Main:z from

Percentage Accurate: 91.7% → 99.8%

Time: 50.0s

Alternatives: 19

Speedup: 2.0×

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: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{1}{\sqrt{1 + z} + \sqrt{z}}\\ t_2 := \sqrt{y + 1}\\ t_3 := t\_2 - \sqrt{y}\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;t\_3 \leq 0.9998:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + t\_2} + t\_1\right) + \frac{1}{t\_4 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_4 - \sqrt{x}\right) + \left(t\_3 + \left(t\_1 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \end{array} \end{array} \]

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 (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (- t_2 (sqrt y)))
        (t_4 (sqrt (+ 1.0 x))))
   (if (<= t_3 0.9998)
     (+ (+ (/ 1.0 (+ (sqrt y) t_2)) t_1) (/ 1.0 (+ t_4 (sqrt x))))
     (+
      (- t_4 (sqrt x))
      (+ t_3 (+ t_1 (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (sqrt((1.0 + z)) + sqrt(z));
	double t_2 = sqrt((y + 1.0));
	double t_3 = t_2 - sqrt(y);
	double t_4 = sqrt((1.0 + x));
	double tmp;
	if (t_3 <= 0.9998) {
		tmp = ((1.0 / (sqrt(y) + t_2)) + t_1) + (1.0 / (t_4 + sqrt(x)));
	} else {
		tmp = (t_4 - sqrt(x)) + (t_3 + (t_1 + (1.0 / (sqrt((1.0 + t)) + 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) :: tmp
    t_1 = 1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))
    t_2 = sqrt((y + 1.0d0))
    t_3 = t_2 - sqrt(y)
    t_4 = sqrt((1.0d0 + x))
    if (t_3 <= 0.9998d0) then
        tmp = ((1.0d0 / (sqrt(y) + t_2)) + t_1) + (1.0d0 / (t_4 + sqrt(x)))
    else
        tmp = (t_4 - sqrt(x)) + (t_3 + (t_1 + (1.0d0 / (sqrt((1.0d0 + t)) + 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 = 1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z));
	double t_2 = Math.sqrt((y + 1.0));
	double t_3 = t_2 - Math.sqrt(y);
	double t_4 = Math.sqrt((1.0 + x));
	double tmp;
	if (t_3 <= 0.9998) {
		tmp = ((1.0 / (Math.sqrt(y) + t_2)) + t_1) + (1.0 / (t_4 + Math.sqrt(x)));
	} else {
		tmp = (t_4 - Math.sqrt(x)) + (t_3 + (t_1 + (1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t)))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))
	t_2 = math.sqrt((y + 1.0))
	t_3 = t_2 - math.sqrt(y)
	t_4 = math.sqrt((1.0 + x))
	tmp = 0
	if t_3 <= 0.9998:
		tmp = ((1.0 / (math.sqrt(y) + t_2)) + t_1) + (1.0 / (t_4 + math.sqrt(x)))
	else:
		tmp = (t_4 - math.sqrt(x)) + (t_3 + (t_1 + (1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t)))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = Float64(t_2 - sqrt(y))
	t_4 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_3 <= 0.9998)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(y) + t_2)) + t_1) + Float64(1.0 / Float64(t_4 + sqrt(x))));
	else
		tmp = Float64(Float64(t_4 - sqrt(x)) + Float64(t_3 + Float64(t_1 + Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + 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 = 1.0 / (sqrt((1.0 + z)) + sqrt(z));
	t_2 = sqrt((y + 1.0));
	t_3 = t_2 - sqrt(y);
	t_4 = sqrt((1.0 + x));
	tmp = 0.0;
	if (t_3 <= 0.9998)
		tmp = ((1.0 / (sqrt(y) + t_2)) + t_1) + (1.0 / (t_4 + sqrt(x)));
	else
		tmp = (t_4 - sqrt(x)) + (t_3 + (t_1 + (1.0 / (sqrt((1.0 + t)) + 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[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.9998], N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(1.0 / N[(t$95$4 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)) < 0.99980000000000002

   1. Initial program 83.4%

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

      1. associate-+l+ 83.4%

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

      2. associate-+l+ 83.4%

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

      3. +-commutative 83.4%

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

      4. +-commutative 83.4%

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

      5. associate-+l- 65.4%

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

      6. +-commutative 65.4%

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

      7. +-commutative 65.4%

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

   3. Simplified 65.4%

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

      1. flip-- 65.7%

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

      2. div-inv 65.7%

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

      3. add-sqr-sqrt 54.3%

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

      4. +-commutative 54.3%

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

      5. add-sqr-sqrt 65.7%

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

      6. +-commutative 65.7%

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

   6. Applied egg-rr 65.7%

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

      1. associate-*r/ 65.7%

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

      2. *-rgt-identity 65.7%

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

      3. remove-double-neg 65.7%

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

      4. sub-neg 65.7%

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

      5. div-sub 65.5%

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

      6. rem-square-sqrt 54.0%

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

      7. sqr-neg 54.0%

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

      8. div-sub 54.3%

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

      9. sqr-neg 54.3%

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

      10. rem-square-sqrt 65.7%

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

      11. associate--l+ 70.5%

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

      12. +-inverses 70.5%

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

      13. metadata-eval 70.5%

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

      14. sub-neg 70.5%

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

      15. remove-double-neg 70.5%

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

   8. Simplified 70.5%

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

      1. flip-- 70.9%

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

      2. div-inv 70.9%

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

      3. add-sqr-sqrt 45.4%

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

      4. add-sqr-sqrt 71.4%

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

   10. Applied egg-rr 71.4%

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

       1. associate-*r/ 71.4%

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

       2. *-rgt-identity 71.4%

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

       3. remove-double-neg 71.4%

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

       4. sub-neg 71.4%

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

       5. div-sub 70.6%

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

       6. rem-square-sqrt 47.0%

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

       7. sqr-neg 47.0%

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

       8. div-sub 45.4%

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

       9. sqr-neg 45.4%

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

       10. rem-square-sqrt 71.4%

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

       11. associate--l+ 74.4%

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

       12. +-inverses 74.4%

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

       13. metadata-eval 74.4%

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

       14. sub-neg 74.4%

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

       15. remove-double-neg 74.4%

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

   12. Simplified 74.4%

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

   13. Taylor expanded in t around inf 51.3%

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

       1. flip-- 51.3%

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

       2. div-inv 51.3%

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

       3. add-sqr-sqrt 39.9%

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

       4. add-sqr-sqrt 51.3%

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

   15. Applied egg-rr 51.3%

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

       1. associate-*r/ 51.3%

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

       2. *-rgt-identity 51.3%

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

       3. remove-double-neg 51.3%

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

       4. sub-neg 51.3%

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

       5. div-sub 51.3%

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

       6. rem-square-sqrt 40.5%

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

       7. sqr-neg 40.5%

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

       8. div-sub 39.9%

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

       9. sqr-neg 39.9%

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

       10. rem-square-sqrt 51.3%

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

       11. associate--l+ 54.6%

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

       12. +-inverses 54.6%

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

       13. metadata-eval 54.6%

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

       14. sub-neg 54.6%

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

       15. remove-double-neg 54.6%

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

   17. Simplified 54.6%

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

   if 0.99980000000000002 < (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))

   1. Initial program 96.9%

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

      1. associate-+l+ 96.9%

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

      2. associate-+l+ 96.9%

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

      3. +-commutative 96.9%

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

      4. add0 96.9%

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

      5. +-commutative 96.9%

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

      6. add0 96.9%

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

      7. +-commutative 96.9%

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

   3. Simplified 96.9%

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

      1. flip-- 62.7%

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

      2. div-inv 62.7%

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

      3. add-sqr-sqrt 50.4%

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

      4. add-sqr-sqrt 62.7%

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

   6. Applied egg-rr 97.4%

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

      1. associate-*r/ 62.7%

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

      2. *-rgt-identity 62.7%

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

      3. remove-double-neg 62.7%

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

      4. sub-neg 62.7%

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

      5. div-sub 62.7%

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

      6. rem-square-sqrt 50.5%

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

      7. sqr-neg 50.5%

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

      8. div-sub 50.4%

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

      9. sqr-neg 50.4%

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

      10. rem-square-sqrt 62.7%

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

      11. associate--l+ 63.3%

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

      12. +-inverses 63.3%

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

      13. metadata-eval 63.3%

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

      14. sub-neg 63.3%

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

      15. remove-double-neg 63.3%

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

   8. Simplified 97.9%

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

      1. flip-- 97.9%

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

      2. div-inv 97.9%

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

      3. add-sqr-sqrt 71.3%

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

      4. add-sqr-sqrt 97.9%

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

   10. Applied egg-rr 97.9%

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

       1. associate-*r/ 97.9%

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

       2. *-rgt-identity 97.9%

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

       3. remove-double-neg 97.9%

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

       4. sub-neg 97.9%

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

       5. div-sub 97.9%

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

       6. rem-square-sqrt 73.9%

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

       7. sqr-neg 73.9%

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

       8. div-sub 71.3%

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

       9. sqr-neg 71.3%

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

       10. rem-square-sqrt 97.9%

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

       11. associate--l+ 98.7%

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

       12. +-inverses 98.7%

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

       13. metadata-eval 98.7%

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

       14. sub-neg 98.7%

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

       15. remove-double-neg 98.7%

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

   12. Simplified 98.7%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{y + 1} - \sqrt{y} \leq 0.9998:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := t\_1 - \sqrt{y}\\ t_3 := \sqrt{1 + z}\\ t_4 := t\_3 - \sqrt{z}\\ t_5 := \sqrt{1 + x}\\ t_6 := t\_5 - \sqrt{x}\\ \mathbf{if}\;t\_4 + \left(t\_6 + t\_2\right) \leq 2.5:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + t\_1} + \frac{1}{t\_3 + \sqrt{z}}\right) + \frac{1}{t\_5 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_6 + \left(t\_2 + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + t\_4\right)\right)\\ \end{array} \end{array} \]

