt(max(x, y));
	t_10 = min(t_6, t_2);
	tmp = 0.0;
	if (t_5 <= 4e-6)
		tmp = (0.5 / t_4) - ((t_8 + t_9) - (sqrt((1.0 + t_7)) + sqrt((1.0 + max(x, y)))));
	else
		tmp = ((t_5 + (sqrt((max(x, y) + 1.0)) - t_9)) + (sqrt((t_10 + 1.0)) - sqrt(t_10))) + (sqrt((t_7 + 1.0)) - 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[Min[t$95$1, t], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(t$95$3 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$6, t$95$2], $MachinePrecision]}, Block[{t$95$8 = N[Sqrt[t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Sqrt[N[Max[x, y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$10 = N[Min[t$95$6, t$95$2], $MachinePrecision]}, If[LessEqual[t$95$5, 4e-6], N[(N[(0.5 / t$95$4), $MachinePrecision] - N[(N[(t$95$8 + t$95$9), $MachinePrecision] - N[(N[Sqrt[N[(1.0 + t$95$7), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$5 + N[(N[Sqrt[N[(N[Max[x, y], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$9), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$10 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$10], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$7 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$8), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 3.9999999999999998e-6

   1. Initial program 91.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 47.4%

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

   4. Applied rewrites 47.4%

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

   6. Applied rewrites 47.4%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 15.6%

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

   9. Applied rewrites 15.6%

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

   if 3.9999999999999998e-6 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 91.5%

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

Alternative 6: 95.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_3 := \mathsf{max}\left(t\_2, t\right)\\ t_4 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_3\right)\\ t_5 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_3\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{min}\left(t\_1, t\_4\right)\\ t_8 := \sqrt{t\_7}\\ t_9 := \mathsf{max}\left(t\_1, t\_4\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \sqrt{t\_9 + 1} - t\_10\\ t_12 := \mathsf{min}\left(t\_2, t\right)\\ t_13 := \sqrt{t\_12}\\ t_14 := \sqrt{t\_7 + 1} - t\_8\\ t_15 := \left(\left(\left(\sqrt{t\_12 + 1} - t\_13\right) + \left(\sqrt{t\_5 + 1} - t\_6\right)\right) + t\_14\right) + t\_11\\ \mathbf{if}\;t\_15 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.5}{t\_13} - \left(\left(t\_10 + t\_6\right) - \left(\sqrt{1 + t\_9} + \sqrt{1 + t\_5}\right)\right)\\ \mathbf{elif}\;t\_15 \leq 3:\\ \;\;\;\;\sqrt{t\_12 - -1} - \left(\left(t\_13 - \left(\sqrt{t\_5 - -1} - t\_6\right)\right) - \left(\sqrt{1 + t\_7} - t\_8\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(1 + 0.5 \cdot t\_12\right) - t\_13\right) + \left(1 - t\_6\right)\right) + t\_14\right) + t\_11\\ \end{array} \]

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 (fmin (fmax x y) t_3))
       (t_6 (sqrt t_5))
       (t_7 (fmin t_1 t_4))
       (t_8 (sqrt t_7))
       (t_9 (fmax t_1 t_4))
       (t_10 (sqrt t_9))
       (t_11 (- (sqrt (+ t_9 1.0)) t_10))
       (t_12 (fmin t_2 t))
       (t_13 (sqrt t_12))
       (t_14 (- (sqrt (+ t_7 1.0)) t_8))
       (t_15
        (+
         (+
          (+ (- (sqrt (+ t_12 1.0)) t_13) (- (sqrt (+ t_5 1.0)) t_6))
          t_14)
         t_11)))
  (if (<= t_15 4e-6)
    (-
     (/ 0.5 t_13)
     (- (+ t_10 t_6) (+ (sqrt (+ 1.0 t_9)) (sqrt (+ 1.0 t_5)))))
    (if (<= t_15 3.0)
      (-
       (sqrt (- t_12 -1.0))
       (-
        (- t_13 (- (sqrt (- t_5 -1.0)) t_6))
        (- (sqrt (+ 1.0 t_7)) t_8)))
      (+
       (+ (+ (- (+ 1.0 (* 0.5 t_12)) t_13) (- 1.0 t_6)) t_14)
       t_11)))))

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 = fmin(fmax(x, y), t_3);
	double t_6 = sqrt(t_5);
	double t_7 = fmin(t_1, t_4);
	double t_8 = sqrt(t_7);
	double t_9 = fmax(t_1, t_4);
	double t_10 = sqrt(t_9);
	double t_11 = sqrt((t_9 + 1.0)) - t_10;
	double t_12 = fmin(t_2, t);
	double t_13 = sqrt(t_12);
	double t_14 = sqrt((t_7 + 1.0)) - t_8;
	double t_15 = (((sqrt((t_12 + 1.0)) - t_13) + (sqrt((t_5 + 1.0)) - t_6)) + t_14) + t_11;
	double tmp;
	if (t_15 <= 4e-6) {
		tmp = (0.5 / t_13) - ((t_10 + t_6) - (sqrt((1.0 + t_9)) + sqrt((1.0 + t_5))));
	} else if (t_15 <= 3.0) {
		tmp = sqrt((t_12 - -1.0)) - ((t_13 - (sqrt((t_5 - -1.0)) - t_6)) - (sqrt((1.0 + t_7)) - t_8));
	} else {
		tmp = ((((1.0 + (0.5 * t_12)) - t_13) + (1.0 - t_6)) + t_14) + t_11;
	}
	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 = fmin(fmin(x, y), z)
    t_3 = fmax(t_2, t)
    t_4 = fmax(fmax(x, y), t_3)
    t_5 = fmin(fmax(x, y), t_3)
    t_6 = sqrt(t_5)
    t_7 = fmin(t_1, t_4)
    t_8 = sqrt(t_7)
    t_9 = fmax(t_1, t_4)
    t_10 = sqrt(t_9)
    t_11 = sqrt((t_9 + 1.0d0)) - t_10
    t_12 = fmin(t_2, t)
    t_13 = sqrt(t_12)
    t_14 = sqrt((t_7 + 1.0d0)) - t_8
    t_15 = (((sqrt((t_12 + 1.0d0)) - t_13) + (sqrt((t_5 + 1.0d0)) - t_6)) + t_14) + t_11
    if (t_15 <= 4d-6) then
        tmp = (0.5d0 / t_13) - ((t_10 + t_6) - (sqrt((1.0d0 + t_9)) + sqrt((1.0d0 + t_5))))
    else if (t_15 <= 3.0d0) then
        tmp = sqrt((t_12 - (-1.0d0))) - ((t_13 - (sqrt((t_5 - (-1.0d0))) - t_6)) - (sqrt((1.0d0 + t_7)) - t_8))
    else
        tmp = ((((1.0d0 + (0.5d0 * t_12)) - t_13) + (1.0d0 - t_6)) + t_14) + t_11
    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 = fmin(fmax(x, y), t_3);
	double t_6 = Math.sqrt(t_5);
	double t_7 = fmin(t_1, t_4);
	double t_8 = Math.sqrt(t_7);
	double t_9 = fmax(t_1, t_4);
	double t_10 = Math.sqrt(t_9);
	double t_11 = Math.sqrt((t_9 + 1.0)) - t_10;
	double t_12 = fmin(t_2, t);
	double t_13 = Math.sqrt(t_12);
	double t_14 = Math.sqrt((t_7 + 1.0)) - t_8;
	double t_15 = (((Math.sqrt((t_12 + 1.0)) - t_13) + (Math.sqrt((t_5 + 1.0)) - t_6)) + t_14) + t_11;
	double tmp;
	if (t_15 <= 4e-6) {
		tmp = (0.5 / t_13) - ((t_10 + t_6) - (Math.sqrt((1.0 + t_9)) + Math.sqrt((1.0 + t_5))));
	} else if (t_15 <= 3.0) {
		tmp = Math.sqrt((t_12 - -1.0)) - ((t_13 - (Math.sqrt((t_5 - -1.0)) - t_6)) - (Math.sqrt((1.0 + t_7)) - t_8));
	} else {
		tmp = ((((1.0 + (0.5 * t_12)) - t_13) + (1.0 - t_6)) + t_14) + t_11;
	}
	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 = fmin(fmax(x, y), t_3)
	t_6 = math.sqrt(t_5)
	t_7 = fmin(t_1, t_4)
	t_8 = math.sqrt(t_7)
	t_9 = fmax(t_1, t_4)
	t_10 = math.sqrt(t_9)
	t_11 = math.sqrt((t_9 + 1.0)) - t_10
	t_12 = fmin(t_2, t)
	t_13 = math.sqrt(t_12)
	t_14 = math.sqrt((t_7 + 1.0)) - t_8
	t_15 = (((math.sqrt((t_12 + 1.0)) - t_13) + (math.sqrt((t_5 + 1.0)) - t_6)) + t_14) + t_11
	tmp = 0
	if t_15 <= 4e-6:
		tmp = (0.5 / t_13) - ((t_10 + t_6) - (math.sqrt((1.0 + t_9)) + math.sqrt((1.0 + t_5))))
	elif t_15 <= 3.0:
		tmp = math.sqrt((t_12 - -1.0)) - ((t_13 - (math.sqrt((t_5 - -1.0)) - t_6)) - (math.sqrt((1.0 + t_7)) - t_8))
	else:
		tmp = ((((1.0 + (0.5 * t_12)) - t_13) + (1.0 - t_6)) + t_14) + t_11
	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 = fmin(fmax(x, y), t_3)
	t_6 = sqrt(t_5)
	t_7 = fmin(t_1, t_4)
	t_8 = sqrt(t_7)
	t_9 = fmax(t_1, t_4)
	t_10 = sqrt(t_9)
	t_11 = Float64(sqrt(Float64(t_9 + 1.0)) - t_10)
	t_12 = fmin(t_2, t)
	t_13 = sqrt(t_12)
	t_14 = Float64(sqrt(Float64(t_7 + 1.0)) - t_8)
	t_15 = Float64(Float64(Float64(Float64(sqrt(Float64(t_12 + 1.0)) - t_13) + Float64(sqrt(Float64(t_5 + 1.0)) - t_6)) + t_14) + t_11)
	tmp = 0.0
	if (t_15 <= 4e-6)
		tmp = Float64(Float64(0.5 / t_13) - Float64(Float64(t_10 + t_6) - Float64(sqrt(Float64(1.0 + t_9)) + sqrt(Float64(1.0 + t_5)))));
	elseif (t_15 <= 3.0)
		tmp = Float64(sqrt(Float64(t_12 - -1.0)) - Float64(Float64(t_13 - Float64(sqrt(Float64(t_5 - -1.0)) - t_6)) - Float64(sqrt(Float64(1.0 + t_7)) - t_8)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.5 * t_12)) - t_13) + Float64(1.0 - t_6)) + t_14) + t_11);
	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 = min(max(x, y), t_3);
	t_6 = sqrt(t_5);
	t_7 = min(t_1, t_4);
	t_8 = sqrt(t_7);
	t_9 = max(t_1, t_4);
	t_10 = sqrt(t_9);
	t_11 = sqrt((t_9 + 1.0)) - t_10;
	t_12 = min(t_2, t);
	t_13 = sqrt(t_12);
	t_14 = sqrt((t_7 + 1.0)) - t_8;
	t_15 = (((sqrt((t_12 + 1.0)) - t_13) + (sqrt((t_5 + 1.0)) - t_6)) + t_14) + t_11;
	tmp = 0.0;
	if (t_15 <= 4e-6)
		tmp = (0.5 / t_13) - ((t_10 + t_6) - (sqrt((1.0 + t_9)) + sqrt((1.0 + t_5))));
	elseif (t_15 <= 3.0)
		tmp = sqrt((t_12 - -1.0)) - ((t_13 - (sqrt((t_5 - -1.0)) - t_6)) - (sqrt((1.0 + t_7)) - t_8));
	else
		tmp = ((((1.0 + (0.5 * t_12)) - t_13) + (1.0 - t_6)) + t_14) + t_11;
	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[Min[N[Max[x, y], $MachinePrecision], t$95$3], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Min[t$95$1, t$95$4], $MachinePrecision]}, Block[{t$95$8 = N[Sqrt[t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Max[t$95$1, t$95$4], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]}, Block[{t$95$12 = N[Min[t$95$2, t], $MachinePrecision]}, Block[{t$95$13 = N[Sqrt[t$95$12], $MachinePrecision]}, Block[{t$95$14 = N[(N[Sqrt[N[(t$95$7 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$8), $MachinePrecision]}, Block[{t$95$15 = N[(N[(N[(N[(N[Sqrt[N[(t$95$12 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$13), $MachinePrecision] + N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision]), $MachinePrecision] + t$95$14), $MachinePrecision] + t$95$11), $MachinePrecision]}, If[LessEqual[t$95$15, 4e-6], N[(N[(0.5 / t$95$13), $MachinePrecision] - N[(N[(t$95$10 + t$95$6), $MachinePrecision] - N[(N[Sqrt[N[(1.0 + t$95$9), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$15, 3.0], N[(N[Sqrt[N[(t$95$12 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(t$95$13 - N[(N[Sqrt[N[(t$95$5 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[(1.0 + t$95$7), $MachinePrecision]], $MachinePrecision] - t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(1.0 + N[(0.5 * t$95$12), $MachinePrecision]), $MachinePrecision] - t$95$13), $MachinePrecision] + N[(1.0 - t$95$6), $MachinePrecision]), $MachinePrecision] + t$95$14), $MachinePrecision] + t$95$11), $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))) < 3.9999999999999998e-6

