Main:z from

Percentage Accurate: 91.9% → 96.9%

Time: 15.6s

Alternatives: 16

Speedup: 0.2×

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: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \mathsf{max}\left(t\_4, t\right)\\ t_7 := \mathsf{max}\left(t\_3, t\_6\right)\\ t_8 := \mathsf{min}\left(t\_2, t\_7\right)\\ t_9 := \sqrt{t\_8}\\ t_10 := \sqrt{t\_8 + 1} - t\_9\\ t_11 := t\_8 - -1\\ t_12 := \mathsf{min}\left(t\_3, t\_6\right)\\ t_13 := \sqrt{t\_12 + 1} - \sqrt{t\_12}\\ t_14 := \left(\sqrt{t\_5 + 1} - \sqrt{t\_5}\right) + t\_13\\ t_15 := \mathsf{max}\left(t\_2, t\_7\right)\\ t_16 := \sqrt{t\_15 + 1} - \sqrt{t\_15}\\ t_17 := \left(t\_14 + t\_10\right) + t\_16\\ \mathbf{if}\;t\_17 \leq 0:\\ \;\;\;\;\left(\left(\frac{0.5}{t\_5 \cdot \sqrt{\frac{1}{t\_5}}} + t\_13\right) + t\_10\right) + t\_16\\ \mathbf{elif}\;t\_17 \leq 2.00002:\\ \;\;\;\;\left(t\_14 + \frac{0.5}{t\_8 \cdot \sqrt{\frac{1}{t\_8}}}\right) + \frac{0.5}{t\_15 \cdot \sqrt{\frac{1}{t\_15}}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_14 + \frac{t\_11 - t\_9 \cdot t\_9}{\sqrt{t\_11} + t\_9}\right) + t\_16\\ \end{array} \]

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 (fmax t_4 t))
        (t_7 (fmax t_3 t_6))
        (t_8 (fmin t_2 t_7))
        (t_9 (sqrt t_8))
        (t_10 (- (sqrt (+ t_8 1.0)) t_9))
        (t_11 (- t_8 -1.0))
        (t_12 (fmin t_3 t_6))
        (t_13 (- (sqrt (+ t_12 1.0)) (sqrt t_12)))
        (t_14 (+ (- (sqrt (+ t_5 1.0)) (sqrt t_5)) t_13))
        (t_15 (fmax t_2 t_7))
        (t_16 (- (sqrt (+ t_15 1.0)) (sqrt t_15)))
        (t_17 (+ (+ t_14 t_10) t_16)))
   (if (<= t_17 0.0)
     (+ (+ (+ (/ 0.5 (* t_5 (sqrt (/ 1.0 t_5)))) t_13) t_10) t_16)
     (if (<= t_17 2.00002)
       (+
        (+ t_14 (/ 0.5 (* t_8 (sqrt (/ 1.0 t_8)))))
        (/ 0.5 (* t_15 (sqrt (/ 1.0 t_15)))))
       (+ (+ t_14 (/ (- t_11 (* t_9 t_9)) (+ (sqrt t_11) t_9))) t_16)))))

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 = fmax(t_4, t);
	double t_7 = fmax(t_3, t_6);
	double t_8 = fmin(t_2, t_7);
	double t_9 = sqrt(t_8);
	double t_10 = sqrt((t_8 + 1.0)) - t_9;
	double t_11 = t_8 - -1.0;
	double t_12 = fmin(t_3, t_6);
	double t_13 = sqrt((t_12 + 1.0)) - sqrt(t_12);
	double t_14 = (sqrt((t_5 + 1.0)) - sqrt(t_5)) + t_13;
	double t_15 = fmax(t_2, t_7);
	double t_16 = sqrt((t_15 + 1.0)) - sqrt(t_15);
	double t_17 = (t_14 + t_10) + t_16;
	double tmp;
	if (t_17 <= 0.0) {
		tmp = (((0.5 / (t_5 * sqrt((1.0 / t_5)))) + t_13) + t_10) + t_16;
	} else if (t_17 <= 2.00002) {
		tmp = (t_14 + (0.5 / (t_8 * sqrt((1.0 / t_8))))) + (0.5 / (t_15 * sqrt((1.0 / t_15))));
	} else {
		tmp = (t_14 + ((t_11 - (t_9 * t_9)) / (sqrt(t_11) + t_9))) + t_16;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_14
    real(8) :: t_15
    real(8) :: t_16
    real(8) :: t_17
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmin(t_4, t)
    t_6 = fmax(t_4, t)
    t_7 = fmax(t_3, t_6)
    t_8 = fmin(t_2, t_7)
    t_9 = sqrt(t_8)
    t_10 = sqrt((t_8 + 1.0d0)) - t_9
    t_11 = t_8 - (-1.0d0)
    t_12 = fmin(t_3, t_6)
    t_13 = sqrt((t_12 + 1.0d0)) - sqrt(t_12)
    t_14 = (sqrt((t_5 + 1.0d0)) - sqrt(t_5)) + t_13
    t_15 = fmax(t_2, t_7)
    t_16 = sqrt((t_15 + 1.0d0)) - sqrt(t_15)
    t_17 = (t_14 + t_10) + t_16
    if (t_17 <= 0.0d0) then
        tmp = (((0.5d0 / (t_5 * sqrt((1.0d0 / t_5)))) + t_13) + t_10) + t_16
    else if (t_17 <= 2.00002d0) then
        tmp = (t_14 + (0.5d0 / (t_8 * sqrt((1.0d0 / t_8))))) + (0.5d0 / (t_15 * sqrt((1.0d0 / t_15))))
    else
        tmp = (t_14 + ((t_11 - (t_9 * t_9)) / (sqrt(t_11) + t_9))) + t_16
    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 = fmax(t_4, t);
	double t_7 = fmax(t_3, t_6);
	double t_8 = fmin(t_2, t_7);
	double t_9 = Math.sqrt(t_8);
	double t_10 = Math.sqrt((t_8 + 1.0)) - t_9;
	double t_11 = t_8 - -1.0;
	double t_12 = fmin(t_3, t_6);
	double t_13 = Math.sqrt((t_12 + 1.0)) - Math.sqrt(t_12);
	double t_14 = (Math.sqrt((t_5 + 1.0)) - Math.sqrt(t_5)) + t_13;
	double t_15 = fmax(t_2, t_7);
	double t_16 = Math.sqrt((t_15 + 1.0)) - Math.sqrt(t_15);
	double t_17 = (t_14 + t_10) + t_16;
	double tmp;
	if (t_17 <= 0.0) {
		tmp = (((0.5 / (t_5 * Math.sqrt((1.0 / t_5)))) + t_13) + t_10) + t_16;
	} else if (t_17 <= 2.00002) {
		tmp = (t_14 + (0.5 / (t_8 * Math.sqrt((1.0 / t_8))))) + (0.5 / (t_15 * Math.sqrt((1.0 / t_15))));
	} else {
		tmp = (t_14 + ((t_11 - (t_9 * t_9)) / (Math.sqrt(t_11) + t_9))) + t_16;
	}
	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 = fmax(t_4, t)
	t_7 = fmax(t_3, t_6)
	t_8 = fmin(t_2, t_7)
	t_9 = math.sqrt(t_8)
	t_10 = math.sqrt((t_8 + 1.0)) - t_9
	t_11 = t_8 - -1.0
	t_12 = fmin(t_3, t_6)
	t_13 = math.sqrt((t_12 + 1.0)) - math.sqrt(t_12)
	t_14 = (math.sqrt((t_5 + 1.0)) - math.sqrt(t_5)) + t_13
	t_15 = fmax(t_2, t_7)
	t_16 = math.sqrt((t_15 + 1.0)) - math.sqrt(t_15)
	t_17 = (t_14 + t_10) + t_16
	tmp = 0
	if t_17 <= 0.0:
		tmp = (((0.5 / (t_5 * math.sqrt((1.0 / t_5)))) + t_13) + t_10) + t_16
	elif t_17 <= 2.00002:
		tmp = (t_14 + (0.5 / (t_8 * math.sqrt((1.0 / t_8))))) + (0.5 / (t_15 * math.sqrt((1.0 / t_15))))
	else:
		tmp = (t_14 + ((t_11 - (t_9 * t_9)) / (math.sqrt(t_11) + t_9))) + t_16
	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 = fmax(t_4, t)
	t_7 = fmax(t_3, t_6)
	t_8 = fmin(t_2, t_7)
	t_9 = sqrt(t_8)
	t_10 = Float64(sqrt(Float64(t_8 + 1.0)) - t_9)
	t_11 = Float64(t_8 - -1.0)
	t_12 = fmin(t_3, t_6)
	t_13 = Float64(sqrt(Float64(t_12 + 1.0)) - sqrt(t_12))
	t_14 = Float64(Float64(sqrt(Float64(t_5 + 1.0)) - sqrt(t_5)) + t_13)
	t_15 = fmax(t_2, t_7)
	t_16 = Float64(sqrt(Float64(t_15 + 1.0)) - sqrt(t_15))
	t_17 = Float64(Float64(t_14 + t_10) + t_16)
	tmp = 0.0
	if (t_17 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(0.5 / Float64(t_5 * sqrt(Float64(1.0 / t_5)))) + t_13) + t_10) + t_16);
	elseif (t_17 <= 2.00002)
		tmp = Float64(Float64(t_14 + Float64(0.5 / Float64(t_8 * sqrt(Float64(1.0 / t_8))))) + Float64(0.5 / Float64(t_15 * sqrt(Float64(1.0 / t_15)))));
	else
		tmp = Float64(Float64(t_14 + Float64(Float64(t_11 - Float64(t_9 * t_9)) / Float64(sqrt(t_11) + t_9))) + t_16);
	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 = max(t_4, t);
	t_7 = max(t_3, t_6);
	t_8 = min(t_2, t_7);
	t_9 = sqrt(t_8);
	t_10 = sqrt((t_8 + 1.0)) - t_9;
	t_11 = t_8 - -1.0;
	t_12 = min(t_3, t_6);
	t_13 = sqrt((t_12 + 1.0)) - sqrt(t_12);
	t_14 = (sqrt((t_5 + 1.0)) - sqrt(t_5)) + t_13;
	t_15 = max(t_2, t_7);
	t_16 = sqrt((t_15 + 1.0)) - sqrt(t_15);
	t_17 = (t_14 + t_10) + t_16;
	tmp = 0.0;
	if (t_17 <= 0.0)
		tmp = (((0.5 / (t_5 * sqrt((1.0 / t_5)))) + t_13) + t_10) + t_16;
	elseif (t_17 <= 2.00002)
		tmp = (t_14 + (0.5 / (t_8 * sqrt((1.0 / t_8))))) + (0.5 / (t_15 * sqrt((1.0 / t_15))));
	else
		tmp = (t_14 + ((t_11 - (t_9 * t_9)) / (sqrt(t_11) + t_9))) + t_16;
	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[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$3, t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[Min[t$95$2, t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Sqrt[t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[(N[Sqrt[N[(t$95$8 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$9), $MachinePrecision]}, Block[{t$95$11 = N[(t$95$8 - -1.0), $MachinePrecision]}, Block[{t$95$12 = N[Min[t$95$3, t$95$6], $MachinePrecision]}, Block[{t$95$13 = N[(N[Sqrt[N[(t$95$12 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$12], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$14 = N[(N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision] + t$95$13), $MachinePrecision]}, Block[{t$95$15 = N[Max[t$95$2, t$95$7], $MachinePrecision]}, Block[{t$95$16 = N[(N[Sqrt[N[(t$95$15 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$15], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$17 = N[(N[(t$95$14 + t$95$10), $MachinePrecision] + t$95$16), $MachinePrecision]}, If[LessEqual[t$95$17, 0.0], N[(N[(N[(N[(0.5 / N[(t$95$5 * N[Sqrt[N[(1.0 / t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$13), $MachinePrecision] + t$95$10), $MachinePrecision] + t$95$16), $MachinePrecision], If[LessEqual[t$95$17, 2.00002], N[(N[(t$95$14 + N[(0.5 / N[(t$95$8 * N[Sqrt[N[(1.0 / t$95$8), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(t$95$15 * N[Sqrt[N[(1.0 / t$95$15), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$14 + N[(N[(t$95$11 - N[(t$95$9 * t$95$9), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$11], $MachinePrecision] + t$95$9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$16), $MachinePrecision]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 91.9%

