Main:z from

Percentage Accurate: 91.7% → 98.4%

Time: 1.2min

Alternatives: 22

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    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: 91.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    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: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + t}\\ t_3 := \sqrt{x + 1} - \sqrt{x}\\ t_4 := \sqrt{1 + y}\\ t_5 := t\_4 - \sqrt{y}\\ t_6 := \left(t\_3 + t\_5\right) + t\_1\\ t_7 := 0.5 \cdot \sqrt{\frac{1}{z}} + \left(t\_2 - \sqrt{t}\right)\\ \mathbf{if}\;t\_6 \leq 0:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{x}} + t\_5\right) + t\_7\\ \mathbf{elif}\;t\_6 \leq 2.00005:\\ \;\;\;\;t\_7 + \left(t\_3 + \frac{1}{t\_4 + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_5 + \left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right)\right) + \left(t\_1 + \frac{1}{t\_2 + \sqrt{t}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_2 (sqrt (+ 1.0 t)))
        (t_3 (- (sqrt (+ x 1.0)) (sqrt x)))
        (t_4 (sqrt (+ 1.0 y)))
        (t_5 (- t_4 (sqrt y)))
        (t_6 (+ (+ t_3 t_5) t_1))
        (t_7 (+ (* 0.5 (sqrt (/ 1.0 z))) (- t_2 (sqrt t)))))
   (if (<= t_6 0.0)
     (+ (+ (* 0.5 (sqrt (/ 1.0 x))) t_5) t_7)
     (if (<= t_6 2.00005)
       (+ t_7 (+ t_3 (/ 1.0 (+ t_4 (sqrt y)))))
       (+
        (+ t_5 (+ 1.0 (- (* x (+ 0.5 (* x -0.125))) (sqrt x))))
        (+ t_1 (/ 1.0 (+ t_2 (sqrt t)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z)) - sqrt(z);
	double t_2 = sqrt((1.0 + t));
	double t_3 = sqrt((x + 1.0)) - sqrt(x);
	double t_4 = sqrt((1.0 + y));
	double t_5 = t_4 - sqrt(y);
	double t_6 = (t_3 + t_5) + t_1;
	double t_7 = (0.5 * sqrt((1.0 / z))) + (t_2 - sqrt(t));
	double tmp;
	if (t_6 <= 0.0) {
		tmp = ((0.5 * sqrt((1.0 / x))) + t_5) + t_7;
	} else if (t_6 <= 2.00005) {
		tmp = t_7 + (t_3 + (1.0 / (t_4 + sqrt(y))));
	} else {
		tmp = (t_5 + (1.0 + ((x * (0.5 + (x * -0.125))) - sqrt(x)))) + (t_1 + (1.0 / (t_2 + sqrt(t))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z)) - sqrt(z)
    t_2 = sqrt((1.0d0 + t))
    t_3 = sqrt((x + 1.0d0)) - sqrt(x)
    t_4 = sqrt((1.0d0 + y))
    t_5 = t_4 - sqrt(y)
    t_6 = (t_3 + t_5) + t_1
    t_7 = (0.5d0 * sqrt((1.0d0 / z))) + (t_2 - sqrt(t))
    if (t_6 <= 0.0d0) then
        tmp = ((0.5d0 * sqrt((1.0d0 / x))) + t_5) + t_7
    else if (t_6 <= 2.00005d0) then
        tmp = t_7 + (t_3 + (1.0d0 / (t_4 + sqrt(y))))
    else
        tmp = (t_5 + (1.0d0 + ((x * (0.5d0 + (x * (-0.125d0)))) - sqrt(x)))) + (t_1 + (1.0d0 / (t_2 + sqrt(t))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double t_2 = Math.sqrt((1.0 + t));
	double t_3 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double t_4 = Math.sqrt((1.0 + y));
	double t_5 = t_4 - Math.sqrt(y);
	double t_6 = (t_3 + t_5) + t_1;
	double t_7 = (0.5 * Math.sqrt((1.0 / z))) + (t_2 - Math.sqrt(t));
	double tmp;
	if (t_6 <= 0.0) {
		tmp = ((0.5 * Math.sqrt((1.0 / x))) + t_5) + t_7;
	} else if (t_6 <= 2.00005) {
		tmp = t_7 + (t_3 + (1.0 / (t_4 + Math.sqrt(y))));
	} else {
		tmp = (t_5 + (1.0 + ((x * (0.5 + (x * -0.125))) - Math.sqrt(x)))) + (t_1 + (1.0 / (t_2 + Math.sqrt(t))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z)) - math.sqrt(z)
	t_2 = math.sqrt((1.0 + t))
	t_3 = math.sqrt((x + 1.0)) - math.sqrt(x)
	t_4 = math.sqrt((1.0 + y))
	t_5 = t_4 - math.sqrt(y)
	t_6 = (t_3 + t_5) + t_1
	t_7 = (0.5 * math.sqrt((1.0 / z))) + (t_2 - math.sqrt(t))
	tmp = 0
	if t_6 <= 0.0:
		tmp = ((0.5 * math.sqrt((1.0 / x))) + t_5) + t_7
	elif t_6 <= 2.00005:
		tmp = t_7 + (t_3 + (1.0 / (t_4 + math.sqrt(y))))
	else:
		tmp = (t_5 + (1.0 + ((x * (0.5 + (x * -0.125))) - math.sqrt(x)))) + (t_1 + (1.0 / (t_2 + math.sqrt(t))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_2 = sqrt(Float64(1.0 + t))
	t_3 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_4 = sqrt(Float64(1.0 + y))
	t_5 = Float64(t_4 - sqrt(y))
	t_6 = Float64(Float64(t_3 + t_5) + t_1)
	t_7 = Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) + Float64(t_2 - sqrt(t)))
	tmp = 0.0
	if (t_6 <= 0.0)
		tmp = Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + t_5) + t_7);
	elseif (t_6 <= 2.00005)
		tmp = Float64(t_7 + Float64(t_3 + Float64(1.0 / Float64(t_4 + sqrt(y)))));
	else
		tmp = Float64(Float64(t_5 + Float64(1.0 + Float64(Float64(x * Float64(0.5 + Float64(x * -0.125))) - sqrt(x)))) + Float64(t_1 + Float64(1.0 / Float64(t_2 + sqrt(t)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z)) - sqrt(z);
	t_2 = sqrt((1.0 + t));
	t_3 = sqrt((x + 1.0)) - sqrt(x);
	t_4 = sqrt((1.0 + y));
	t_5 = t_4 - sqrt(y);
	t_6 = (t_3 + t_5) + t_1;
	t_7 = (0.5 * sqrt((1.0 / z))) + (t_2 - sqrt(t));
	tmp = 0.0;
	if (t_6 <= 0.0)
		tmp = ((0.5 * sqrt((1.0 / x))) + t_5) + t_7;
	elseif (t_6 <= 2.00005)
		tmp = t_7 + (t_3 + (1.0 / (t_4 + sqrt(y))));
	else
		tmp = (t_5 + (1.0 + ((x * (0.5 + (x * -0.125))) - sqrt(x)))) + (t_1 + (1.0 / (t_2 + sqrt(t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$3 + t$95$5), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$7 = N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 0.0], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$7), $MachinePrecision], If[LessEqual[t$95$6, 2.00005], N[(t$95$7 + N[(t$95$3 + N[(1.0 / N[(t$95$4 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$5 + N[(1.0 + N[(N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(1.0 / N[(t$95$2 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.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))) < 0.0

   1. Initial program 62.9%

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

      1. associate-+l+ 62.9%

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

      2. sub-neg 62.9%

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

      3. sub-neg 62.9%

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

      4. +-commutative 62.9%

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

      5. +-commutative 62.9%

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

      6. +-commutative 62.9%

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

   3. Simplified 62.9%

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

   5. Taylor expanded in z around inf 71.7%

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

   6. Taylor expanded in x around inf 82.2%

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

   if 0.0 < (+.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))) < 2.0000499999999999

   1. Initial program 96.5%

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

      1. associate-+l+ 96.5%

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

      2. sub-neg 96.5%

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

      3. sub-neg 96.5%

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

      4. +-commutative 96.5%

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

      5. +-commutative 96.5%

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

      6. +-commutative 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in z around inf 54.9%

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

      1. flip-- 55.1%

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

      2. div-inv 55.1%

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

      3. add-sqr-sqrt 48.5%

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

      4. add-sqr-sqrt 55.4%

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

   7. Applied egg-rr 55.4%

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

      1. associate-*r/ 55.4%

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

      2. *-rgt-identity 55.4%

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

      3. associate--l+ 55.4%

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

      4. +-inverses 55.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

      5. metadata-eval 55.4%

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

   9. Simplified 55.4%

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

   if 2.0000499999999999 < (+.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)))

   1. Initial program 98.7%

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

      1. associate-+l+ 98.7%

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

      2. sub-neg 98.7%

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

      3. sub-neg 98.7%

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

      4. +-commutative 98.7%

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

      5. +-commutative 98.7%

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

      6. +-commutative 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in x around 0 96.5%

      \[\leadsto \left(\color{blue}{\left(\left(1 + x \cdot \left(0.5 + -0.125 \cdot x\right)\right) - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   6. Step-by-step derivation

      1. associate--l+ 96.5%

         \[\leadsto \left(\color{blue}{\left(1 + \left(x \cdot \left(0.5 + -0.125 \cdot x\right) - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

      2. *-commutative 96.5%

         \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + \color{blue}{x \cdot -0.125}\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   7. Simplified 96.5%

      \[\leadsto \left(\color{blue}{\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   8. Step-by-step derivation

      1. flip-- 97.0%

         \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

      2. div-inv 97.0%

         \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\left(\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

      3. add-sqr-sqrt 75.6%

         \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\color{blue}{\left(1 + t\right)} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]

      4. add-sqr-sqrt 97.0%

         \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 + t\right) - \color{blue}{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]

   9. Applied egg-rr 97.0%

      \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\left(\left(1 + t\right) - t\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]
   10. Step-by-step derivation

       1. associate-*r/ 97.0%

          \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{\left(\left(1 + t\right) - t\right) \cdot 1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

       2. *-rgt-identity 97.0%

          \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\color{blue}{\left(1 + t\right) - t}}{\sqrt{1 + t} + \sqrt{t}}\right) \]

       3. associate--l+ 97.0%

          \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\color{blue}{1 + \left(t - t\right)}}{\sqrt{1 + t} + \sqrt{t}}\right) \]

       4. +-inverses 97.0%

          \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1 + \color{blue}{0}}{\sqrt{1 + t} + \sqrt{t}}\right) \]

       5. metadata-eval 97.0%

          \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\color{blue}{1}}{\sqrt{1 + t} + \sqrt{t}}\right) \]

   11. Simplified 97.0%

       \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 0:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2.00005:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := t\_1 - \sqrt{y}\\ t_3 := \sqrt{x + 1} - \sqrt{x}\\ t_4 := 0.5 \cdot \sqrt{\frac{1}{z}}\\ t_5 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{x}} + t\_2\right) + \left(t\_4 + t\_5\right)\\ \mathbf{elif}\;t\_3 \leq 1:\\ \;\;\;\;\left(t\_3 + t\_2\right) + \left(t\_4 + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_5 + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(1 + \left(t\_1 - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (- t_1 (sqrt y)))
        (t_3 (- (sqrt (+ x 1.0)) (sqrt x)))
        (t_4 (* 0.5 (sqrt (/ 1.0 z))))
        (t_5 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_3 0.0)
     (+ (+ (* 0.5 (sqrt (/ 1.0 x))) t_2) (+ t_4 t_5))
     (if (<= t_3 1.0)
       (+ (+ t_3 t_2) (+ t_4 (* 0.5 (sqrt (/ 1.0 t)))))
       (+
        (+ t_5 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (+ 1.0 (- t_1 (+ (sqrt x) (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = t_1 - sqrt(y);
	double t_3 = sqrt((x + 1.0)) - sqrt(x);
	double t_4 = 0.5 * sqrt((1.0 / z));
	double t_5 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = ((0.5 * sqrt((1.0 / x))) + t_2) + (t_4 + t_5);
	} else if (t_3 <= 1.0) {
		tmp = (t_3 + t_2) + (t_4 + (0.5 * sqrt((1.0 / t))));
	} else {
		tmp = (t_5 + (sqrt((1.0 + z)) - sqrt(z))) + (1.0 + (t_1 - (sqrt(x) + sqrt(y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = t_1 - sqrt(y)
    t_3 = sqrt((x + 1.0d0)) - sqrt(x)
    t_4 = 0.5d0 * sqrt((1.0d0 / z))
    t_5 = sqrt((1.0d0 + t)) - sqrt(t)
    if (t_3 <= 0.0d0) then
        tmp = ((0.5d0 * sqrt((1.0d0 / x))) + t_2) + (t_4 + t_5)
    else if (t_3 <= 1.0d0) then
        tmp = (t_3 + t_2) + (t_4 + (0.5d0 * sqrt((1.0d0 / t))))
    else
        tmp = (t_5 + (sqrt((1.0d0 + z)) - sqrt(z))) + (1.0d0 + (t_1 - (sqrt(x) + sqrt(y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = t_1 - Math.sqrt(y);
	double t_3 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double t_4 = 0.5 * Math.sqrt((1.0 / z));
	double t_5 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = ((0.5 * Math.sqrt((1.0 / x))) + t_2) + (t_4 + t_5);
	} else if (t_3 <= 1.0) {
		tmp = (t_3 + t_2) + (t_4 + (0.5 * Math.sqrt((1.0 / t))));
	} else {
		tmp = (t_5 + (Math.sqrt((1.0 + z)) - Math.sqrt(z))) + (1.0 + (t_1 - (Math.sqrt(x) + Math.sqrt(y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = t_1 - math.sqrt(y)
	t_3 = math.sqrt((x + 1.0)) - math.sqrt(x)
	t_4 = 0.5 * math.sqrt((1.0 / z))
	t_5 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if t_3 <= 0.0:
		tmp = ((0.5 * math.sqrt((1.0 / x))) + t_2) + (t_4 + t_5)
	elif t_3 <= 1.0:
		tmp = (t_3 + t_2) + (t_4 + (0.5 * math.sqrt((1.0 / t))))
	else:
		tmp = (t_5 + (math.sqrt((1.0 + z)) - math.sqrt(z))) + (1.0 + (t_1 - (math.sqrt(x) + math.sqrt(y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(t_1 - sqrt(y))
	t_3 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_4 = Float64(0.5 * sqrt(Float64(1.0 / z)))
	t_5 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + t_2) + Float64(t_4 + t_5));
	elseif (t_3 <= 1.0)
		tmp = Float64(Float64(t_3 + t_2) + Float64(t_4 + Float64(0.5 * sqrt(Float64(1.0 / t)))));
	else
		tmp = Float64(Float64(t_5 + Float64(sqrt(Float64(1.0 + z)) - sqrt(z))) + Float64(1.0 + Float64(t_1 - Float64(sqrt(x) + sqrt(y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = t_1 - sqrt(y);
	t_3 = sqrt((x + 1.0)) - sqrt(x);
	t_4 = 0.5 * sqrt((1.0 / z));
	t_5 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = ((0.5 * sqrt((1.0 / x))) + t_2) + (t_4 + t_5);
	elseif (t_3 <= 1.0)
		tmp = (t_3 + t_2) + (t_4 + (0.5 * sqrt((1.0 / t))));
	else
		tmp = (t_5 + (sqrt((1.0 + z)) - sqrt(z))) + (1.0 + (t_1 - (sqrt(x) + sqrt(y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(t$95$4 + t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1.0], N[(N[(t$95$3 + t$95$2), $MachinePrecision] + N[(t$95$4 + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$5 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 88.7%