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 (+ y 1.0)))
        (t_2 (- t_1 (sqrt y)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4 (- t_3 (sqrt z)))
        (t_5 (sqrt (+ 1.0 x)))
        (t_6 (- t_5 (sqrt x))))
   (if (<= (+ t_4 (+ t_6 t_2)) 2.5)
     (+
      (+ (/ 1.0 (+ (sqrt y) t_1)) (/ 1.0 (+ t_3 (sqrt z))))
      (/ 1.0 (+ t_5 (sqrt x))))
     (+ t_6 (+ t_2 (+ (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t))) t_4))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = t_1 - sqrt(y);
	double t_3 = sqrt((1.0 + z));
	double t_4 = t_3 - sqrt(z);
	double t_5 = sqrt((1.0 + x));
	double t_6 = t_5 - sqrt(x);
	double tmp;
	if ((t_4 + (t_6 + t_2)) <= 2.5) {
		tmp = ((1.0 / (sqrt(y) + t_1)) + (1.0 / (t_3 + sqrt(z)))) + (1.0 / (t_5 + sqrt(x)));
	} else {
		tmp = t_6 + (t_2 + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + t_4));
	}
	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) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = t_1 - sqrt(y)
    t_3 = sqrt((1.0d0 + z))
    t_4 = t_3 - sqrt(z)
    t_5 = sqrt((1.0d0 + x))
    t_6 = t_5 - sqrt(x)
    if ((t_4 + (t_6 + t_2)) <= 2.5d0) then
        tmp = ((1.0d0 / (sqrt(y) + t_1)) + (1.0d0 / (t_3 + sqrt(z)))) + (1.0d0 / (t_5 + sqrt(x)))
    else
        tmp = t_6 + (t_2 + ((1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t))) + t_4))
    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((y + 1.0));
	double t_2 = t_1 - Math.sqrt(y);
	double t_3 = Math.sqrt((1.0 + z));
	double t_4 = t_3 - Math.sqrt(z);
	double t_5 = Math.sqrt((1.0 + x));
	double t_6 = t_5 - Math.sqrt(x);
	double tmp;
	if ((t_4 + (t_6 + t_2)) <= 2.5) {
		tmp = ((1.0 / (Math.sqrt(y) + t_1)) + (1.0 / (t_3 + Math.sqrt(z)))) + (1.0 / (t_5 + Math.sqrt(x)));
	} else {
		tmp = t_6 + (t_2 + ((1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))) + t_4));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = t_1 - math.sqrt(y)
	t_3 = math.sqrt((1.0 + z))
	t_4 = t_3 - math.sqrt(z)
	t_5 = math.sqrt((1.0 + x))
	t_6 = t_5 - math.sqrt(x)
	tmp = 0
	if (t_4 + (t_6 + t_2)) <= 2.5:
		tmp = ((1.0 / (math.sqrt(y) + t_1)) + (1.0 / (t_3 + math.sqrt(z)))) + (1.0 / (t_5 + math.sqrt(x)))
	else:
		tmp = t_6 + (t_2 + ((1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))) + t_4))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = Float64(t_1 - sqrt(y))
	t_3 = sqrt(Float64(1.0 + z))
	t_4 = Float64(t_3 - sqrt(z))
	t_5 = sqrt(Float64(1.0 + x))
	t_6 = Float64(t_5 - sqrt(x))
	tmp = 0.0
	if (Float64(t_4 + Float64(t_6 + t_2)) <= 2.5)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) + Float64(1.0 / Float64(t_3 + sqrt(z)))) + Float64(1.0 / Float64(t_5 + sqrt(x))));
	else
		tmp = Float64(t_6 + Float64(t_2 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))) + t_4)));
	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((y + 1.0));
	t_2 = t_1 - sqrt(y);
	t_3 = sqrt((1.0 + z));
	t_4 = t_3 - sqrt(z);
	t_5 = sqrt((1.0 + x));
	t_6 = t_5 - sqrt(x);
	tmp = 0.0;
	if ((t_4 + (t_6 + t_2)) <= 2.5)
		tmp = ((1.0 / (sqrt(y) + t_1)) + (1.0 / (t_3 + sqrt(z)))) + (1.0 / (t_5 + sqrt(x)));
	else
		tmp = t_6 + (t_2 + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + t_4));
	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[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$4 + N[(t$95$6 + t$95$2), $MachinePrecision]), $MachinePrecision], 2.5], N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$3 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$5 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$6 + N[(t$95$2 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))) < 2.5

   1. Initial program 87.4%

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

      1. associate-+l+ 87.4%

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

      2. associate-+l+ 87.4%

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

      3. +-commutative 87.4%

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

      4. +-commutative 87.4%

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

      5. associate-+l- 70.3%

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

      6. +-commutative 70.3%

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

      7. +-commutative 70.3%

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

   3. Simplified 70.3%

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

      1. flip-- 70.6%

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

      2. div-inv 70.6%

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

      3. add-sqr-sqrt 55.2%

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

      4. +-commutative 55.2%

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

      5. add-sqr-sqrt 70.6%

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

      6. +-commutative 70.6%

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

   6. Applied egg-rr 70.6%

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

      1. associate-*r/ 70.6%

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

      2. *-rgt-identity 70.6%

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

      3. remove-double-neg 70.6%

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

      4. sub-neg 70.6%

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

      5. div-sub 70.4%

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

      6. rem-square-sqrt 55.0%

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

      7. sqr-neg 55.0%

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

      8. div-sub 55.2%

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

      9. sqr-neg 55.2%

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

      10. rem-square-sqrt 70.6%

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

      11. associate--l+ 74.0%

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

      12. +-inverses 74.0%

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

      13. metadata-eval 74.0%

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

      14. sub-neg 74.0%

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

      15. remove-double-neg 74.0%

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

   8. Simplified 74.0%

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

      1. flip-- 74.3%

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

      2. div-inv 74.3%

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

      3. add-sqr-sqrt 57.2%

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

      4. add-sqr-sqrt 74.6%

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

   10. Applied egg-rr 74.6%

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

       1. associate-*r/ 74.6%

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

       2. *-rgt-identity 74.6%

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

       3. remove-double-neg 74.6%

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

       4. sub-neg 74.6%

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

       5. div-sub 74.1%

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

       6. rem-square-sqrt 58.3%

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

       7. sqr-neg 58.3%

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

       8. div-sub 57.2%

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

       9. sqr-neg 57.2%

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

       10. rem-square-sqrt 74.6%

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

       11. associate--l+ 76.6%

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

       12. +-inverses 76.6%

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

       13. metadata-eval 76.6%

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

       14. sub-neg 76.6%

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

       15. remove-double-neg 76.6%

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

   12. Simplified 76.6%

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

   13. Taylor expanded in t around inf 54.9%

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

       1. flip-- 54.9%

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

       2. div-inv 54.9%

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

       3. add-sqr-sqrt 41.3%

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

       4. add-sqr-sqrt 54.9%

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

   15. Applied egg-rr 54.9%

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

       1. associate-*r/ 54.9%

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

       2. *-rgt-identity 54.9%

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

       3. remove-double-neg 54.9%

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

       4. sub-neg 54.9%

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

       5. div-sub 54.9%

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

       6. rem-square-sqrt 41.8%

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

       7. sqr-neg 41.8%

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

       8. div-sub 41.3%

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

       9. sqr-neg 41.3%

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

       10. rem-square-sqrt 54.9%

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

       11. associate--l+ 57.4%

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

       12. +-inverses 57.4%

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

       13. metadata-eval 57.4%

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

       14. sub-neg 57.4%

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

       15. remove-double-neg 57.4%

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

   17. Simplified 57.4%

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

   if 2.5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z)))