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

   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%

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

   if 0.0 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.0000200000000001

   1. Initial program 91.9%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 48.7

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 26.5

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

   7. Applied rewrites 26.5%

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

   if 2.0000200000000001 < (+.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 91.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

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

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

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

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

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

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

      15. lower-unsound-+.f64 72.9

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 72.9

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

   3. Applied rewrites 72.9%

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

Alternative 2: 95.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{max}\left(t\_4, t\right)\\ t_8 := \mathsf{max}\left(t\_2, t\_7\right)\\ t_9 := \mathsf{min}\left(t\_3, t\_8\right)\\ t_10 := \sqrt{t\_9 - -1}\\ t_11 := \sqrt{t\_9}\\ t_12 := \sqrt{t\_9 + 1} - t\_11\\ t_13 := \mathsf{min}\left(t\_2, t\_7\right)\\ t_14 := \mathsf{max}\left(t\_3, t\_8\right)\\ t_15 := \sqrt{t\_14 + 1} - \sqrt{t\_14}\\ t_16 := \sqrt{t\_13}\\ t_17 := \sqrt{t\_13 + 1} - t\_16\\ t_18 := \sqrt{t\_5 - -1}\\ \mathbf{if}\;\left(\left(\left(\sqrt{t\_5 + 1} - t\_6\right) + t\_17\right) + t\_12\right) + t\_15 \leq 0:\\ \;\;\;\;\left(\left(\frac{0.5}{t\_5 \cdot \sqrt{\frac{1}{t\_5}}} + t\_17\right) + t\_12\right) + t\_15\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \frac{t\_6 - \left(\sqrt{t\_13 - -1} - t\_16\right)}{t\_18}\right) \cdot t\_18 + \frac{t\_10 \cdot t\_10 - t\_11 \cdot t\_11}{t\_10 + t\_11}\right) + t\_15\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) z))
        (t_2 (fmin (fmax x y) t_1))
        (t_3 (fmax (fmax x y) t_1))
        (t_4 (fmin (fmin x y) z))
        (t_5 (fmin t_4 t))
        (t_6 (sqrt t_5))
        (t_7 (fmax t_4 t))
        (t_8 (fmax t_2 t_7))
        (t_9 (fmin t_3 t_8))
        (t_10 (sqrt (- t_9 -1.0)))
        (t_11 (sqrt t_9))
        (t_12 (- (sqrt (+ t_9 1.0)) t_11))
        (t_13 (fmin t_2 t_7))
        (t_14 (fmax t_3 t_8))
        (t_15 (- (sqrt (+ t_14 1.0)) (sqrt t_14)))
        (t_16 (sqrt t_13))
        (t_17 (- (sqrt (+ t_13 1.0)) t_16))
        (t_18 (sqrt (- t_5 -1.0))))
   (if (<= (+ (+ (+ (- (sqrt (+ t_5 1.0)) t_6) t_17) t_12) t_15) 0.0)
     (+ (+ (+ (/ 0.5 (* t_5 (sqrt (/ 1.0 t_5)))) t_17) t_12) t_15)
     (+
      (+
       (* (- 1.0 (/ (- t_6 (- (sqrt (- t_13 -1.0)) t_16)) t_18)) t_18)
       (/ (- (* t_10 t_10) (* t_11 t_11)) (+ t_10 t_11)))
      t_15))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_2, t_7);
	double t_9 = fmin(t_3, t_8);
	double t_10 = sqrt((t_9 - -1.0));
	double t_11 = sqrt(t_9);
	double t_12 = sqrt((t_9 + 1.0)) - t_11;
	double t_13 = fmin(t_2, t_7);
	double t_14 = fmax(t_3, t_8);
	double t_15 = sqrt((t_14 + 1.0)) - sqrt(t_14);
	double t_16 = sqrt(t_13);
	double t_17 = sqrt((t_13 + 1.0)) - t_16;
	double t_18 = sqrt((t_5 - -1.0));
	double tmp;
	if (((((sqrt((t_5 + 1.0)) - t_6) + t_17) + t_12) + t_15) <= 0.0) {
		tmp = (((0.5 / (t_5 * sqrt((1.0 / t_5)))) + t_17) + t_12) + t_15;
	} else {
		tmp = (((1.0 - ((t_6 - (sqrt((t_13 - -1.0)) - t_16)) / t_18)) * t_18) + (((t_10 * t_10) - (t_11 * t_11)) / (t_10 + t_11))) + t_15;
	}
	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 = fmin(fmax(x, y), t_1)
    t_3 = fmax(fmax(x, y), t_1)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmin(t_4, t)
    t_6 = sqrt(t_5)
    t_7 = fmax(t_4, t)
    t_8 = fmax(t_2, t_7)
    t_9 = fmin(t_3, t_8)
    t_10 = sqrt((t_9 - (-1.0d0)))
    t_11 = sqrt(t_9)
    t_12 = sqrt((t_9 + 1.0d0)) - t_11
    t_13 = fmin(t_2, t_7)
    t_14 = fmax(t_3, t_8)
    t_15 = sqrt((t_14 + 1.0d0)) - sqrt(t_14)
    t_16 = sqrt(t_13)
    t_17 = sqrt((t_13 + 1.0d0)) - t_16
    t_18 = sqrt((t_5 - (-1.0d0)))
    if (((((sqrt((t_5 + 1.0d0)) - t_6) + t_17) + t_12) + t_15) <= 0.0d0) then
        tmp = (((0.5d0 / (t_5 * sqrt((1.0d0 / t_5)))) + t_17) + t_12) + t_15
    else
        tmp = (((1.0d0 - ((t_6 - (sqrt((t_13 - (-1.0d0))) - t_16)) / t_18)) * t_18) + (((t_10 * t_10) - (t_11 * t_11)) / (t_10 + t_11))) + t_15
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = Math.sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_2, t_7);
	double t_9 = fmin(t_3, t_8);
	double t_10 = Math.sqrt((t_9 - -1.0));
	double t_11 = Math.sqrt(t_9);
	double t_12 = Math.sqrt((t_9 + 1.0)) - t_11;
	double t_13 = fmin(t_2, t_7);
	double t_14 = fmax(t_3, t_8);
	double t_15 = Math.sqrt((t_14 + 1.0)) - Math.sqrt(t_14);
	double t_16 = Math.sqrt(t_13);
	double t_17 = Math.sqrt((t_13 + 1.0)) - t_16;
	double t_18 = Math.sqrt((t_5 - -1.0));
	double tmp;
	if (((((Math.sqrt((t_5 + 1.0)) - t_6) + t_17) + t_12) + t_15) <= 0.0) {
		tmp = (((0.5 / (t_5 * Math.sqrt((1.0 / t_5)))) + t_17) + t_12) + t_15;
	} else {
		tmp = (((1.0 - ((t_6 - (Math.sqrt((t_13 - -1.0)) - t_16)) / t_18)) * t_18) + (((t_10 * t_10) - (t_11 * t_11)) / (t_10 + t_11))) + t_15;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = math.sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_2, t_7)
	t_9 = fmin(t_3, t_8)
	t_10 = math.sqrt((t_9 - -1.0))
	t_11 = math.sqrt(t_9)
	t_12 = math.sqrt((t_9 + 1.0)) - t_11
	t_13 = fmin(t_2, t_7)
	t_14 = fmax(t_3, t_8)
	t_15 = math.sqrt((t_14 + 1.0)) - math.sqrt(t_14)
	t_16 = math.sqrt(t_13)
	t_17 = math.sqrt((t_13 + 1.0)) - t_16
	t_18 = math.sqrt((t_5 - -1.0))
	tmp = 0
	if ((((math.sqrt((t_5 + 1.0)) - t_6) + t_17) + t_12) + t_15) <= 0.0:
		tmp = (((0.5 / (t_5 * math.sqrt((1.0 / t_5)))) + t_17) + t_12) + t_15
	else:
		tmp = (((1.0 - ((t_6 - (math.sqrt((t_13 - -1.0)) - t_16)) / t_18)) * t_18) + (((t_10 * t_10) - (t_11 * t_11)) / (t_10 + t_11))) + t_15
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_2, t_7)
	t_9 = fmin(t_3, t_8)
	t_10 = sqrt(Float64(t_9 - -1.0))
	t_11 = sqrt(t_9)
	t_12 = Float64(sqrt(Float64(t_9 + 1.0)) - t_11)
	t_13 = fmin(t_2, t_7)
	t_14 = fmax(t_3, t_8)
	t_15 = Float64(sqrt(Float64(t_14 + 1.0)) - sqrt(t_14))
	t_16 = sqrt(t_13)
	t_17 = Float64(sqrt(Float64(t_13 + 1.0)) - t_16)
	t_18 = sqrt(Float64(t_5 - -1.0))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + t_17) + t_12) + t_15) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(0.5 / Float64(t_5 * sqrt(Float64(1.0 / t_5)))) + t_17) + t_12) + t_15);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - Float64(Float64(t_6 - Float64(sqrt(Float64(t_13 - -1.0)) - t_16)) / t_18)) * t_18) + Float64(Float64(Float64(t_10 * t_10) - Float64(t_11 * t_11)) / Float64(t_10 + t_11))) + t_15);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = min(max(x, y), t_1);
	t_3 = max(max(x, y), t_1);
	t_4 = min(min(x, y), z);
	t_5 = min(t_4, t);
	t_6 = sqrt(t_5);
	t_7 = max(t_4, t);
	t_8 = max(t_2, t_7);
	t_9 = min(t_3, t_8);
	t_10 = sqrt((t_9 - -1.0));
	t_11 = sqrt(t_9);
	t_12 = sqrt((t_9 + 1.0)) - t_11;
	t_13 = min(t_2, t_7);
	t_14 = max(t_3, t_8);
	t_15 = sqrt((t_14 + 1.0)) - sqrt(t_14);
	t_16 = sqrt(t_13);
	t_17 = sqrt((t_13 + 1.0)) - t_16;
	t_18 = sqrt((t_5 - -1.0));
	tmp = 0.0;
	if (((((sqrt((t_5 + 1.0)) - t_6) + t_17) + t_12) + t_15) <= 0.0)
		tmp = (((0.5 / (t_5 * sqrt((1.0 / t_5)))) + t_17) + t_12) + t_15;
	else
		tmp = (((1.0 - ((t_6 - (sqrt((t_13 - -1.0)) - t_16)) / t_18)) * t_18) + (((t_10 * t_10) - (t_11 * t_11)) / (t_10 + t_11))) + t_15;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$4, t], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$2, t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$3, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[N[(t$95$9 - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$11 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$12 = N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$11), $MachinePrecision]}, Block[{t$95$13 = N[Min[t$95$2, t$95$7], $MachinePrecision]}, Block[{t$95$14 = N[Max[t$95$3, t$95$8], $MachinePrecision]}, Block[{t$95$15 = N[(N[Sqrt[N[(t$95$14 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$14], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$16 = N[Sqrt[t$95$13], $MachinePrecision]}, Block[{t$95$17 = N[(N[Sqrt[N[(t$95$13 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$16), $MachinePrecision]}, Block[{t$95$18 = N[Sqrt[N[(t$95$5 - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + t$95$17), $MachinePrecision] + t$95$12), $MachinePrecision] + t$95$15), $MachinePrecision], 0.0], N[(N[(N[(N[(0.5 / N[(t$95$5 * N[Sqrt[N[(1.0 / t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$17), $MachinePrecision] + t$95$12), $MachinePrecision] + t$95$15), $MachinePrecision], N[(N[(N[(N[(1.0 - N[(N[(t$95$6 - N[(N[Sqrt[N[(t$95$13 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$16), $MachinePrecision]), $MachinePrecision] / t$95$18), $MachinePrecision]), $MachinePrecision] * t$95$18), $MachinePrecision] + N[(N[(N[(t$95$10 * t$95$10), $MachinePrecision] - N[(t$95$11 * t$95$11), $MachinePrecision]), $MachinePrecision] / N[(t$95$10 + t$95$11), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$15), $MachinePrecision]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.9%