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

      1. associate-+l+ 88.7%

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

      2. sub-neg 88.7%

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

      3. sub-neg 88.7%

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

      4. +-commutative 88.7%

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

      5. +-commutative 88.7%

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

      6. +-commutative 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in z around inf 49.5%

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

   6. Taylor expanded in x around inf 52.2%

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

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

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in z around inf 48.1%

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

   6. Taylor expanded in t around inf 27.6%

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

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

   1. Initial program 93.4%

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

      1. associate-+l+ 93.4%

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

      2. sub-neg 93.4%

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

      3. sub-neg 93.4%

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

      4. +-commutative 93.4%

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

      5. +-commutative 93.4%

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

      6. +-commutative 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in x around 0 42.5%

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

      1. associate--l+ 52.8%

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

   7. Simplified 52.8%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{elif}\;\sqrt{x + 1} - \sqrt{x} \leq 1:\\ \;\;\;\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(0.5 \cdot \sqrt{\frac{1}{z}} + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y} - \sqrt{y}\\ t_2 := \sqrt{x + 1} - \sqrt{x}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{x}} + t\_1\right) + \left(0.5 \cdot \sqrt{\frac{1}{z}} + t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + t\_1\right) + \left(t\_3 + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 y)) (sqrt y)))
        (t_2 (- (sqrt (+ x 1.0)) (sqrt x)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_2 0.0)
     (+ (+ (* 0.5 (sqrt (/ 1.0 x))) t_1) (+ (* 0.5 (sqrt (/ 1.0 z))) t_3))
     (+ (+ t_2 t_1) (+ t_3 (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y)) - sqrt(y);
	double t_2 = sqrt((x + 1.0)) - sqrt(x);
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = ((0.5 * sqrt((1.0 / x))) + t_1) + ((0.5 * sqrt((1.0 / z))) + t_3);
	} else {
		tmp = (t_2 + t_1) + (t_3 + (1.0 / (sqrt((1.0 + z)) + sqrt(z))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y)) - sqrt(y)
    t_2 = sqrt((x + 1.0d0)) - sqrt(x)
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    if (t_2 <= 0.0d0) then
        tmp = ((0.5d0 * sqrt((1.0d0 / x))) + t_1) + ((0.5d0 * sqrt((1.0d0 / z))) + t_3)
    else
        tmp = (t_2 + t_1) + (t_3 + (1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y)) - Math.sqrt(y);
	double t_2 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = ((0.5 * Math.sqrt((1.0 / x))) + t_1) + ((0.5 * Math.sqrt((1.0 / z))) + t_3);
	} else {
		tmp = (t_2 + t_1) + (t_3 + (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y)) - math.sqrt(y)
	t_2 = math.sqrt((x + 1.0)) - math.sqrt(x)
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if t_2 <= 0.0:
		tmp = ((0.5 * math.sqrt((1.0 / x))) + t_1) + ((0.5 * math.sqrt((1.0 / z))) + t_3)
	else:
		tmp = (t_2 + t_1) + (t_3 + (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + y)) - sqrt(y))
	t_2 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + t_1) + Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) + t_3));
	else
		tmp = Float64(Float64(t_2 + t_1) + Float64(t_3 + Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y)) - sqrt(y);
	t_2 = sqrt((x + 1.0)) - sqrt(x);
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = ((0.5 * sqrt((1.0 / x))) + t_1) + ((0.5 * sqrt((1.0 / z))) + t_3);
	else
		tmp = (t_2 + t_1) + (t_3 + (1.0 / (sqrt((1.0 + z)) + sqrt(z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + t$95$1), $MachinePrecision] + N[(t$95$3 + N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.7%

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

      1. associate-+l+ 88.7%

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

      2. sub-neg 88.7%

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

      3. sub-neg 88.7%

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

      4. +-commutative 88.7%

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

      5. +-commutative 88.7%

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

      6. +-commutative 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in z around inf 49.5%

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

   6. Taylor expanded in x around inf 52.2%

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

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

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

      1. flip-- 97.3%

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

      2. div-inv 97.3%

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

      3. add-sqr-sqrt 79.8%

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

      4. add-sqr-sqrt 97.7%

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

   6. Applied egg-rr 97.7%

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

      1. associate-*r/ 97.7%

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

      2. *-rgt-identity 97.7%

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

      3. associate--l+ 97.9%

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

      4. +-inverses 97.9%

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

      5. metadata-eval 97.9%

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

   8. Simplified 97.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;z \leq 2.15 \cdot 10^{+15}:\\ \;\;\;\;\left(t\_2 + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\left(1 + t\_1\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{z}} + t\_2\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{t\_1 + \sqrt{y}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))) (t_2 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= z 2.15e+15)
     (+
      (+ t_2 (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))))
      (- (+ 1.0 t_1) (+ (sqrt x) (sqrt y))))
     (+
      (+ (* 0.5 (sqrt (/ 1.0 z))) t_2)
      (+ (- (sqrt (+ x 1.0)) (sqrt x)) (/ 1.0 (+ t_1 (sqrt y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (z <= 2.15e+15) {
		tmp = (t_2 + (1.0 / (sqrt((1.0 + z)) + sqrt(z)))) + ((1.0 + t_1) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = ((0.5 * sqrt((1.0 / z))) + t_2) + ((sqrt((x + 1.0)) - sqrt(x)) + (1.0 / (t_1 + sqrt(y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + t)) - sqrt(t)
    if (z <= 2.15d+15) then
        tmp = (t_2 + (1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z)))) + ((1.0d0 + t_1) - (sqrt(x) + sqrt(y)))
    else
        tmp = ((0.5d0 * sqrt((1.0d0 / z))) + t_2) + ((sqrt((x + 1.0d0)) - sqrt(x)) + (1.0d0 / (t_1 + sqrt(y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (z <= 2.15e+15) {
		tmp = (t_2 + (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z)))) + ((1.0 + t_1) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = ((0.5 * Math.sqrt((1.0 / z))) + t_2) + ((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (1.0 / (t_1 + Math.sqrt(y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if z <= 2.15e+15:
		tmp = (t_2 + (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z)))) + ((1.0 + t_1) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = ((0.5 * math.sqrt((1.0 / z))) + t_2) + ((math.sqrt((x + 1.0)) - math.sqrt(x)) + (1.0 / (t_1 + math.sqrt(y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (z <= 2.15e+15)
		tmp = Float64(Float64(t_2 + Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))) + Float64(Float64(1.0 + t_1) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) + t_2) + Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(1.0 / Float64(t_1 + sqrt(y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (z <= 2.15e+15)
		tmp = (t_2 + (1.0 / (sqrt((1.0 + z)) + sqrt(z)))) + ((1.0 + t_1) - (sqrt(x) + sqrt(y)));
	else
		tmp = ((0.5 * sqrt((1.0 / z))) + t_2) + ((sqrt((x + 1.0)) - sqrt(x)) + (1.0 / (t_1 + sqrt(y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 2.15e+15], N[(N[(t$95$2 + N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + t$95$1), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 2.15e15

   1. Initial program 97.2%

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

      1. associate-+l+ 97.2%

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

      2. sub-neg 97.2%

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

      3. sub-neg 97.2%

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

      4. +-commutative 97.2%

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

      5. +-commutative 97.2%

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

      6. +-commutative 97.2%

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

   3. Simplified 97.2%

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

      1. flip-- 97.5%

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

      2. div-inv 97.5%

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

      3. add-sqr-sqrt 97.6%

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

      4. add-sqr-sqrt 97.9%

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

   6. Applied egg-rr 97.9%

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

      1. associate-*r/ 97.9%

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

      2. *-rgt-identity 97.9%

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

      3. associate--l+ 97.9%

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

      4. +-inverses 97.9%

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

      5. metadata-eval 97.9%

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

   8. Simplified 97.9%

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

   9. Taylor expanded in x around 0 45.7%

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

   if 2.15e15 < z

   1. Initial program 89.1%

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

      1. associate-+l+ 89.1%

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

      2. sub-neg 89.1%

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

      3. sub-neg 89.1%

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

      4. +-commutative 89.1%

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

      5. +-commutative 89.1%

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

      6. +-commutative 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in z around inf 92.2%

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

      1. flip-- 92.4%

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

      2. div-inv 92.4%

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

      3. add-sqr-sqrt 75.6%

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

      4. add-sqr-sqrt 92.9%

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

   7. Applied egg-rr 92.9%

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

      1. associate-*r/ 92.9%

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

      2. *-rgt-identity 92.9%

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

      3. associate--l+ 95.7%

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

      4. +-inverses 95.7%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

      5. metadata-eval 95.7%

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

   9. Simplified 95.7%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.15 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;y \leq 63000:\\ \;\;\;\;\left(t\_1 + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{z}} + t\_1\right) + \left(\left(\sqrt{x + 1} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= y 63000.0)
     (+
      (+ t_1 (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))))
      (- (+ 1.0 (sqrt (+ 1.0 y))) (+ (sqrt x) (sqrt y))))
     (+
      (+ (* 0.5 (sqrt (/ 1.0 z))) t_1)
      (- (+ (sqrt (+ x 1.0)) (* 0.5 (sqrt (/ 1.0 y)))) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (y <= 63000.0) {
		tmp = (t_1 + (1.0 / (sqrt((1.0 + z)) + sqrt(z)))) + ((1.0 + sqrt((1.0 + y))) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = ((0.5 * sqrt((1.0 / z))) + t_1) + ((sqrt((x + 1.0)) + (0.5 * sqrt((1.0 / y)))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + t)) - sqrt(t)
    if (y <= 63000.0d0) then
        tmp = (t_1 + (1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z)))) + ((1.0d0 + sqrt((1.0d0 + y))) - (sqrt(x) + sqrt(y)))
    else
        tmp = ((0.5d0 * sqrt((1.0d0 / z))) + t_1) + ((sqrt((x + 1.0d0)) + (0.5d0 * sqrt((1.0d0 / y)))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (y <= 63000.0) {
		tmp = (t_1 + (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z)))) + ((1.0 + Math.sqrt((1.0 + y))) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = ((0.5 * Math.sqrt((1.0 / z))) + t_1) + ((Math.sqrt((x + 1.0)) + (0.5 * Math.sqrt((1.0 / y)))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if y <= 63000.0:
		tmp = (t_1 + (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z)))) + ((1.0 + math.sqrt((1.0 + y))) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = ((0.5 * math.sqrt((1.0 / z))) + t_1) + ((math.sqrt((x + 1.0)) + (0.5 * math.sqrt((1.0 / y)))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (y <= 63000.0)
		tmp = Float64(Float64(t_1 + Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))) + Float64(Float64(1.0 + sqrt(Float64(1.0 + y))) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) + t_1) + Float64(Float64(sqrt(Float64(x + 1.0)) + Float64(0.5 * sqrt(Float64(1.0 / y)))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (y <= 63000.0)
		tmp = (t_1 + (1.0 / (sqrt((1.0 + z)) + sqrt(z)))) + ((1.0 + sqrt((1.0 + y))) - (sqrt(x) + sqrt(y)));
	else
		tmp = ((0.5 * sqrt((1.0 / z))) + t_1) + ((sqrt((x + 1.0)) + (0.5 * sqrt((1.0 / y)))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 63000.0], N[(N[(t$95$1 + N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 63000