   1. Initial program 91.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 47.4%

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

   4. Applied rewrites 47.4%

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

   6. Applied rewrites 47.4%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 15.6%

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

   9. Applied rewrites 15.6%

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

   if 3.9999999999999998e-6 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3

   1. Initial program 91.5%

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

      1. lift--.f64 N/A

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

      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 72.6%

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

      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 72.6%

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

   3. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt 91.9%

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 32.8%

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

   10. Applied rewrites 32.8%

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

   if 3 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 26.5%

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

   7. Applied rewrites 26.5%

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

Alternative 7: 95.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_3 := \mathsf{max}\left(t\_2, t\right)\\ t_4 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_3\right)\\ t_5 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_3\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{min}\left(t\_2, t\right)\\ t_8 := \sqrt{t\_7}\\ t_9 := \mathsf{min}\left(t\_1, t\_4\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \mathsf{max}\left(t\_1, t\_4\right)\\ t_12 := \sqrt{t\_11}\\ t_13 := \left(\left(\left(\sqrt{t\_7 + 1} - t\_8\right) + \left(\sqrt{t\_5 + 1} - t\_6\right)\right) + \left(\sqrt{t\_9 + 1} - t\_10\right)\right) + \left(\sqrt{t\_11 + 1} - t\_12\right)\\ t_14 := \sqrt{t\_5 - -1}\\ \mathbf{if}\;t\_13 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.5}{t\_8} - \left(\left(t\_12 + t\_6\right) - \left(\sqrt{1 + t\_11} + \sqrt{1 + t\_5}\right)\right)\\ \mathbf{elif}\;t\_13 \leq 3.5:\\ \;\;\;\;\sqrt{t\_7 - -1} - \left(\left(t\_8 - \left(t\_14 - t\_6\right)\right) - \left(\sqrt{1 + t\_9} - t\_10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\_8\right) - \left(\left(t\_6 - t\_14\right) - \left(\left(\sqrt{t\_9 - -1} - t\_10\right) - \left(t\_12 - 1\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmin x y) z))
       (t_3 (fmax t_2 t))
       (t_4 (fmax (fmax x y) t_3))
       (t_5 (fmin (fmax x y) t_3))
       (t_6 (sqrt t_5))
       (t_7 (fmin t_2 t))
       (t_8 (sqrt t_7))
       (t_9 (fmin t_1 t_4))
       (t_10 (sqrt t_9))
       (t_11 (fmax t_1 t_4))
       (t_12 (sqrt t_11))
       (t_13
        (+
         (+
          (+ (- (sqrt (+ t_7 1.0)) t_8) (- (sqrt (+ t_5 1.0)) t_6))
          (- (sqrt (+ t_9 1.0)) t_10))
         (- (sqrt (+ t_11 1.0)) t_12)))
       (t_14 (sqrt (- t_5 -1.0))))
  (if (<= t_13 4e-6)
    (-
     (/ 0.5 t_8)
     (- (+ t_12 t_6) (+ (sqrt (+ 1.0 t_11)) (sqrt (+ 1.0 t_5)))))
    (if (<= t_13 3.5)
      (-
       (sqrt (- t_7 -1.0))
       (- (- t_8 (- t_14 t_6)) (- (sqrt (+ 1.0 t_9)) t_10)))
      (-
       (- 1.0 t_8)
       (-
        (- t_6 t_14)
        (- (- (sqrt (- t_9 -1.0)) t_10) (- t_12 1.0))))))))