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

   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%

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

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

   1. Initial program 91.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-to-mult N/A

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

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

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

   3. Applied rewrites 72.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

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

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

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

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

      6. lift-+.f64 N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lift--.f64 N/A

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

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

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

      15. lower-unsound-+.f64 72.3

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lift--.f64 72.3

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

   5. Applied rewrites 72.3%

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

Alternative 3: 95.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \sqrt{t\_2}\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_6 := \mathsf{max}\left(t\_4, t\right)\\ t_7 := \sqrt{t\_6 + 1} - \sqrt{t\_6}\\ t_8 := \sqrt{t\_2 + 1} - t\_3\\ t_9 := \mathsf{min}\left(t\_4, t\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \sqrt{t\_9 - -1}\\ t_12 := \sqrt{t\_5 + 1} - \sqrt{t\_5}\\ \mathbf{if}\;\left(\left(\left(\sqrt{t\_9 + 1} - t\_10\right) + t\_8\right) + t\_12\right) + t\_7 \leq 0:\\ \;\;\;\;\left(\left(\frac{0.5}{t\_9 \cdot \sqrt{\frac{1}{t\_9}}} + t\_8\right) + t\_12\right) + t\_7\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\left(t\_11 - t\_10\right) - \left(t\_3 - \sqrt{t\_2 - -1}\right)\right) \cdot t\_11}{t\_11} + t\_12\right) + t\_7\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) z))
        (t_2 (fmin (fmax x y) t_1))
        (t_3 (sqrt t_2))
        (t_4 (fmin (fmin x y) z))
        (t_5 (fmax (fmax x y) t_1))
        (t_6 (fmax t_4 t))
        (t_7 (- (sqrt (+ t_6 1.0)) (sqrt t_6)))
        (t_8 (- (sqrt (+ t_2 1.0)) t_3))
        (t_9 (fmin t_4 t))
        (t_10 (sqrt t_9))
        (t_11 (sqrt (- t_9 -1.0)))
        (t_12 (- (sqrt (+ t_5 1.0)) (sqrt t_5))))
   (if (<= (+ (+ (+ (- (sqrt (+ t_9 1.0)) t_10) t_8) t_12) t_7) 0.0)
     (+ (+ (+ (/ 0.5 (* t_9 (sqrt (/ 1.0 t_9)))) t_8) t_12) t_7)
     (+
      (+ (/ (* (- (- t_11 t_10) (- t_3 (sqrt (- t_2 -1.0)))) t_11) t_11) t_12)
      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(fmax(x, y), t_1);
	double t_3 = sqrt(t_2);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmax(fmax(x, y), t_1);
	double t_6 = fmax(t_4, t);
	double t_7 = sqrt((t_6 + 1.0)) - sqrt(t_6);
	double t_8 = sqrt((t_2 + 1.0)) - t_3;
	double t_9 = fmin(t_4, t);
	double t_10 = sqrt(t_9);
	double t_11 = sqrt((t_9 - -1.0));
	double t_12 = sqrt((t_5 + 1.0)) - sqrt(t_5);
	double tmp;
	if (((((sqrt((t_9 + 1.0)) - t_10) + t_8) + t_12) + t_7) <= 0.0) {
		tmp = (((0.5 / (t_9 * sqrt((1.0 / t_9)))) + t_8) + t_12) + t_7;
	} else {
		tmp = (((((t_11 - t_10) - (t_3 - sqrt((t_2 - -1.0)))) * t_11) / t_11) + t_12) + t_7;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmin(fmax(x, y), t_1)
    t_3 = sqrt(t_2)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmax(fmax(x, y), t_1)
    t_6 = fmax(t_4, t)
    t_7 = sqrt((t_6 + 1.0d0)) - sqrt(t_6)
    t_8 = sqrt((t_2 + 1.0d0)) - t_3
    t_9 = fmin(t_4, t)
    t_10 = sqrt(t_9)
    t_11 = sqrt((t_9 - (-1.0d0)))
    t_12 = sqrt((t_5 + 1.0d0)) - sqrt(t_5)
    if (((((sqrt((t_9 + 1.0d0)) - t_10) + t_8) + t_12) + t_7) <= 0.0d0) then
        tmp = (((0.5d0 / (t_9 * sqrt((1.0d0 / t_9)))) + t_8) + t_12) + t_7
    else
        tmp = (((((t_11 - t_10) - (t_3 - sqrt((t_2 - (-1.0d0))))) * t_11) / t_11) + t_12) + 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(fmax(x, y), t_1);
	double t_3 = Math.sqrt(t_2);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmax(fmax(x, y), t_1);
	double t_6 = fmax(t_4, t);
	double t_7 = Math.sqrt((t_6 + 1.0)) - Math.sqrt(t_6);
	double t_8 = Math.sqrt((t_2 + 1.0)) - t_3;
	double t_9 = fmin(t_4, t);
	double t_10 = Math.sqrt(t_9);
	double t_11 = Math.sqrt((t_9 - -1.0));
	double t_12 = Math.sqrt((t_5 + 1.0)) - Math.sqrt(t_5);
	double tmp;
	if (((((Math.sqrt((t_9 + 1.0)) - t_10) + t_8) + t_12) + t_7) <= 0.0) {
		tmp = (((0.5 / (t_9 * Math.sqrt((1.0 / t_9)))) + t_8) + t_12) + t_7;
	} else {
		tmp = (((((t_11 - t_10) - (t_3 - Math.sqrt((t_2 - -1.0)))) * t_11) / t_11) + t_12) + t_7;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = math.sqrt(t_2)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmax(fmax(x, y), t_1)
	t_6 = fmax(t_4, t)
	t_7 = math.sqrt((t_6 + 1.0)) - math.sqrt(t_6)
	t_8 = math.sqrt((t_2 + 1.0)) - t_3
	t_9 = fmin(t_4, t)
	t_10 = math.sqrt(t_9)
	t_11 = math.sqrt((t_9 - -1.0))
	t_12 = math.sqrt((t_5 + 1.0)) - math.sqrt(t_5)
	tmp = 0
	if ((((math.sqrt((t_9 + 1.0)) - t_10) + t_8) + t_12) + t_7) <= 0.0:
		tmp = (((0.5 / (t_9 * math.sqrt((1.0 / t_9)))) + t_8) + t_12) + t_7
	else:
		tmp = (((((t_11 - t_10) - (t_3 - math.sqrt((t_2 - -1.0)))) * t_11) / t_11) + t_12) + t_7
	return tmp

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.9%

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

   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%

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

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

   1. Initial program 91.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-to-mult N/A

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

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

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

   3. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-to-fraction N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 91.9%

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

Alternative 4: 95.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \sqrt{t\_2 + 1} - \sqrt{t\_2}\\ t_4 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_5 := \sqrt{t\_4 + 1} - \sqrt{t\_4}\\ t_6 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_7 := \mathsf{min}\left(t\_6, t\right)\\ t_8 := \left(\sqrt{t\_7 + 1} - \sqrt{t\_7}\right) + t\_3\\ t_9 := \mathsf{max}\left(t\_6, t\right)\\ t_10 := \sqrt{t\_9 + 1} - \sqrt{t\_9}\\ \mathbf{if}\;t\_8 \leq 0:\\ \;\;\;\;\left(\left(\frac{0.5}{t\_7 \cdot \sqrt{\frac{1}{t\_7}}} + t\_3\right) + t\_5\right) + t\_10\\ \mathbf{else}:\\ \;\;\;\;\left(t\_8 + t\_5\right) + t\_10\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) z))
        (t_2 (fmin (fmax x y) t_1))
        (t_3 (- (sqrt (+ t_2 1.0)) (sqrt t_2)))
        (t_4 (fmax (fmax x y) t_1))
        (t_5 (- (sqrt (+ t_4 1.0)) (sqrt t_4)))
        (t_6 (fmin (fmin x y) z))
        (t_7 (fmin t_6 t))
        (t_8 (+ (- (sqrt (+ t_7 1.0)) (sqrt t_7)) t_3))
        (t_9 (fmax t_6 t))
        (t_10 (- (sqrt (+ t_9 1.0)) (sqrt t_9))))
   (if (<= t_8 0.0)
     (+ (+ (+ (/ 0.5 (* t_7 (sqrt (/ 1.0 t_7)))) t_3) t_5) t_10)
     (+ (+ t_8 t_5) t_10))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = sqrt((t_2 + 1.0)) - sqrt(t_2);
	double t_4 = fmax(fmax(x, y), t_1);
	double t_5 = sqrt((t_4 + 1.0)) - sqrt(t_4);
	double t_6 = fmin(fmin(x, y), z);
	double t_7 = fmin(t_6, t);
	double t_8 = (sqrt((t_7 + 1.0)) - sqrt(t_7)) + t_3;
	double t_9 = fmax(t_6, t);
	double t_10 = sqrt((t_9 + 1.0)) - sqrt(t_9);
	double tmp;
	if (t_8 <= 0.0) {
		tmp = (((0.5 / (t_7 * sqrt((1.0 / t_7)))) + t_3) + t_5) + t_10;
	} else {
		tmp = (t_8 + t_5) + t_10;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmin(fmax(x, y), t_1)
    t_3 = sqrt((t_2 + 1.0d0)) - sqrt(t_2)
    t_4 = fmax(fmax(x, y), t_1)
    t_5 = sqrt((t_4 + 1.0d0)) - sqrt(t_4)
    t_6 = fmin(fmin(x, y), z)
    t_7 = fmin(t_6, t)
    t_8 = (sqrt((t_7 + 1.0d0)) - sqrt(t_7)) + t_3
    t_9 = fmax(t_6, t)
    t_10 = sqrt((t_9 + 1.0d0)) - sqrt(t_9)
    if (t_8 <= 0.0d0) then
        tmp = (((0.5d0 / (t_7 * sqrt((1.0d0 / t_7)))) + t_3) + t_5) + t_10
    else
        tmp = (t_8 + t_5) + t_10
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = Math.sqrt((t_2 + 1.0)) - Math.sqrt(t_2);
	double t_4 = fmax(fmax(x, y), t_1);
	double t_5 = Math.sqrt((t_4 + 1.0)) - Math.sqrt(t_4);
	double t_6 = fmin(fmin(x, y), z);
	double t_7 = fmin(t_6, t);
	double t_8 = (Math.sqrt((t_7 + 1.0)) - Math.sqrt(t_7)) + t_3;
	double t_9 = fmax(t_6, t);
	double t_10 = Math.sqrt((t_9 + 1.0)) - Math.sqrt(t_9);
	double tmp;
	if (t_8 <= 0.0) {
		tmp = (((0.5 / (t_7 * Math.sqrt((1.0 / t_7)))) + t_3) + t_5) + t_10;
	} else {
		tmp = (t_8 + t_5) + t_10;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = math.sqrt((t_2 + 1.0)) - math.sqrt(t_2)
	t_4 = fmax(fmax(x, y), t_1)
	t_5 = math.sqrt((t_4 + 1.0)) - math.sqrt(t_4)
	t_6 = fmin(fmin(x, y), z)
	t_7 = fmin(t_6, t)
	t_8 = (math.sqrt((t_7 + 1.0)) - math.sqrt(t_7)) + t_3
	t_9 = fmax(t_6, t)
	t_10 = math.sqrt((t_9 + 1.0)) - math.sqrt(t_9)
	tmp = 0
	if t_8 <= 0.0:
		tmp = (((0.5 / (t_7 * math.sqrt((1.0 / t_7)))) + t_3) + t_5) + t_10
	else:
		tmp = (t_8 + t_5) + t_10
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $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[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$7 = N[Min[t$95$6, t], $MachinePrecision]}, Block[{t$95$8 = N[(N[(N[Sqrt[N[(t$95$7 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$7], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$9 = N[Max[t$95$6, t], $MachinePrecision]}, Block[{t$95$10 = N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$9], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$8, 0.0], N[(N[(N[(N[(0.5 / N[(t$95$7 * N[Sqrt[N[(1.0 / t$95$7), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$10), $MachinePrecision], N[(N[(t$95$8 + t$95$5), $MachinePrecision] + t$95$10), $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))) < 0.0