   1. Initial program 97.2%

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

      1. associate-+l+ 97.2%

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

      2. sub-neg 97.2%

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

      3. sub-neg 97.2%

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

      4. +-commutative 97.2%

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

      5. +-commutative 97.2%

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

      6. +-commutative 97.2%

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

   3. Simplified 97.2%

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

      1. flip-- 97.5%

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

      2. div-inv 97.5%

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

      3. add-sqr-sqrt 75.5%

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

      4. add-sqr-sqrt 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. associate-*r/ 98.0%

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

      2. *-rgt-identity 98.0%

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

      3. associate--l+ 98.7%

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

      4. +-inverses 98.7%

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

      5. metadata-eval 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in x around 0 56.7%

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

   if 63000 < y

   1. Initial program 89.0%

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

      1. associate-+l+ 89.0%

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

      2. sub-neg 89.0%

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

      3. sub-neg 89.0%

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

      4. +-commutative 89.0%

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

      5. +-commutative 89.0%

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

      6. +-commutative 89.0%

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

   3. Simplified 89.0%

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

   5. Taylor expanded in z around inf 39.5%

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

   6. Taylor expanded in y around inf 40.1%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 63000:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(\sqrt{x + 1} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{x + 1}\\ \mathbf{if}\;z \leq 4.5 \cdot 10^{-28}:\\ \;\;\;\;3 + \left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 2400000000:\\ \;\;\;\;\left(t\_2 + \left(t\_1 + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_2 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + \left(0.5 \cdot \sqrt{\frac{1}{z}} + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))) (t_2 (sqrt (+ x 1.0))))
   (if (<= z 4.5e-28)
     (+
      3.0
      (- (sqrt (+ 1.0 t)) (+ (sqrt t) (+ (sqrt y) (+ (sqrt x) (sqrt z))))))
     (if (<= z 2400000000.0)
       (-
        (+ t_2 (+ t_1 (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z)))))
        (+ (sqrt x) (sqrt y)))
       (+
        (+ (- t_2 (sqrt x)) (- t_1 (sqrt y)))
        (+ (* 0.5 (sqrt (/ 1.0 z))) (* 0.5 (sqrt (/ 1.0 t)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((x + 1.0));
	double tmp;
	if (z <= 4.5e-28) {
		tmp = 3.0 + (sqrt((1.0 + t)) - (sqrt(t) + (sqrt(y) + (sqrt(x) + sqrt(z)))));
	} else if (z <= 2400000000.0) {
		tmp = (t_2 + (t_1 + (1.0 / (sqrt((1.0 + z)) + sqrt(z))))) - (sqrt(x) + sqrt(y));
	} else {
		tmp = ((t_2 - sqrt(x)) + (t_1 - sqrt(y))) + ((0.5 * sqrt((1.0 / z))) + (0.5 * sqrt((1.0 / t))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((x + 1.0d0))
    if (z <= 4.5d-28) then
        tmp = 3.0d0 + (sqrt((1.0d0 + t)) - (sqrt(t) + (sqrt(y) + (sqrt(x) + sqrt(z)))))
    else if (z <= 2400000000.0d0) then
        tmp = (t_2 + (t_1 + (1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))))) - (sqrt(x) + sqrt(y))
    else
        tmp = ((t_2 - sqrt(x)) + (t_1 - sqrt(y))) + ((0.5d0 * sqrt((1.0d0 / z))) + (0.5d0 * sqrt((1.0d0 / t))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((x + 1.0));
	double tmp;
	if (z <= 4.5e-28) {
		tmp = 3.0 + (Math.sqrt((1.0 + t)) - (Math.sqrt(t) + (Math.sqrt(y) + (Math.sqrt(x) + Math.sqrt(z)))));
	} else if (z <= 2400000000.0) {
		tmp = (t_2 + (t_1 + (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))))) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = ((t_2 - Math.sqrt(x)) + (t_1 - Math.sqrt(y))) + ((0.5 * Math.sqrt((1.0 / z))) + (0.5 * Math.sqrt((1.0 / t))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((x + 1.0))
	tmp = 0
	if z <= 4.5e-28:
		tmp = 3.0 + (math.sqrt((1.0 + t)) - (math.sqrt(t) + (math.sqrt(y) + (math.sqrt(x) + math.sqrt(z)))))
	elif z <= 2400000000.0:
		tmp = (t_2 + (t_1 + (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))))) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = ((t_2 - math.sqrt(x)) + (t_1 - math.sqrt(y))) + ((0.5 * math.sqrt((1.0 / z))) + (0.5 * math.sqrt((1.0 / t))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (z <= 4.5e-28)
		tmp = Float64(3.0 + Float64(sqrt(Float64(1.0 + t)) - Float64(sqrt(t) + Float64(sqrt(y) + Float64(sqrt(x) + sqrt(z))))));
	elseif (z <= 2400000000.0)
		tmp = Float64(Float64(t_2 + Float64(t_1 + Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))))) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(t_2 - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) + Float64(0.5 * sqrt(Float64(1.0 / t)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((x + 1.0));
	tmp = 0.0;
	if (z <= 4.5e-28)
		tmp = 3.0 + (sqrt((1.0 + t)) - (sqrt(t) + (sqrt(y) + (sqrt(x) + sqrt(z)))));
	elseif (z <= 2400000000.0)
		tmp = (t_2 + (t_1 + (1.0 / (sqrt((1.0 + z)) + sqrt(z))))) - (sqrt(x) + sqrt(y));
	else
		tmp = ((t_2 - sqrt(x)) + (t_1 - sqrt(y))) + ((0.5 * sqrt((1.0 / z))) + (0.5 * sqrt((1.0 / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 4.5e-28], N[(3.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2400000000.0], N[(N[(t$95$2 + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 4.4999999999999998e-28

   1. Initial program 97.9%

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

      1. associate-+l+ 97.9%

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

      2. sub-neg 97.9%

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

      3. sub-neg 97.9%

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

      4. +-commutative 97.9%

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

      5. +-commutative 97.9%

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

      6. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in y around 0 17.4%

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

      1. associate--l+ 29.5%

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

      2. +-commutative 29.5%

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

   7. Simplified 29.5%

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

   8. Taylor expanded in x around 0 16.8%

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

      1. +-commutative 16.8%

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

   10. Simplified 16.8%

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

   11. Taylor expanded in z around 0 16.8%

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

       1. associate--l+ 32.3%

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

       2. +-commutative 32.3%

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

       3. associate-+r+ 32.3%

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

       4. +-commutative 32.3%

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

       5. associate-+r+ 32.3%

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

   13. Simplified 32.3%

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

   if 4.4999999999999998e-28 < z < 2.4e9

   1. Initial program 93.9%

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

      1. associate-+l+ 93.9%

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

      2. sub-neg 93.9%

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

      3. sub-neg 93.9%

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

      4. +-commutative 93.9%

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

      5. +-commutative 93.9%

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

      6. +-commutative 93.9%

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

   3. Simplified 93.9%

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

      1. flip-- 96.0%

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

      2. div-inv 96.0%

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

      3. add-sqr-sqrt 95.4%

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

      4. add-sqr-sqrt 97.8%

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

   6. Applied egg-rr 97.8%

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

      1. associate-*r/ 97.8%

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

      2. *-rgt-identity 97.8%

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

      3. associate--l+ 97.7%

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

      4. +-inverses 97.7%

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

      5. metadata-eval 97.7%

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

   8. Simplified 97.7%

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

   9. Taylor expanded in t around inf 23.1%

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

   if 2.4e9 < z

   1. Initial program 89.0%

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

      1. associate-+l+ 89.0%

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

      2. sub-neg 89.0%

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

      3. sub-neg 89.0%

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

      4. +-commutative 89.0%

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

      5. +-commutative 89.0%

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

      6. +-commutative 89.0%

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

   3. Simplified 89.0%

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

   5. Taylor expanded in z around inf 92.2%

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

   6. Taylor expanded in t around inf 47.4%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{-28}:\\ \;\;\;\;3 + \left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 2400000000:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(0.5 \cdot \sqrt{\frac{1}{z}} + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 44000000:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) + \left(\left(x \cdot \left(0.5 + x \cdot -0.125\right) + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(0.5 \cdot \sqrt{\frac{1}{z}} + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 44000000.0)
   (+
    (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t))))
    (- (+ (* x (+ 0.5 (* x -0.125))) 2.0) (+ (sqrt x) (sqrt y))))
   (+
    (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y)))
    (+ (* 0.5 (sqrt (/ 1.0 z))) (* 0.5 (sqrt (/ 1.0 t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 44000000.0) {
		tmp = ((sqrt((1.0 + z)) - sqrt(z)) + (1.0 / (sqrt((1.0 + t)) + sqrt(t)))) + (((x * (0.5 + (x * -0.125))) + 2.0) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + ((0.5 * sqrt((1.0 / z))) + (0.5 * sqrt((1.0 / t))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 44000000.0d0) then
        tmp = ((sqrt((1.0d0 + z)) - sqrt(z)) + (1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t)))) + (((x * (0.5d0 + (x * (-0.125d0)))) + 2.0d0) - (sqrt(x) + sqrt(y)))
    else
        tmp = ((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((1.0d0 + y)) - sqrt(y))) + ((0.5d0 * sqrt((1.0d0 / z))) + (0.5d0 * sqrt((1.0d0 / t))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 44000000.0) {
		tmp = ((Math.sqrt((1.0 + z)) - Math.sqrt(z)) + (1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t)))) + (((x * (0.5 + (x * -0.125))) + 2.0) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = ((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((1.0 + y)) - Math.sqrt(y))) + ((0.5 * Math.sqrt((1.0 / z))) + (0.5 * Math.sqrt((1.0 / t))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 44000000.0:
		tmp = ((math.sqrt((1.0 + z)) - math.sqrt(z)) + (1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t)))) + (((x * (0.5 + (x * -0.125))) + 2.0) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = ((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((1.0 + y)) - math.sqrt(y))) + ((0.5 * math.sqrt((1.0 / z))) + (0.5 * math.sqrt((1.0 / t))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 44000000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t)))) + Float64(Float64(Float64(x * Float64(0.5 + Float64(x * -0.125))) + 2.0) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))) + Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) + Float64(0.5 * sqrt(Float64(1.0 / t)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 44000000.0)
		tmp = ((sqrt((1.0 + z)) - sqrt(z)) + (1.0 / (sqrt((1.0 + t)) + sqrt(t)))) + (((x * (0.5 + (x * -0.125))) + 2.0) - (sqrt(x) + sqrt(y)));
	else
		tmp = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + ((0.5 * sqrt((1.0 / z))) + (0.5 * sqrt((1.0 / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 44000000.0], N[(N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.4e7

   1. Initial program 97.7%

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

      1. associate-+l+ 97.7%

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

      2. sub-neg 97.7%

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

      3. sub-neg 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0 56.8%

      \[\leadsto \left(\color{blue}{\left(\left(1 + x \cdot \left(0.5 + -0.125 \cdot x\right)\right) - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   6. Step-by-step derivation

      1. associate--l+ 56.8%

         \[\leadsto \left(\color{blue}{\left(1 + \left(x \cdot \left(0.5 + -0.125 \cdot x\right) - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

      2. *-commutative 56.8%

         \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + \color{blue}{x \cdot -0.125}\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   7. Simplified 56.8%

      \[\leadsto \left(\color{blue}{\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   8. Step-by-step derivation

      1. flip-- 57.2%

         \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

      2. div-inv 57.2%

         \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\left(\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

      3. add-sqr-sqrt 47.0%

         \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\color{blue}{\left(1 + t\right)} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]

      4. add-sqr-sqrt 57.2%

         \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 + t\right) - \color{blue}{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]

   9. Applied egg-rr 57.2%

      \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\left(\left(1 + t\right) - t\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]
   10. Step-by-step derivation

       1. associate-*r/ 57.2%

          \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{\left(\left(1 + t\right) - t\right) \cdot 1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

       2. *-rgt-identity 57.2%

          \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\color{blue}{\left(1 + t\right) - t}}{\sqrt{1 + t} + \sqrt{t}}\right) \]

       3. associate--l+ 57.2%

          \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\color{blue}{1 + \left(t - t\right)}}{\sqrt{1 + t} + \sqrt{t}}\right) \]

       4. +-inverses 57.2%

          \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1 + \color{blue}{0}}{\sqrt{1 + t} + \sqrt{t}}\right) \]

       5. metadata-eval 57.2%

          \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\color{blue}{1}}{\sqrt{1 + t} + \sqrt{t}}\right) \]

   11. Simplified 57.2%

       \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

   12. Taylor expanded in y around 0 30.2%

       \[\leadsto \color{blue}{\left(\left(2 + x \cdot \left(0.5 + -0.125 \cdot x\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]

   if 4.4e7 < z

   1. Initial program 88.9%

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

      1. associate-+l+ 88.9%

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

      2. sub-neg 88.9%

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

      3. sub-neg 88.9%

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

      4. +-commutative 88.9%

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

      5. +-commutative 88.9%

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

      6. +-commutative 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in z around inf 92.2%