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 = fmin(fmax(x, y), t_3);
	double t_6 = sqrt(t_5);
	double t_7 = fmin(t_2, t);
	double t_8 = sqrt(t_7);
	double t_9 = fmin(t_1, t_4);
	double t_10 = sqrt(t_9);
	double t_11 = fmax(t_1, t_4);
	double t_12 = sqrt(t_11);
	double t_13 = (((sqrt((t_7 + 1.0)) - t_8) + (sqrt((t_5 + 1.0)) - t_6)) + (sqrt((t_9 + 1.0)) - t_10)) + (sqrt((t_11 + 1.0)) - t_12);
	double t_14 = sqrt((t_5 - -1.0));
	double tmp;
	if (t_13 <= 4e-6) {
		tmp = (0.5 / t_8) - ((t_12 + t_6) - (sqrt((1.0 + t_11)) + sqrt((1.0 + t_5))));
	} else if (t_13 <= 3.5) {
		tmp = sqrt((t_7 - -1.0)) - ((t_8 - (t_14 - t_6)) - (sqrt((1.0 + t_9)) - t_10));
	} else {
		tmp = (1.0 - t_8) - ((t_6 - t_14) - ((sqrt((t_9 - -1.0)) - t_10) - (t_12 - 1.0)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_14
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmin(fmin(x, y), z)
    t_3 = fmax(t_2, t)
    t_4 = fmax(fmax(x, y), t_3)
    t_5 = fmin(fmax(x, y), t_3)
    t_6 = sqrt(t_5)
    t_7 = fmin(t_2, t)
    t_8 = sqrt(t_7)
    t_9 = fmin(t_1, t_4)
    t_10 = sqrt(t_9)
    t_11 = fmax(t_1, t_4)
    t_12 = sqrt(t_11)
    t_13 = (((sqrt((t_7 + 1.0d0)) - t_8) + (sqrt((t_5 + 1.0d0)) - t_6)) + (sqrt((t_9 + 1.0d0)) - t_10)) + (sqrt((t_11 + 1.0d0)) - t_12)
    t_14 = sqrt((t_5 - (-1.0d0)))
    if (t_13 <= 4d-6) then
        tmp = (0.5d0 / t_8) - ((t_12 + t_6) - (sqrt((1.0d0 + t_11)) + sqrt((1.0d0 + t_5))))
    else if (t_13 <= 3.5d0) then
        tmp = sqrt((t_7 - (-1.0d0))) - ((t_8 - (t_14 - t_6)) - (sqrt((1.0d0 + t_9)) - t_10))
    else
        tmp = (1.0d0 - t_8) - ((t_6 - t_14) - ((sqrt((t_9 - (-1.0d0))) - t_10) - (t_12 - 1.0d0)))
    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 = fmin(fmax(x, y), t_3);
	double t_6 = Math.sqrt(t_5);
	double t_7 = fmin(t_2, t);
	double t_8 = Math.sqrt(t_7);
	double t_9 = fmin(t_1, t_4);
	double t_10 = Math.sqrt(t_9);
	double t_11 = fmax(t_1, t_4);
	double t_12 = Math.sqrt(t_11);
	double t_13 = (((Math.sqrt((t_7 + 1.0)) - t_8) + (Math.sqrt((t_5 + 1.0)) - t_6)) + (Math.sqrt((t_9 + 1.0)) - t_10)) + (Math.sqrt((t_11 + 1.0)) - t_12);
	double t_14 = Math.sqrt((t_5 - -1.0));
	double tmp;
	if (t_13 <= 4e-6) {
		tmp = (0.5 / t_8) - ((t_12 + t_6) - (Math.sqrt((1.0 + t_11)) + Math.sqrt((1.0 + t_5))));
	} else if (t_13 <= 3.5) {
		tmp = Math.sqrt((t_7 - -1.0)) - ((t_8 - (t_14 - t_6)) - (Math.sqrt((1.0 + t_9)) - t_10));
	} else {
		tmp = (1.0 - t_8) - ((t_6 - t_14) - ((Math.sqrt((t_9 - -1.0)) - t_10) - (t_12 - 1.0)));
	}
	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 = fmin(fmax(x, y), t_3)
	t_6 = math.sqrt(t_5)
	t_7 = fmin(t_2, t)
	t_8 = math.sqrt(t_7)
	t_9 = fmin(t_1, t_4)
	t_10 = math.sqrt(t_9)
	t_11 = fmax(t_1, t_4)
	t_12 = math.sqrt(t_11)
	t_13 = (((math.sqrt((t_7 + 1.0)) - t_8) + (math.sqrt((t_5 + 1.0)) - t_6)) + (math.sqrt((t_9 + 1.0)) - t_10)) + (math.sqrt((t_11 + 1.0)) - t_12)
	t_14 = math.sqrt((t_5 - -1.0))
	tmp = 0
	if t_13 <= 4e-6:
		tmp = (0.5 / t_8) - ((t_12 + t_6) - (math.sqrt((1.0 + t_11)) + math.sqrt((1.0 + t_5))))
	elif t_13 <= 3.5:
		tmp = math.sqrt((t_7 - -1.0)) - ((t_8 - (t_14 - t_6)) - (math.sqrt((1.0 + t_9)) - t_10))
	else:
		tmp = (1.0 - t_8) - ((t_6 - t_14) - ((math.sqrt((t_9 - -1.0)) - t_10) - (t_12 - 1.0)))
	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 = fmin(fmax(x, y), t_3)
	t_6 = sqrt(t_5)
	t_7 = fmin(t_2, t)
	t_8 = sqrt(t_7)
	t_9 = fmin(t_1, t_4)
	t_10 = sqrt(t_9)
	t_11 = fmax(t_1, t_4)
	t_12 = sqrt(t_11)
	t_13 = Float64(Float64(Float64(Float64(sqrt(Float64(t_7 + 1.0)) - t_8) + Float64(sqrt(Float64(t_5 + 1.0)) - t_6)) + Float64(sqrt(Float64(t_9 + 1.0)) - t_10)) + Float64(sqrt(Float64(t_11 + 1.0)) - t_12))
	t_14 = sqrt(Float64(t_5 - -1.0))
	tmp = 0.0
	if (t_13 <= 4e-6)
		tmp = Float64(Float64(0.5 / t_8) - Float64(Float64(t_12 + t_6) - Float64(sqrt(Float64(1.0 + t_11)) + sqrt(Float64(1.0 + t_5)))));
	elseif (t_13 <= 3.5)
		tmp = Float64(sqrt(Float64(t_7 - -1.0)) - Float64(Float64(t_8 - Float64(t_14 - t_6)) - Float64(sqrt(Float64(1.0 + t_9)) - t_10)));
	else
		tmp = Float64(Float64(1.0 - t_8) - Float64(Float64(t_6 - t_14) - Float64(Float64(sqrt(Float64(t_9 - -1.0)) - t_10) - Float64(t_12 - 1.0))));
	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 = min(max(x, y), t_3);
	t_6 = sqrt(t_5);
	t_7 = min(t_2, t);
	t_8 = sqrt(t_7);
	t_9 = min(t_1, t_4);
	t_10 = sqrt(t_9);
	t_11 = max(t_1, t_4);
	t_12 = sqrt(t_11);
	t_13 = (((sqrt((t_7 + 1.0)) - t_8) + (sqrt((t_5 + 1.0)) - t_6)) + (sqrt((t_9 + 1.0)) - t_10)) + (sqrt((t_11 + 1.0)) - t_12);
	t_14 = sqrt((t_5 - -1.0));
	tmp = 0.0;
	if (t_13 <= 4e-6)
		tmp = (0.5 / t_8) - ((t_12 + t_6) - (sqrt((1.0 + t_11)) + sqrt((1.0 + t_5))));
	elseif (t_13 <= 3.5)
		tmp = sqrt((t_7 - -1.0)) - ((t_8 - (t_14 - t_6)) - (sqrt((1.0 + t_9)) - t_10));
	else
		tmp = (1.0 - t_8) - ((t_6 - t_14) - ((sqrt((t_9 - -1.0)) - t_10) - (t_12 - 1.0)));
	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[Min[N[Max[x, y], $MachinePrecision], t$95$3], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Min[t$95$2, t], $MachinePrecision]}, Block[{t$95$8 = N[Sqrt[t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$1, t$95$4], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Max[t$95$1, t$95$4], $MachinePrecision]}, Block[{t$95$12 = N[Sqrt[t$95$11], $MachinePrecision]}, Block[{t$95$13 = N[(N[(N[(N[(N[Sqrt[N[(t$95$7 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$8), $MachinePrecision] + N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$11 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$12), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$14 = N[Sqrt[N[(t$95$5 - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$13, 4e-6], N[(N[(0.5 / t$95$8), $MachinePrecision] - N[(N[(t$95$12 + t$95$6), $MachinePrecision] - N[(N[Sqrt[N[(1.0 + t$95$11), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$13, 3.5], N[(N[Sqrt[N[(t$95$7 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(t$95$8 - N[(t$95$14 - t$95$6), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[(1.0 + t$95$9), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$8), $MachinePrecision] - N[(N[(t$95$6 - t$95$14), $MachinePrecision] - N[(N[(N[Sqrt[N[(t$95$9 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision] - N[(t$95$12 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.9999999999999998e-6

   1. Initial program 91.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 47.4%

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

   4. Applied rewrites 47.4%

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

   6. Applied rewrites 47.4%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 15.6%

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

   9. Applied rewrites 15.6%

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

   if 3.9999999999999998e-6 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.5

   1. Initial program 91.5%

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

      1. lift--.f64 N/A

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

      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 72.6%

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

      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 72.6%

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

   3. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt 91.9%

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 32.8%

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

   10. Applied rewrites 32.8%

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

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

   1. Initial program 91.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 47.4%

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

   4. Applied rewrites 47.4%

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

   6. Applied rewrites 47.4%

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

   7. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 27.5%

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

   9. Applied rewrites 27.5%

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

   10. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-sqrt.f64 24.1%

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

   12. Applied rewrites 24.1%

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

Alternative 8: 90.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_3 := \mathsf{min}\left(t\_2, t\right)\\ t_4 := \sqrt{t\_3}\\ t_5 := \mathsf{max}\left(t\_2, t\right)\\ t_6 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_5\right)\\ t_7 := \mathsf{min}\left(t\_1, t\_6\right)\\ t_8 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_5\right)\\ t_9 := \sqrt{t\_8}\\ t_10 := \mathsf{max}\left(t\_1, t\_6\right)\\ \mathbf{if}\;t\_3 \leq 42000000000:\\ \;\;\;\;\sqrt{t\_3 - -1} - \left(\left(t\_4 - \left(\sqrt{t\_8 - -1} - t\_9\right)\right) - \left(\sqrt{1 + t\_7} - \sqrt{t\_7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{t\_4} - \left(\left(\sqrt{t\_10} + t\_9\right) - \left(\sqrt{1 + t\_10} + \sqrt{1 + t\_8}\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmin x y) z))
       (t_3 (fmin t_2 t))
       (t_4 (sqrt t_3))
       (t_5 (fmax t_2 t))
       (t_6 (fmax (fmax x y) t_5))
       (t_7 (fmin t_1 t_6))
       (t_8 (fmin (fmax x y) t_5))
       (t_9 (sqrt t_8))
       (t_10 (fmax t_1 t_6)))
  (if (<= t_3 42000000000.0)
    (-
     (sqrt (- t_3 -1.0))
     (-
      (- t_4 (- (sqrt (- t_8 -1.0)) t_9))
      (- (sqrt (+ 1.0 t_7)) (sqrt t_7))))
    (-
     (/ 0.5 t_4)
     (-
      (+ (sqrt t_10) t_9)
      (+ (sqrt (+ 1.0 t_10)) (sqrt (+ 1.0 t_8))))))))