   1. Initial program 91.9%

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

   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%

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

   if 0.0 < (+.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 91.9%

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

Alternative 5: 93.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \sqrt{t\_5 + 1} - t\_6\\ t_8 := \mathsf{max}\left(t\_4, t\right)\\ t_9 := \mathsf{max}\left(t\_3, t\_8\right)\\ t_10 := \mathsf{min}\left(t\_2, t\_9\right)\\ t_11 := \sqrt{t\_10}\\ t_12 := \sqrt{t\_10 + 1} - t\_11\\ t_13 := \mathsf{min}\left(t\_3, t\_8\right)\\ t_14 := \mathsf{max}\left(t\_2, t\_9\right)\\ t_15 := \sqrt{t\_14}\\ t_16 := \sqrt{t\_14 + 1} - t\_15\\ t_17 := \sqrt{t\_13}\\ t_18 := \left(\left(t\_7 + \left(\sqrt{t\_13 + 1} - t\_17\right)\right) + t\_12\right) + t\_16\\ \mathbf{if}\;t\_18 \leq 1:\\ \;\;\;\;\left(\left(\sqrt{1 + t\_5} - t\_6\right) + \frac{0.5}{t\_10 \cdot \sqrt{\frac{1}{t\_10}}}\right) + \frac{0.5}{t\_14 \cdot \sqrt{\frac{1}{t\_14}}}\\ \mathbf{elif}\;t\_18 \leq 2.0002:\\ \;\;\;\;\left(\left(\sqrt{t\_13 - -1} - \left(t\_6 - \sqrt{t\_5 - -1}\right)\right) - t\_17\right) + \left(\frac{0.5}{t\_11} - \left(t\_15 - \sqrt{t\_14 - -1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_7 + \left(1 - t\_17\right)\right) + t\_12\right) + t\_16\\ \end{array} \]

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 (+ t_5 1.0)) t_6))
        (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 (- (sqrt (+ t_10 1.0)) t_11))
        (t_13 (fmin t_3 t_8))
        (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_13))
        (t_18 (+ (+ (+ t_7 (- (sqrt (+ t_13 1.0)) t_17)) t_12) t_16)))
   (if (<= t_18 1.0)
     (+
      (+ (- (sqrt (+ 1.0 t_5)) t_6) (/ 0.5 (* t_10 (sqrt (/ 1.0 t_10)))))
      (/ 0.5 (* t_14 (sqrt (/ 1.0 t_14)))))
     (if (<= t_18 2.0002)
       (+
        (- (- (sqrt (- t_13 -1.0)) (- t_6 (sqrt (- t_5 -1.0)))) t_17)
        (- (/ 0.5 t_11) (- t_15 (sqrt (- t_14 -1.0)))))
       (+ (+ (+ t_7 (- 1.0 t_17)) t_12) t_16)))))

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((t_5 + 1.0)) - t_6;
	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 = sqrt((t_10 + 1.0)) - t_11;
	double t_13 = fmin(t_3, t_8);
	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_13);
	double t_18 = ((t_7 + (sqrt((t_13 + 1.0)) - t_17)) + t_12) + t_16;
	double tmp;
	if (t_18 <= 1.0) {
		tmp = ((sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_10 * sqrt((1.0 / t_10))))) + (0.5 / (t_14 * sqrt((1.0 / t_14))));
	} else if (t_18 <= 2.0002) {
		tmp = ((sqrt((t_13 - -1.0)) - (t_6 - sqrt((t_5 - -1.0)))) - t_17) + ((0.5 / t_11) - (t_15 - sqrt((t_14 - -1.0))));
	} else {
		tmp = ((t_7 + (1.0 - t_17)) + t_12) + t_16;
	}
	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((t_5 + 1.0d0)) - t_6
    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 = sqrt((t_10 + 1.0d0)) - t_11
    t_13 = fmin(t_3, t_8)
    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_13)
    t_18 = ((t_7 + (sqrt((t_13 + 1.0d0)) - t_17)) + t_12) + t_16
    if (t_18 <= 1.0d0) then
        tmp = ((sqrt((1.0d0 + t_5)) - t_6) + (0.5d0 / (t_10 * sqrt((1.0d0 / t_10))))) + (0.5d0 / (t_14 * sqrt((1.0d0 / t_14))))
    else if (t_18 <= 2.0002d0) then
        tmp = ((sqrt((t_13 - (-1.0d0))) - (t_6 - sqrt((t_5 - (-1.0d0))))) - t_17) + ((0.5d0 / t_11) - (t_15 - sqrt((t_14 - (-1.0d0)))))
    else
        tmp = ((t_7 + (1.0d0 - t_17)) + t_12) + t_16
    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((t_5 + 1.0)) - t_6;
	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 = Math.sqrt((t_10 + 1.0)) - t_11;
	double t_13 = fmin(t_3, t_8);
	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_13);
	double t_18 = ((t_7 + (Math.sqrt((t_13 + 1.0)) - t_17)) + t_12) + t_16;
	double tmp;
	if (t_18 <= 1.0) {
		tmp = ((Math.sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_10 * Math.sqrt((1.0 / t_10))))) + (0.5 / (t_14 * Math.sqrt((1.0 / t_14))));
	} else if (t_18 <= 2.0002) {
		tmp = ((Math.sqrt((t_13 - -1.0)) - (t_6 - Math.sqrt((t_5 - -1.0)))) - t_17) + ((0.5 / t_11) - (t_15 - Math.sqrt((t_14 - -1.0))));
	} else {
		tmp = ((t_7 + (1.0 - t_17)) + t_12) + t_16;
	}
	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((t_5 + 1.0)) - t_6
	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 = math.sqrt((t_10 + 1.0)) - t_11
	t_13 = fmin(t_3, t_8)
	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_13)
	t_18 = ((t_7 + (math.sqrt((t_13 + 1.0)) - t_17)) + t_12) + t_16
	tmp = 0
	if t_18 <= 1.0:
		tmp = ((math.sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_10 * math.sqrt((1.0 / t_10))))) + (0.5 / (t_14 * math.sqrt((1.0 / t_14))))
	elif t_18 <= 2.0002:
		tmp = ((math.sqrt((t_13 - -1.0)) - (t_6 - math.sqrt((t_5 - -1.0)))) - t_17) + ((0.5 / t_11) - (t_15 - math.sqrt((t_14 - -1.0))))
	else:
		tmp = ((t_7 + (1.0 - t_17)) + t_12) + t_16
	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 = Float64(sqrt(Float64(t_5 + 1.0)) - t_6)
	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 = Float64(sqrt(Float64(t_10 + 1.0)) - t_11)
	t_13 = fmin(t_3, t_8)
	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_13)
	t_18 = Float64(Float64(Float64(t_7 + Float64(sqrt(Float64(t_13 + 1.0)) - t_17)) + t_12) + t_16)
	tmp = 0.0
	if (t_18 <= 1.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t_5)) - t_6) + Float64(0.5 / Float64(t_10 * sqrt(Float64(1.0 / t_10))))) + Float64(0.5 / Float64(t_14 * sqrt(Float64(1.0 / t_14)))));
	elseif (t_18 <= 2.0002)
		tmp = Float64(Float64(Float64(sqrt(Float64(t_13 - -1.0)) - Float64(t_6 - sqrt(Float64(t_5 - -1.0)))) - t_17) + Float64(Float64(0.5 / t_11) - Float64(t_15 - sqrt(Float64(t_14 - -1.0)))));
	else
		tmp = Float64(Float64(Float64(t_7 + Float64(1.0 - t_17)) + t_12) + t_16);
	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((t_5 + 1.0)) - t_6;
	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 = sqrt((t_10 + 1.0)) - t_11;
	t_13 = min(t_3, t_8);
	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_13);
	t_18 = ((t_7 + (sqrt((t_13 + 1.0)) - t_17)) + t_12) + t_16;
	tmp = 0.0;
	if (t_18 <= 1.0)
		tmp = ((sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_10 * sqrt((1.0 / t_10))))) + (0.5 / (t_14 * sqrt((1.0 / t_14))));
	elseif (t_18 <= 2.0002)
		tmp = ((sqrt((t_13 - -1.0)) - (t_6 - sqrt((t_5 - -1.0)))) - t_17) + ((0.5 / t_11) - (t_15 - sqrt((t_14 - -1.0))));
	else
		tmp = ((t_7 + (1.0 - t_17)) + t_12) + t_16;
	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[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $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[(N[Sqrt[N[(t$95$10 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$11), $MachinePrecision]}, Block[{t$95$13 = N[Min[t$95$3, t$95$8], $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$13], $MachinePrecision]}, Block[{t$95$18 = N[(N[(N[(t$95$7 + N[(N[Sqrt[N[(t$95$13 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$17), $MachinePrecision]), $MachinePrecision] + t$95$12), $MachinePrecision] + t$95$16), $MachinePrecision]}, If[LessEqual[t$95$18, 1.0], N[(N[(N[(N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + N[(0.5 / N[(t$95$10 * N[Sqrt[N[(1.0 / t$95$10), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(t$95$14 * N[Sqrt[N[(1.0 / t$95$14), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$18, 2.0002], N[(N[(N[(N[Sqrt[N[(t$95$13 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(t$95$6 - N[Sqrt[N[(t$95$5 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$17), $MachinePrecision] + N[(N[(0.5 / t$95$11), $MachinePrecision] - N[(t$95$15 - N[Sqrt[N[(t$95$14 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$7 + N[(1.0 - t$95$17), $MachinePrecision]), $MachinePrecision] + t$95$12), $MachinePrecision] + t$95$16), $MachinePrecision]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 91.9%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 48.7

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 27.3

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

   7. Applied rewrites 27.3%

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

   8. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 15.6

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

   10. Applied rewrites 15.6%

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

   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))) < 2.00019999999999998

   1. Initial program 91.9%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--l+ N/A

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

      6. flip-+ N/A

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

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

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

   3. Applied rewrites 42.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 22.7

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

   6. Applied rewrites 22.7%

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

   8. Applied rewrites 37.8%

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

   9. Taylor expanded in z around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 37.8

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

   11. Applied rewrites 37.8%

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

   if 2.00019999999999998 < (+.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 91.9%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 49.5