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

   6. Taylor expanded in t around inf 46.8%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 44000000:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) + \left(\left(x \cdot \left(0.5 + x \cdot -0.125\right) + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(0.5 \cdot \sqrt{\frac{1}{z}} + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 44000000:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\left(x \cdot \left(0.5 + x \cdot -0.125\right) + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(0.5 \cdot \sqrt{\frac{1}{z}} + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 44000000.0)
   (+
    (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (- (sqrt (+ 1.0 z)) (sqrt z)))
    (- (+ (* x (+ 0.5 (* x -0.125))) 2.0) (+ (sqrt x) (sqrt y))))
   (+
    (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y)))
    (+ (* 0.5 (sqrt (/ 1.0 z))) (* 0.5 (sqrt (/ 1.0 t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 44000000.0) {
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) + (sqrt((1.0 + z)) - sqrt(z))) + (((x * (0.5 + (x * -0.125))) + 2.0) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + ((0.5 * sqrt((1.0 / z))) + (0.5 * sqrt((1.0 / t))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 44000000.0d0) then
        tmp = ((sqrt((1.0d0 + t)) - sqrt(t)) + (sqrt((1.0d0 + z)) - sqrt(z))) + (((x * (0.5d0 + (x * (-0.125d0)))) + 2.0d0) - (sqrt(x) + sqrt(y)))
    else
        tmp = ((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((1.0d0 + y)) - sqrt(y))) + ((0.5d0 * sqrt((1.0d0 / z))) + (0.5d0 * sqrt((1.0d0 / t))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 44000000.0) {
		tmp = ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (Math.sqrt((1.0 + z)) - Math.sqrt(z))) + (((x * (0.5 + (x * -0.125))) + 2.0) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = ((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((1.0 + y)) - Math.sqrt(y))) + ((0.5 * Math.sqrt((1.0 / z))) + (0.5 * Math.sqrt((1.0 / t))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 44000000.0:
		tmp = ((math.sqrt((1.0 + t)) - math.sqrt(t)) + (math.sqrt((1.0 + z)) - math.sqrt(z))) + (((x * (0.5 + (x * -0.125))) + 2.0) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = ((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((1.0 + y)) - math.sqrt(y))) + ((0.5 * math.sqrt((1.0 / z))) + (0.5 * math.sqrt((1.0 / t))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 44000000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(sqrt(Float64(1.0 + z)) - sqrt(z))) + Float64(Float64(Float64(x * Float64(0.5 + Float64(x * -0.125))) + 2.0) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))) + Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) + Float64(0.5 * sqrt(Float64(1.0 / t)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 44000000.0)
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) + (sqrt((1.0 + z)) - sqrt(z))) + (((x * (0.5 + (x * -0.125))) + 2.0) - (sqrt(x) + sqrt(y)));
	else
		tmp = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + ((0.5 * sqrt((1.0 / z))) + (0.5 * sqrt((1.0 / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 44000000.0], N[(N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.4e7

   1. Initial program 97.7%

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

      1. associate-+l+ 97.7%

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

      2. sub-neg 97.7%

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

      3. sub-neg 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0 56.8%

      \[\leadsto \left(\color{blue}{\left(\left(1 + x \cdot \left(0.5 + -0.125 \cdot x\right)\right) - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   6. Step-by-step derivation

      1. associate--l+ 56.8%

         \[\leadsto \left(\color{blue}{\left(1 + \left(x \cdot \left(0.5 + -0.125 \cdot x\right) - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

      2. *-commutative 56.8%

         \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + \color{blue}{x \cdot -0.125}\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   7. Simplified 56.8%

      \[\leadsto \left(\color{blue}{\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   8. Taylor expanded in y around 0 30.0%

      \[\leadsto \color{blue}{\left(\left(2 + x \cdot \left(0.5 + -0.125 \cdot x\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   if 4.4e7 < z

   1. Initial program 88.9%

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

      1. associate-+l+ 88.9%

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

      2. sub-neg 88.9%

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

      3. sub-neg 88.9%

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

      4. +-commutative 88.9%

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

      5. +-commutative 88.9%

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

      6. +-commutative 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in z around inf 92.2%

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

   6. Taylor expanded in t around inf 46.8%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 44000000:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\left(x \cdot \left(0.5 + x \cdot -0.125\right) + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(0.5 \cdot \sqrt{\frac{1}{z}} + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{x} + \sqrt{y}\\ t_3 := \sqrt{x + 1}\\ \mathbf{if}\;z \leq 3.8 \cdot 10^{-27}:\\ \;\;\;\;3 + \left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 2400000000:\\ \;\;\;\;\left(t\_3 + \left(t\_1 + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\right) - t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_1 + \left(0.5 \cdot \sqrt{\frac{1}{z}} - t\_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (+ (sqrt x) (sqrt y)))
        (t_3 (sqrt (+ x 1.0))))
   (if (<= z 3.8e-27)
     (+
      3.0
      (- (sqrt (+ 1.0 t)) (+ (sqrt t) (+ (sqrt y) (+ (sqrt x) (sqrt z))))))
     (if (<= z 2400000000.0)
       (- (+ t_3 (+ t_1 (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))))) t_2)
       (+ t_3 (+ t_1 (- (* 0.5 (sqrt (/ 1.0 z))) t_2)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt(x) + sqrt(y);
	double t_3 = sqrt((x + 1.0));
	double tmp;
	if (z <= 3.8e-27) {
		tmp = 3.0 + (sqrt((1.0 + t)) - (sqrt(t) + (sqrt(y) + (sqrt(x) + sqrt(z)))));
	} else if (z <= 2400000000.0) {
		tmp = (t_3 + (t_1 + (1.0 / (sqrt((1.0 + z)) + sqrt(z))))) - t_2;
	} else {
		tmp = t_3 + (t_1 + ((0.5 * sqrt((1.0 / z))) - t_2));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt(x) + sqrt(y)
    t_3 = sqrt((x + 1.0d0))
    if (z <= 3.8d-27) then
        tmp = 3.0d0 + (sqrt((1.0d0 + t)) - (sqrt(t) + (sqrt(y) + (sqrt(x) + sqrt(z)))))
    else if (z <= 2400000000.0d0) then
        tmp = (t_3 + (t_1 + (1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))))) - t_2
    else
        tmp = t_3 + (t_1 + ((0.5d0 * sqrt((1.0d0 / z))) - t_2))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt(x) + Math.sqrt(y);
	double t_3 = Math.sqrt((x + 1.0));
	double tmp;
	if (z <= 3.8e-27) {
		tmp = 3.0 + (Math.sqrt((1.0 + t)) - (Math.sqrt(t) + (Math.sqrt(y) + (Math.sqrt(x) + Math.sqrt(z)))));
	} else if (z <= 2400000000.0) {
		tmp = (t_3 + (t_1 + (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))))) - t_2;
	} else {
		tmp = t_3 + (t_1 + ((0.5 * Math.sqrt((1.0 / z))) - t_2));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt(x) + math.sqrt(y)
	t_3 = math.sqrt((x + 1.0))
	tmp = 0
	if z <= 3.8e-27:
		tmp = 3.0 + (math.sqrt((1.0 + t)) - (math.sqrt(t) + (math.sqrt(y) + (math.sqrt(x) + math.sqrt(z)))))
	elif z <= 2400000000.0:
		tmp = (t_3 + (t_1 + (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))))) - t_2
	else:
		tmp = t_3 + (t_1 + ((0.5 * math.sqrt((1.0 / z))) - t_2))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(x) + sqrt(y))
	t_3 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (z <= 3.8e-27)
		tmp = Float64(3.0 + Float64(sqrt(Float64(1.0 + t)) - Float64(sqrt(t) + Float64(sqrt(y) + Float64(sqrt(x) + sqrt(z))))));
	elseif (z <= 2400000000.0)
		tmp = Float64(Float64(t_3 + Float64(t_1 + Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))))) - t_2);
	else
		tmp = Float64(t_3 + Float64(t_1 + Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) - t_2)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt(x) + sqrt(y);
	t_3 = sqrt((x + 1.0));
	tmp = 0.0;
	if (z <= 3.8e-27)
		tmp = 3.0 + (sqrt((1.0 + t)) - (sqrt(t) + (sqrt(y) + (sqrt(x) + sqrt(z)))));
	elseif (z <= 2400000000.0)
		tmp = (t_3 + (t_1 + (1.0 / (sqrt((1.0 + z)) + sqrt(z))))) - t_2;
	else
		tmp = t_3 + (t_1 + ((0.5 * sqrt((1.0 / z))) - t_2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 3.8e-27], N[(3.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2400000000.0], N[(N[(t$95$3 + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(t$95$3 + N[(t$95$1 + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.8e-27

   1. Initial program 98.0%

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

      1. associate-+l+ 98.0%

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

      2. sub-neg 98.0%

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

      3. sub-neg 98.0%

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

      4. +-commutative 98.0%

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

      5. +-commutative 98.0%

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

      6. +-commutative 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in y around 0 17.2%

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

      1. associate--l+ 29.7%

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

      2. +-commutative 29.7%

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

   7. Simplified 29.7%

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

   8. Taylor expanded in x around 0 16.5%

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

      1. +-commutative 16.5%

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

   10. Simplified 16.5%

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

   11. Taylor expanded in z around 0 16.5%

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

       1. associate--l+ 31.8%

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

       2. +-commutative 31.8%

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

       3. associate-+r+ 31.8%

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

       4. +-commutative 31.8%

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

       5. associate-+r+ 31.8%

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

   13. Simplified 31.8%

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

   if 3.8e-27 < z < 2.4e9

   1. Initial program 93.2%

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

      1. associate-+l+ 93.2%

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

      2. sub-neg 93.2%

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

      3. sub-neg 93.2%

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

      4. +-commutative 93.2%

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

      5. +-commutative 93.2%

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

      6. +-commutative 93.2%

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

   3. Simplified 93.2%

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

      1. flip-- 95.7%

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

      2. div-inv 95.7%

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

      3. add-sqr-sqrt 95.0%

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

      4. add-sqr-sqrt 97.6%

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

   6. Applied egg-rr 97.6%

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

      1. associate-*r/ 97.6%

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

      2. *-rgt-identity 97.6%

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

      3. associate--l+ 97.5%

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

      4. +-inverses 97.5%

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

      5. metadata-eval 97.5%

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

   8. Simplified 97.5%

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

   9. Taylor expanded in t around inf 25.2%

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

   if 2.4e9 < z

   1. Initial program 89.0%

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

      1. associate-+l+ 89.0%

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

      2. sub-neg 89.0%

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

      3. sub-neg 89.0%

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

      4. +-commutative 89.0%

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

      5. +-commutative 89.0%

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

      6. +-commutative 89.0%

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

   3. Simplified 89.0%

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

   5. Taylor expanded in t around inf 4.4%

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

      1. associate--l+ 17.1%

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

   7. Simplified 17.1%

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

   8. Taylor expanded in z around inf 25.4%

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

      1. associate--l+ 32.4%

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

      2. associate--l+ 32.4%

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

   10. Simplified 32.4%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{-27}:\\ \;\;\;\;3 + \left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 2400000000:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{1 + y} + \left(0.5 \cdot \sqrt{\frac{1}{z}} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} + \sqrt{y}\\ \mathbf{if}\;z \leq 3.7 \cdot 10^{-26}:\\ \;\;\;\;3 + \left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 350000000000:\\ \;\;\;\;\left(\left(x \cdot \left(0.5 + x \cdot -0.125\right) + 2\right) - t\_1\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{1 + y} + \left(0.5 \cdot \sqrt{\frac{1}{z}} - t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt x) (sqrt y))))
   (if (<= z 3.7e-26)
     (+
      3.0
      (- (sqrt (+ 1.0 t)) (+ (sqrt t) (+ (sqrt y) (+ (sqrt x) (sqrt z))))))
     (if (<= z 350000000000.0)
       (+
        (- (+ (* x (+ 0.5 (* x -0.125))) 2.0) t_1)
        (+ (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))) (* 0.5 (sqrt (/ 1.0 t)))))
       (+
        (sqrt (+ x 1.0))
        (+ (sqrt (+ 1.0 y)) (- (* 0.5 (sqrt (/ 1.0 z))) t_1)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(x) + sqrt(y);
	double tmp;
	if (z <= 3.7e-26) {
		tmp = 3.0 + (sqrt((1.0 + t)) - (sqrt(t) + (sqrt(y) + (sqrt(x) + sqrt(z)))));
	} else if (z <= 350000000000.0) {
		tmp = (((x * (0.5 + (x * -0.125))) + 2.0) - t_1) + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (0.5 * sqrt((1.0 / t))));
	} else {
		tmp = sqrt((x + 1.0)) + (sqrt((1.0 + y)) + ((0.5 * sqrt((1.0 / z))) - t_1));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(x) + sqrt(y)
    if (z <= 3.7d-26) then
        tmp = 3.0d0 + (sqrt((1.0d0 + t)) - (sqrt(t) + (sqrt(y) + (sqrt(x) + sqrt(z)))))
    else if (z <= 350000000000.0d0) then
        tmp = (((x * (0.5d0 + (x * (-0.125d0)))) + 2.0d0) - t_1) + ((1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + (0.5d0 * sqrt((1.0d0 / t))))
    else
        tmp = sqrt((x + 1.0d0)) + (sqrt((1.0d0 + y)) + ((0.5d0 * sqrt((1.0d0 / z))) - t_1))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(x) + Math.sqrt(y);
	double tmp;
	if (z <= 3.7e-26) {
		tmp = 3.0 + (Math.sqrt((1.0 + t)) - (Math.sqrt(t) + (Math.sqrt(y) + (Math.sqrt(x) + Math.sqrt(z)))));
	} else if (z <= 350000000000.0) {
		tmp = (((x * (0.5 + (x * -0.125))) + 2.0) - t_1) + ((1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + (0.5 * Math.sqrt((1.0 / t))));
	} else {
		tmp = Math.sqrt((x + 1.0)) + (Math.sqrt((1.0 + y)) + ((0.5 * Math.sqrt((1.0 / z))) - t_1));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(x) + math.sqrt(y)
	tmp = 0
	if z <= 3.7e-26:
		tmp = 3.0 + (math.sqrt((1.0 + t)) - (math.sqrt(t) + (math.sqrt(y) + (math.sqrt(x) + math.sqrt(z)))))
	elif z <= 350000000000.0:
		tmp = (((x * (0.5 + (x * -0.125))) + 2.0) - t_1) + ((1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + (0.5 * math.sqrt((1.0 / t))))
	else:
		tmp = math.sqrt((x + 1.0)) + (math.sqrt((1.0 + y)) + ((0.5 * math.sqrt((1.0 / z))) - t_1))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) + sqrt(y))
	tmp = 0.0
	if (z <= 3.7e-26)
		tmp = Float64(3.0 + Float64(sqrt(Float64(1.0 + t)) - Float64(sqrt(t) + Float64(sqrt(y) + Float64(sqrt(x) + sqrt(z))))));
	elseif (z <= 350000000000.0)
		tmp = Float64(Float64(Float64(Float64(x * Float64(0.5 + Float64(x * -0.125))) + 2.0) - t_1) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + Float64(0.5 * sqrt(Float64(1.0 / t)))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(sqrt(Float64(1.0 + y)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) - t_1)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(x) + sqrt(y);
	tmp = 0.0;
	if (z <= 3.7e-26)
		tmp = 3.0 + (sqrt((1.0 + t)) - (sqrt(t) + (sqrt(y) + (sqrt(x) + sqrt(z)))));
	elseif (z <= 350000000000.0)
		tmp = (((x * (0.5 + (x * -0.125))) + 2.0) - t_1) + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (0.5 * sqrt((1.0 / t))));
	else
		tmp = sqrt((x + 1.0)) + (sqrt((1.0 + y)) + ((0.5 * sqrt((1.0 / z))) - t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 3.7e-26], N[(3.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 350000000000.0], N[(N[(N[(N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] - t$95$1), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.6999999999999999e-26