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. add0 100.0%

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

      5. +-commutative 100.0%

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

      6. add0 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. flip-- 100.0%

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

      2. div-inv 100.0%

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

      3. add-sqr-sqrt 68.0%

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

      4. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. remove-double-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. div-sub 100.0%

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

      6. rem-square-sqrt 70.6%

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

      7. sqr-neg 70.6%

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

      8. div-sub 68.0%

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

      9. sqr-neg 68.0%

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

      10. rem-square-sqrt 100.0%

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

      11. associate--l+ 100.0%

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

      12. +-inverses 100.0%

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

      13. metadata-eval 100.0%

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

      14. sub-neg 100.0%

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

      15. remove-double-neg 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 2.5:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + x}\\ \mathbf{if}\;t\_2 - \sqrt{z} \leq 0.9999999999999994:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + t\_1} + \frac{1}{t\_2 + \sqrt{z}}\right) + \frac{1}{t\_3 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_3 - \sqrt{x}\right) + \left(\left(t\_1 - \sqrt{y}\right) + \left(1 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \end{array} \end{array} \]

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 (+ y 1.0))) (t_2 (sqrt (+ 1.0 z))) (t_3 (sqrt (+ 1.0 x))))
   (if (<= (- t_2 (sqrt z)) 0.9999999999999994)
     (+
      (+ (/ 1.0 (+ (sqrt y) t_1)) (/ 1.0 (+ t_2 (sqrt z))))
      (/ 1.0 (+ t_3 (sqrt x))))
     (+
      (- t_3 (sqrt x))
      (+ (- t_1 (sqrt y)) (+ 1.0 (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + x));
	double tmp;
	if ((t_2 - sqrt(z)) <= 0.9999999999999994) {
		tmp = ((1.0 / (sqrt(y) + t_1)) + (1.0 / (t_2 + sqrt(z)))) + (1.0 / (t_3 + sqrt(x)));
	} else {
		tmp = (t_3 - sqrt(x)) + ((t_1 - sqrt(y)) + (1.0 + (1.0 / (sqrt((1.0 + t)) + 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) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((1.0d0 + z))
    t_3 = sqrt((1.0d0 + x))
    if ((t_2 - sqrt(z)) <= 0.9999999999999994d0) then
        tmp = ((1.0d0 / (sqrt(y) + t_1)) + (1.0d0 / (t_2 + sqrt(z)))) + (1.0d0 / (t_3 + sqrt(x)))
    else
        tmp = (t_3 - sqrt(x)) + ((t_1 - sqrt(y)) + (1.0d0 + (1.0d0 / (sqrt((1.0d0 + t)) + 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((y + 1.0));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = Math.sqrt((1.0 + x));
	double tmp;
	if ((t_2 - Math.sqrt(z)) <= 0.9999999999999994) {
		tmp = ((1.0 / (Math.sqrt(y) + t_1)) + (1.0 / (t_2 + Math.sqrt(z)))) + (1.0 / (t_3 + Math.sqrt(x)));
	} else {
		tmp = (t_3 - Math.sqrt(x)) + ((t_1 - Math.sqrt(y)) + (1.0 + (1.0 / (Math.sqrt((1.0 + t)) + 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((y + 1.0))
	t_2 = math.sqrt((1.0 + z))
	t_3 = math.sqrt((1.0 + x))
	tmp = 0
	if (t_2 - math.sqrt(z)) <= 0.9999999999999994:
		tmp = ((1.0 / (math.sqrt(y) + t_1)) + (1.0 / (t_2 + math.sqrt(z)))) + (1.0 / (t_3 + math.sqrt(x)))
	else:
		tmp = (t_3 - math.sqrt(x)) + ((t_1 - math.sqrt(y)) + (1.0 + (1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t)))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_2 - sqrt(z)) <= 0.9999999999999994)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) + Float64(1.0 / Float64(t_2 + sqrt(z)))) + Float64(1.0 / Float64(t_3 + sqrt(x))));
	else
		tmp = Float64(Float64(t_3 - sqrt(x)) + Float64(Float64(t_1 - sqrt(y)) + Float64(1.0 + Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + 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((y + 1.0));
	t_2 = sqrt((1.0 + z));
	t_3 = sqrt((1.0 + x));
	tmp = 0.0;
	if ((t_2 - sqrt(z)) <= 0.9999999999999994)
		tmp = ((1.0 / (sqrt(y) + t_1)) + (1.0 / (t_2 + sqrt(z)))) + (1.0 / (t_3 + sqrt(x)));
	else
		tmp = (t_3 - sqrt(x)) + ((t_1 - sqrt(y)) + (1.0 + (1.0 / (sqrt((1.0 + t)) + 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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.9999999999999994], N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$3 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z)) < 0.999999999999999445

   1. Initial program 81.6%

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

      1. associate-+l+ 81.6%

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

      2. associate-+l+ 81.6%

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

      3. +-commutative 81.6%

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

      4. +-commutative 81.6%

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

      5. associate-+l- 77.8%

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

      6. +-commutative 77.8%

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

      7. +-commutative 77.8%

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

   3. Simplified 77.8%

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

      1. flip-- 78.0%

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

      2. div-inv 78.0%

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

      3. add-sqr-sqrt 59.8%

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

      4. +-commutative 59.8%

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

      5. add-sqr-sqrt 78.0%

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

      6. +-commutative 78.0%

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

   6. Applied egg-rr 78.0%

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

      1. associate-*r/ 78.0%

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

      2. *-rgt-identity 78.0%

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

      3. remove-double-neg 78.0%

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

      4. sub-neg 78.0%

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

      5. div-sub 77.9%

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

      6. rem-square-sqrt 59.8%

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

      7. sqr-neg 59.8%

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

      8. div-sub 59.8%

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

      9. sqr-neg 59.8%

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

      10. rem-square-sqrt 78.0%

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

      11. associate--l+ 83.4%

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

      12. +-inverses 83.4%

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

      13. metadata-eval 83.4%

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

      14. sub-neg 83.4%

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

      15. remove-double-neg 83.4%

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

   8. Simplified 83.4%

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

      1. flip-- 83.6%

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

      2. div-inv 83.6%

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

      3. add-sqr-sqrt 68.9%

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

      4. add-sqr-sqrt 83.9%

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

   10. Applied egg-rr 83.9%

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

       1. associate-*r/ 83.9%

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

       2. *-rgt-identity 83.9%

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

       3. remove-double-neg 83.9%

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

       4. sub-neg 83.9%

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

       5. div-sub 83.4%

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

       6. rem-square-sqrt 69.3%

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

       7. sqr-neg 69.3%

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

       8. div-sub 68.9%

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

       9. sqr-neg 68.9%

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

       10. rem-square-sqrt 83.9%

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

       11. associate--l+ 87.1%

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

       12. +-inverses 87.1%

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

       13. metadata-eval 87.1%

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

       14. sub-neg 87.1%

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

       15. remove-double-neg 87.1%

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

   12. Simplified 87.1%

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

   13. Taylor expanded in t around inf 52.2%

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

       1. flip-- 52.2%

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

       2. div-inv 52.2%

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

       3. add-sqr-sqrt 28.8%

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

       4. add-sqr-sqrt 52.2%

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

   15. Applied egg-rr 52.2%

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

       1. associate-*r/ 52.2%

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

       2. *-rgt-identity 52.2%

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

       3. remove-double-neg 52.2%

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

       4. sub-neg 52.2%

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

       5. div-sub 52.2%

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

       6. rem-square-sqrt 29.6%

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

       7. sqr-neg 29.6%

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

       8. div-sub 28.8%

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

       9. sqr-neg 28.8%

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

       10. rem-square-sqrt 52.2%

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

       11. associate--l+ 56.5%

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

       12. +-inverses 56.5%

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

       13. metadata-eval 56.5%

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

       14. sub-neg 56.5%

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

       15. remove-double-neg 56.5%

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

   17. Simplified 56.5%

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

   if 0.999999999999999445 < (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))

   1. Initial program 96.6%

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

      1. associate-+l+ 96.6%

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

      2. associate-+l+ 96.6%

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

      3. +-commutative 96.6%

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

      4. +-commutative 96.6%

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

      5. associate-+l- 59.1%

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

      6. +-commutative 59.1%

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

      7. +-commutative 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in z around 0 59.1%