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

   4. Applied rewrites 49.5%

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

Alternative 6: 92.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{max}\left(x, z\right), \mathsf{max}\left(y, t\right)\right)\\ t_2 := \sqrt{t\_1}\\ t_3 := \sqrt{\mathsf{min}\left(x, z\right) + 1} - \sqrt{\mathsf{min}\left(x, z\right)}\\ t_4 := \mathsf{min}\left(\mathsf{max}\left(x, z\right), \mathsf{max}\left(y, t\right)\right)\\ t_5 := \sqrt{t\_4 + 1} - \sqrt{t\_4}\\ t_6 := \sqrt{t\_1 + 1} - t\_2\\ t_7 := \sqrt{\mathsf{min}\left(y, t\right)}\\ t_8 := \left(t\_3 + \left(\sqrt{\mathsf{min}\left(y, t\right) + 1} - t\_7\right)\right) + t\_5\\ \mathbf{if}\;t\_8 + t\_6 \leq 3.5:\\ \;\;\;\;t\_8 + \frac{0.5}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_3 + \left(1 - t\_7\right)\right) + t\_5\right) + t\_6\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmax x z) (fmax y t)))
        (t_2 (sqrt t_1))
        (t_3 (- (sqrt (+ (fmin x z) 1.0)) (sqrt (fmin x z))))
        (t_4 (fmin (fmax x z) (fmax y t)))
        (t_5 (- (sqrt (+ t_4 1.0)) (sqrt t_4)))
        (t_6 (- (sqrt (+ t_1 1.0)) t_2))
        (t_7 (sqrt (fmin y t)))
        (t_8 (+ (+ t_3 (- (sqrt (+ (fmin y t) 1.0)) t_7)) t_5)))
   (if (<= (+ t_8 t_6) 3.5)
     (+ t_8 (/ 0.5 t_2))
     (+ (+ (+ t_3 (- 1.0 t_7)) t_5) t_6))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmax(x, z), fmax(y, t));
	double t_2 = sqrt(t_1);
	double t_3 = sqrt((fmin(x, z) + 1.0)) - sqrt(fmin(x, z));
	double t_4 = fmin(fmax(x, z), fmax(y, t));
	double t_5 = sqrt((t_4 + 1.0)) - sqrt(t_4);
	double t_6 = sqrt((t_1 + 1.0)) - t_2;
	double t_7 = sqrt(fmin(y, t));
	double t_8 = (t_3 + (sqrt((fmin(y, t) + 1.0)) - t_7)) + t_5;
	double tmp;
	if ((t_8 + t_6) <= 3.5) {
		tmp = t_8 + (0.5 / t_2);
	} else {
		tmp = ((t_3 + (1.0 - t_7)) + t_5) + t_6;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: tmp
    t_1 = fmax(fmax(x, z), fmax(y, t))
    t_2 = sqrt(t_1)
    t_3 = sqrt((fmin(x, z) + 1.0d0)) - sqrt(fmin(x, z))
    t_4 = fmin(fmax(x, z), fmax(y, t))
    t_5 = sqrt((t_4 + 1.0d0)) - sqrt(t_4)
    t_6 = sqrt((t_1 + 1.0d0)) - t_2
    t_7 = sqrt(fmin(y, t))
    t_8 = (t_3 + (sqrt((fmin(y, t) + 1.0d0)) - t_7)) + t_5
    if ((t_8 + t_6) <= 3.5d0) then
        tmp = t_8 + (0.5d0 / t_2)
    else
        tmp = ((t_3 + (1.0d0 - t_7)) + t_5) + t_6
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmax(x, z), fmax(y, t));
	double t_2 = Math.sqrt(t_1);
	double t_3 = Math.sqrt((fmin(x, z) + 1.0)) - Math.sqrt(fmin(x, z));
	double t_4 = fmin(fmax(x, z), fmax(y, t));
	double t_5 = Math.sqrt((t_4 + 1.0)) - Math.sqrt(t_4);
	double t_6 = Math.sqrt((t_1 + 1.0)) - t_2;
	double t_7 = Math.sqrt(fmin(y, t));
	double t_8 = (t_3 + (Math.sqrt((fmin(y, t) + 1.0)) - t_7)) + t_5;
	double tmp;
	if ((t_8 + t_6) <= 3.5) {
		tmp = t_8 + (0.5 / t_2);
	} else {
		tmp = ((t_3 + (1.0 - t_7)) + t_5) + t_6;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmax(x, z), fmax(y, t))
	t_2 = math.sqrt(t_1)
	t_3 = math.sqrt((fmin(x, z) + 1.0)) - math.sqrt(fmin(x, z))
	t_4 = fmin(fmax(x, z), fmax(y, t))
	t_5 = math.sqrt((t_4 + 1.0)) - math.sqrt(t_4)
	t_6 = math.sqrt((t_1 + 1.0)) - t_2
	t_7 = math.sqrt(fmin(y, t))
	t_8 = (t_3 + (math.sqrt((fmin(y, t) + 1.0)) - t_7)) + t_5
	tmp = 0
	if (t_8 + t_6) <= 3.5:
		tmp = t_8 + (0.5 / t_2)
	else:
		tmp = ((t_3 + (1.0 - t_7)) + t_5) + t_6
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmax(x, z), fmax(y, t))
	t_2 = sqrt(t_1)
	t_3 = Float64(sqrt(Float64(fmin(x, z) + 1.0)) - sqrt(fmin(x, z)))
	t_4 = fmin(fmax(x, z), fmax(y, t))
	t_5 = Float64(sqrt(Float64(t_4 + 1.0)) - sqrt(t_4))
	t_6 = Float64(sqrt(Float64(t_1 + 1.0)) - t_2)
	t_7 = sqrt(fmin(y, t))
	t_8 = Float64(Float64(t_3 + Float64(sqrt(Float64(fmin(y, t) + 1.0)) - t_7)) + t_5)
	tmp = 0.0
	if (Float64(t_8 + t_6) <= 3.5)
		tmp = Float64(t_8 + Float64(0.5 / t_2));
	else
		tmp = Float64(Float64(Float64(t_3 + Float64(1.0 - t_7)) + t_5) + t_6);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(max(x, z), max(y, t));
	t_2 = sqrt(t_1);
	t_3 = sqrt((min(x, z) + 1.0)) - sqrt(min(x, z));
	t_4 = min(max(x, z), max(y, t));
	t_5 = sqrt((t_4 + 1.0)) - sqrt(t_4);
	t_6 = sqrt((t_1 + 1.0)) - t_2;
	t_7 = sqrt(min(y, t));
	t_8 = (t_3 + (sqrt((min(y, t) + 1.0)) - t_7)) + t_5;
	tmp = 0.0;
	if ((t_8 + t_6) <= 3.5)
		tmp = t_8 + (0.5 / t_2);
	else
		tmp = ((t_3 + (1.0 - t_7)) + t_5) + t_6;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Max[x, z], $MachinePrecision], N[Max[y, t], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[Min[x, z], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[N[Min[x, z], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Max[x, z], $MachinePrecision], N[Max[y, t], $MachinePrecision]], $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[(N[Sqrt[N[(t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$2), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[Min[y, t], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$8 = N[(N[(t$95$3 + N[(N[Sqrt[N[(N[Min[y, t], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$7), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]}, If[LessEqual[N[(t$95$8 + t$95$6), $MachinePrecision], 3.5], N[(t$95$8 + N[(0.5 / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$3 + N[(1.0 - t$95$7), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$6), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.9%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 48.7

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 48.7

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

   7. Applied rewrites 48.7%

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

   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 91.9%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 49.5

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

   4. Applied rewrites 49.5%

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

Alternative 7: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \sqrt{t\_5 + 1} - t\_6\\ t_8 := \mathsf{max}\left(t\_4, t\right)\\ t_9 := \mathsf{max}\left(t\_3, t\_8\right)\\ t_10 := \mathsf{min}\left(t\_2, t\_9\right)\\ t_11 := \sqrt{t\_10}\\ t_12 := \sqrt{t\_10 + 1} - t\_11\\ t_13 := \mathsf{min}\left(t\_3, t\_8\right)\\ t_14 := \mathsf{max}\left(t\_2, t\_9\right)\\ t_15 := \sqrt{t\_14 + 1} - \sqrt{t\_14}\\ t_16 := \sqrt{t\_13}\\ t_17 := \left(\left(t\_7 + \left(\sqrt{t\_13 + 1} - t\_16\right)\right) + t\_12\right) + t\_15\\ \mathbf{if}\;t\_17 \leq 1:\\ \;\;\;\;\left(\left(\sqrt{1 + t\_5} - t\_6\right) + \frac{0.5}{t\_10 \cdot \sqrt{\frac{1}{t\_10}}}\right) + \frac{0.5}{t\_14 \cdot \sqrt{\frac{1}{t\_14}}}\\ \mathbf{elif}\;t\_17 \leq 3.5:\\ \;\;\;\;\left(\left(\sqrt{t\_13 - -1} - \left(t\_6 - \sqrt{t\_5 - -1}\right)\right) - t\_16\right) + \left(\sqrt{1 + t\_10} - t\_11\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_7 + \left(1 - t\_16\right)\right) + t\_12\right) + t\_15\\ \end{array} \]

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 (+ t_5 1.0)) t_6))
        (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 (- (sqrt (+ t_10 1.0)) t_11))
        (t_13 (fmin t_3 t_8))
        (t_14 (fmax t_2 t_9))
        (t_15 (- (sqrt (+ t_14 1.0)) (sqrt t_14)))
        (t_16 (sqrt t_13))
        (t_17 (+ (+ (+ t_7 (- (sqrt (+ t_13 1.0)) t_16)) t_12) t_15)))
   (if (<= t_17 1.0)
     (+
      (+ (- (sqrt (+ 1.0 t_5)) t_6) (/ 0.5 (* t_10 (sqrt (/ 1.0 t_10)))))
      (/ 0.5 (* t_14 (sqrt (/ 1.0 t_14)))))
     (if (<= t_17 3.5)
       (+
        (- (- (sqrt (- t_13 -1.0)) (- t_6 (sqrt (- t_5 -1.0)))) t_16)
        (- (sqrt (+ 1.0 t_10)) t_11))
       (+ (+ (+ t_7 (- 1.0 t_16)) t_12) t_15)))))