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 = fmin(t_1, t_6);
	double t_8 = fmin(fmax(x, y), t_5);
	double t_9 = sqrt(t_8);
	double t_10 = fmax(t_1, t_6);
	double tmp;
	if (t_3 <= 42000000000.0) {
		tmp = sqrt((t_3 - -1.0)) - ((t_4 - (sqrt((t_8 - -1.0)) - t_9)) - (sqrt((1.0 + t_7)) - sqrt(t_7)));
	} else {
		tmp = (0.5 / t_4) - ((sqrt(t_10) + t_9) - (sqrt((1.0 + t_10)) + sqrt((1.0 + 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_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 = fmin(t_1, t_6)
    t_8 = fmin(fmax(x, y), t_5)
    t_9 = sqrt(t_8)
    t_10 = fmax(t_1, t_6)
    if (t_3 <= 42000000000.0d0) then
        tmp = sqrt((t_3 - (-1.0d0))) - ((t_4 - (sqrt((t_8 - (-1.0d0))) - t_9)) - (sqrt((1.0d0 + t_7)) - sqrt(t_7)))
    else
        tmp = (0.5d0 / t_4) - ((sqrt(t_10) + t_9) - (sqrt((1.0d0 + t_10)) + sqrt((1.0d0 + t_8))))
    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 = fmin(t_1, t_6);
	double t_8 = fmin(fmax(x, y), t_5);
	double t_9 = Math.sqrt(t_8);
	double t_10 = fmax(t_1, t_6);
	double tmp;
	if (t_3 <= 42000000000.0) {
		tmp = Math.sqrt((t_3 - -1.0)) - ((t_4 - (Math.sqrt((t_8 - -1.0)) - t_9)) - (Math.sqrt((1.0 + t_7)) - Math.sqrt(t_7)));
	} else {
		tmp = (0.5 / t_4) - ((Math.sqrt(t_10) + t_9) - (Math.sqrt((1.0 + t_10)) + Math.sqrt((1.0 + t_8))));
	}
	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 = fmin(t_1, t_6)
	t_8 = fmin(fmax(x, y), t_5)
	t_9 = math.sqrt(t_8)
	t_10 = fmax(t_1, t_6)
	tmp = 0
	if t_3 <= 42000000000.0:
		tmp = math.sqrt((t_3 - -1.0)) - ((t_4 - (math.sqrt((t_8 - -1.0)) - t_9)) - (math.sqrt((1.0 + t_7)) - math.sqrt(t_7)))
	else:
		tmp = (0.5 / t_4) - ((math.sqrt(t_10) + t_9) - (math.sqrt((1.0 + t_10)) + math.sqrt((1.0 + t_8))))
	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 = fmin(t_1, t_6)
	t_8 = fmin(fmax(x, y), t_5)
	t_9 = sqrt(t_8)
	t_10 = fmax(t_1, t_6)
	tmp = 0.0
	if (t_3 <= 42000000000.0)
		tmp = Float64(sqrt(Float64(t_3 - -1.0)) - Float64(Float64(t_4 - Float64(sqrt(Float64(t_8 - -1.0)) - t_9)) - Float64(sqrt(Float64(1.0 + t_7)) - sqrt(t_7))));
	else
		tmp = Float64(Float64(0.5 / t_4) - Float64(Float64(sqrt(t_10) + t_9) - Float64(sqrt(Float64(1.0 + t_10)) + sqrt(Float64(1.0 + t_8)))));
	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 = min(t_1, t_6);
	t_8 = min(max(x, y), t_5);
	t_9 = sqrt(t_8);
	t_10 = max(t_1, t_6);
	tmp = 0.0;
	if (t_3 <= 42000000000.0)
		tmp = sqrt((t_3 - -1.0)) - ((t_4 - (sqrt((t_8 - -1.0)) - t_9)) - (sqrt((1.0 + t_7)) - sqrt(t_7)));
	else
		tmp = (0.5 / t_4) - ((sqrt(t_10) + t_9) - (sqrt((1.0 + t_10)) + sqrt((1.0 + t_8))));
	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[Min[t$95$1, 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[Max[t$95$1, t$95$6], $MachinePrecision]}, If[LessEqual[t$95$3, 42000000000.0], N[(N[Sqrt[N[(t$95$3 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(t$95$4 - N[(N[Sqrt[N[(t$95$8 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$9), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[(1.0 + t$95$7), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$7], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / t$95$4), $MachinePrecision] - N[(N[(N[Sqrt[t$95$10], $MachinePrecision] + t$95$9), $MachinePrecision] - N[(N[Sqrt[N[(1.0 + t$95$10), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + t$95$8), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2e10

   1. Initial program 91.5%

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

      1. lift--.f64 N/A

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

      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 72.6%

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

      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 72.6%

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

   3. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt 91.9%

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 32.8%

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

   10. Applied rewrites 32.8%

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

   if 4.2e10 < x

   1. Initial program 91.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 47.4%

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

   4. Applied rewrites 47.4%

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

   6. Applied rewrites 47.4%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 15.6%

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

   9. Applied rewrites 15.6%

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

Alternative 9: 90.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \sqrt{1 + t\_5}\\ t_8 := \mathsf{max}\left(t\_4, t\right)\\ t_9 := \mathsf{max}\left(t\_3, t\_8\right)\\ t_10 := \mathsf{min}\left(t\_2, t\_9\right)\\ t_11 := \sqrt{t\_10}\\ t_12 := \mathsf{min}\left(t\_3, t\_8\right)\\ t_13 := \sqrt{1 + t\_12}\\ t_14 := \mathsf{max}\left(t\_2, t\_9\right)\\ t_15 := \sqrt{t\_14}\\ t_16 := \sqrt{t\_14 + 1} - t\_15\\ t_17 := \sqrt{t\_12}\\ t_18 := \left(\left(\left(\sqrt{t\_5 + 1} - t\_6\right) + \left(\sqrt{t\_12 + 1} - t\_17\right)\right) + \left(\sqrt{t\_10 + 1} - t\_11\right)\right) + t\_16\\ \mathbf{if}\;t\_18 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.5}{t\_6} - \left(\left(t\_15 + t\_17\right) - \left(\sqrt{1 + t\_14} + t\_13\right)\right)\\ \mathbf{elif}\;t\_18 \leq 1:\\ \;\;\;\;\left(t\_7 - t\_6\right) + t\_16\\ \mathbf{elif}\;t\_18 \leq 2:\\ \;\;\;\;\left(\left(1 + t\_13\right) - \left(t\_6 + t\_17\right)\right) + t\_16\\ \mathbf{else}:\\ \;\;\;\;\left(t\_7 + \left(t\_13 + \sqrt{1 + t\_10}\right)\right) - \left(t\_6 + \left(t\_17 + t\_11\right)\right)\\ \end{array} \]

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 (sqrt (+ 1.0 t_5)))
       (t_8 (fmax t_4 t))
       (t_9 (fmax t_3 t_8))
       (t_10 (fmin t_2 t_9))
       (t_11 (sqrt t_10))
       (t_12 (fmin t_3 t_8))
       (t_13 (sqrt (+ 1.0 t_12)))
       (t_14 (fmax t_2 t_9))
       (t_15 (sqrt t_14))
       (t_16 (- (sqrt (+ t_14 1.0)) t_15))
       (t_17 (sqrt t_12))
       (t_18
        (+
         (+
          (+ (- (sqrt (+ t_5 1.0)) t_6) (- (sqrt (+ t_12 1.0)) t_17))
          (- (sqrt (+ t_10 1.0)) t_11))
         t_16)))
  (if (<= t_18 4e-6)
    (- (/ 0.5 t_6) (- (+ t_15 t_17) (+ (sqrt (+ 1.0 t_14)) t_13)))
    (if (<= t_18 1.0)
      (+ (- t_7 t_6) t_16)
      (if (<= t_18 2.0)
        (+ (- (+ 1.0 t_13) (+ t_6 t_17)) t_16)
        (-
         (+ t_7 (+ t_13 (sqrt (+ 1.0 t_10))))
         (+ t_6 (+ t_17 t_11))))))))