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

      1. associate--l+ 96.6%

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

   7. Simplified 96.6%

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

      1. flip-- 96.8%

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

      2. div-inv 96.8%

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

      3. add-sqr-sqrt 70.7%

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

      4. add-sqr-sqrt 97.2%

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

   9. Applied egg-rr 97.2%

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

       1. associate-*r/ 97.2%

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

       2. *-rgt-identity 97.2%

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

       3. remove-double-neg 97.2%

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

       4. sub-neg 97.2%

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

       5. div-sub 96.6%

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

       6. rem-square-sqrt 71.2%

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

       7. sqr-neg 71.2%

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

       8. div-sub 70.7%

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

       9. sqr-neg 70.7%

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

       10. rem-square-sqrt 97.2%

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

       11. associate--l+ 97.7%

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

       12. +-inverses 97.7%

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

       13. metadata-eval 97.7%

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

       14. sub-neg 97.7%

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

       15. remove-double-neg 97.7%

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

   11. Simplified 97.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 0.9999999999999994:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(1 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := t\_1 - \sqrt{x}\\ t_3 := \sqrt{y + 1}\\ t_4 := \frac{1}{\sqrt{y} + t\_3}\\ \mathbf{if}\;z \leq 2.15 \cdot 10^{-32}:\\ \;\;\;\;t\_2 + \left(\left(t\_3 - \sqrt{y}\right) + \left(1 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+30}:\\ \;\;\;\;\left(t\_4 + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_4 + \frac{1}{t\_1 + \sqrt{x}}\\ \end{array} \end{array} \]

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 (+ 1.0 x)))
        (t_2 (- t_1 (sqrt x)))
        (t_3 (sqrt (+ y 1.0)))
        (t_4 (/ 1.0 (+ (sqrt y) t_3))))
   (if (<= z 2.15e-32)
     (+ t_2 (+ (- t_3 (sqrt y)) (+ 1.0 (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t))))))
     (if (<= z 8.2e+30)
       (+ (+ t_4 (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z)))) t_2)
       (+ t_4 (/ 1.0 (+ t_1 (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 + x));
	double t_2 = t_1 - sqrt(x);
	double t_3 = sqrt((y + 1.0));
	double t_4 = 1.0 / (sqrt(y) + t_3);
	double tmp;
	if (z <= 2.15e-32) {
		tmp = t_2 + ((t_3 - sqrt(y)) + (1.0 + (1.0 / (sqrt((1.0 + t)) + sqrt(t)))));
	} else if (z <= 8.2e+30) {
		tmp = (t_4 + (1.0 / (sqrt((1.0 + z)) + sqrt(z)))) + t_2;
	} else {
		tmp = t_4 + (1.0 / (t_1 + 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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = t_1 - sqrt(x)
    t_3 = sqrt((y + 1.0d0))
    t_4 = 1.0d0 / (sqrt(y) + t_3)
    if (z <= 2.15d-32) then
        tmp = t_2 + ((t_3 - sqrt(y)) + (1.0d0 + (1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t)))))
    else if (z <= 8.2d+30) then
        tmp = (t_4 + (1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z)))) + t_2
    else
        tmp = t_4 + (1.0d0 / (t_1 + 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 + x));
	double t_2 = t_1 - Math.sqrt(x);
	double t_3 = Math.sqrt((y + 1.0));
	double t_4 = 1.0 / (Math.sqrt(y) + t_3);
	double tmp;
	if (z <= 2.15e-32) {
		tmp = t_2 + ((t_3 - Math.sqrt(y)) + (1.0 + (1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t)))));
	} else if (z <= 8.2e+30) {
		tmp = (t_4 + (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z)))) + t_2;
	} else {
		tmp = t_4 + (1.0 / (t_1 + 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 + x))
	t_2 = t_1 - math.sqrt(x)
	t_3 = math.sqrt((y + 1.0))
	t_4 = 1.0 / (math.sqrt(y) + t_3)
	tmp = 0
	if z <= 2.15e-32:
		tmp = t_2 + ((t_3 - math.sqrt(y)) + (1.0 + (1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t)))))
	elif z <= 8.2e+30:
		tmp = (t_4 + (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z)))) + t_2
	else:
		tmp = t_4 + (1.0 / (t_1 + math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = Float64(t_1 - sqrt(x))
	t_3 = sqrt(Float64(y + 1.0))
	t_4 = Float64(1.0 / Float64(sqrt(y) + t_3))
	tmp = 0.0
	if (z <= 2.15e-32)
		tmp = Float64(t_2 + Float64(Float64(t_3 - sqrt(y)) + Float64(1.0 + Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))))));
	elseif (z <= 8.2e+30)
		tmp = Float64(Float64(t_4 + Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))) + t_2);
	else
		tmp = Float64(t_4 + Float64(1.0 / Float64(t_1 + 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 + x));
	t_2 = t_1 - sqrt(x);
	t_3 = sqrt((y + 1.0));
	t_4 = 1.0 / (sqrt(y) + t_3);
	tmp = 0.0;
	if (z <= 2.15e-32)
		tmp = t_2 + ((t_3 - sqrt(y)) + (1.0 + (1.0 / (sqrt((1.0 + t)) + sqrt(t)))));
	elseif (z <= 8.2e+30)
		tmp = (t_4 + (1.0 / (sqrt((1.0 + z)) + sqrt(z)))) + t_2;
	else
		tmp = t_4 + (1.0 / (t_1 + 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[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 2.15e-32], N[(t$95$2 + N[(N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.2e+30], N[(N[(t$95$4 + N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(t$95$4 + N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.14999999999999995e-32

   1. Initial program 96.6%

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

      1. associate-+l+ 96.6%

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

      2. associate-+l+ 96.6%

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

      3. +-commutative 96.6%

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

      4. +-commutative 96.6%

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

      5. associate-+l- 58.8%

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

      6. +-commutative 58.8%

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

      7. +-commutative 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in z around 0 58.8%

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

      1. associate--l+ 96.6%

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

   7. Simplified 96.6%

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

      1. flip-- 96.8%

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

      2. div-inv 96.8%

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

      3. add-sqr-sqrt 70.5%

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

      4. add-sqr-sqrt 97.2%

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

   9. Applied egg-rr 97.2%

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

       1. associate-*r/ 97.2%

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

       2. *-rgt-identity 97.2%

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

       3. remove-double-neg 97.2%

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

       4. sub-neg 97.2%

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

       5. div-sub 96.6%

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

       6. rem-square-sqrt 71.0%

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

       7. sqr-neg 71.0%

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

       8. div-sub 70.5%

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

       9. sqr-neg 70.5%

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

       10. rem-square-sqrt 97.2%

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

       11. associate--l+ 97.7%

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

       12. +-inverses 97.7%

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

       13. metadata-eval 97.7%

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

       14. sub-neg 97.7%

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

       15. remove-double-neg 97.7%

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

   11. Simplified 97.7%

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

   if 2.14999999999999995e-32 < z < 8.20000000000000011e30

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. associate-+l+ 82.8%

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

      3. +-commutative 82.8%

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

      4. add0 82.8%

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

      5. +-commutative 82.8%

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

      6. add0 82.8%

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

      7. +-commutative 82.8%

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

   3. Simplified 82.8%

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

      1. flip-- 41.2%

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

      2. div-inv 41.2%

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

      3. add-sqr-sqrt 38.9%

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

      4. add-sqr-sqrt 41.3%

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

   6. Applied egg-rr 84.2%

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

      1. associate-*r/ 41.3%

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

      2. *-rgt-identity 41.3%

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

      3. remove-double-neg 41.3%

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

      4. sub-neg 41.3%

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

      5. div-sub 41.3%

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

      6. rem-square-sqrt 38.8%

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

      7. sqr-neg 38.8%

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

      8. div-sub 38.9%

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

      9. sqr-neg 38.9%

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

      10. rem-square-sqrt 41.3%

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

      11. associate--l+ 51.5%

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

      12. +-inverses 51.5%

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

      13. metadata-eval 51.5%

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

      14. sub-neg 51.5%

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

      15. remove-double-neg 51.5%

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

   8. Simplified 96.6%

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

   9. Taylor expanded in t around inf 50.6%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{0}\right)\right) \]
   10. Step-by-step derivation

       1. flip-- 68.7%

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

       2. div-inv 68.7%

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

       3. add-sqr-sqrt 54.8%

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

       4. add-sqr-sqrt 68.7%

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

   11. Applied egg-rr 51.0%

       \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\left(\left(1 + y\right) - y\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + 0\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r/ 68.7%