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((t_5 + 1.0)) - t_6;
	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 = sqrt((t_10 + 1.0)) - t_11;
	double t_13 = fmin(t_3, t_8);
	double t_14 = fmax(t_2, t_9);
	double t_15 = sqrt((t_14 + 1.0)) - sqrt(t_14);
	double t_16 = sqrt(t_13);
	double t_17 = ((t_7 + (sqrt((t_13 + 1.0)) - t_16)) + t_12) + t_15;
	double tmp;
	if (t_17 <= 1.0) {
		tmp = ((sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_10 * sqrt((1.0 / t_10))))) + (0.5 / (t_14 * sqrt((1.0 / t_14))));
	} else if (t_17 <= 3.5) {
		tmp = ((sqrt((t_13 - -1.0)) - (t_6 - sqrt((t_5 - -1.0)))) - t_16) + (sqrt((1.0 + t_10)) - t_11);
	} else {
		tmp = ((t_7 + (1.0 - t_16)) + t_12) + t_15;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_14
    real(8) :: t_15
    real(8) :: t_16
    real(8) :: t_17
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmin(t_4, t)
    t_6 = sqrt(t_5)
    t_7 = sqrt((t_5 + 1.0d0)) - t_6
    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 = sqrt((t_10 + 1.0d0)) - t_11
    t_13 = fmin(t_3, t_8)
    t_14 = fmax(t_2, t_9)
    t_15 = sqrt((t_14 + 1.0d0)) - sqrt(t_14)
    t_16 = sqrt(t_13)
    t_17 = ((t_7 + (sqrt((t_13 + 1.0d0)) - t_16)) + t_12) + t_15
    if (t_17 <= 1.0d0) then
        tmp = ((sqrt((1.0d0 + t_5)) - t_6) + (0.5d0 / (t_10 * sqrt((1.0d0 / t_10))))) + (0.5d0 / (t_14 * sqrt((1.0d0 / t_14))))
    else if (t_17 <= 3.5d0) then
        tmp = ((sqrt((t_13 - (-1.0d0))) - (t_6 - sqrt((t_5 - (-1.0d0))))) - t_16) + (sqrt((1.0d0 + t_10)) - t_11)
    else
        tmp = ((t_7 + (1.0d0 - t_16)) + t_12) + t_15
    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((t_5 + 1.0)) - t_6;
	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 = Math.sqrt((t_10 + 1.0)) - t_11;
	double t_13 = fmin(t_3, t_8);
	double t_14 = fmax(t_2, t_9);
	double t_15 = Math.sqrt((t_14 + 1.0)) - Math.sqrt(t_14);
	double t_16 = Math.sqrt(t_13);
	double t_17 = ((t_7 + (Math.sqrt((t_13 + 1.0)) - t_16)) + t_12) + t_15;
	double tmp;
	if (t_17 <= 1.0) {
		tmp = ((Math.sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_10 * Math.sqrt((1.0 / t_10))))) + (0.5 / (t_14 * Math.sqrt((1.0 / t_14))));
	} else if (t_17 <= 3.5) {
		tmp = ((Math.sqrt((t_13 - -1.0)) - (t_6 - Math.sqrt((t_5 - -1.0)))) - t_16) + (Math.sqrt((1.0 + t_10)) - t_11);
	} else {
		tmp = ((t_7 + (1.0 - t_16)) + t_12) + t_15;
	}
	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((t_5 + 1.0)) - t_6
	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 = math.sqrt((t_10 + 1.0)) - t_11
	t_13 = fmin(t_3, t_8)
	t_14 = fmax(t_2, t_9)
	t_15 = math.sqrt((t_14 + 1.0)) - math.sqrt(t_14)
	t_16 = math.sqrt(t_13)
	t_17 = ((t_7 + (math.sqrt((t_13 + 1.0)) - t_16)) + t_12) + t_15
	tmp = 0
	if t_17 <= 1.0:
		tmp = ((math.sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_10 * math.sqrt((1.0 / t_10))))) + (0.5 / (t_14 * math.sqrt((1.0 / t_14))))
	elif t_17 <= 3.5:
		tmp = ((math.sqrt((t_13 - -1.0)) - (t_6 - math.sqrt((t_5 - -1.0)))) - t_16) + (math.sqrt((1.0 + t_10)) - t_11)
	else:
		tmp = ((t_7 + (1.0 - t_16)) + t_12) + t_15
	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 = Float64(sqrt(Float64(t_5 + 1.0)) - t_6)
	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 = Float64(sqrt(Float64(t_10 + 1.0)) - t_11)
	t_13 = fmin(t_3, t_8)
	t_14 = fmax(t_2, t_9)
	t_15 = Float64(sqrt(Float64(t_14 + 1.0)) - sqrt(t_14))
	t_16 = sqrt(t_13)
	t_17 = Float64(Float64(Float64(t_7 + Float64(sqrt(Float64(t_13 + 1.0)) - t_16)) + t_12) + t_15)
	tmp = 0.0
	if (t_17 <= 1.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t_5)) - t_6) + Float64(0.5 / Float64(t_10 * sqrt(Float64(1.0 / t_10))))) + Float64(0.5 / Float64(t_14 * sqrt(Float64(1.0 / t_14)))));
	elseif (t_17 <= 3.5)
		tmp = Float64(Float64(Float64(sqrt(Float64(t_13 - -1.0)) - Float64(t_6 - sqrt(Float64(t_5 - -1.0)))) - t_16) + Float64(sqrt(Float64(1.0 + t_10)) - t_11));
	else
		tmp = Float64(Float64(Float64(t_7 + Float64(1.0 - t_16)) + t_12) + t_15);
	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((t_5 + 1.0)) - t_6;
	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 = sqrt((t_10 + 1.0)) - t_11;
	t_13 = min(t_3, t_8);
	t_14 = max(t_2, t_9);
	t_15 = sqrt((t_14 + 1.0)) - sqrt(t_14);
	t_16 = sqrt(t_13);
	t_17 = ((t_7 + (sqrt((t_13 + 1.0)) - t_16)) + t_12) + t_15;
	tmp = 0.0;
	if (t_17 <= 1.0)
		tmp = ((sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_10 * sqrt((1.0 / t_10))))) + (0.5 / (t_14 * sqrt((1.0 / t_14))));
	elseif (t_17 <= 3.5)
		tmp = ((sqrt((t_13 - -1.0)) - (t_6 - sqrt((t_5 - -1.0)))) - t_16) + (sqrt((1.0 + t_10)) - t_11);
	else
		tmp = ((t_7 + (1.0 - t_16)) + t_12) + t_15;
	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[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $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[(N[Sqrt[N[(t$95$10 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$11), $MachinePrecision]}, Block[{t$95$13 = N[Min[t$95$3, t$95$8], $MachinePrecision]}, Block[{t$95$14 = N[Max[t$95$2, t$95$9], $MachinePrecision]}, Block[{t$95$15 = N[(N[Sqrt[N[(t$95$14 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$14], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$16 = N[Sqrt[t$95$13], $MachinePrecision]}, Block[{t$95$17 = N[(N[(N[(t$95$7 + N[(N[Sqrt[N[(t$95$13 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$16), $MachinePrecision]), $MachinePrecision] + t$95$12), $MachinePrecision] + t$95$15), $MachinePrecision]}, If[LessEqual[t$95$17, 1.0], N[(N[(N[(N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + N[(0.5 / N[(t$95$10 * N[Sqrt[N[(1.0 / t$95$10), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(t$95$14 * N[Sqrt[N[(1.0 / t$95$14), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$17, 3.5], N[(N[(N[(N[Sqrt[N[(t$95$13 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(t$95$6 - N[Sqrt[N[(t$95$5 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$16), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t$95$10), $MachinePrecision]], $MachinePrecision] - t$95$11), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$7 + N[(1.0 - t$95$16), $MachinePrecision]), $MachinePrecision] + t$95$12), $MachinePrecision] + t$95$15), $MachinePrecision]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 91.9%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 48.7

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 27.3

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

   7. Applied rewrites 27.3%

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

   8. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 15.6

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

   10. Applied rewrites 15.6%

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

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

   1. Initial program 91.9%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--l+ N/A

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

      6. flip-+ N/A

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

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

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

   3. Applied rewrites 42.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 22.7

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

   6. Applied rewrites 22.7%

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

   8. Applied rewrites 37.8%

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

   9. Taylor expanded in t around inf

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

       1. lower--.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 40.8

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

   11. Applied rewrites 40.8%

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

   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 91.9%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 49.5