   1. Initial program 98.0%

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

      1. associate-+l+ 98.0%

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

      2. sub-neg 98.0%

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

      3. sub-neg 98.0%

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

      4. +-commutative 98.0%

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

      5. +-commutative 98.0%

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

      6. +-commutative 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in y around 0 17.2%

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

      1. associate--l+ 29.7%

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

      2. +-commutative 29.7%

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

   7. Simplified 29.7%

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

   8. Taylor expanded in x around 0 16.5%

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

      1. +-commutative 16.5%

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

   10. Simplified 16.5%

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

   11. Taylor expanded in z around 0 16.5%

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

       1. associate--l+ 31.8%

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

       2. +-commutative 31.8%

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

       3. associate-+r+ 31.8%

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

       4. +-commutative 31.8%

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

       5. associate-+r+ 31.8%

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

   13. Simplified 31.8%

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

   if 3.6999999999999999e-26 < z < 3.5e11

   1. Initial program 92.4%

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

      1. associate-+l+ 92.4%

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

      2. sub-neg 92.4%

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

      3. sub-neg 92.4%

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

      4. +-commutative 92.4%

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

      5. +-commutative 92.4%

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

      6. +-commutative 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in x around 0 47.6%

      \[\leadsto \left(\color{blue}{\left(\left(1 + x \cdot \left(0.5 + -0.125 \cdot x\right)\right) - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   6. Step-by-step derivation

      1. associate--l+ 47.6%

         \[\leadsto \left(\color{blue}{\left(1 + \left(x \cdot \left(0.5 + -0.125 \cdot x\right) - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

      2. *-commutative 47.6%

         \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + \color{blue}{x \cdot -0.125}\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   7. Simplified 47.6%

      \[\leadsto \left(\color{blue}{\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   8. Taylor expanded in t around inf 30.7%

      \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}}\right) \]
   9. Step-by-step derivation

      1. flip-- 94.7%

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

      2. div-inv 94.7%

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

      3. add-sqr-sqrt 95.2%

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

      4. add-sqr-sqrt 97.7%

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

   10. Applied egg-rr 31.8%

       \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(\left(1 + z\right) - z\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}} + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 97.7%

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

       2. *-rgt-identity 97.7%

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

       3. associate--l+ 97.7%

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

       4. +-inverses 97.7%

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

       5. metadata-eval 97.7%

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

   12. Simplified 31.8%

       \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]

   13. Taylor expanded in y around 0 21.3%

       \[\leadsto \color{blue}{\left(\left(2 + x \cdot \left(0.5 + -0.125 \cdot x\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]

   if 3.5e11 < z

   1. Initial program 89.1%

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

      1. associate-+l+ 89.1%

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

      2. sub-neg 89.1%

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

      3. sub-neg 89.1%

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

      4. +-commutative 89.1%

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

      5. +-commutative 89.1%

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

      6. +-commutative 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in t around inf 4.4%

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

      1. associate--l+ 17.2%

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

   7. Simplified 17.2%

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

   8. Taylor expanded in z around inf 25.5%

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

      1. associate--l+ 32.6%

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

      2. associate--l+ 32.6%

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

   10. Simplified 32.6%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{-26}:\\ \;\;\;\;3 + \left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 350000000000:\\ \;\;\;\;\left(\left(x \cdot \left(0.5 + x \cdot -0.125\right) + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{1 + y} + \left(0.5 \cdot \sqrt{\frac{1}{z}} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-28}:\\ \;\;\;\;3 + \left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 44000000:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + z} + 0.5 \cdot y\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{1 + y} + \left(0.5 \cdot \sqrt{\frac{1}{z}} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 4e-28)
   (+ 3.0 (- (sqrt (+ 1.0 t)) (+ (sqrt t) (+ (sqrt y) (+ (sqrt x) (sqrt z))))))
   (if (<= z 44000000.0)
     (+
      2.0
      (- (+ (sqrt (+ 1.0 z)) (* 0.5 y)) (+ (sqrt x) (+ (sqrt y) (sqrt z)))))
     (+
      (sqrt (+ x 1.0))
      (+
       (sqrt (+ 1.0 y))
       (- (* 0.5 (sqrt (/ 1.0 z))) (+ (sqrt x) (sqrt y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4e-28) {
		tmp = 3.0 + (sqrt((1.0 + t)) - (sqrt(t) + (sqrt(y) + (sqrt(x) + sqrt(z)))));
	} else if (z <= 44000000.0) {
		tmp = 2.0 + ((sqrt((1.0 + z)) + (0.5 * y)) - (sqrt(x) + (sqrt(y) + sqrt(z))));
	} else {
		tmp = sqrt((x + 1.0)) + (sqrt((1.0 + y)) + ((0.5 * sqrt((1.0 / z))) - (sqrt(x) + sqrt(y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 4d-28) then
        tmp = 3.0d0 + (sqrt((1.0d0 + t)) - (sqrt(t) + (sqrt(y) + (sqrt(x) + sqrt(z)))))
    else if (z <= 44000000.0d0) then
        tmp = 2.0d0 + ((sqrt((1.0d0 + z)) + (0.5d0 * y)) - (sqrt(x) + (sqrt(y) + sqrt(z))))
    else
        tmp = sqrt((x + 1.0d0)) + (sqrt((1.0d0 + y)) + ((0.5d0 * sqrt((1.0d0 / z))) - (sqrt(x) + sqrt(y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4e-28) {
		tmp = 3.0 + (Math.sqrt((1.0 + t)) - (Math.sqrt(t) + (Math.sqrt(y) + (Math.sqrt(x) + Math.sqrt(z)))));
	} else if (z <= 44000000.0) {
		tmp = 2.0 + ((Math.sqrt((1.0 + z)) + (0.5 * y)) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z))));
	} else {
		tmp = Math.sqrt((x + 1.0)) + (Math.sqrt((1.0 + y)) + ((0.5 * Math.sqrt((1.0 / z))) - (Math.sqrt(x) + Math.sqrt(y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 4e-28:
		tmp = 3.0 + (math.sqrt((1.0 + t)) - (math.sqrt(t) + (math.sqrt(y) + (math.sqrt(x) + math.sqrt(z)))))
	elif z <= 44000000.0:
		tmp = 2.0 + ((math.sqrt((1.0 + z)) + (0.5 * y)) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z))))
	else:
		tmp = math.sqrt((x + 1.0)) + (math.sqrt((1.0 + y)) + ((0.5 * math.sqrt((1.0 / z))) - (math.sqrt(x) + math.sqrt(y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 4e-28)
		tmp = Float64(3.0 + Float64(sqrt(Float64(1.0 + t)) - Float64(sqrt(t) + Float64(sqrt(y) + Float64(sqrt(x) + sqrt(z))))));
	elseif (z <= 44000000.0)
		tmp = Float64(2.0 + Float64(Float64(sqrt(Float64(1.0 + z)) + Float64(0.5 * y)) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(sqrt(Float64(1.0 + y)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) - Float64(sqrt(x) + sqrt(y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 4e-28)
		tmp = 3.0 + (sqrt((1.0 + t)) - (sqrt(t) + (sqrt(y) + (sqrt(x) + sqrt(z)))));
	elseif (z <= 44000000.0)
		tmp = 2.0 + ((sqrt((1.0 + z)) + (0.5 * y)) - (sqrt(x) + (sqrt(y) + sqrt(z))));
	else
		tmp = sqrt((x + 1.0)) + (sqrt((1.0 + y)) + ((0.5 * sqrt((1.0 / z))) - (sqrt(x) + sqrt(y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 4e-28], N[(3.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 44000000.0], N[(2.0 + N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(0.5 * y), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.99999999999999988e-28

   1. Initial program 97.9%

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

      1. associate-+l+ 97.9%

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

      2. sub-neg 97.9%

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

      3. sub-neg 97.9%

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

      4. +-commutative 97.9%

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

      5. +-commutative 97.9%

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

      6. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in y around 0 17.4%

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

      1. associate--l+ 29.5%

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

      2. +-commutative 29.5%

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

   7. Simplified 29.5%

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

   8. Taylor expanded in x around 0 16.8%

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

      1. +-commutative 16.8%

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

   10. Simplified 16.8%

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

   11. Taylor expanded in z around 0 16.8%

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

       1. associate--l+ 32.3%

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

       2. +-commutative 32.3%

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

       3. associate-+r+ 32.3%

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

       4. +-commutative 32.3%

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

       5. associate-+r+ 32.3%

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

   13. Simplified 32.3%

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

   if 3.99999999999999988e-28 < z < 4.4e7

   1. Initial program 96.4%

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

      1. associate-+l+ 96.4%

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

      2. sub-neg 96.4%

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

      3. sub-neg 96.4%

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

      4. +-commutative 96.4%

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

      5. +-commutative 96.4%

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

      6. +-commutative 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in t around inf 21.0%

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

      1. associate--l+ 25.2%

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

   7. Simplified 25.2%

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

   8. Taylor expanded in x around 0 20.4%

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

      1. associate--l+ 28.0%

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

      2. associate-+r- 27.9%

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

      3. +-commutative 27.9%

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

   10. Simplified 27.9%

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

   11. Taylor expanded in y around 0 21.2%

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

       1. associate--l+ 21.3%

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

       2. *-commutative 21.3%

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

   13. Simplified 21.3%

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

   if 4.4e7 < z

   1. Initial program 88.9%

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

      1. associate-+l+ 88.9%

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

      2. sub-neg 88.9%

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

      3. sub-neg 88.9%

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

      4. +-commutative 88.9%

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

      5. +-commutative 88.9%

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

      6. +-commutative 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in t around inf 5.2%