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

       2. *-rgt-identity 68.7%

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

       3. remove-double-neg 68.7%

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

       4. sub-neg 68.7%

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

       5. div-sub 68.7%

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

       6. rem-square-sqrt 57.0%

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

       7. sqr-neg 57.0%

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

       8. div-sub 54.8%

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

       9. sqr-neg 54.8%

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

       10. rem-square-sqrt 68.7%

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

       11. associate--l+ 68.8%

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

       12. +-inverses 68.8%

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

       13. metadata-eval 68.8%

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

       14. sub-neg 68.8%

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

       15. remove-double-neg 68.8%

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

   13. Simplified 51.0%

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

   if 8.20000000000000011e30 < z

   1. Initial program 81.4%

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

      1. associate-+l+ 81.4%

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

      2. associate-+l+ 81.4%

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

      3. +-commutative 81.4%

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

      4. +-commutative 81.4%

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

      5. associate-+l- 81.4%

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

      6. +-commutative 81.4%

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

      7. +-commutative 81.4%

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

   3. Simplified 81.4%

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

      1. flip-- 81.6%

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

      2. div-inv 81.6%

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

      3. add-sqr-sqrt 63.8%

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

      4. +-commutative 63.8%

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

      5. add-sqr-sqrt 81.7%

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

      6. +-commutative 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. associate-*r/ 81.7%

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

      2. *-rgt-identity 81.7%

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

      3. remove-double-neg 81.7%

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

      4. sub-neg 81.7%

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

      5. div-sub 81.5%

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

      6. rem-square-sqrt 63.8%

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

      7. sqr-neg 63.8%

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

      8. div-sub 63.8%

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

      9. sqr-neg 63.8%

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

      10. rem-square-sqrt 81.7%

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

      11. associate--l+ 88.2%

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

      12. +-inverses 88.2%

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

      13. metadata-eval 88.2%

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

      14. sub-neg 88.2%

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

      15. remove-double-neg 88.2%

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

   8. Simplified 88.2%

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

      1. flip-- 88.5%

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

      2. div-inv 88.5%

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

      3. add-sqr-sqrt 73.7%

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

      4. add-sqr-sqrt 88.8%

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

   10. Applied egg-rr 88.8%

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

       1. associate-*r/ 88.8%

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

       2. *-rgt-identity 88.8%

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

       3. remove-double-neg 88.8%

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

       4. sub-neg 88.8%

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

       5. div-sub 88.2%

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

       6. rem-square-sqrt 73.4%

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

       7. sqr-neg 73.4%

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

       8. div-sub 73.7%

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

       9. sqr-neg 73.7%

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

       10. rem-square-sqrt 88.8%

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

       11. associate--l+ 93.0%

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

       12. +-inverses 93.0%

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

       13. metadata-eval 93.0%

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

       14. sub-neg 93.0%

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

       15. remove-double-neg 93.0%

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

   12. Simplified 93.0%

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

   13. Taylor expanded in t around inf 55.3%

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

   14. Taylor expanded in z around inf 55.3%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.15 \cdot 10^{-32}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(1 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+30}:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := t\_1 - \sqrt{y}\\ t_3 := \sqrt{1 + x}\\ t_4 := t\_3 - \sqrt{x}\\ \mathbf{if}\;z \leq 2.15 \cdot 10^{-32}:\\ \;\;\;\;t\_4 + \left(t\_2 + \left(1 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(t\_4 + t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y} + t\_1} + \frac{1}{t\_3 + \sqrt{x}}\\ \end{array} \end{array} \]

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 (+ y 1.0)))
        (t_2 (- t_1 (sqrt y)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (- t_3 (sqrt x))))
   (if (<= z 2.15e-32)
     (+ t_4 (+ t_2 (+ 1.0 (- (sqrt (+ 1.0 t)) (sqrt t)))))
     (if (<= z 1.35e+36)
       (+ (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))) (+ t_4 t_2))
       (+ (/ 1.0 (+ (sqrt y) t_1)) (/ 1.0 (+ t_3 (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = t_1 - sqrt(y);
	double t_3 = sqrt((1.0 + x));
	double t_4 = t_3 - sqrt(x);
	double tmp;
	if (z <= 2.15e-32) {
		tmp = t_4 + (t_2 + (1.0 + (sqrt((1.0 + t)) - sqrt(t))));
	} else if (z <= 1.35e+36) {
		tmp = (1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (t_4 + t_2);
	} else {
		tmp = (1.0 / (sqrt(y) + t_1)) + (1.0 / (t_3 + 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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = t_1 - sqrt(y)
    t_3 = sqrt((1.0d0 + x))
    t_4 = t_3 - sqrt(x)
    if (z <= 2.15d-32) then
        tmp = t_4 + (t_2 + (1.0d0 + (sqrt((1.0d0 + t)) - sqrt(t))))
    else if (z <= 1.35d+36) then
        tmp = (1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + (t_4 + t_2)
    else
        tmp = (1.0d0 / (sqrt(y) + t_1)) + (1.0d0 / (t_3 + 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((y + 1.0));
	double t_2 = t_1 - Math.sqrt(y);
	double t_3 = Math.sqrt((1.0 + x));
	double t_4 = t_3 - Math.sqrt(x);
	double tmp;
	if (z <= 2.15e-32) {
		tmp = t_4 + (t_2 + (1.0 + (Math.sqrt((1.0 + t)) - Math.sqrt(t))));
	} else if (z <= 1.35e+36) {
		tmp = (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + (t_4 + t_2);
	} else {
		tmp = (1.0 / (Math.sqrt(y) + t_1)) + (1.0 / (t_3 + 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((y + 1.0))
	t_2 = t_1 - math.sqrt(y)
	t_3 = math.sqrt((1.0 + x))
	t_4 = t_3 - math.sqrt(x)
	tmp = 0
	if z <= 2.15e-32:
		tmp = t_4 + (t_2 + (1.0 + (math.sqrt((1.0 + t)) - math.sqrt(t))))
	elif z <= 1.35e+36:
		tmp = (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + (t_4 + t_2)
	else:
		tmp = (1.0 / (math.sqrt(y) + t_1)) + (1.0 / (t_3 + math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = Float64(t_1 - sqrt(y))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = Float64(t_3 - sqrt(x))
	tmp = 0.0
	if (z <= 2.15e-32)
		tmp = Float64(t_4 + Float64(t_2 + Float64(1.0 + Float64(sqrt(Float64(1.0 + t)) - sqrt(t)))));
	elseif (z <= 1.35e+36)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + Float64(t_4 + t_2));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) + Float64(1.0 / Float64(t_3 + 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((y + 1.0));
	t_2 = t_1 - sqrt(y);
	t_3 = sqrt((1.0 + x));
	t_4 = t_3 - sqrt(x);
	tmp = 0.0;
	if (z <= 2.15e-32)
		tmp = t_4 + (t_2 + (1.0 + (sqrt((1.0 + t)) - sqrt(t))));
	elseif (z <= 1.35e+36)
		tmp = (1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (t_4 + t_2);
	else
		tmp = (1.0 / (sqrt(y) + t_1)) + (1.0 / (t_3 + 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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 2.15e-32], N[(t$95$4 + N[(t$95$2 + N[(1.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.35e+36], N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$3 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.14999999999999995e-32

   1. Initial program 96.6%

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

      1. associate-+l+ 96.6%

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

      2. associate-+l+ 96.6%

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

      3. +-commutative 96.6%

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

      4. +-commutative 96.6%

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

      5. associate-+l- 58.8%

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

      6. +-commutative 58.8%

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

      7. +-commutative 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in z around 0 58.8%

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

      1. associate--l+ 96.6%

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

   7. Simplified 96.6%

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

   if 2.14999999999999995e-32 < z < 1.35e36

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. associate-+l+ 82.8%

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

      3. +-commutative 82.8%

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

      4. add0 82.8%

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

      5. +-commutative 82.8%

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

      6. add0 82.8%

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

      7. +-commutative 82.8%

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

   3. Simplified 82.8%

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

      1. flip-- 41.2%

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

      2. div-inv 41.2%

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

      3. add-sqr-sqrt 38.9%

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

      4. add-sqr-sqrt 41.3%

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

   6. Applied egg-rr 84.2%

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

      1. associate-*r/ 41.3%

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

      2. *-rgt-identity 41.3%

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

      3. remove-double-neg 41.3%

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

      4. sub-neg 41.3%

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

      5. div-sub 41.3%

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

      6. rem-square-sqrt 38.8%

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

      7. sqr-neg 38.8%

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

      8. div-sub 38.9%

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

      9. sqr-neg 38.9%

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

      10. rem-square-sqrt 41.3%

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

      11. associate--l+ 51.5%

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

      12. +-inverses 51.5%

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

      13. metadata-eval 51.5%

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

      14. sub-neg 51.5%

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

      15. remove-double-neg 51.5%

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

   8. Simplified 96.6%

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

   9. Taylor expanded in t around inf 50.6%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{0}\right)\right) \]
   10. Step-by-step derivation