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

   4. Applied rewrites 49.5%

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

Alternative 8: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{max}\left(t\_4, t\right)\\ t_8 := \mathsf{max}\left(t\_3, t\_7\right)\\ t_9 := \mathsf{min}\left(t\_2, t\_8\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \mathsf{min}\left(t\_3, t\_7\right)\\ t_12 := \sqrt{t\_11 - -1}\\ t_13 := \mathsf{max}\left(t\_2, t\_8\right)\\ t_14 := \sqrt{t\_13}\\ t_15 := \sqrt{t\_11}\\ t_16 := \left(\left(\left(\sqrt{t\_5 + 1} - t\_6\right) + \left(\sqrt{t\_11 + 1} - t\_15\right)\right) + \left(\sqrt{t\_9 + 1} - t\_10\right)\right) + \left(\sqrt{t\_13 + 1} - t\_14\right)\\ \mathbf{if}\;t\_16 \leq 1:\\ \;\;\;\;\left(\left(\sqrt{1 + t\_5} - t\_6\right) + \frac{0.5}{t\_9 \cdot \sqrt{\frac{1}{t\_9}}}\right) + \frac{0.5}{t\_13 \cdot \sqrt{\frac{1}{t\_13}}}\\ \mathbf{elif}\;t\_16 \leq 3.5:\\ \;\;\;\;\left(\left(t\_12 - \left(t\_6 - \sqrt{t\_5 - -1}\right)\right) - t\_15\right) + \left(\sqrt{1 + t\_9} - t\_10\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t\_13 - -1} - \left(\left(t\_15 - \left(\sqrt{t\_9 - -1} - t\_10\right)\right) - \left(t\_12 - \left(t\_6 - 1\right)\right)\right)\right) - t\_14\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) z))
        (t_2 (fmax (fmax x y) t_1))
        (t_3 (fmin (fmax x y) t_1))
        (t_4 (fmin (fmin x y) z))
        (t_5 (fmin t_4 t))
        (t_6 (sqrt t_5))
        (t_7 (fmax t_4 t))
        (t_8 (fmax t_3 t_7))
        (t_9 (fmin t_2 t_8))
        (t_10 (sqrt t_9))
        (t_11 (fmin t_3 t_7))
        (t_12 (sqrt (- t_11 -1.0)))
        (t_13 (fmax t_2 t_8))
        (t_14 (sqrt t_13))
        (t_15 (sqrt t_11))
        (t_16
         (+
          (+
           (+ (- (sqrt (+ t_5 1.0)) t_6) (- (sqrt (+ t_11 1.0)) t_15))
           (- (sqrt (+ t_9 1.0)) t_10))
          (- (sqrt (+ t_13 1.0)) t_14))))
   (if (<= t_16 1.0)
     (+
      (+ (- (sqrt (+ 1.0 t_5)) t_6) (/ 0.5 (* t_9 (sqrt (/ 1.0 t_9)))))
      (/ 0.5 (* t_13 (sqrt (/ 1.0 t_13)))))
     (if (<= t_16 3.5)
       (+
        (- (- t_12 (- t_6 (sqrt (- t_5 -1.0)))) t_15)
        (- (sqrt (+ 1.0 t_9)) t_10))
       (-
        (-
         (sqrt (- t_13 -1.0))
         (- (- t_15 (- (sqrt (- t_9 -1.0)) t_10)) (- t_12 (- t_6 1.0))))
        t_14)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = sqrt((t_11 - -1.0));
	double t_13 = fmax(t_2, t_8);
	double t_14 = sqrt(t_13);
	double t_15 = sqrt(t_11);
	double t_16 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_15)) + (sqrt((t_9 + 1.0)) - t_10)) + (sqrt((t_13 + 1.0)) - t_14);
	double tmp;
	if (t_16 <= 1.0) {
		tmp = ((sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_9 * sqrt((1.0 / t_9))))) + (0.5 / (t_13 * sqrt((1.0 / t_13))));
	} else if (t_16 <= 3.5) {
		tmp = ((t_12 - (t_6 - sqrt((t_5 - -1.0)))) - t_15) + (sqrt((1.0 + t_9)) - t_10);
	} else {
		tmp = (sqrt((t_13 - -1.0)) - ((t_15 - (sqrt((t_9 - -1.0)) - t_10)) - (t_12 - (t_6 - 1.0)))) - t_14;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_14
    real(8) :: t_15
    real(8) :: t_16
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmin(t_4, t)
    t_6 = sqrt(t_5)
    t_7 = fmax(t_4, t)
    t_8 = fmax(t_3, t_7)
    t_9 = fmin(t_2, t_8)
    t_10 = sqrt(t_9)
    t_11 = fmin(t_3, t_7)
    t_12 = sqrt((t_11 - (-1.0d0)))
    t_13 = fmax(t_2, t_8)
    t_14 = sqrt(t_13)
    t_15 = sqrt(t_11)
    t_16 = (((sqrt((t_5 + 1.0d0)) - t_6) + (sqrt((t_11 + 1.0d0)) - t_15)) + (sqrt((t_9 + 1.0d0)) - t_10)) + (sqrt((t_13 + 1.0d0)) - t_14)
    if (t_16 <= 1.0d0) then
        tmp = ((sqrt((1.0d0 + t_5)) - t_6) + (0.5d0 / (t_9 * sqrt((1.0d0 / t_9))))) + (0.5d0 / (t_13 * sqrt((1.0d0 / t_13))))
    else if (t_16 <= 3.5d0) then
        tmp = ((t_12 - (t_6 - sqrt((t_5 - (-1.0d0))))) - t_15) + (sqrt((1.0d0 + t_9)) - t_10)
    else
        tmp = (sqrt((t_13 - (-1.0d0))) - ((t_15 - (sqrt((t_9 - (-1.0d0))) - t_10)) - (t_12 - (t_6 - 1.0d0)))) - t_14
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = Math.sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = Math.sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = Math.sqrt((t_11 - -1.0));
	double t_13 = fmax(t_2, t_8);
	double t_14 = Math.sqrt(t_13);
	double t_15 = Math.sqrt(t_11);
	double t_16 = (((Math.sqrt((t_5 + 1.0)) - t_6) + (Math.sqrt((t_11 + 1.0)) - t_15)) + (Math.sqrt((t_9 + 1.0)) - t_10)) + (Math.sqrt((t_13 + 1.0)) - t_14);
	double tmp;
	if (t_16 <= 1.0) {
		tmp = ((Math.sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_9 * Math.sqrt((1.0 / t_9))))) + (0.5 / (t_13 * Math.sqrt((1.0 / t_13))));
	} else if (t_16 <= 3.5) {
		tmp = ((t_12 - (t_6 - Math.sqrt((t_5 - -1.0)))) - t_15) + (Math.sqrt((1.0 + t_9)) - t_10);
	} else {
		tmp = (Math.sqrt((t_13 - -1.0)) - ((t_15 - (Math.sqrt((t_9 - -1.0)) - t_10)) - (t_12 - (t_6 - 1.0)))) - t_14;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = math.sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = math.sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = math.sqrt((t_11 - -1.0))
	t_13 = fmax(t_2, t_8)
	t_14 = math.sqrt(t_13)
	t_15 = math.sqrt(t_11)
	t_16 = (((math.sqrt((t_5 + 1.0)) - t_6) + (math.sqrt((t_11 + 1.0)) - t_15)) + (math.sqrt((t_9 + 1.0)) - t_10)) + (math.sqrt((t_13 + 1.0)) - t_14)
	tmp = 0
	if t_16 <= 1.0:
		tmp = ((math.sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_9 * math.sqrt((1.0 / t_9))))) + (0.5 / (t_13 * math.sqrt((1.0 / t_13))))
	elif t_16 <= 3.5:
		tmp = ((t_12 - (t_6 - math.sqrt((t_5 - -1.0)))) - t_15) + (math.sqrt((1.0 + t_9)) - t_10)
	else:
		tmp = (math.sqrt((t_13 - -1.0)) - ((t_15 - (math.sqrt((t_9 - -1.0)) - t_10)) - (t_12 - (t_6 - 1.0)))) - t_14
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = sqrt(Float64(t_11 - -1.0))
	t_13 = fmax(t_2, t_8)
	t_14 = sqrt(t_13)
	t_15 = sqrt(t_11)
	t_16 = Float64(Float64(Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + Float64(sqrt(Float64(t_11 + 1.0)) - t_15)) + Float64(sqrt(Float64(t_9 + 1.0)) - t_10)) + Float64(sqrt(Float64(t_13 + 1.0)) - t_14))
	tmp = 0.0
	if (t_16 <= 1.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t_5)) - t_6) + Float64(0.5 / Float64(t_9 * sqrt(Float64(1.0 / t_9))))) + Float64(0.5 / Float64(t_13 * sqrt(Float64(1.0 / t_13)))));
	elseif (t_16 <= 3.5)
		tmp = Float64(Float64(Float64(t_12 - Float64(t_6 - sqrt(Float64(t_5 - -1.0)))) - t_15) + Float64(sqrt(Float64(1.0 + t_9)) - t_10));
	else
		tmp = Float64(Float64(sqrt(Float64(t_13 - -1.0)) - Float64(Float64(t_15 - Float64(sqrt(Float64(t_9 - -1.0)) - t_10)) - Float64(t_12 - Float64(t_6 - 1.0)))) - t_14);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = max(max(x, y), t_1);
	t_3 = min(max(x, y), t_1);
	t_4 = min(min(x, y), z);
	t_5 = min(t_4, t);
	t_6 = sqrt(t_5);
	t_7 = max(t_4, t);
	t_8 = max(t_3, t_7);
	t_9 = min(t_2, t_8);
	t_10 = sqrt(t_9);
	t_11 = min(t_3, t_7);
	t_12 = sqrt((t_11 - -1.0));
	t_13 = max(t_2, t_8);
	t_14 = sqrt(t_13);
	t_15 = sqrt(t_11);
	t_16 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_15)) + (sqrt((t_9 + 1.0)) - t_10)) + (sqrt((t_13 + 1.0)) - t_14);
	tmp = 0.0;
	if (t_16 <= 1.0)
		tmp = ((sqrt((1.0 + t_5)) - t_6) + (0.5 / (t_9 * sqrt((1.0 / t_9))))) + (0.5 / (t_13 * sqrt((1.0 / t_13))));
	elseif (t_16 <= 3.5)
		tmp = ((t_12 - (t_6 - sqrt((t_5 - -1.0)))) - t_15) + (sqrt((1.0 + t_9)) - t_10);
	else
		tmp = (sqrt((t_13 - -1.0)) - ((t_15 - (sqrt((t_9 - -1.0)) - t_10)) - (t_12 - (t_6 - 1.0)))) - t_14;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$4, t], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Min[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$12 = N[Sqrt[N[(t$95$11 - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$13 = N[Max[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$14 = N[Sqrt[t$95$13], $MachinePrecision]}, Block[{t$95$15 = N[Sqrt[t$95$11], $MachinePrecision]}, Block[{t$95$16 = N[(N[(N[(N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + N[(N[Sqrt[N[(t$95$11 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$15), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$13 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$14), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$16, 1.0], N[(N[(N[(N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + N[(0.5 / N[(t$95$9 * N[Sqrt[N[(1.0 / t$95$9), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(t$95$13 * N[Sqrt[N[(1.0 / t$95$13), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$16, 3.5], N[(N[(N[(t$95$12 - N[(t$95$6 - N[Sqrt[N[(t$95$5 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$15), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t$95$9), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(t$95$13 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(t$95$15 - N[(N[Sqrt[N[(t$95$9 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision] - N[(t$95$12 - N[(t$95$6 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$14), $MachinePrecision]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 91.9%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 48.7

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 27.3

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

   7. Applied rewrites 27.3%

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

   8. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 15.6

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

   10. Applied rewrites 15.6%

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

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

   1. Initial program 91.9%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--l+ N/A

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

      6. flip-+ N/A

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

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

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

   3. Applied rewrites 42.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 22.7

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

   6. Applied rewrites 22.7%

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

   8. Applied rewrites 37.8%

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

   9. Taylor expanded in t around inf

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

       1. lower--.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 40.8

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

   11. Applied rewrites 40.8%

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

   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 91.9%

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

   3. Applied rewrites 38.8%

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

   4. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 21.3

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

   6. Applied rewrites 21.3%

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

Alternative 9: 91.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (((sqrt((fmin(x, z) + 1.0)) - sqrt(fmin(x, z))) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((fmax(x, z) + 1.0)) - sqrt(fmax(x, 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((fmin(x, z) + 1.0d0)) - sqrt(fmin(x, z))) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((fmax(x, z) + 1.0d0)) - sqrt(fmax(x, 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((fmin(x, z) + 1.0)) - Math.sqrt(fmin(x, z))) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((fmax(x, z) + 1.0)) - Math.sqrt(fmax(x, z)))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

Alternative 10: 86.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.9%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 48.7

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 27.3

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

   7. Applied rewrites 27.3%

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

   8. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 15.6

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

   10. Applied rewrites 15.6%

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

   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 91.9%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--l+ N/A

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

      6. flip-+ N/A

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

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

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

   3. Applied rewrites 42.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 22.7

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

   6. Applied rewrites 22.7%

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

   8. Applied rewrites 37.8%

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

   9. Taylor expanded in t around inf

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

       1. lower--.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 40.8

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

   11. Applied rewrites 40.8%

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

Alternative 11: 86.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.9%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--l+ N/A

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

      6. flip-+ N/A

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

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

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

   3. Applied rewrites 42.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 22.7

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

   6. Applied rewrites 22.7%

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

   8. Applied rewrites 37.8%

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

   9. Taylor expanded in y around inf

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

       1. lower--.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 26.7

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

   11. Applied rewrites 26.7%

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

   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 91.9%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--l+ N/A