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

      1. associate--l+ 17.5%

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

   7. Simplified 17.5%

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

   8. Taylor expanded in z around inf 25.5%

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

      1. associate--l+ 32.4%

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

      2. associate--l+ 32.5%

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

   10. Simplified 32.5%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-28}:\\ \;\;\;\;3 + \left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 44000000:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + z} + 0.5 \cdot y\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{1 + y} + \left(0.5 \cdot \sqrt{\frac{1}{z}} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{-28}:\\ \;\;\;\;3 + \left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 44000000:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + z} + 0.5 \cdot y\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(0.5 \cdot \sqrt{\frac{1}{z}} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.7e-28)
   (+ 3.0 (- (sqrt (+ 1.0 t)) (+ (sqrt t) (+ (sqrt y) (+ (sqrt x) (sqrt z))))))
   (if (<= z 44000000.0)
     (+
      2.0
      (- (+ (sqrt (+ 1.0 z)) (* 0.5 y)) (+ (sqrt x) (+ (sqrt y) (sqrt z)))))
     (+
      1.0
      (+
       (sqrt (+ 1.0 y))
       (- (* 0.5 (sqrt (/ 1.0 z))) (+ (sqrt x) (sqrt y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.7e-28) {
		tmp = 3.0 + (sqrt((1.0 + t)) - (sqrt(t) + (sqrt(y) + (sqrt(x) + sqrt(z)))));
	} else if (z <= 44000000.0) {
		tmp = 2.0 + ((sqrt((1.0 + z)) + (0.5 * y)) - (sqrt(x) + (sqrt(y) + sqrt(z))));
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) + ((0.5 * sqrt((1.0 / z))) - (sqrt(x) + sqrt(y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 3.7d-28) then
        tmp = 3.0d0 + (sqrt((1.0d0 + t)) - (sqrt(t) + (sqrt(y) + (sqrt(x) + sqrt(z)))))
    else if (z <= 44000000.0d0) then
        tmp = 2.0d0 + ((sqrt((1.0d0 + z)) + (0.5d0 * y)) - (sqrt(x) + (sqrt(y) + sqrt(z))))
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) + ((0.5d0 * sqrt((1.0d0 / z))) - (sqrt(x) + sqrt(y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.7e-28) {
		tmp = 3.0 + (Math.sqrt((1.0 + t)) - (Math.sqrt(t) + (Math.sqrt(y) + (Math.sqrt(x) + Math.sqrt(z)))));
	} else if (z <= 44000000.0) {
		tmp = 2.0 + ((Math.sqrt((1.0 + z)) + (0.5 * y)) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z))));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) + ((0.5 * Math.sqrt((1.0 / z))) - (Math.sqrt(x) + Math.sqrt(y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 3.7e-28:
		tmp = 3.0 + (math.sqrt((1.0 + t)) - (math.sqrt(t) + (math.sqrt(y) + (math.sqrt(x) + math.sqrt(z)))))
	elif z <= 44000000.0:
		tmp = 2.0 + ((math.sqrt((1.0 + z)) + (0.5 * y)) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z))))
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) + ((0.5 * math.sqrt((1.0 / z))) - (math.sqrt(x) + math.sqrt(y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.7e-28)
		tmp = Float64(3.0 + Float64(sqrt(Float64(1.0 + t)) - Float64(sqrt(t) + Float64(sqrt(y) + Float64(sqrt(x) + sqrt(z))))));
	elseif (z <= 44000000.0)
		tmp = Float64(2.0 + Float64(Float64(sqrt(Float64(1.0 + z)) + Float64(0.5 * y)) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) - Float64(sqrt(x) + sqrt(y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3.7e-28)
		tmp = 3.0 + (sqrt((1.0 + t)) - (sqrt(t) + (sqrt(y) + (sqrt(x) + sqrt(z)))));
	elseif (z <= 44000000.0)
		tmp = 2.0 + ((sqrt((1.0 + z)) + (0.5 * y)) - (sqrt(x) + (sqrt(y) + sqrt(z))));
	else
		tmp = 1.0 + (sqrt((1.0 + y)) + ((0.5 * sqrt((1.0 / z))) - (sqrt(x) + sqrt(y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 3.7e-28], N[(3.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 44000000.0], N[(2.0 + N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(0.5 * y), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.7000000000000002e-28

   1. Initial program 97.9%

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

      1. associate-+l+ 97.9%

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

      2. sub-neg 97.9%

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

      3. sub-neg 97.9%

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

      4. +-commutative 97.9%

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

      5. +-commutative 97.9%

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

      6. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in y around 0 17.4%

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

      1. associate--l+ 29.5%

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

      2. +-commutative 29.5%

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

   7. Simplified 29.5%

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

   8. Taylor expanded in x around 0 16.8%

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

      1. +-commutative 16.8%

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

   10. Simplified 16.8%

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

   11. Taylor expanded in z around 0 16.8%

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

       1. associate--l+ 32.3%

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

       2. +-commutative 32.3%

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

       3. associate-+r+ 32.3%

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

       4. +-commutative 32.3%

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

       5. associate-+r+ 32.3%

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

   13. Simplified 32.3%

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

   if 3.7000000000000002e-28 < z < 4.4e7

   1. Initial program 96.4%

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

      1. associate-+l+ 96.4%

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

      2. sub-neg 96.4%

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

      3. sub-neg 96.4%

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

      4. +-commutative 96.4%

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

      5. +-commutative 96.4%

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

      6. +-commutative 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in t around inf 21.0%

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

      1. associate--l+ 25.2%

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

   7. Simplified 25.2%

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

   8. Taylor expanded in x around 0 20.4%

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

      1. associate--l+ 28.0%

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

      2. associate-+r- 27.9%

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

      3. +-commutative 27.9%

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

   10. Simplified 27.9%

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

   11. Taylor expanded in y around 0 21.2%

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

       1. associate--l+ 21.3%

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

       2. *-commutative 21.3%

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

   13. Simplified 21.3%

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

   if 4.4e7 < z

   1. Initial program 88.9%

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

      1. associate-+l+ 88.9%

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

      2. sub-neg 88.9%

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

      3. sub-neg 88.9%

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

      4. +-commutative 88.9%

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

      5. +-commutative 88.9%

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

      6. +-commutative 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in t around inf 5.2%

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

      1. associate--l+ 17.5%

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

   7. Simplified 17.5%

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

   8. Taylor expanded in x around 0 4.3%

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

      1. associate--l+ 26.2%

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

      2. associate-+r- 32.5%

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

      3. +-commutative 32.5%

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

   10. Simplified 32.5%

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

   11. Taylor expanded in z around inf 34.2%

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

       1. +-commutative 34.2%

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

   13. Simplified 34.2%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{-28}:\\ \;\;\;\;3 + \left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 44000000:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + z} + 0.5 \cdot y\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(0.5 \cdot \sqrt{\frac{1}{z}} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} + \sqrt{y}\\ t_2 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 8.6 \cdot 10^{-29}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + t\_1\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;t\_2 + \left(\sqrt{1 + y} - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt x) (sqrt y))) (t_2 (sqrt (+ x 1.0))))
   (if (<= y 8.6e-29)
     (+ 2.0 (- (sqrt (+ 1.0 z)) (+ (sqrt z) t_1)))
     (if (<= y 5e+14) (+ t_2 (- (sqrt (+ 1.0 y)) t_1)) (- t_2 (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(x) + sqrt(y);
	double t_2 = sqrt((x + 1.0));
	double tmp;
	if (y <= 8.6e-29) {
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(z) + t_1));
	} else if (y <= 5e+14) {
		tmp = t_2 + (sqrt((1.0 + y)) - t_1);
	} else {
		tmp = t_2 - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt(x) + sqrt(y)
    t_2 = sqrt((x + 1.0d0))
    if (y <= 8.6d-29) then
        tmp = 2.0d0 + (sqrt((1.0d0 + z)) - (sqrt(z) + t_1))
    else if (y <= 5d+14) then
        tmp = t_2 + (sqrt((1.0d0 + y)) - t_1)
    else
        tmp = t_2 - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(x) + Math.sqrt(y);
	double t_2 = Math.sqrt((x + 1.0));
	double tmp;
	if (y <= 8.6e-29) {
		tmp = 2.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(z) + t_1));
	} else if (y <= 5e+14) {
		tmp = t_2 + (Math.sqrt((1.0 + y)) - t_1);
	} else {
		tmp = t_2 - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(x) + math.sqrt(y)
	t_2 = math.sqrt((x + 1.0))
	tmp = 0
	if y <= 8.6e-29:
		tmp = 2.0 + (math.sqrt((1.0 + z)) - (math.sqrt(z) + t_1))
	elif y <= 5e+14:
		tmp = t_2 + (math.sqrt((1.0 + y)) - t_1)
	else:
		tmp = t_2 - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) + sqrt(y))
	t_2 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (y <= 8.6e-29)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(z) + t_1)));
	elseif (y <= 5e+14)
		tmp = Float64(t_2 + Float64(sqrt(Float64(1.0 + y)) - t_1));
	else
		tmp = Float64(t_2 - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(x) + sqrt(y);
	t_2 = sqrt((x + 1.0));
	tmp = 0.0;
	if (y <= 8.6e-29)
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(z) + t_1));
	elseif (y <= 5e+14)
		tmp = t_2 + (sqrt((1.0 + y)) - t_1);
	else
		tmp = t_2 - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 8.6e-29], N[(2.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+14], N[(t$95$2 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.5999999999999996e-29

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in t around inf 22.9%

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

      1. associate--l+ 27.5%

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

   7. Simplified 27.5%

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

   8. Taylor expanded in x around 0 20.2%

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

      1. associate--l+ 27.1%

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

      2. associate-+r- 40.0%

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

      3. +-commutative 40.0%

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

   10. Simplified 40.0%

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

   11. Taylor expanded in y around 0 20.2%

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

       1. associate--l+ 40.0%

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

       2. associate-+r+ 40.0%

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

       3. +-commutative 40.0%

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

   13. Simplified 40.0%

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

   if 8.5999999999999996e-29 < y < 5e14

   1. Initial program 94.4%

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

      1. associate-+l+ 94.3%

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

      2. sub-neg 94.3%

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

      3. sub-neg 94.3%

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

      4. +-commutative 94.3%

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

      5. +-commutative 94.3%

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

      6. +-commutative 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in t around inf 28.8%

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

      1. associate--l+ 33.5%

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

   7. Simplified 33.5%

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

   8. Taylor expanded in z around inf 22.2%

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

      1. associate--l+ 22.2%

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

   10. Simplified 22.2%

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

   if 5e14 < y

   1. Initial program 89.4%

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

      1. associate-+l+ 89.4%

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

      2. sub-neg 89.4%

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

      3. sub-neg 89.4%

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

      4. +-commutative 89.4%

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

      5. +-commutative 89.4%

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

      6. +-commutative 89.4%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in t around inf 3.7%

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

      1. associate--l+ 16.6%

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

   7. Simplified 16.6%

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

   8. Taylor expanded in x around inf 16.4%

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

      1. mul-1-neg 16.4%

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

   10. Simplified 16.4%

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

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-29}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 44000000:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + z} + 0.5 \cdot y\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(0.5 \cdot \sqrt{\frac{1}{z}} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 44000000.0)
   (+
    2.0
    (- (+ (sqrt (+ 1.0 z)) (* 0.5 y)) (+ (sqrt x) (+ (sqrt y) (sqrt z)))))
   (+
    1.0
    (+ (sqrt (+ 1.0 y)) (- (* 0.5 (sqrt (/ 1.0 z))) (+ (sqrt x) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 44000000.0) {
		tmp = 2.0 + ((sqrt((1.0 + z)) + (0.5 * y)) - (sqrt(x) + (sqrt(y) + sqrt(z))));
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) + ((0.5 * sqrt((1.0 / z))) - (sqrt(x) + sqrt(y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 44000000.0d0) then
        tmp = 2.0d0 + ((sqrt((1.0d0 + z)) + (0.5d0 * y)) - (sqrt(x) + (sqrt(y) + sqrt(z))))
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) + ((0.5d0 * sqrt((1.0d0 / z))) - (sqrt(x) + sqrt(y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 44000000.0) {
		tmp = 2.0 + ((Math.sqrt((1.0 + z)) + (0.5 * y)) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z))));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) + ((0.5 * Math.sqrt((1.0 / z))) - (Math.sqrt(x) + Math.sqrt(y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 44000000.0:
		tmp = 2.0 + ((math.sqrt((1.0 + z)) + (0.5 * y)) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z))))
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) + ((0.5 * math.sqrt((1.0 / z))) - (math.sqrt(x) + math.sqrt(y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 44000000.0)
		tmp = Float64(2.0 + Float64(Float64(sqrt(Float64(1.0 + z)) + Float64(0.5 * y)) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) - Float64(sqrt(x) + sqrt(y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 44000000.0)
		tmp = 2.0 + ((sqrt((1.0 + z)) + (0.5 * y)) - (sqrt(x) + (sqrt(y) + sqrt(z))));
	else
		tmp = 1.0 + (sqrt((1.0 + y)) + ((0.5 * sqrt((1.0 / z))) - (sqrt(x) + sqrt(y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 44000000.0], N[(2.0 + N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(0.5 * y), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.4e7

   1. Initial program 97.7%

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

      1. associate-+l+ 97.7%

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

      2. sub-neg 97.7%

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

      3. sub-neg 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in t around inf 24.2%

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

      1. associate--l+ 28.5%

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

   7. Simplified 28.5%

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

   8. Taylor expanded in x around 0 22.0%

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

      1. associate--l+ 28.6%

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

      2. associate-+r- 28.6%

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

      3. +-commutative 28.6%

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

   10. Simplified 28.6%

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

   11. Taylor expanded in y around 0 20.6%

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

       1. associate--l+ 20.6%

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

       2. *-commutative 20.6%

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

   13. Simplified 20.6%

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

   if 4.4e7 < z

   1. Initial program 88.9%

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

      1. associate-+l+ 88.9%

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

      2. sub-neg 88.9%

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

      3. sub-neg 88.9%

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

      4. +-commutative 88.9%

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

      5. +-commutative 88.9%

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

      6. +-commutative 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in t around inf 5.2%