       1. *-un-lft-identity 50.6%

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

       2. add0 50.6%

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

       3. associate-+r+ 50.6%

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

       4. +-commutative 50.6%

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

   11. Applied egg-rr 50.6%

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

   if 1.35e36 < z

   1. Initial program 81.4%

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

      1. associate-+l+ 81.4%

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

      2. associate-+l+ 81.4%

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

      3. +-commutative 81.4%

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

      4. +-commutative 81.4%

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

      5. associate-+l- 81.4%

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

      6. +-commutative 81.4%

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

      7. +-commutative 81.4%

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

   3. Simplified 81.4%

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

      1. flip-- 81.6%

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

      2. div-inv 81.6%

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

      3. add-sqr-sqrt 63.8%

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

      4. +-commutative 63.8%

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

      5. add-sqr-sqrt 81.7%

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

      6. +-commutative 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. associate-*r/ 81.7%

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

      2. *-rgt-identity 81.7%

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

      3. remove-double-neg 81.7%

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

      4. sub-neg 81.7%

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

      5. div-sub 81.5%

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

      6. rem-square-sqrt 63.8%

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

      7. sqr-neg 63.8%

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

      8. div-sub 63.8%

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

      9. sqr-neg 63.8%

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

      10. rem-square-sqrt 81.7%

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

      11. associate--l+ 88.2%

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

      12. +-inverses 88.2%

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

      13. metadata-eval 88.2%

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

      14. sub-neg 88.2%

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

      15. remove-double-neg 88.2%

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

   8. Simplified 88.2%

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

      1. flip-- 88.5%

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

      2. div-inv 88.5%

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

      3. add-sqr-sqrt 73.7%

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

      4. add-sqr-sqrt 88.8%

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

   10. Applied egg-rr 88.8%

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

       1. associate-*r/ 88.8%

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

       2. *-rgt-identity 88.8%

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

       3. remove-double-neg 88.8%

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

       4. sub-neg 88.8%

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

       5. div-sub 88.2%

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

       6. rem-square-sqrt 73.4%

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

       7. sqr-neg 73.4%

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

       8. div-sub 73.7%

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

       9. sqr-neg 73.7%

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

       10. rem-square-sqrt 88.8%

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

       11. associate--l+ 93.0%

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

       12. +-inverses 93.0%

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

       13. metadata-eval 93.0%

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

       14. sub-neg 93.0%

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

       15. remove-double-neg 93.0%

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

   12. Simplified 93.0%

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

   13. Taylor expanded in t around inf 55.3%

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

   14. Taylor expanded in z around inf 55.3%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.15 \cdot 10^{-32}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(1 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := t\_1 - \sqrt{y}\\ t_3 := \sqrt{1 + x}\\ t_4 := t\_3 - \sqrt{x}\\ \mathbf{if}\;z \leq 5 \cdot 10^{-33}:\\ \;\;\;\;t\_4 + \left(t\_2 + \left(1 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(t\_4 + t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y} + t\_1} + \frac{1}{t\_3 + \sqrt{x}}\\ \end{array} \end{array} \]

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 (+ y 1.0)))
        (t_2 (- t_1 (sqrt y)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (- t_3 (sqrt x))))
   (if (<= z 5e-33)
     (+ t_4 (+ t_2 (+ 1.0 (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t))))))
     (if (<= z 6.5e+36)
       (+ (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))) (+ t_4 t_2))
       (+ (/ 1.0 (+ (sqrt y) t_1)) (/ 1.0 (+ t_3 (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = t_1 - sqrt(y);
	double t_3 = sqrt((1.0 + x));
	double t_4 = t_3 - sqrt(x);
	double tmp;
	if (z <= 5e-33) {
		tmp = t_4 + (t_2 + (1.0 + (1.0 / (sqrt((1.0 + t)) + sqrt(t)))));
	} else if (z <= 6.5e+36) {
		tmp = (1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (t_4 + t_2);
	} else {
		tmp = (1.0 / (sqrt(y) + t_1)) + (1.0 / (t_3 + 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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = t_1 - sqrt(y)
    t_3 = sqrt((1.0d0 + x))
    t_4 = t_3 - sqrt(x)
    if (z <= 5d-33) then
        tmp = t_4 + (t_2 + (1.0d0 + (1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t)))))
    else if (z <= 6.5d+36) then
        tmp = (1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + (t_4 + t_2)
    else
        tmp = (1.0d0 / (sqrt(y) + t_1)) + (1.0d0 / (t_3 + 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((y + 1.0));
	double t_2 = t_1 - Math.sqrt(y);
	double t_3 = Math.sqrt((1.0 + x));
	double t_4 = t_3 - Math.sqrt(x);
	double tmp;
	if (z <= 5e-33) {
		tmp = t_4 + (t_2 + (1.0 + (1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t)))));
	} else if (z <= 6.5e+36) {
		tmp = (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + (t_4 + t_2);
	} else {
		tmp = (1.0 / (Math.sqrt(y) + t_1)) + (1.0 / (t_3 + 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((y + 1.0))
	t_2 = t_1 - math.sqrt(y)
	t_3 = math.sqrt((1.0 + x))
	t_4 = t_3 - math.sqrt(x)
	tmp = 0
	if z <= 5e-33:
		tmp = t_4 + (t_2 + (1.0 + (1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t)))))
	elif z <= 6.5e+36:
		tmp = (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + (t_4 + t_2)
	else:
		tmp = (1.0 / (math.sqrt(y) + t_1)) + (1.0 / (t_3 + math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = Float64(t_1 - sqrt(y))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = Float64(t_3 - sqrt(x))
	tmp = 0.0
	if (z <= 5e-33)
		tmp = Float64(t_4 + Float64(t_2 + Float64(1.0 + Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))))));
	elseif (z <= 6.5e+36)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + Float64(t_4 + t_2));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) + Float64(1.0 / Float64(t_3 + 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((y + 1.0));
	t_2 = t_1 - sqrt(y);
	t_3 = sqrt((1.0 + x));
	t_4 = t_3 - sqrt(x);
	tmp = 0.0;
	if (z <= 5e-33)
		tmp = t_4 + (t_2 + (1.0 + (1.0 / (sqrt((1.0 + t)) + sqrt(t)))));
	elseif (z <= 6.5e+36)
		tmp = (1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (t_4 + t_2);
	else
		tmp = (1.0 / (sqrt(y) + t_1)) + (1.0 / (t_3 + 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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 5e-33], N[(t$95$4 + N[(t$95$2 + N[(1.0 + N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.5e+36], N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$3 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.00000000000000028e-33

   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. associate-+l+ 96.5%

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

      3. +-commutative 96.5%

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

      4. +-commutative 96.5%

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

      5. associate-+l- 59.1%

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

      6. +-commutative 59.1%

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

      7. +-commutative 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in z around 0 59.1%

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

      1. associate--l+ 96.5%

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

   7. Simplified 96.5%

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

      1. flip-- 96.7%

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

      2. div-inv 96.7%

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

      3. add-sqr-sqrt 71.1%

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

      4. add-sqr-sqrt 97.1%

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

   9. Applied egg-rr 97.1%

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

       1. associate-*r/ 97.1%

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

       2. *-rgt-identity 97.1%

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

       3. remove-double-neg 97.1%

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

       4. sub-neg 97.1%

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

       5. div-sub 96.6%

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

       6. rem-square-sqrt 71.5%

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

       7. sqr-neg 71.5%

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

       8. div-sub 71.1%

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

       9. sqr-neg 71.1%

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

       10. rem-square-sqrt 97.1%

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

       11. associate--l+ 97.7%

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

       12. +-inverses 97.7%

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

       13. metadata-eval 97.7%

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

       14. sub-neg 97.7%

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

       15. remove-double-neg 97.7%

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

   11. Simplified 97.7%

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

   if 5.00000000000000028e-33 < z < 6.4999999999999998e36

   1. Initial program 83.3%

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

      1. associate-+l+ 83.3%

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

      2. associate-+l+ 83.3%

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

      3. +-commutative 83.3%

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

      4. add0 83.3%

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

      5. +-commutative 83.3%

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

      6. add0 83.3%

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

      7. +-commutative 83.3%

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

   3. Simplified 83.3%

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

      1. flip-- 43.1%

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

      2. div-inv 43.1%

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

      3. add-sqr-sqrt 40.8%

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

      4. add-sqr-sqrt 43.1%

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

   6. Applied egg-rr 84.7%

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

      1. associate-*r/ 43.1%

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

      2. *-rgt-identity 43.1%

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

      3. remove-double-neg 43.1%

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

      4. sub-neg 43.1%

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

      5. div-sub 43.1%

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

      6. rem-square-sqrt 40.7%

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

      7. sqr-neg 40.7%

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

      8. div-sub 40.8%

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

      9. sqr-neg 40.8%

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

      10. rem-square-sqrt 43.1%

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

      11. associate--l+ 53.0%

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

      12. +-inverses 53.0%

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

      13. metadata-eval 53.0%

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

      14. sub-neg 53.0%

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

      15. remove-double-neg 53.0%

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

   8. Simplified 96.7%

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

   9. Taylor expanded in t around inf 52.1%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{0}\right)\right) \]
   10. Step-by-step derivation