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 = sqrt((1.0 + t_5));
	double t_8 = fmax(t_4, t);
	double t_9 = fmax(t_3, t_8);
	double t_10 = fmin(t_2, t_9);
	double t_11 = sqrt(t_10);
	double t_12 = fmin(t_3, t_8);
	double t_13 = sqrt((1.0 + t_12));
	double t_14 = fmax(t_2, t_9);
	double t_15 = sqrt(t_14);
	double t_16 = sqrt((t_14 + 1.0)) - t_15;
	double t_17 = sqrt(t_12);
	double t_18 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_12 + 1.0)) - t_17)) + (sqrt((t_10 + 1.0)) - t_11)) + t_16;
	double tmp;
	if (t_18 <= 4e-6) {
		tmp = (0.5 / t_6) - ((t_15 + t_17) - (sqrt((1.0 + t_14)) + t_13));
	} else if (t_18 <= 1.0) {
		tmp = (t_7 - t_6) + t_16;
	} else if (t_18 <= 2.0) {
		tmp = ((1.0 + t_13) - (t_6 + t_17)) + t_16;
	} else {
		tmp = (t_7 + (t_13 + sqrt((1.0 + t_10)))) - (t_6 + (t_17 + t_11));
	}
	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_18
    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 = sqrt((1.0d0 + t_5))
    t_8 = fmax(t_4, t)
    t_9 = fmax(t_3, t_8)
    t_10 = fmin(t_2, t_9)
    t_11 = sqrt(t_10)
    t_12 = fmin(t_3, t_8)
    t_13 = sqrt((1.0d0 + t_12))
    t_14 = fmax(t_2, t_9)
    t_15 = sqrt(t_14)
    t_16 = sqrt((t_14 + 1.0d0)) - t_15
    t_17 = sqrt(t_12)
    t_18 = (((sqrt((t_5 + 1.0d0)) - t_6) + (sqrt((t_12 + 1.0d0)) - t_17)) + (sqrt((t_10 + 1.0d0)) - t_11)) + t_16
    if (t_18 <= 4d-6) then
        tmp = (0.5d0 / t_6) - ((t_15 + t_17) - (sqrt((1.0d0 + t_14)) + t_13))
    else if (t_18 <= 1.0d0) then
        tmp = (t_7 - t_6) + t_16
    else if (t_18 <= 2.0d0) then
        tmp = ((1.0d0 + t_13) - (t_6 + t_17)) + t_16
    else
        tmp = (t_7 + (t_13 + sqrt((1.0d0 + t_10)))) - (t_6 + (t_17 + t_11))
    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 = Math.sqrt((1.0 + t_5));
	double t_8 = fmax(t_4, t);
	double t_9 = fmax(t_3, t_8);
	double t_10 = fmin(t_2, t_9);
	double t_11 = Math.sqrt(t_10);
	double t_12 = fmin(t_3, t_8);
	double t_13 = Math.sqrt((1.0 + t_12));
	double t_14 = fmax(t_2, t_9);
	double t_15 = Math.sqrt(t_14);
	double t_16 = Math.sqrt((t_14 + 1.0)) - t_15;
	double t_17 = Math.sqrt(t_12);
	double t_18 = (((Math.sqrt((t_5 + 1.0)) - t_6) + (Math.sqrt((t_12 + 1.0)) - t_17)) + (Math.sqrt((t_10 + 1.0)) - t_11)) + t_16;
	double tmp;
	if (t_18 <= 4e-6) {
		tmp = (0.5 / t_6) - ((t_15 + t_17) - (Math.sqrt((1.0 + t_14)) + t_13));
	} else if (t_18 <= 1.0) {
		tmp = (t_7 - t_6) + t_16;
	} else if (t_18 <= 2.0) {
		tmp = ((1.0 + t_13) - (t_6 + t_17)) + t_16;
	} else {
		tmp = (t_7 + (t_13 + Math.sqrt((1.0 + t_10)))) - (t_6 + (t_17 + t_11));
	}
	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 = math.sqrt((1.0 + t_5))
	t_8 = fmax(t_4, t)
	t_9 = fmax(t_3, t_8)
	t_10 = fmin(t_2, t_9)
	t_11 = math.sqrt(t_10)
	t_12 = fmin(t_3, t_8)
	t_13 = math.sqrt((1.0 + t_12))
	t_14 = fmax(t_2, t_9)
	t_15 = math.sqrt(t_14)
	t_16 = math.sqrt((t_14 + 1.0)) - t_15
	t_17 = math.sqrt(t_12)
	t_18 = (((math.sqrt((t_5 + 1.0)) - t_6) + (math.sqrt((t_12 + 1.0)) - t_17)) + (math.sqrt((t_10 + 1.0)) - t_11)) + t_16
	tmp = 0
	if t_18 <= 4e-6:
		tmp = (0.5 / t_6) - ((t_15 + t_17) - (math.sqrt((1.0 + t_14)) + t_13))
	elif t_18 <= 1.0:
		tmp = (t_7 - t_6) + t_16
	elif t_18 <= 2.0:
		tmp = ((1.0 + t_13) - (t_6 + t_17)) + t_16
	else:
		tmp = (t_7 + (t_13 + math.sqrt((1.0 + t_10)))) - (t_6 + (t_17 + t_11))
	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 = sqrt(Float64(1.0 + t_5))
	t_8 = fmax(t_4, t)
	t_9 = fmax(t_3, t_8)
	t_10 = fmin(t_2, t_9)
	t_11 = sqrt(t_10)
	t_12 = fmin(t_3, t_8)
	t_13 = sqrt(Float64(1.0 + t_12))
	t_14 = fmax(t_2, t_9)
	t_15 = sqrt(t_14)
	t_16 = Float64(sqrt(Float64(t_14 + 1.0)) - t_15)
	t_17 = sqrt(t_12)
	t_18 = Float64(Float64(Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + Float64(sqrt(Float64(t_12 + 1.0)) - t_17)) + Float64(sqrt(Float64(t_10 + 1.0)) - t_11)) + t_16)
	tmp = 0.0
	if (t_18 <= 4e-6)
		tmp = Float64(Float64(0.5 / t_6) - Float64(Float64(t_15 + t_17) - Float64(sqrt(Float64(1.0 + t_14)) + t_13)));
	elseif (t_18 <= 1.0)
		tmp = Float64(Float64(t_7 - t_6) + t_16);
	elseif (t_18 <= 2.0)
		tmp = Float64(Float64(Float64(1.0 + t_13) - Float64(t_6 + t_17)) + t_16);
	else
		tmp = Float64(Float64(t_7 + Float64(t_13 + sqrt(Float64(1.0 + t_10)))) - Float64(t_6 + Float64(t_17 + t_11)));
	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 = sqrt((1.0 + t_5));
	t_8 = max(t_4, t);
	t_9 = max(t_3, t_8);
	t_10 = min(t_2, t_9);
	t_11 = sqrt(t_10);
	t_12 = min(t_3, t_8);
	t_13 = sqrt((1.0 + t_12));
	t_14 = max(t_2, t_9);
	t_15 = sqrt(t_14);
	t_16 = sqrt((t_14 + 1.0)) - t_15;
	t_17 = sqrt(t_12);
	t_18 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_12 + 1.0)) - t_17)) + (sqrt((t_10 + 1.0)) - t_11)) + t_16;
	tmp = 0.0;
	if (t_18 <= 4e-6)
		tmp = (0.5 / t_6) - ((t_15 + t_17) - (sqrt((1.0 + t_14)) + t_13));
	elseif (t_18 <= 1.0)
		tmp = (t_7 - t_6) + t_16;
	elseif (t_18 <= 2.0)
		tmp = ((1.0 + t_13) - (t_6 + t_17)) + t_16;
	else
		tmp = (t_7 + (t_13 + sqrt((1.0 + t_10)))) - (t_6 + (t_17 + t_11));
	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[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$9 = N[Max[t$95$3, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Min[t$95$2, t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Sqrt[t$95$10], $MachinePrecision]}, Block[{t$95$12 = N[Min[t$95$3, t$95$8], $MachinePrecision]}, Block[{t$95$13 = N[Sqrt[N[(1.0 + t$95$12), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$14 = N[Max[t$95$2, t$95$9], $MachinePrecision]}, Block[{t$95$15 = N[Sqrt[t$95$14], $MachinePrecision]}, Block[{t$95$16 = N[(N[Sqrt[N[(t$95$14 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$15), $MachinePrecision]}, Block[{t$95$17 = N[Sqrt[t$95$12], $MachinePrecision]}, Block[{t$95$18 = N[(N[(N[(N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + N[(N[Sqrt[N[(t$95$12 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$17), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$10 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$11), $MachinePrecision]), $MachinePrecision] + t$95$16), $MachinePrecision]}, If[LessEqual[t$95$18, 4e-6], N[(N[(0.5 / t$95$6), $MachinePrecision] - N[(N[(t$95$15 + t$95$17), $MachinePrecision] - N[(N[Sqrt[N[(1.0 + t$95$14), $MachinePrecision]], $MachinePrecision] + t$95$13), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$18, 1.0], N[(N[(t$95$7 - t$95$6), $MachinePrecision] + t$95$16), $MachinePrecision], If[LessEqual[t$95$18, 2.0], N[(N[(N[(1.0 + t$95$13), $MachinePrecision] - N[(t$95$6 + t$95$17), $MachinePrecision]), $MachinePrecision] + t$95$16), $MachinePrecision], N[(N[(t$95$7 + N[(t$95$13 + N[Sqrt[N[(1.0 + t$95$10), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$6 + N[(t$95$17 + t$95$11), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.9999999999999998e-6

   1. Initial program 91.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 47.4%

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

   4. Applied rewrites 47.4%

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

   6. Applied rewrites 47.4%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 15.6%

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

   9. Applied rewrites 15.6%

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

   if 3.9999999999999998e-6 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in z around 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 29.7%

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

   7. Applied rewrites 29.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 21.7%

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

   10. Applied rewrites 21.7%

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

   11. Taylor expanded in y around inf

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

       1. lower--.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 29.5%

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

   13. Applied rewrites 29.5%

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

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

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in z around 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 29.7%

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

   7. Applied rewrites 29.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 21.7%

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

   10. Applied rewrites 21.7%

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

   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 91.5%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 12.0%