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

      1. associate--l+ 17.5%

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

   7. Simplified 17.5%

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

   8. Taylor expanded in x around 0 4.3%

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

      1. associate--l+ 26.2%

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

      2. associate-+r- 32.5%

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

      3. +-commutative 32.5%

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

   10. Simplified 32.5%

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

   11. Taylor expanded in z around inf 34.2%

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

       1. +-commutative 34.2%

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

   13. Simplified 34.2%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 44000000:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + z} + 0.5 \cdot y\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(0.5 \cdot \sqrt{\frac{1}{z}} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} + \sqrt{y}\\ \mathbf{if}\;z \leq 44000000:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(0.5 \cdot \sqrt{\frac{1}{z}} - t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt x) (sqrt y))))
   (if (<= z 44000000.0)
     (+ 2.0 (- (sqrt (+ 1.0 z)) (+ (sqrt z) t_1)))
     (+ 1.0 (+ (sqrt (+ 1.0 y)) (- (* 0.5 (sqrt (/ 1.0 z))) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(x) + sqrt(y);
	double tmp;
	if (z <= 44000000.0) {
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(z) + t_1));
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) + ((0.5 * sqrt((1.0 / z))) - t_1));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(x) + sqrt(y)
    if (z <= 44000000.0d0) then
        tmp = 2.0d0 + (sqrt((1.0d0 + z)) - (sqrt(z) + t_1))
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) + ((0.5d0 * sqrt((1.0d0 / z))) - t_1))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(x) + Math.sqrt(y);
	double tmp;
	if (z <= 44000000.0) {
		tmp = 2.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(z) + t_1));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) + ((0.5 * Math.sqrt((1.0 / z))) - t_1));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(x) + math.sqrt(y)
	tmp = 0
	if z <= 44000000.0:
		tmp = 2.0 + (math.sqrt((1.0 + z)) - (math.sqrt(z) + t_1))
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) + ((0.5 * math.sqrt((1.0 / z))) - t_1))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) + sqrt(y))
	tmp = 0.0
	if (z <= 44000000.0)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(z) + t_1)));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) - t_1)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(x) + sqrt(y);
	tmp = 0.0;
	if (z <= 44000000.0)
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(z) + t_1));
	else
		tmp = 1.0 + (sqrt((1.0 + y)) + ((0.5 * sqrt((1.0 / z))) - t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 44000000.0], N[(2.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 4.4e7

   1. Initial program 97.7%

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

      1. associate-+l+ 97.7%

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

      2. sub-neg 97.7%

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

      3. sub-neg 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in t around inf 24.2%

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

      1. associate--l+ 28.5%

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

   7. Simplified 28.5%

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

   8. Taylor expanded in x around 0 22.0%

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

      1. associate--l+ 28.6%

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

      2. associate-+r- 28.6%

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

      3. +-commutative 28.6%

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

   10. Simplified 28.6%

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

   11. Taylor expanded in y around 0 18.9%

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

       1. associate--l+ 18.9%

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

       2. associate-+r+ 18.9%

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

       3. +-commutative 18.9%

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

   13. Simplified 18.9%

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

   if 4.4e7 < z

   1. Initial program 88.9%

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

      1. associate-+l+ 88.9%

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

      2. sub-neg 88.9%

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

      3. sub-neg 88.9%

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

      4. +-commutative 88.9%

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

      5. +-commutative 88.9%

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

      6. +-commutative 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in t around inf 5.2%

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

      1. associate--l+ 17.5%

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

   7. Simplified 17.5%

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

   8. Taylor expanded in x around 0 4.3%

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

      1. associate--l+ 26.2%

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

      2. associate-+r- 32.5%

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

      3. +-commutative 32.5%

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

   10. Simplified 32.5%

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

   11. Taylor expanded in z around inf 34.2%

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

       1. +-commutative 34.2%

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

   13. Simplified 34.2%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 44000000:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(0.5 \cdot \sqrt{\frac{1}{z}} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 85.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} + \sqrt{y}\\ \mathbf{if}\;y \leq 8.6 \cdot 10^{-29}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + t\_1\right)\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt x) (sqrt y))))
   (if (<= y 8.6e-29)
     (+ 2.0 (- (sqrt (+ 1.0 z)) (+ (sqrt z) t_1)))
     (if (<= y 4e+15)
       (+ 1.0 (- (sqrt (+ 1.0 y)) t_1))
       (- (sqrt (+ x 1.0)) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(x) + sqrt(y);
	double tmp;
	if (y <= 8.6e-29) {
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(z) + t_1));
	} else if (y <= 4e+15) {
		tmp = 1.0 + (sqrt((1.0 + y)) - t_1);
	} else {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(x) + sqrt(y)
    if (y <= 8.6d-29) then
        tmp = 2.0d0 + (sqrt((1.0d0 + z)) - (sqrt(z) + t_1))
    else if (y <= 4d+15) then
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - t_1)
    else
        tmp = sqrt((x + 1.0d0)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(x) + Math.sqrt(y);
	double tmp;
	if (y <= 8.6e-29) {
		tmp = 2.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(z) + t_1));
	} else if (y <= 4e+15) {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - t_1);
	} else {
		tmp = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(x) + math.sqrt(y)
	tmp = 0
	if y <= 8.6e-29:
		tmp = 2.0 + (math.sqrt((1.0 + z)) - (math.sqrt(z) + t_1))
	elif y <= 4e+15:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - t_1)
	else:
		tmp = math.sqrt((x + 1.0)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) + sqrt(y))
	tmp = 0.0
	if (y <= 8.6e-29)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(z) + t_1)));
	elseif (y <= 4e+15)
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - t_1));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(x) + sqrt(y);
	tmp = 0.0;
	if (y <= 8.6e-29)
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(z) + t_1));
	elseif (y <= 4e+15)
		tmp = 1.0 + (sqrt((1.0 + y)) - t_1);
	else
		tmp = sqrt((x + 1.0)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 8.6e-29], N[(2.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+15], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.5999999999999996e-29

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in t around inf 22.9%

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

      1. associate--l+ 27.5%

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

   7. Simplified 27.5%

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

   8. Taylor expanded in x around 0 20.2%

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

      1. associate--l+ 27.1%

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

      2. associate-+r- 40.0%

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

      3. +-commutative 40.0%

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

   10. Simplified 40.0%

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

   11. Taylor expanded in y around 0 20.2%

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

       1. associate--l+ 40.0%

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

       2. associate-+r+ 40.0%

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

       3. +-commutative 40.0%

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

   13. Simplified 40.0%

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

   if 8.5999999999999996e-29 < y < 4e15

   1. Initial program 93.0%

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

      1. associate-+l+ 93.0%

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

      2. sub-neg 93.0%

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

      3. sub-neg 93.0%

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

      4. +-commutative 93.0%

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

      5. +-commutative 93.0%

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

      6. +-commutative 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in t around inf 27.7%

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

      1. associate--l+ 32.9%

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

   7. Simplified 32.9%

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

   8. Taylor expanded in x around 0 27.1%

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

      1. associate--l+ 34.0%

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

      2. associate-+r- 33.5%

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

      3. +-commutative 33.5%

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

   10. Simplified 33.5%

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

   11. Taylor expanded in z around inf 21.3%

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

       1. associate--l+ 21.3%

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

       2. sub-neg 21.3%

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

       3. +-commutative 21.3%

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

       4. +-commutative 21.3%

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

       5. sub-neg 21.3%

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

   13. Simplified 21.3%

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

   if 4e15 < y

   1. Initial program 89.6%

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

      1. associate-+l+ 89.6%

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

      2. sub-neg 89.6%

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

      3. sub-neg 89.6%

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

      4. +-commutative 89.6%

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

      5. +-commutative 89.6%

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

      6. +-commutative 89.6%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in t around inf 3.7%

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

      1. associate--l+ 16.5%

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

   7. Simplified 16.5%

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

   8. Taylor expanded in x around inf 16.3%

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

      1. mul-1-neg 16.3%

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

   10. Simplified 16.3%

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

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-29}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 85.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-29}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 8.6e-29)
   (+ 2.0 (- (sqrt (+ 1.0 z)) (+ (sqrt x) (sqrt z))))
   (if (<= y 4.5e+15)
     (+ 1.0 (- (sqrt (+ 1.0 y)) (+ (sqrt x) (sqrt y))))
     (- (sqrt (+ x 1.0)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 8.6e-29) {
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(x) + sqrt(z)));
	} else if (y <= 4.5e+15) {
		tmp = 1.0 + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 8.6d-29) then
        tmp = 2.0d0 + (sqrt((1.0d0 + z)) - (sqrt(x) + sqrt(z)))
    else if (y <= 4.5d+15) then
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - (sqrt(x) + sqrt(y)))
    else
        tmp = sqrt((x + 1.0d0)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 8.6e-29) {
		tmp = 2.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(x) + Math.sqrt(z)));
	} else if (y <= 4.5e+15) {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 8.6e-29:
		tmp = 2.0 + (math.sqrt((1.0 + z)) - (math.sqrt(x) + math.sqrt(z)))
	elif y <= 4.5e+15:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = math.sqrt((x + 1.0)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 8.6e-29)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(x) + sqrt(z))));
	elseif (y <= 4.5e+15)
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 8.6e-29)
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(x) + sqrt(z)));
	elseif (y <= 4.5e+15)
		tmp = 1.0 + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	else
		tmp = sqrt((x + 1.0)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 8.6e-29], N[(2.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e+15], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.5999999999999996e-29

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in t around inf 22.9%

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

      1. associate--l+ 27.5%

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

   7. Simplified 27.5%

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

   8. Taylor expanded in x around 0 20.2%

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

      1. associate--l+ 27.1%

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

      2. associate-+r- 40.0%

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

      3. +-commutative 40.0%

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

   10. Simplified 40.0%

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

   11. Taylor expanded in y around 0 20.2%

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

       1. associate--l+ 40.0%

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

       2. associate-+r+ 40.0%

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

       3. +-commutative 40.0%

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

   13. Simplified 40.0%

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

   14. Taylor expanded in x around inf 40.0%

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

   if 8.5999999999999996e-29 < y < 4.5e15

   1. Initial program 93.0%

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

      1. associate-+l+ 93.0%

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

      2. sub-neg 93.0%

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

      3. sub-neg 93.0%

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

      4. +-commutative 93.0%

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

      5. +-commutative 93.0%

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

      6. +-commutative 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in t around inf 27.7%

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

      1. associate--l+ 32.9%

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

   7. Simplified 32.9%

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

   8. Taylor expanded in x around 0 27.1%

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

      1. associate--l+ 34.0%

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

      2. associate-+r- 33.5%

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

      3. +-commutative 33.5%

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

   10. Simplified 33.5%

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

   11. Taylor expanded in z around inf 21.3%

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

       1. associate--l+ 21.3%

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

       2. sub-neg 21.3%

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

       3. +-commutative 21.3%

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

       4. +-commutative 21.3%

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

       5. sub-neg 21.3%

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

   13. Simplified 21.3%

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

   if 4.5e15 < y

   1. Initial program 89.6%

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

      1. associate-+l+ 89.6%

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

      2. sub-neg 89.6%

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

      3. sub-neg 89.6%

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

      4. +-commutative 89.6%

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

      5. +-commutative 89.6%

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

      6. +-commutative 89.6%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in t around inf 3.7%

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

      1. associate--l+ 16.5%

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

   7. Simplified 16.5%

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

   8. Taylor expanded in x around inf 16.3%

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

      1. mul-1-neg 16.3%

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

   10. Simplified 16.3%

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

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-29}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 81.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.102:\\ \;\;\;\;3 - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 0.102)
   (- 3.0 (+ (sqrt y) (+ (sqrt x) (sqrt z))))
   (+ 1.0 (- (sqrt (+ 1.0 y)) (+ (sqrt x) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.102) {
		tmp = 3.0 - (sqrt(y) + (sqrt(x) + sqrt(z)));
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 0.102d0) then
        tmp = 3.0d0 - (sqrt(y) + (sqrt(x) + sqrt(z)))
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - (sqrt(x) + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.102) {
		tmp = 3.0 - (Math.sqrt(y) + (Math.sqrt(x) + Math.sqrt(z)));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - (Math.sqrt(x) + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 0.102:
		tmp = 3.0 - (math.sqrt(y) + (math.sqrt(x) + math.sqrt(z)))
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - (math.sqrt(x) + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 0.102)
		tmp = Float64(3.0 - Float64(sqrt(y) + Float64(sqrt(x) + sqrt(z))));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - Float64(sqrt(x) + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 0.102)
		tmp = 3.0 - (sqrt(y) + (sqrt(x) + sqrt(z)));
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 0.102], N[(3.0 - N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.101999999999999993

   1. Initial program 97.8%

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

      1. associate-+l+ 97.8%

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

      2. sub-neg 97.8%

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

      3. sub-neg 97.8%

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

      4. +-commutative 97.8%

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

      5. +-commutative 97.8%

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

      6. +-commutative 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in t around inf 24.9%

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

      1. associate--l+ 29.1%

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

   7. Simplified 29.1%

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

   8. Taylor expanded in x around 0 22.6%

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

      1. associate--l+ 28.9%

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

      2. associate-+r- 28.9%

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

      3. +-commutative 28.9%

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

   10. Simplified 28.9%

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

   11. Taylor expanded in y around 0 19.5%

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

       1. associate--l+ 19.5%

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

       2. associate-+r+ 19.5%

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

       3. +-commutative 19.5%

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

   13. Simplified 19.5%

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

   14. Taylor expanded in z around 0 19.1%

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

       1. associate-+r+ 19.1%

          \[\leadsto 3 - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

       2. +-commutative 19.1%

          \[\leadsto 3 - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right) \]

       3. associate-+r+ 19.1%

          \[\leadsto 3 - \color{blue}{\left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)} \]