       1. *-un-lft-identity 52.1%

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

       2. add0 52.1%

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

       3. associate-+r+ 52.2%

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

       4. +-commutative 52.2%

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

   11. Applied egg-rr 52.2%

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

   if 6.4999999999999998e36 < z

   1. Initial program 81.4%

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

      1. associate-+l+ 81.4%

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

      2. associate-+l+ 81.4%

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

      3. +-commutative 81.4%

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

      4. +-commutative 81.4%

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

      5. associate-+l- 81.4%

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

      6. +-commutative 81.4%

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

      7. +-commutative 81.4%

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

   3. Simplified 81.4%

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

      1. flip-- 81.6%

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

      2. div-inv 81.6%

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

      3. add-sqr-sqrt 63.8%

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

      4. +-commutative 63.8%

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

      5. add-sqr-sqrt 81.7%

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

      6. +-commutative 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. associate-*r/ 81.7%

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

      2. *-rgt-identity 81.7%

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

      3. remove-double-neg 81.7%

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

      4. sub-neg 81.7%

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

      5. div-sub 81.5%

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

      6. rem-square-sqrt 63.8%

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

      7. sqr-neg 63.8%

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

      8. div-sub 63.8%

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

      9. sqr-neg 63.8%

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

      10. rem-square-sqrt 81.7%

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

      11. associate--l+ 88.2%

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

      12. +-inverses 88.2%

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

      13. metadata-eval 88.2%

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

      14. sub-neg 88.2%

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

      15. remove-double-neg 88.2%

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

   8. Simplified 88.2%

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

      1. flip-- 88.5%

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

      2. div-inv 88.5%

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

      3. add-sqr-sqrt 73.7%

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

      4. add-sqr-sqrt 88.8%

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

   10. Applied egg-rr 88.8%

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

       1. associate-*r/ 88.8%

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

       2. *-rgt-identity 88.8%

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

       3. remove-double-neg 88.8%

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

       4. sub-neg 88.8%

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

       5. div-sub 88.2%

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

       6. rem-square-sqrt 73.4%

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

       7. sqr-neg 73.4%

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

       8. div-sub 73.7%

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

       9. sqr-neg 73.7%

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

       10. rem-square-sqrt 88.8%

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

       11. associate--l+ 93.0%

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

       12. +-inverses 93.0%

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

       13. metadata-eval 93.0%

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

       14. sub-neg 93.0%

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

       15. remove-double-neg 93.0%

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

   12. Simplified 93.0%

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

   13. Taylor expanded in t around inf 55.3%

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

   14. Taylor expanded in z around inf 55.3%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-33}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(1 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;z \leq 2.15 \cdot 10^{-32}:\\ \;\;\;\;\left(t\_2 - \sqrt{x}\right) + \left(\left(t\_1 - \sqrt{y}\right) + \left(1 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\left(1 + t\_1\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y} + t\_1} + \frac{1}{t\_2 + \sqrt{x}}\\ \end{array} \end{array} \]

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 (+ y 1.0))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= z 2.15e-32)
     (+
      (- t_2 (sqrt x))
      (+ (- t_1 (sqrt y)) (+ 1.0 (- (sqrt (+ 1.0 t)) (sqrt t)))))
     (if (<= z 3e+30)
       (+ (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))) (- (+ 1.0 t_1) (sqrt y)))
       (+ (/ 1.0 (+ (sqrt y) t_1)) (/ 1.0 (+ 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((y + 1.0));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (z <= 2.15e-32) {
		tmp = (t_2 - sqrt(x)) + ((t_1 - sqrt(y)) + (1.0 + (sqrt((1.0 + t)) - sqrt(t))));
	} else if (z <= 3e+30) {
		tmp = (1.0 / (sqrt((1.0 + z)) + sqrt(z))) + ((1.0 + t_1) - sqrt(y));
	} else {
		tmp = (1.0 / (sqrt(y) + t_1)) + (1.0 / (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((y + 1.0d0))
    t_2 = sqrt((1.0d0 + x))
    if (z <= 2.15d-32) then
        tmp = (t_2 - sqrt(x)) + ((t_1 - sqrt(y)) + (1.0d0 + (sqrt((1.0d0 + t)) - sqrt(t))))
    else if (z <= 3d+30) then
        tmp = (1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + ((1.0d0 + t_1) - sqrt(y))
    else
        tmp = (1.0d0 / (sqrt(y) + t_1)) + (1.0d0 / (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((y + 1.0));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (z <= 2.15e-32) {
		tmp = (t_2 - Math.sqrt(x)) + ((t_1 - Math.sqrt(y)) + (1.0 + (Math.sqrt((1.0 + t)) - Math.sqrt(t))));
	} else if (z <= 3e+30) {
		tmp = (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + ((1.0 + t_1) - Math.sqrt(y));
	} else {
		tmp = (1.0 / (Math.sqrt(y) + t_1)) + (1.0 / (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((y + 1.0))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if z <= 2.15e-32:
		tmp = (t_2 - math.sqrt(x)) + ((t_1 - math.sqrt(y)) + (1.0 + (math.sqrt((1.0 + t)) - math.sqrt(t))))
	elif z <= 3e+30:
		tmp = (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + ((1.0 + t_1) - math.sqrt(y))
	else:
		tmp = (1.0 / (math.sqrt(y) + t_1)) + (1.0 / (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 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (z <= 2.15e-32)
		tmp = Float64(Float64(t_2 - sqrt(x)) + Float64(Float64(t_1 - sqrt(y)) + Float64(1.0 + Float64(sqrt(Float64(1.0 + t)) - sqrt(t)))));
	elseif (z <= 3e+30)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + Float64(Float64(1.0 + t_1) - sqrt(y)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) + Float64(1.0 / 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((y + 1.0));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (z <= 2.15e-32)
		tmp = (t_2 - sqrt(x)) + ((t_1 - sqrt(y)) + (1.0 + (sqrt((1.0 + t)) - sqrt(t))));
	elseif (z <= 3e+30)
		tmp = (1.0 / (sqrt((1.0 + z)) + sqrt(z))) + ((1.0 + t_1) - sqrt(y));
	else
		tmp = (1.0 / (sqrt(y) + t_1)) + (1.0 / (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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 2.15e-32], N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3e+30], N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + t$95$1), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.14999999999999995e-32

   1. Initial program 96.6%

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

      1. associate-+l+ 96.6%

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

      2. associate-+l+ 96.6%

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

      3. +-commutative 96.6%

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

      4. +-commutative 96.6%

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

      5. associate-+l- 58.8%

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

      6. +-commutative 58.8%

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

      7. +-commutative 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in z around 0 58.8%

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

      1. associate--l+ 96.6%

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

   7. Simplified 96.6%

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

   if 2.14999999999999995e-32 < z < 2.99999999999999978e30

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. associate-+l+ 82.8%

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

      3. +-commutative 82.8%

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

      4. add0 82.8%

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

      5. +-commutative 82.8%

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

      6. add0 82.8%

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

      7. +-commutative 82.8%

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

   3. Simplified 82.8%

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

      1. flip-- 41.2%

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

      2. div-inv 41.2%

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

      3. add-sqr-sqrt 38.9%

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

      4. add-sqr-sqrt 41.3%

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

   6. Applied egg-rr 84.2%

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

      1. associate-*r/ 41.3%

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

      2. *-rgt-identity 41.3%

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

      3. remove-double-neg 41.3%

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

      4. sub-neg 41.3%

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

      5. div-sub 41.3%

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

      6. rem-square-sqrt 38.8%

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

      7. sqr-neg 38.8%

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

      8. div-sub 38.9%

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

      9. sqr-neg 38.9%

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

      10. rem-square-sqrt 41.3%

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

      11. associate--l+ 51.5%

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

      12. +-inverses 51.5%

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

      13. metadata-eval 51.5%

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

      14. sub-neg 51.5%

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

      15. remove-double-neg 51.5%

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

   8. Simplified 96.6%

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

   9. Taylor expanded in t around inf 50.6%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{0}\right)\right) \]
   10. Step-by-step derivation

       1. *-un-lft-identity 50.6%

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

       2. add0 50.6%

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

       3. associate-+r+ 50.6%

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

       4. +-commutative 50.6%

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

   11. Applied egg-rr 50.6%

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

   12. Taylor expanded in x around 0 49.1%

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

   if 2.99999999999999978e30 < z

   1. Initial program 81.4%

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

      1. associate-+l+ 81.4%

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

      2. associate-+l+ 81.4%

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

      3. +-commutative 81.4%

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

      4. +-commutative 81.4%

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

      5. associate-+l- 81.4%

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

      6. +-commutative 81.4%

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

      7. +-commutative 81.4%

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

   3. Simplified 81.4%

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

      1. flip-- 81.6%

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

      2. div-inv 81.6%

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

      3. add-sqr-sqrt 63.8%

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

      4. +-commutative 63.8%

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

      5. add-sqr-sqrt 81.7%

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

      6. +-commutative 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. associate-*r/ 81.7%