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

      6. flip-+ N/A

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

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

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

   3. Applied rewrites 42.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 22.7

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

   6. Applied rewrites 22.7%

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

   8. Applied rewrites 37.8%

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

   9. Taylor expanded in t around inf

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

       1. lower--.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 40.8

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

   11. Applied rewrites 40.8%

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

Alternative 12: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_2\right)\\ t_4 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_2\right)\\ t_5 := \mathsf{max}\left(t\_4, t\right)\\ t_6 := \mathsf{min}\left(t\_3, t\_5\right)\\ t_7 := \sqrt{t\_6}\\ t_8 := \mathsf{min}\left(t\_4, t\right)\\ t_9 := \sqrt{t\_1}\\ \mathbf{if}\;t\_8 \leq 9500000000000:\\ \;\;\;\;\left(\left(\sqrt{t\_8 - -1} - \left(t\_9 - \sqrt{t\_1 - -1}\right)\right) - \sqrt{t\_8}\right) + \left(\sqrt{1 + t\_6} - t\_7\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + t\_1} - t\_9\right) + \left(\sqrt{t\_6 + 1} - t\_7\right)\right) + \frac{0.5}{\sqrt{\mathsf{max}\left(t\_3, t\_5\right)}}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmin (fmin x y) z))
        (t_2 (fmax (fmin x y) z))
        (t_3 (fmax (fmax x y) t_2))
        (t_4 (fmin (fmax x y) t_2))
        (t_5 (fmax t_4 t))
        (t_6 (fmin t_3 t_5))
        (t_7 (sqrt t_6))
        (t_8 (fmin t_4 t))
        (t_9 (sqrt t_1)))
   (if (<= t_8 9500000000000.0)
     (+
      (- (- (sqrt (- t_8 -1.0)) (- t_9 (sqrt (- t_1 -1.0)))) (sqrt t_8))
      (- (sqrt (+ 1.0 t_6)) t_7))
     (+
      (+ (- (sqrt (+ 1.0 t_1)) t_9) (- (sqrt (+ t_6 1.0)) t_7))
      (/ 0.5 (sqrt (fmax t_3 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(fmin(x, y), z);
	double t_3 = fmax(fmax(x, y), t_2);
	double t_4 = fmin(fmax(x, y), t_2);
	double t_5 = fmax(t_4, t);
	double t_6 = fmin(t_3, t_5);
	double t_7 = sqrt(t_6);
	double t_8 = fmin(t_4, t);
	double t_9 = sqrt(t_1);
	double tmp;
	if (t_8 <= 9500000000000.0) {
		tmp = ((sqrt((t_8 - -1.0)) - (t_9 - sqrt((t_1 - -1.0)))) - sqrt(t_8)) + (sqrt((1.0 + t_6)) - t_7);
	} else {
		tmp = ((sqrt((1.0 + t_1)) - t_9) + (sqrt((t_6 + 1.0)) - t_7)) + (0.5 / sqrt(fmax(t_3, t_5)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmin(fmin(x, y), z)
    t_2 = fmax(fmin(x, y), z)
    t_3 = fmax(fmax(x, y), t_2)
    t_4 = fmin(fmax(x, y), t_2)
    t_5 = fmax(t_4, t)
    t_6 = fmin(t_3, t_5)
    t_7 = sqrt(t_6)
    t_8 = fmin(t_4, t)
    t_9 = sqrt(t_1)
    if (t_8 <= 9500000000000.0d0) then
        tmp = ((sqrt((t_8 - (-1.0d0))) - (t_9 - sqrt((t_1 - (-1.0d0))))) - sqrt(t_8)) + (sqrt((1.0d0 + t_6)) - t_7)
    else
        tmp = ((sqrt((1.0d0 + t_1)) - t_9) + (sqrt((t_6 + 1.0d0)) - t_7)) + (0.5d0 / sqrt(fmax(t_3, 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(fmin(x, y), z);
	double t_3 = fmax(fmax(x, y), t_2);
	double t_4 = fmin(fmax(x, y), t_2);
	double t_5 = fmax(t_4, t);
	double t_6 = fmin(t_3, t_5);
	double t_7 = Math.sqrt(t_6);
	double t_8 = fmin(t_4, t);
	double t_9 = Math.sqrt(t_1);
	double tmp;
	if (t_8 <= 9500000000000.0) {
		tmp = ((Math.sqrt((t_8 - -1.0)) - (t_9 - Math.sqrt((t_1 - -1.0)))) - Math.sqrt(t_8)) + (Math.sqrt((1.0 + t_6)) - t_7);
	} else {
		tmp = ((Math.sqrt((1.0 + t_1)) - t_9) + (Math.sqrt((t_6 + 1.0)) - t_7)) + (0.5 / Math.sqrt(fmax(t_3, t_5)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(fmin(x, y), z)
	t_3 = fmax(fmax(x, y), t_2)
	t_4 = fmin(fmax(x, y), t_2)
	t_5 = fmax(t_4, t)
	t_6 = fmin(t_3, t_5)
	t_7 = math.sqrt(t_6)
	t_8 = fmin(t_4, t)
	t_9 = math.sqrt(t_1)
	tmp = 0
	if t_8 <= 9500000000000.0:
		tmp = ((math.sqrt((t_8 - -1.0)) - (t_9 - math.sqrt((t_1 - -1.0)))) - math.sqrt(t_8)) + (math.sqrt((1.0 + t_6)) - t_7)
	else:
		tmp = ((math.sqrt((1.0 + t_1)) - t_9) + (math.sqrt((t_6 + 1.0)) - t_7)) + (0.5 / math.sqrt(fmax(t_3, t_5)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(fmin(x, y), z)
	t_3 = fmax(fmax(x, y), t_2)
	t_4 = fmin(fmax(x, y), t_2)
	t_5 = fmax(t_4, t)
	t_6 = fmin(t_3, t_5)
	t_7 = sqrt(t_6)
	t_8 = fmin(t_4, t)
	t_9 = sqrt(t_1)
	tmp = 0.0
	if (t_8 <= 9500000000000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(t_8 - -1.0)) - Float64(t_9 - sqrt(Float64(t_1 - -1.0)))) - sqrt(t_8)) + Float64(sqrt(Float64(1.0 + t_6)) - t_7));
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t_1)) - t_9) + Float64(sqrt(Float64(t_6 + 1.0)) - t_7)) + Float64(0.5 / sqrt(fmax(t_3, 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(min(x, y), z);
	t_3 = max(max(x, y), t_2);
	t_4 = min(max(x, y), t_2);
	t_5 = max(t_4, t);
	t_6 = min(t_3, t_5);
	t_7 = sqrt(t_6);
	t_8 = min(t_4, t);
	t_9 = sqrt(t_1);
	tmp = 0.0;
	if (t_8 <= 9500000000000.0)
		tmp = ((sqrt((t_8 - -1.0)) - (t_9 - sqrt((t_1 - -1.0)))) - sqrt(t_8)) + (sqrt((1.0 + t_6)) - t_7);
	else
		tmp = ((sqrt((1.0 + t_1)) - t_9) + (sqrt((t_6 + 1.0)) - t_7)) + (0.5 / sqrt(max(t_3, 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[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Max[x, y], $MachinePrecision], t$95$2], $MachinePrecision]}, Block[{t$95$5 = N[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$6 = N[Min[t$95$3, t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[Min[t$95$4, t], $MachinePrecision]}, Block[{t$95$9 = N[Sqrt[t$95$1], $MachinePrecision]}, If[LessEqual[t$95$8, 9500000000000.0], N[(N[(N[(N[Sqrt[N[(t$95$8 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(t$95$9 - N[Sqrt[N[(t$95$1 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[t$95$8], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t$95$6), $MachinePrecision]], $MachinePrecision] - t$95$7), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(1.0 + t$95$1), $MachinePrecision]], $MachinePrecision] - t$95$9), $MachinePrecision] + N[(N[Sqrt[N[(t$95$6 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$7), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Sqrt[N[Max[t$95$3, t$95$5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 9.5e12

   1. Initial program 91.9%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--l+ N/A

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

      6. flip-+ N/A

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

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

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

   3. Applied rewrites 42.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 22.7

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

   6. Applied rewrites 22.7%

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

   8. Applied rewrites 37.8%

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

   9. Taylor expanded in t around inf

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

       1. lower--.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 40.8

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

   11. Applied rewrites 40.8%

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

   if 9.5e12 < y

   1. Initial program 91.9%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 48.7

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 27.3

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

   7. Applied rewrites 27.3%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 27.3

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

   10. Applied rewrites 27.3%

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

Alternative 13: 70.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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((fmin(x, y) - (-1.0d0))) - sqrt(fmin(x, y))) - (sqrt(z) - sqrt((z - (-1.0d0))))) - (sqrt(t) - sqrt((t - (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

2. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-sqrt.f64 50.2

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

4. Applied rewrites 50.2%

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

6. Applied rewrites 50.2%

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

Alternative 14: 59.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), t\right)\\ t_2 := \mathsf{min}\left(z, t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), t\right)\\ \left(\left(\sqrt{1 + t\_3} - \sqrt{t\_3}\right) + \left(\sqrt{t\_2 + 1} - \sqrt{t\_2}\right)\right) + \frac{0.5}{\sqrt{\mathsf{max}\left(z, t\_1\right)}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) t)) (t_2 (fmin z t_1)) (t_3 (fmin (fmin x y) t)))
   (+
    (+ (- (sqrt (+ 1.0 t_3)) (sqrt t_3)) (- (sqrt (+ t_2 1.0)) (sqrt t_2)))
    (/ 0.5 (sqrt (fmax z t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), t);
	double t_2 = fmin(z, t_1);
	double t_3 = fmin(fmin(x, y), t);
	return ((sqrt((1.0 + t_3)) - sqrt(t_3)) + (sqrt((t_2 + 1.0)) - sqrt(t_2))) + (0.5 / sqrt(fmax(z, t_1)));
}

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 = fmax(fmin(x, y), t)
    t_2 = fmin(z, t_1)
    t_3 = fmin(fmin(x, y), t)
    code = ((sqrt((1.0d0 + t_3)) - sqrt(t_3)) + (sqrt((t_2 + 1.0d0)) - sqrt(t_2))) + (0.5d0 / sqrt(fmax(z, t_1)))
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), t);
	double t_2 = fmin(z, t_1);
	double t_3 = fmin(fmin(x, y), t);
	return ((Math.sqrt((1.0 + t_3)) - Math.sqrt(t_3)) + (Math.sqrt((t_2 + 1.0)) - Math.sqrt(t_2))) + (0.5 / Math.sqrt(fmax(z, t_1)));
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), t)
	t_2 = fmin(z, t_1)
	t_3 = fmin(fmin(x, y), t)
	return ((math.sqrt((1.0 + t_3)) - math.sqrt(t_3)) + (math.sqrt((t_2 + 1.0)) - math.sqrt(t_2))) + (0.5 / math.sqrt(fmax(z, t_1)))

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), t)
	t_2 = fmin(z, t_1)
	t_3 = fmin(fmin(x, y), t)
	return Float64(Float64(Float64(sqrt(Float64(1.0 + t_3)) - sqrt(t_3)) + Float64(sqrt(Float64(t_2 + 1.0)) - sqrt(t_2))) + Float64(0.5 / sqrt(fmax(z, t_1))))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = max(min(x, y), t);
	t_2 = min(z, t_1);
	t_3 = min(min(x, y), t);
	tmp = ((sqrt((1.0 + t_3)) - sqrt(t_3)) + (sqrt((t_2 + 1.0)) - sqrt(t_2))) + (0.5 / sqrt(max(z, t_1)));
end

Wolfram

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

Derivation

1. Initial program 91.9%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 48.7

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

4. Applied rewrites 48.7%

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

5. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-sqrt.f64 27.3

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

7. Applied rewrites 27.3%

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

8. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 27.3

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

10. Applied rewrites 27.3%

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

Alternative 15: 34.5% accurate, 1.6× speedup

Math

\[\frac{{\left(\sqrt{\mathsf{max}\left(z, t\right)}\right)}^{2}}{\mathsf{max}\left(z, t\right)} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ (pow (sqrt (fmax z t)) 2.0) (fmax z t)))

C

double code(double x, double y, double z, double t) {
	return pow(sqrt(fmax(z, t)), 2.0) / fmax(z, 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(fmax(z, t)) ** 2.0d0) / fmax(z, t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.pow(Math.sqrt(fmax(z, t)), 2.0) / fmax(z, t);
}

Python

def code(x, y, z, t):
	return math.pow(math.sqrt(fmax(z, t)), 2.0) / fmax(z, t)

Julia

function code(x, y, z, t)
	return Float64((sqrt(fmax(z, t)) ^ 2.0) / fmax(z, t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (sqrt(max(z, t)) ^ 2.0) / max(z, t);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Power[N[Sqrt[N[Max[z, t], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Max[z, t], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.9%

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

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

   3. +-commutative N/A

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

   4. sum-to-mult N/A

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

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

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

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

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

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

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

   8. lift-+.f64 N/A

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

   9. add-flip N/A

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

   10. lower--.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-neg.f64 91.9

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

3. Applied rewrites 91.9%

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

5. Applied rewrites 62.4%

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

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{{\left(\sqrt{t}\right)}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\left(\sqrt{t}\right)}^{2}}{\color{blue}{t}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\left(\sqrt{t}\right)}^{2}}{t} \]

   3. lower-sqrt.f64 34.5

      \[\leadsto \frac{{\left(\sqrt{t}\right)}^{2}}{t} \]

8. Applied rewrites 34.5%

   \[\leadsto \color{blue}{\frac{{\left(\sqrt{t}\right)}^{2}}{t}} \]
9. Add Preprocessing

Alternative 16: 6.9% accurate, 5.1× speedup

Math

\[0.5 \cdot \sqrt{\mathsf{min}\left(x, t\right)} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return 0.5 * sqrt(fmin(x, 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 * sqrt(fmin(x, t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. associate--l+ N/A

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

   6. flip-+ N/A

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

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

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

3. Applied rewrites 42.1%

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

4. Taylor expanded in x around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 6.9

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

6. Applied rewrites 6.9%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{x}} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 6.9

      \[\leadsto 0.5 \cdot \sqrt{x} \]

9. Applied rewrites 6.9%

   \[\leadsto 0.5 \cdot \color{blue}{\sqrt{x}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025175 
(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))))