   16. Simplified 19.1%

       \[\leadsto \color{blue}{3 - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)} \]

   if 0.101999999999999993 < z

   1. Initial program 89.0%

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

      1. associate-+l+ 89.0%

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

      2. sub-neg 89.0%

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

      3. sub-neg 89.0%

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

      4. +-commutative 89.0%

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

      5. +-commutative 89.0%

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

      6. +-commutative 89.0%

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

   3. Simplified 89.0%

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

   5. Taylor expanded in t around inf 5.1%

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

      1. associate--l+ 17.3%

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

   7. Simplified 17.3%

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

   8. Taylor expanded in x around 0 4.2%

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

      1. associate--l+ 26.0%

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

      2. associate-+r- 32.0%

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

      3. +-commutative 32.0%

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

   10. Simplified 32.0%

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

   11. Taylor expanded in z around inf 21.6%

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

       1. associate--l+ 33.2%

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

       2. sub-neg 33.2%

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

       3. +-commutative 33.2%

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

       4. +-commutative 33.2%

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

       5. sub-neg 33.2%

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

   13. Simplified 33.2%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.102:\\ \;\;\;\;3 - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 63.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-174}:\\ \;\;\;\;3 - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 0.86:\\ \;\;\;\;2 - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.2e-174)
   (- 3.0 (+ (sqrt y) (+ (sqrt x) (sqrt z))))
   (if (<= y 0.86)
     (- 2.0 (+ (sqrt x) (sqrt y)))
     (- (sqrt (+ x 1.0)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.2e-174) {
		tmp = 3.0 - (sqrt(y) + (sqrt(x) + sqrt(z)));
	} else if (y <= 0.86) {
		tmp = 2.0 - (sqrt(x) + sqrt(y));
	} else {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 2.2d-174) then
        tmp = 3.0d0 - (sqrt(y) + (sqrt(x) + sqrt(z)))
    else if (y <= 0.86d0) then
        tmp = 2.0d0 - (sqrt(x) + sqrt(y))
    else
        tmp = sqrt((x + 1.0d0)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.2e-174) {
		tmp = 3.0 - (Math.sqrt(y) + (Math.sqrt(x) + Math.sqrt(z)));
	} else if (y <= 0.86) {
		tmp = 2.0 - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 2.2e-174:
		tmp = 3.0 - (math.sqrt(y) + (math.sqrt(x) + math.sqrt(z)))
	elif y <= 0.86:
		tmp = 2.0 - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = math.sqrt((x + 1.0)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.2e-174)
		tmp = Float64(3.0 - Float64(sqrt(y) + Float64(sqrt(x) + sqrt(z))));
	elseif (y <= 0.86)
		tmp = Float64(2.0 - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.2e-174)
		tmp = 3.0 - (sqrt(y) + (sqrt(x) + sqrt(z)));
	elseif (y <= 0.86)
		tmp = 2.0 - (sqrt(x) + sqrt(y));
	else
		tmp = sqrt((x + 1.0)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 2.2e-174], N[(3.0 - N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.86], N[(2.0 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.20000000000000022e-174

   1. Initial program 96.4%

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

      1. associate-+l+ 96.4%

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

      2. sub-neg 96.4%

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

      3. sub-neg 96.4%

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

      4. +-commutative 96.4%

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

      5. +-commutative 96.4%

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

      6. +-commutative 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in t around inf 22.2%

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

      1. associate--l+ 26.4%

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

   7. Simplified 26.4%

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

   8. Taylor expanded in x around 0 19.0%

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

      1. associate--l+ 24.7%

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

      2. associate-+r- 38.1%

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

      3. +-commutative 38.1%

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

   10. Simplified 38.1%

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

   11. Taylor expanded in y around 0 19.0%

       \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   12. Step-by-step derivation

       1. associate--l+ 38.1%

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

       2. associate-+r+ 38.1%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

       3. +-commutative 38.1%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right) \]

   13. Simplified 38.1%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)\right)} \]

   14. Taylor expanded in z around 0 15.2%

       \[\leadsto \color{blue}{3 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   15. Step-by-step derivation

       1. associate-+r+ 15.2%

          \[\leadsto 3 - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

       2. +-commutative 15.2%

          \[\leadsto 3 - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right) \]

       3. associate-+r+ 15.2%

          \[\leadsto 3 - \color{blue}{\left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)} \]

   16. Simplified 15.2%

       \[\leadsto \color{blue}{3 - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)} \]

   if 2.20000000000000022e-174 < y < 0.859999999999999987

   1. Initial program 98.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 98.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 98.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 98.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 98.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 98.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 98.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 24.4%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 28.9%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   7. Simplified 28.9%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in x around 0 22.5%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   9. Step-by-step derivation

      1. associate--l+ 30.5%

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate-+r- 40.6%

         \[\leadsto 1 + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. +-commutative 40.6%

         \[\leadsto 1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   10. Simplified 40.6%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)} \]

   11. Taylor expanded in y around 0 22.5%

       \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   12. Step-by-step derivation

       1. associate--l+ 40.6%

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

       2. associate-+r+ 40.6%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

       3. +-commutative 40.6%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right) \]

   13. Simplified 40.6%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)\right)} \]

   14. Taylor expanded in z around inf 27.4%

       \[\leadsto \color{blue}{2 - \left(\sqrt{x} + \sqrt{y}\right)} \]
   15. Step-by-step derivation

       1. +-commutative 27.4%

          \[\leadsto 2 - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

   16. Simplified 27.4%

       \[\leadsto \color{blue}{2 - \left(\sqrt{y} + \sqrt{x}\right)} \]

   if 0.859999999999999987 < y

   1. Initial program 89.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 89.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 5.8%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 18.2%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   7. Simplified 18.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in x around inf 16.4%

      \[\leadsto \sqrt{1 + x} + \color{blue}{-1 \cdot \sqrt{x}} \]
   9. Step-by-step derivation

      1. mul-1-neg 16.4%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(-\sqrt{x}\right)} \]

   10. Simplified 16.4%

       \[\leadsto \sqrt{1 + x} + \color{blue}{\left(-\sqrt{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 19.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-174}:\\ \;\;\;\;3 - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 0.86:\\ \;\;\;\;2 - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 62.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.95:\\ \;\;\;\;2 - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 0.95) (- 2.0 (+ (sqrt x) (sqrt y))) (- (sqrt (+ x 1.0)) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.95) {
		tmp = 2.0 - (sqrt(x) + sqrt(y));
	} else {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 0.95d0) then
        tmp = 2.0d0 - (sqrt(x) + sqrt(y))
    else
        tmp = sqrt((x + 1.0d0)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.95) {
		tmp = 2.0 - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 0.95:
		tmp = 2.0 - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = math.sqrt((x + 1.0)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 0.95)
		tmp = Float64(2.0 - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 0.95)
		tmp = 2.0 - (sqrt(x) + sqrt(y));
	else
		tmp = sqrt((x + 1.0)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 0.95], N[(2.0 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.94999999999999996

   1. Initial program 97.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 97.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 97.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 97.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 97.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 97.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 97.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 23.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 27.7%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   7. Simplified 27.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in x around 0 20.7%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   9. Step-by-step derivation

      1. associate--l+ 27.6%

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate-+r- 39.3%

         \[\leadsto 1 + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. +-commutative 39.3%

         \[\leadsto 1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   10. Simplified 39.3%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)} \]

   11. Taylor expanded in y around 0 20.7%

       \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   12. Step-by-step derivation

       1. associate--l+ 39.3%

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

       2. associate-+r+ 39.3%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

       3. +-commutative 39.3%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right) \]

   13. Simplified 39.3%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)\right)} \]

   14. Taylor expanded in z around inf 24.5%

       \[\leadsto \color{blue}{2 - \left(\sqrt{x} + \sqrt{y}\right)} \]
   15. Step-by-step derivation

       1. +-commutative 24.5%

          \[\leadsto 2 - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

   16. Simplified 24.5%

       \[\leadsto \color{blue}{2 - \left(\sqrt{y} + \sqrt{x}\right)} \]

   if 0.94999999999999996 < y

   1. Initial program 89.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 89.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 5.8%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 18.2%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   7. Simplified 18.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in x around inf 16.4%

      \[\leadsto \sqrt{1 + x} + \color{blue}{-1 \cdot \sqrt{x}} \]
   9. Step-by-step derivation

      1. mul-1-neg 16.4%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(-\sqrt{x}\right)} \]

   10. Simplified 16.4%

       \[\leadsto \sqrt{1 + x} + \color{blue}{\left(-\sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.95:\\ \;\;\;\;2 - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 41.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.98:\\ \;\;\;\;2 - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 0.98) (- 2.0 (+ (sqrt x) (sqrt y))) (sqrt y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.98) {
		tmp = 2.0 - (sqrt(x) + sqrt(y));
	} else {
		tmp = sqrt(y);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 0.98d0) then
        tmp = 2.0d0 - (sqrt(x) + sqrt(y))
    else
        tmp = sqrt(y)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.98) {
		tmp = 2.0 - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 0.98:
		tmp = 2.0 - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 0.98)
		tmp = Float64(2.0 - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = sqrt(y);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 0.98)
		tmp = 2.0 - (sqrt(x) + sqrt(y));
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 0.98], N[(2.0 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.97999999999999998

   1. Initial program 97.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 97.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 97.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 97.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 97.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 97.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 97.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 23.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 27.7%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   7. Simplified 27.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in x around 0 20.7%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   9. Step-by-step derivation

      1. associate--l+ 27.6%

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate-+r- 39.3%

         \[\leadsto 1 + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. +-commutative 39.3%

         \[\leadsto 1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   10. Simplified 39.3%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)} \]

   11. Taylor expanded in y around 0 20.7%

       \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   12. Step-by-step derivation

       1. associate--l+ 39.3%

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

       2. associate-+r+ 39.3%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

       3. +-commutative 39.3%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right) \]

   13. Simplified 39.3%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)\right)} \]

   14. Taylor expanded in z around inf 24.5%

       \[\leadsto \color{blue}{2 - \left(\sqrt{x} + \sqrt{y}\right)} \]
   15. Step-by-step derivation

       1. +-commutative 24.5%

          \[\leadsto 2 - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

   16. Simplified 24.5%

       \[\leadsto \color{blue}{2 - \left(\sqrt{y} + \sqrt{x}\right)} \]

   if 0.97999999999999998 < y

   1. Initial program 89.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 89.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 2.1%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 19.5%

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. +-commutative 19.5%

         \[\leadsto 1 + \left(\left(\sqrt{1 + t} + \color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + x}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   7. Simplified 19.5%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + z} + \sqrt{1 + x}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   8. Taylor expanded in y around inf 1.3%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{y}} \]
   9. Step-by-step derivation

      1. mul-1-neg 1.3%

         \[\leadsto \color{blue}{-\sqrt{y}} \]

   10. Simplified 1.3%

       \[\leadsto \color{blue}{-\sqrt{y}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \color{blue}{\sqrt{-\sqrt{y}} \cdot \sqrt{-\sqrt{y}}} \]

       2. sqrt-unprod 6.8%

          \[\leadsto \color{blue}{\sqrt{\left(-\sqrt{y}\right) \cdot \left(-\sqrt{y}\right)}} \]

       3. sqr-neg 6.8%

          \[\leadsto \sqrt{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

       4. add-sqr-sqrt 6.8%

          \[\leadsto \sqrt{\color{blue}{y}} \]

       5. pow1/2 6.8%

          \[\leadsto \color{blue}{{y}^{0.5}} \]

   12. Applied egg-rr 6.8%

       \[\leadsto \color{blue}{{y}^{0.5}} \]

   13. Taylor expanded in y around 0 6.8%

       \[\leadsto \color{blue}{\sqrt{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 16.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.98:\\ \;\;\;\;2 - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 7.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{y} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (sqrt y))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt(y);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt(y)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.sqrt(y);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt(y)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return sqrt(y)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt(y);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[Sqrt[y], $MachinePrecision]

Derivation

1. Initial program 93.4%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 93.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. sub-neg 93.4%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. sub-neg 93.4%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. +-commutative 93.4%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   5. +-commutative 93.4%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   6. +-commutative 93.4%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

3. Simplified 93.4%

   \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 10.4%

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate--l+ 30.0%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   2. +-commutative 30.0%

      \[\leadsto 1 + \left(\left(\sqrt{1 + t} + \color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + x}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

7. Simplified 30.0%

   \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + z} + \sqrt{1 + x}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

8. Taylor expanded in y around inf 1.6%

   \[\leadsto \color{blue}{-1 \cdot \sqrt{y}} \]
9. Step-by-step derivation

   1. mul-1-neg 1.6%

      \[\leadsto \color{blue}{-\sqrt{y}} \]

10. Simplified 1.6%

    \[\leadsto \color{blue}{-\sqrt{y}} \]
11. Step-by-step derivation

    1. add-sqr-sqrt 0.0%

       \[\leadsto \color{blue}{\sqrt{-\sqrt{y}} \cdot \sqrt{-\sqrt{y}}} \]

    2. sqrt-unprod 6.8%

       \[\leadsto \color{blue}{\sqrt{\left(-\sqrt{y}\right) \cdot \left(-\sqrt{y}\right)}} \]

    3. sqr-neg 6.8%

       \[\leadsto \sqrt{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

    4. add-sqr-sqrt 6.8%

       \[\leadsto \sqrt{\color{blue}{y}} \]

    5. pow1/2 6.8%

       \[\leadsto \color{blue}{{y}^{0.5}} \]

12. Applied egg-rr 6.8%

    \[\leadsto \color{blue}{{y}^{0.5}} \]

13. Taylor expanded in y around 0 6.8%

    \[\leadsto \color{blue}{\sqrt{y}} \]
14. Add Preprocessing

Developer Target 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
    (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y)))) + (1.0d0 / (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 (((1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + (1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y)))) + (1.0 / (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(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(1.0 / 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 = (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024143 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (/ 1 (+ (sqrt (+ x 1)) (sqrt x))) (/ 1 (+ (sqrt (+ y 1)) (sqrt y)))) (/ 1 (+ (sqrt (+ z 1)) (sqrt z)))) (- (sqrt (+ t 1)) (sqrt t))))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))