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

Alternative 10: 85.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(t\_1, t\right)\\ t_3 := \sqrt{t\_2}\\ t_4 := \mathsf{min}\left(t\_1, t\right)\\ t_5 := \sqrt{t\_4}\\ t_6 := \sqrt{t\_2 + 1} - t\_3\\ t_7 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\right)\\ t_8 := \sqrt{1 + t\_7}\\ t_9 := \sqrt{t\_7}\\ t_10 := \left(\sqrt{t\_4 + 1} - t\_5\right) + \left(\sqrt{t\_7 + 1} - t\_9\right)\\ \mathbf{if}\;t\_10 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.5}{t\_5} - \left(\left(t\_3 + t\_9\right) - \left(\sqrt{1 + t\_2} + t\_8\right)\right)\\ \mathbf{elif}\;t\_10 \leq 1:\\ \;\;\;\;\left(\sqrt{1 + t\_4} - t\_5\right) + t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + t\_8\right) - \left(t\_5 + t\_9\right)\right) + t\_6\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin (fmin x y) z))
       (t_2 (fmax t_1 t))
       (t_3 (sqrt t_2))
       (t_4 (fmin t_1 t))
       (t_5 (sqrt t_4))
       (t_6 (- (sqrt (+ t_2 1.0)) t_3))
       (t_7 (fmin (fmax x y) (fmax (fmin x y) z)))
       (t_8 (sqrt (+ 1.0 t_7)))
       (t_9 (sqrt t_7))
       (t_10
        (+ (- (sqrt (+ t_4 1.0)) t_5) (- (sqrt (+ t_7 1.0)) t_9))))
  (if (<= t_10 4e-6)
    (- (/ 0.5 t_5) (- (+ t_3 t_9) (+ (sqrt (+ 1.0 t_2)) t_8)))
    (if (<= t_10 1.0)
      (+ (- (sqrt (+ 1.0 t_4)) t_5) t_6)
      (+ (- (+ 1.0 t_8) (+ t_5 t_9)) t_6)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(t_1, t);
	double t_3 = sqrt(t_2);
	double t_4 = fmin(t_1, t);
	double t_5 = sqrt(t_4);
	double t_6 = sqrt((t_2 + 1.0)) - t_3;
	double t_7 = fmin(fmax(x, y), fmax(fmin(x, y), z));
	double t_8 = sqrt((1.0 + t_7));
	double t_9 = sqrt(t_7);
	double t_10 = (sqrt((t_4 + 1.0)) - t_5) + (sqrt((t_7 + 1.0)) - t_9);
	double tmp;
	if (t_10 <= 4e-6) {
		tmp = (0.5 / t_5) - ((t_3 + t_9) - (sqrt((1.0 + t_2)) + t_8));
	} else if (t_10 <= 1.0) {
		tmp = (sqrt((1.0 + t_4)) - t_5) + t_6;
	} else {
		tmp = ((1.0 + t_8) - (t_5 + t_9)) + t_6;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_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 = fmin(fmin(x, y), z)
    t_2 = fmax(t_1, t)
    t_3 = sqrt(t_2)
    t_4 = fmin(t_1, t)
    t_5 = sqrt(t_4)
    t_6 = sqrt((t_2 + 1.0d0)) - t_3
    t_7 = fmin(fmax(x, y), fmax(fmin(x, y), z))
    t_8 = sqrt((1.0d0 + t_7))
    t_9 = sqrt(t_7)
    t_10 = (sqrt((t_4 + 1.0d0)) - t_5) + (sqrt((t_7 + 1.0d0)) - t_9)
    if (t_10 <= 4d-6) then
        tmp = (0.5d0 / t_5) - ((t_3 + t_9) - (sqrt((1.0d0 + t_2)) + t_8))
    else if (t_10 <= 1.0d0) then
        tmp = (sqrt((1.0d0 + t_4)) - t_5) + t_6
    else
        tmp = ((1.0d0 + t_8) - (t_5 + t_9)) + t_6
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(t_1, t);
	double t_3 = Math.sqrt(t_2);
	double t_4 = fmin(t_1, t);
	double t_5 = Math.sqrt(t_4);
	double t_6 = Math.sqrt((t_2 + 1.0)) - t_3;
	double t_7 = fmin(fmax(x, y), fmax(fmin(x, y), z));
	double t_8 = Math.sqrt((1.0 + t_7));
	double t_9 = Math.sqrt(t_7);
	double t_10 = (Math.sqrt((t_4 + 1.0)) - t_5) + (Math.sqrt((t_7 + 1.0)) - t_9);
	double tmp;
	if (t_10 <= 4e-6) {
		tmp = (0.5 / t_5) - ((t_3 + t_9) - (Math.sqrt((1.0 + t_2)) + t_8));
	} else if (t_10 <= 1.0) {
		tmp = (Math.sqrt((1.0 + t_4)) - t_5) + t_6;
	} else {
		tmp = ((1.0 + t_8) - (t_5 + t_9)) + t_6;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = math.sqrt(t_2)
	t_4 = fmin(t_1, t)
	t_5 = math.sqrt(t_4)
	t_6 = math.sqrt((t_2 + 1.0)) - t_3
	t_7 = fmin(fmax(x, y), fmax(fmin(x, y), z))
	t_8 = math.sqrt((1.0 + t_7))
	t_9 = math.sqrt(t_7)
	t_10 = (math.sqrt((t_4 + 1.0)) - t_5) + (math.sqrt((t_7 + 1.0)) - t_9)
	tmp = 0
	if t_10 <= 4e-6:
		tmp = (0.5 / t_5) - ((t_3 + t_9) - (math.sqrt((1.0 + t_2)) + t_8))
	elif t_10 <= 1.0:
		tmp = (math.sqrt((1.0 + t_4)) - t_5) + t_6
	else:
		tmp = ((1.0 + t_8) - (t_5 + t_9)) + t_6
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = sqrt(t_2)
	t_4 = fmin(t_1, t)
	t_5 = sqrt(t_4)
	t_6 = Float64(sqrt(Float64(t_2 + 1.0)) - t_3)
	t_7 = fmin(fmax(x, y), fmax(fmin(x, y), z))
	t_8 = sqrt(Float64(1.0 + t_7))
	t_9 = sqrt(t_7)
	t_10 = Float64(Float64(sqrt(Float64(t_4 + 1.0)) - t_5) + Float64(sqrt(Float64(t_7 + 1.0)) - t_9))
	tmp = 0.0
	if (t_10 <= 4e-6)
		tmp = Float64(Float64(0.5 / t_5) - Float64(Float64(t_3 + t_9) - Float64(sqrt(Float64(1.0 + t_2)) + t_8)));
	elseif (t_10 <= 1.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t_4)) - t_5) + t_6);
	else
		tmp = Float64(Float64(Float64(1.0 + t_8) - Float64(t_5 + t_9)) + t_6);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if (+.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))) < 3.9999999999999998e-6

   1. Initial program 91.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 47.4%

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

   4. Applied rewrites 47.4%

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

   6. Applied rewrites 47.4%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 15.6%

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

   9. Applied rewrites 15.6%

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

   if 3.9999999999999998e-6 < (+.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))) < 1

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in z around 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 29.7%

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

   7. Applied rewrites 29.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 21.7%

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

   10. Applied rewrites 21.7%

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

   11. Taylor expanded in y around inf

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

       1. lower--.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 29.5%

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

   13. Applied rewrites 29.5%

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

   if 1 < (+.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)))

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in z around 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 29.7%

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

   7. Applied rewrites 29.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 21.7%

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

   10. Applied rewrites 21.7%

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

Alternative 11: 81.0% accurate, 0.0× speedup

Math

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

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 (- (sqrt (+ t_8 1.0)) (sqrt t_8)))
       (t_10 (fmin t_3 t_7))
       (t_11 (sqrt t_10)))
  (if (<=
       (+
        (+
         (+ (- (sqrt (+ t_5 1.0)) t_6) (- (sqrt (+ t_10 1.0)) t_11))
         (- (sqrt (+ t_2 1.0)) (sqrt t_2)))
        t_9)
       1.0)
    (+ (- (sqrt (+ 1.0 t_5)) t_6) t_9)
    (+ (- (+ 1.0 (sqrt (+ 1.0 t_10))) (+ t_6 t_11)) t_9))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = sqrt((t_8 + 1.0)) - sqrt(t_8);
	double t_10 = fmin(t_3, t_7);
	double t_11 = sqrt(t_10);
	double tmp;
	if (((((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_10 + 1.0)) - t_11)) + (sqrt((t_2 + 1.0)) - sqrt(t_2))) + t_9) <= 1.0) {
		tmp = (sqrt((1.0 + t_5)) - t_6) + t_9;
	} else {
		tmp = ((1.0 + sqrt((1.0 + t_10))) - (t_6 + t_11)) + 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_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 = sqrt((t_8 + 1.0d0)) - sqrt(t_8)
    t_10 = fmin(t_3, t_7)
    t_11 = sqrt(t_10)
    if (((((sqrt((t_5 + 1.0d0)) - t_6) + (sqrt((t_10 + 1.0d0)) - t_11)) + (sqrt((t_2 + 1.0d0)) - sqrt(t_2))) + t_9) <= 1.0d0) then
        tmp = (sqrt((1.0d0 + t_5)) - t_6) + t_9
    else
        tmp = ((1.0d0 + sqrt((1.0d0 + t_10))) - (t_6 + t_11)) + t_9
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in z around 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 29.7%

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

   7. Applied rewrites 29.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 21.7%

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

   10. Applied rewrites 21.7%

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

   11. Taylor expanded in y around inf

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

       1. lower--.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 29.5%

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

   13. Applied rewrites 29.5%

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

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

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in z around 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 29.7%

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

   7. Applied rewrites 29.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 21.7%

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

   10. Applied rewrites 21.7%

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

Alternative 12: 78.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\right)\\ t_2 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_3 := \mathsf{min}\left(t\_2, t\right)\\ t_4 := \sqrt{t\_3}\\ t_5 := \mathsf{max}\left(t\_2, t\right)\\ t_6 := \mathsf{min}\left(t\_1, t\_5\right)\\ t_7 := \mathsf{max}\left(t\_1, t\_5\right)\\ t_8 := \sqrt{t\_7 + 1} - \sqrt{t\_7}\\ \mathbf{if}\;t\_6 \leq 1.82:\\ \;\;\;\;\left(\left(2 + 0.5 \cdot t\_6\right) - \left(t\_4 + \sqrt{t\_6}\right)\right) + t\_8\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t\_3} - t\_4\right) + t\_8\\ \end{array} \]

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 (fmin t_2 t))
       (t_4 (sqrt t_3))
       (t_5 (fmax t_2 t))
       (t_6 (fmin t_1 t_5))
       (t_7 (fmax t_1 t_5))
       (t_8 (- (sqrt (+ t_7 1.0)) (sqrt t_7))))
  (if (<= t_6 1.82)
    (+ (- (+ 2.0 (* 0.5 t_6)) (+ t_4 (sqrt t_6))) t_8)
    (+ (- (sqrt (+ 1.0 t_3)) t_4) t_8))))

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 = fmin(t_2, t);
	double t_4 = sqrt(t_3);
	double t_5 = fmax(t_2, t);
	double t_6 = fmin(t_1, t_5);
	double t_7 = fmax(t_1, t_5);
	double t_8 = sqrt((t_7 + 1.0)) - sqrt(t_7);
	double tmp;
	if (t_6 <= 1.82) {
		tmp = ((2.0 + (0.5 * t_6)) - (t_4 + sqrt(t_6))) + t_8;
	} else {
		tmp = (sqrt((1.0 + t_3)) - t_4) + 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_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: tmp
    t_1 = fmin(fmax(x, y), 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 = fmin(t_1, t_5)
    t_7 = fmax(t_1, t_5)
    t_8 = sqrt((t_7 + 1.0d0)) - sqrt(t_7)
    if (t_6 <= 1.82d0) then
        tmp = ((2.0d0 + (0.5d0 * t_6)) - (t_4 + sqrt(t_6))) + t_8
    else
        tmp = (sqrt((1.0d0 + t_3)) - t_4) + 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(fmax(x, y), 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 = fmin(t_1, t_5);
	double t_7 = fmax(t_1, t_5);
	double t_8 = Math.sqrt((t_7 + 1.0)) - Math.sqrt(t_7);
	double tmp;
	if (t_6 <= 1.82) {
		tmp = ((2.0 + (0.5 * t_6)) - (t_4 + Math.sqrt(t_6))) + t_8;
	} else {
		tmp = (Math.sqrt((1.0 + t_3)) - t_4) + t_8;
	}
	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 = fmin(t_2, t)
	t_4 = math.sqrt(t_3)
	t_5 = fmax(t_2, t)
	t_6 = fmin(t_1, t_5)
	t_7 = fmax(t_1, t_5)
	t_8 = math.sqrt((t_7 + 1.0)) - math.sqrt(t_7)
	tmp = 0
	if t_6 <= 1.82:
		tmp = ((2.0 + (0.5 * t_6)) - (t_4 + math.sqrt(t_6))) + t_8
	else:
		tmp = (math.sqrt((1.0 + t_3)) - t_4) + t_8
	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 = fmin(t_2, t)
	t_4 = sqrt(t_3)
	t_5 = fmax(t_2, t)
	t_6 = fmin(t_1, t_5)
	t_7 = fmax(t_1, t_5)
	t_8 = Float64(sqrt(Float64(t_7 + 1.0)) - sqrt(t_7))
	tmp = 0.0
	if (t_6 <= 1.82)
		tmp = Float64(Float64(Float64(2.0 + Float64(0.5 * t_6)) - Float64(t_4 + sqrt(t_6))) + t_8);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t_3)) - t_4) + t_8);
	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 = min(t_2, t);
	t_4 = sqrt(t_3);
	t_5 = max(t_2, t);
	t_6 = min(t_1, t_5);
	t_7 = max(t_1, t_5);
	t_8 = sqrt((t_7 + 1.0)) - sqrt(t_7);
	tmp = 0.0;
	if (t_6 <= 1.82)
		tmp = ((2.0 + (0.5 * t_6)) - (t_4 + sqrt(t_6))) + t_8;
	else
		tmp = (sqrt((1.0 + t_3)) - t_4) + t_8;
	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[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[Min[t$95$1, t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$1, t$95$5], $MachinePrecision]}, Block[{t$95$8 = N[(N[Sqrt[N[(t$95$7 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$7], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 1.82], N[(N[(N[(2.0 + N[(0.5 * t$95$6), $MachinePrecision]), $MachinePrecision] - N[(t$95$4 + N[Sqrt[t$95$6], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$8), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t$95$3), $MachinePrecision]], $MachinePrecision] - t$95$4), $MachinePrecision] + t$95$8), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8200000000000001

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in z around 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 29.7%

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

   7. Applied rewrites 29.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 21.7%

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

   10. Applied rewrites 21.7%

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

   11. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 16.9%

          \[\leadsto \left(\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   13. Applied rewrites 16.9%

       \[\leadsto \left(\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.8200000000000001 < y

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in z around 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 29.7%

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

   7. Applied rewrites 29.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 21.7%

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

   10. Applied rewrites 21.7%

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

   11. Taylor expanded in y around inf

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

       1. lower--.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 29.5%

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

   13. Applied rewrites 29.5%

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

Alternative 13: 78.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\right)\\ t_2 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_3 := \mathsf{min}\left(t\_2, t\right)\\ t_4 := \sqrt{t\_3}\\ t_5 := \mathsf{max}\left(t\_2, t\right)\\ t_6 := \mathsf{min}\left(t\_1, t\_5\right)\\ t_7 := \mathsf{max}\left(t\_1, t\_5\right)\\ \mathbf{if}\;t\_6 \leq 1.82:\\ \;\;\;\;\left(\left(2 + 0.5 \cdot t\_6\right) - \left(t\_4 + \sqrt{t\_6}\right)\right) + \left(\sqrt{t\_7 + 1} - \sqrt{t\_7}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t\_3} + 0.5 \cdot \frac{1}{t\_7 \cdot \sqrt{\frac{1}{t\_7}}}\right) - t\_4\\ \end{array} \]

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 (fmin t_2 t))
       (t_4 (sqrt t_3))
       (t_5 (fmax t_2 t))
       (t_6 (fmin t_1 t_5))
       (t_7 (fmax t_1 t_5)))
  (if (<= t_6 1.82)
    (+
     (- (+ 2.0 (* 0.5 t_6)) (+ t_4 (sqrt t_6)))
     (- (sqrt (+ t_7 1.0)) (sqrt t_7)))
    (-
     (+ (sqrt (+ 1.0 t_3)) (* 0.5 (/ 1.0 (* t_7 (sqrt (/ 1.0 t_7))))))
     t_4))))

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 = fmin(t_2, t);
	double t_4 = sqrt(t_3);
	double t_5 = fmax(t_2, t);
	double t_6 = fmin(t_1, t_5);
	double t_7 = fmax(t_1, t_5);
	double tmp;
	if (t_6 <= 1.82) {
		tmp = ((2.0 + (0.5 * t_6)) - (t_4 + sqrt(t_6))) + (sqrt((t_7 + 1.0)) - sqrt(t_7));
	} else {
		tmp = (sqrt((1.0 + t_3)) + (0.5 * (1.0 / (t_7 * sqrt((1.0 / t_7)))))) - t_4;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = fmin(fmax(x, y), 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 = fmin(t_1, t_5)
    t_7 = fmax(t_1, t_5)
    if (t_6 <= 1.82d0) then
        tmp = ((2.0d0 + (0.5d0 * t_6)) - (t_4 + sqrt(t_6))) + (sqrt((t_7 + 1.0d0)) - sqrt(t_7))
    else
        tmp = (sqrt((1.0d0 + t_3)) + (0.5d0 * (1.0d0 / (t_7 * sqrt((1.0d0 / t_7)))))) - 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(fmax(x, y), 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 = fmin(t_1, t_5);
	double t_7 = fmax(t_1, t_5);
	double tmp;
	if (t_6 <= 1.82) {
		tmp = ((2.0 + (0.5 * t_6)) - (t_4 + Math.sqrt(t_6))) + (Math.sqrt((t_7 + 1.0)) - Math.sqrt(t_7));
	} else {
		tmp = (Math.sqrt((1.0 + t_3)) + (0.5 * (1.0 / (t_7 * Math.sqrt((1.0 / t_7)))))) - t_4;
	}
	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 = fmin(t_2, t)
	t_4 = math.sqrt(t_3)
	t_5 = fmax(t_2, t)
	t_6 = fmin(t_1, t_5)
	t_7 = fmax(t_1, t_5)
	tmp = 0
	if t_6 <= 1.82:
		tmp = ((2.0 + (0.5 * t_6)) - (t_4 + math.sqrt(t_6))) + (math.sqrt((t_7 + 1.0)) - math.sqrt(t_7))
	else:
		tmp = (math.sqrt((1.0 + t_3)) + (0.5 * (1.0 / (t_7 * math.sqrt((1.0 / t_7)))))) - t_4
	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 = fmin(t_2, t)
	t_4 = sqrt(t_3)
	t_5 = fmax(t_2, t)
	t_6 = fmin(t_1, t_5)
	t_7 = fmax(t_1, t_5)
	tmp = 0.0
	if (t_6 <= 1.82)
		tmp = Float64(Float64(Float64(2.0 + Float64(0.5 * t_6)) - Float64(t_4 + sqrt(t_6))) + Float64(sqrt(Float64(t_7 + 1.0)) - sqrt(t_7)));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t_3)) + Float64(0.5 * Float64(1.0 / Float64(t_7 * sqrt(Float64(1.0 / t_7)))))) - t_4);
	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 = min(t_2, t);
	t_4 = sqrt(t_3);
	t_5 = max(t_2, t);
	t_6 = min(t_1, t_5);
	t_7 = max(t_1, t_5);
	tmp = 0.0;
	if (t_6 <= 1.82)
		tmp = ((2.0 + (0.5 * t_6)) - (t_4 + sqrt(t_6))) + (sqrt((t_7 + 1.0)) - sqrt(t_7));
	else
		tmp = (sqrt((1.0 + t_3)) + (0.5 * (1.0 / (t_7 * sqrt((1.0 / t_7)))))) - t_4;
	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[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[Min[t$95$1, t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$1, t$95$5], $MachinePrecision]}, If[LessEqual[t$95$6, 1.82], N[(N[(N[(2.0 + N[(0.5 * t$95$6), $MachinePrecision]), $MachinePrecision] - N[(t$95$4 + N[Sqrt[t$95$6], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$7 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$7], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t$95$3), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[(t$95$7 * N[Sqrt[N[(1.0 / t$95$7), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8200000000000001

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in z around 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 29.7%

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

   7. Applied rewrites 29.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 21.7%

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

   10. Applied rewrites 21.7%

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

   11. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 16.9%

          \[\leadsto \left(\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   13. Applied rewrites 16.9%

       \[\leadsto \left(\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.8200000000000001 < y

   1. Initial program 91.5%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 12.0%

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

   5. Taylor expanded in z 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 13.4%

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

   7. Applied rewrites 13.4%

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

   8. Taylor expanded in t around inf

      \[\leadsto \left(\sqrt{1 + x} + \frac{1}{2} \cdot \frac{1}{t \cdot \sqrt{\frac{1}{t}}}\right) - \sqrt{x} \]
   9. 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 13.5%

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

   10. Applied rewrites 13.5%

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

Alternative 14: 78.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(t\_1, t\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\right)\\ t_4 := \mathsf{max}\left(t\_3, t\_2\right)\\ t_5 := \sqrt{t\_4 + 1} - \sqrt{t\_4}\\ t_6 := \mathsf{min}\left(t\_3, t\_2\right)\\ t_7 := \mathsf{min}\left(t\_1, t\right)\\ t_8 := \sqrt{t\_7}\\ \mathbf{if}\;t\_6 \leq 0.31:\\ \;\;\;\;\left(2 - \left(t\_8 + \sqrt{t\_6}\right)\right) + t\_5\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t\_7} - t\_8\right) + t\_5\\ \end{array} \]

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 (fmax x y) (fmax (fmin x y) z)))
       (t_4 (fmax t_3 t_2))
       (t_5 (- (sqrt (+ t_4 1.0)) (sqrt t_4)))
       (t_6 (fmin t_3 t_2))
       (t_7 (fmin t_1 t))
       (t_8 (sqrt t_7)))
  (if (<= t_6 0.31)
    (+ (- 2.0 (+ t_8 (sqrt t_6))) t_5)
    (+ (- (sqrt (+ 1.0 t_7)) t_8) t_5))))

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(fmax(x, y), fmax(fmin(x, y), z));
	double t_4 = fmax(t_3, t_2);
	double t_5 = sqrt((t_4 + 1.0)) - sqrt(t_4);
	double t_6 = fmin(t_3, t_2);
	double t_7 = fmin(t_1, t);
	double t_8 = sqrt(t_7);
	double tmp;
	if (t_6 <= 0.31) {
		tmp = (2.0 - (t_8 + sqrt(t_6))) + t_5;
	} else {
		tmp = (sqrt((1.0 + t_7)) - t_8) + 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) :: tmp
    t_1 = fmin(fmin(x, y), z)
    t_2 = fmax(t_1, t)
    t_3 = fmin(fmax(x, y), fmax(fmin(x, y), z))
    t_4 = fmax(t_3, t_2)
    t_5 = sqrt((t_4 + 1.0d0)) - sqrt(t_4)
    t_6 = fmin(t_3, t_2)
    t_7 = fmin(t_1, t)
    t_8 = sqrt(t_7)
    if (t_6 <= 0.31d0) then
        tmp = (2.0d0 - (t_8 + sqrt(t_6))) + t_5
    else
        tmp = (sqrt((1.0d0 + t_7)) - t_8) + t_5
    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 = fmin(fmax(x, y), fmax(fmin(x, y), z));
	double t_4 = fmax(t_3, t_2);
	double t_5 = Math.sqrt((t_4 + 1.0)) - Math.sqrt(t_4);
	double t_6 = fmin(t_3, t_2);
	double t_7 = fmin(t_1, t);
	double t_8 = Math.sqrt(t_7);
	double tmp;
	if (t_6 <= 0.31) {
		tmp = (2.0 - (t_8 + Math.sqrt(t_6))) + t_5;
	} else {
		tmp = (Math.sqrt((1.0 + t_7)) - t_8) + t_5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = fmin(fmax(x, y), fmax(fmin(x, y), z))
	t_4 = fmax(t_3, t_2)
	t_5 = math.sqrt((t_4 + 1.0)) - math.sqrt(t_4)
	t_6 = fmin(t_3, t_2)
	t_7 = fmin(t_1, t)
	t_8 = math.sqrt(t_7)
	tmp = 0
	if t_6 <= 0.31:
		tmp = (2.0 - (t_8 + math.sqrt(t_6))) + t_5
	else:
		tmp = (math.sqrt((1.0 + t_7)) - t_8) + t_5
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = fmin(fmax(x, y), fmax(fmin(x, y), z))
	t_4 = fmax(t_3, t_2)
	t_5 = Float64(sqrt(Float64(t_4 + 1.0)) - sqrt(t_4))
	t_6 = fmin(t_3, t_2)
	t_7 = fmin(t_1, t)
	t_8 = sqrt(t_7)
	tmp = 0.0
	if (t_6 <= 0.31)
		tmp = Float64(Float64(2.0 - Float64(t_8 + sqrt(t_6))) + t_5);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t_7)) - t_8) + t_5);
	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 = min(max(x, y), max(min(x, y), z));
	t_4 = max(t_3, t_2);
	t_5 = sqrt((t_4 + 1.0)) - sqrt(t_4);
	t_6 = min(t_3, t_2);
	t_7 = min(t_1, t);
	t_8 = sqrt(t_7);
	tmp = 0.0;
	if (t_6 <= 0.31)
		tmp = (2.0 - (t_8 + sqrt(t_6))) + t_5;
	else
		tmp = (sqrt((1.0 + t_7)) - t_8) + t_5;
	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[Min[N[Max[x, y], $MachinePrecision], N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Max[t$95$3, t$95$2], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(t$95$4 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Min[t$95$3, t$95$2], $MachinePrecision]}, Block[{t$95$7 = N[Min[t$95$1, t], $MachinePrecision]}, Block[{t$95$8 = N[Sqrt[t$95$7], $MachinePrecision]}, If[LessEqual[t$95$6, 0.31], N[(N[(2.0 - N[(t$95$8 + N[Sqrt[t$95$6], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t$95$7), $MachinePrecision]], $MachinePrecision] - t$95$8), $MachinePrecision] + t$95$5), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.31

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in z around 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 29.7%

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

   7. Applied rewrites 29.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 21.7%

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

   10. Applied rewrites 21.7%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 15.2%

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

   if 0.31 < y

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in z around 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 29.7%

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

   7. Applied rewrites 29.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 21.7%

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

   10. Applied rewrites 21.7%

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

   11. Taylor expanded in y around inf

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

       1. lower--.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 29.5%

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

   13. Applied rewrites 29.5%

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

Alternative 15: 50.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ \left(\sqrt{1 + t\_1} - \sqrt{t\_1}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fmin (fmin x y) z)))
  (+ (- (sqrt (+ 1.0 t_1)) (sqrt t_1)) (- (sqrt (+ t 1.0)) (sqrt t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	return (sqrt((1.0 + t_1)) - sqrt(t_1)) + (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
    real(8) :: t_1
    t_1 = fmin(fmin(x, y), z)
    code = (sqrt((1.0d0 + t_1)) - sqrt(t_1)) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

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

Python

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

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	return Float64(Float64(sqrt(Float64(1.0 + t_1)) - sqrt(t_1)) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = min(min(x, y), z);
	tmp = (sqrt((1.0 + t_1)) - sqrt(t_1)) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

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

Derivation

1. Initial program 91.5%

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

2. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 51.2%

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

4. Applied rewrites 51.2%

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

5. Taylor expanded in z around 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 29.7%

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

7. Applied rewrites 29.7%

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

8. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-+.f64 21.7%

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

10. Applied rewrites 21.7%

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

11. Taylor expanded in y around inf

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

    1. lower--.f64 N/A

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

    2. lower-sqrt.f64 N/A

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

    3. lower-+.f64 N/A

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

    4. lower-sqrt.f64 29.5%

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

13. Applied rewrites 29.5%

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

Alternative 16: 49.4% 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 -1.0))
   (- (sqrt (- t_2 -1.0)) (+ (sqrt t_2) (sqrt t_3))))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

2. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-+.f64 N/A

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

4. Applied rewrites 12.0%

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

5. Taylor expanded in z 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 13.4%

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

7. Applied rewrites 13.4%

   \[\leadsto \left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{t} + \sqrt{x}\right)} \]
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. lower--.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. lift--.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 17: 16.3% accurate, 3.0× speedup

Math

\[\frac{0.5}{t \cdot \sqrt{\frac{1}{t}}} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (/ 0.5 (* t (sqrt (/ 1.0 t)))))

C

double code(double x, double y, double z, double t) {
	return 0.5 / (t * sqrt((1.0 / 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 = 0.5d0 / (t * sqrt((1.0d0 / t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.5 / (t * Math.sqrt((1.0 / t)));
}

Python

def code(x, y, z, t):
	return 0.5 / (t * math.sqrt((1.0 / t)))

Julia

function code(x, y, z, t)
	return Float64(0.5 / Float64(t * sqrt(Float64(1.0 / t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.5 / (t * sqrt((1.0 / t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(0.5 / N[(t * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

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

2. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-+.f64 N/A

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

4. Applied rewrites 12.0%

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

5. Taylor expanded in z 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 13.4%

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

7. Applied rewrites 13.4%

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

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 15.6%

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

10. Applied rewrites 15.6%

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

11. Taylor expanded in t around inf

    \[\leadsto \frac{\frac{1}{2}}{t \cdot \sqrt{\frac{1}{t}}} \]
12. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{t \cdot \sqrt{\frac{1}{t}}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{t \cdot \sqrt{\frac{1}{t}}} \]

    3. lower-sqrt.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{t \cdot \sqrt{\frac{1}{t}}} \]

    4. lower-/.f64 8.3%

       \[\leadsto \frac{0.5}{t \cdot \sqrt{\frac{1}{t}}} \]

13. Applied rewrites 8.3%

    \[\leadsto \frac{0.5}{t \cdot \sqrt{\frac{1}{t}}} \]
14. Add Preprocessing

Alternative 18: 14.3% accurate, 5.2× speedup

Math

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

FPCore

(FPCore (x y z t)
  :precision binary64
  (- (+ 1.0 (* 0.5 t)) (sqrt t)))

C

double code(double x, double y, double z, double t) {
	return (1.0 + (0.5 * 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 = (1.0d0 + (0.5d0 * t)) - sqrt(t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (1.0 + (0.5 * t)) - Math.sqrt(t);
}

Python

def code(x, y, z, t):
	return (1.0 + (0.5 * t)) - math.sqrt(t)

Julia

function code(x, y, z, t)
	return Float64(Float64(1.0 + Float64(0.5 * t)) - sqrt(t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (1.0 + (0.5 * t)) - sqrt(t);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(1.0 + N[(0.5 * t), $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

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

2. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-+.f64 N/A

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

4. Applied rewrites 12.0%

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

5. Taylor expanded in z 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 13.4%

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

7. Applied rewrites 13.4%

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

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 15.6%

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

10. Applied rewrites 15.6%

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

11. Taylor expanded in t around 0

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

    1. lower--.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-sqrt.f64 16.3%

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

13. Applied rewrites 16.3%

    \[\leadsto \left(1 + 0.5 \cdot t\right) - \sqrt{t} \]
14. Add Preprocessing

Alternative 19: 8.3% accurate, 8.1× speedup

Math

\[1 - \sqrt{t} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

2. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-+.f64 N/A

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

4. Applied rewrites 12.0%

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

5. Taylor expanded in z 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 13.4%

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

7. Applied rewrites 13.4%

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

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 15.6%

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

10. Applied rewrites 15.6%

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

11. Taylor expanded in t around 0

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

    1. lower--.f64 N/A

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

    2. lower-sqrt.f64 14.3%

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

13. Applied rewrites 14.3%

    \[\leadsto 1 - \sqrt{t} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64
  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))