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

      2. *-rgt-identity 81.7%

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

      3. remove-double-neg 81.7%

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

      4. sub-neg 81.7%

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

      5. div-sub 81.5%

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

      6. rem-square-sqrt 63.8%

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

      7. sqr-neg 63.8%

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

      8. div-sub 63.8%

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

      9. sqr-neg 63.8%

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

      10. rem-square-sqrt 81.7%

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

      11. associate--l+ 88.2%

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

      12. +-inverses 88.2%

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

      13. metadata-eval 88.2%

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

      14. sub-neg 88.2%

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

      15. remove-double-neg 88.2%

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

   8. Simplified 88.2%

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

      1. flip-- 88.5%

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

      2. div-inv 88.5%

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

      3. add-sqr-sqrt 73.7%

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

      4. add-sqr-sqrt 88.8%

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

   10. Applied egg-rr 88.8%

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

       1. associate-*r/ 88.8%

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

       2. *-rgt-identity 88.8%

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

       3. remove-double-neg 88.8%

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

       4. sub-neg 88.8%

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

       5. div-sub 88.2%

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

       6. rem-square-sqrt 73.4%

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

       7. sqr-neg 73.4%

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

       8. div-sub 73.7%

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

       9. sqr-neg 73.7%

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

       10. rem-square-sqrt 88.8%

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

       11. associate--l+ 93.0%

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

       12. +-inverses 93.0%

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

       13. metadata-eval 93.0%

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

       14. sub-neg 93.0%

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

       15. remove-double-neg 93.0%

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

   12. Simplified 93.0%

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

   13. Taylor expanded in t around inf 55.3%

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

   14. Taylor expanded in z around inf 55.3%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.15 \cdot 10^{-32}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(1 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\left(1 + \sqrt{y + 1}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1} - \sqrt{y}\\ \mathbf{if}\;x \leq 2.8 \cdot 10^{-27}:\\ \;\;\;\;1 + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

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 (+ y 1.0)) (sqrt y))))
   (if (<= x 2.8e-27)
     (+ 1.0 (+ (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))) t_1))
     (+ t_1 (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0)) - sqrt(y);
	double tmp;
	if (x <= 2.8e-27) {
		tmp = 1.0 + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + t_1);
	} else {
		tmp = t_1 + (1.0 / (sqrt((1.0 + x)) + 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((y + 1.0d0)) - sqrt(y)
    if (x <= 2.8d-27) then
        tmp = 1.0d0 + ((1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + t_1)
    else
        tmp = t_1 + (1.0d0 / (sqrt((1.0d0 + x)) + 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((y + 1.0)) - Math.sqrt(y);
	double tmp;
	if (x <= 2.8e-27) {
		tmp = 1.0 + ((1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + t_1);
	} else {
		tmp = t_1 + (1.0 / (Math.sqrt((1.0 + x)) + 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((y + 1.0)) - math.sqrt(y)
	tmp = 0
	if x <= 2.8e-27:
		tmp = 1.0 + ((1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + t_1)
	else:
		tmp = t_1 + (1.0 / (math.sqrt((1.0 + x)) + 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(y + 1.0)) - sqrt(y))
	tmp = 0.0
	if (x <= 2.8e-27)
		tmp = Float64(1.0 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + t_1));
	else
		tmp = Float64(t_1 + Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + 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((y + 1.0)) - sqrt(y);
	tmp = 0.0;
	if (x <= 2.8e-27)
		tmp = 1.0 + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + t_1);
	else
		tmp = t_1 + (1.0 / (sqrt((1.0 + x)) + 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[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.8e-27], N[(1.0 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8e-27

   1. Initial program 96.6%

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

      1. associate-+l+ 96.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. associate-+l+ 96.6%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 96.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      4. add0 96.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + 0\right)}\right)\right) \]

      5. +-commutative 96.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + 0\right)\right)\right) \]

      6. add0 96.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)}\right)\right) \]

      7. +-commutative 96.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right)\right) \]

   3. Simplified 96.6%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 59.2%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}}\right) \]

      2. div-inv 59.2%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\left(\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}}\right) \]

      3. add-sqr-sqrt 48.1%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) \]

      4. add-sqr-sqrt 59.2%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(1 + z\right) - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) \]

   6. Applied egg-rr 96.7%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\color{blue}{\left(\left(1 + z\right) - z\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 59.2%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\left(\left(1 + z\right) - z\right) \cdot 1}{\sqrt{1 + z} + \sqrt{z}}}\right) \]

      2. *-rgt-identity 59.2%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{\left(1 + z\right) - z}}{\sqrt{1 + z} + \sqrt{z}}\right) \]

      3. remove-double-neg 59.2%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\left(-\left(-\sqrt{z}\right)\right)}}\right) \]

      4. sub-neg 59.2%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}}\right) \]

      5. div-sub 59.2%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\left(\frac{1 + z}{\sqrt{1 + z} - \left(-\sqrt{z}\right)} - \frac{z}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right)}\right) \]

      6. rem-square-sqrt 48.1%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1 + z}{\sqrt{1 + z} - \left(-\sqrt{z}\right)} - \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right)\right) \]

      7. sqr-neg 48.1%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1 + z}{\sqrt{1 + z} - \left(-\sqrt{z}\right)} - \frac{\color{blue}{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right)\right) \]

      8. div-sub 48.1%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\left(1 + z\right) - \left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}}\right) \]

      9. sqr-neg 48.1%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\left(1 + z\right) - \color{blue}{\sqrt{z} \cdot \sqrt{z}}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right) \]

      10. rem-square-sqrt 59.2%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\left(1 + z\right) - \color{blue}{z}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right) \]

      11. associate--l+ 60.1%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{1 + \left(z - z\right)}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right) \]

      12. +-inverses 60.1%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right) \]

      13. metadata-eval 60.1%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{1}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right) \]

      14. sub-neg 60.1%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{1 + z} + \left(-\left(-\sqrt{z}\right)\right)}}\right) \]

      15. remove-double-neg 60.1%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) \]

   8. Simplified 97.8%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   9. Taylor expanded in t around inf 59.6%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{0}\right)\right) \]

   10. Taylor expanded in x around 0 32.7%

       \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \sqrt{y}} \]
   11. Step-by-step derivation

       1. associate--l+ 48.2%

          \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \sqrt{y}\right)} \]

       2. +-commutative 48.2%

          \[\leadsto 1 + \left(\color{blue}{\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) \]

       3. +-commutative 48.2%

          \[\leadsto 1 + \left(\left(\frac{1}{\color{blue}{\sqrt{1 + z} + \sqrt{z}}} + \sqrt{1 + y}\right) - \sqrt{y}\right) \]

       4. associate-+r- 59.6%

          \[\leadsto 1 + \color{blue}{\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} \]

       5. +-commutative 59.6%

          \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)} \]

   12. Simplified 59.6%

       \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)} \]

   if 2.8e-27 < x

   1. Initial program 82.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 82.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. associate-+l+ 82.3%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 82.3%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      4. +-commutative 82.3%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

      5. associate-+l- 61.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

      6. +-commutative 61.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

      7. +-commutative 61.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

   3. Simplified 61.8%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 62.2%

         \[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      2. div-inv 62.2%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      3. add-sqr-sqrt 36.7%

         \[\leadsto \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      4. +-commutative 36.7%

         \[\leadsto \left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      5. add-sqr-sqrt 62.2%

         \[\leadsto \left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative 62.2%

         \[\leadsto \left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   6. Applied egg-rr 62.2%

      \[\leadsto \color{blue}{\left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 62.2%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + x\right) - x\right) \cdot 1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      2. *-rgt-identity 62.2%

         \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      3. remove-double-neg 62.2%

         \[\leadsto \frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \color{blue}{\left(-\left(-\sqrt{x}\right)\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      4. sub-neg 62.2%

         \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      5. div-sub 61.9%

         \[\leadsto \color{blue}{\left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      6. rem-square-sqrt 36.4%

         \[\leadsto \left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      7. sqr-neg 36.4%

         \[\leadsto \left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{\color{blue}{\left(-\sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      8. div-sub 36.7%

         \[\leadsto \color{blue}{\frac{\left(1 + x\right) - \left(-\sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      9. sqr-neg 36.7%

         \[\leadsto \frac{\left(1 + x\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      10. rem-square-sqrt 62.2%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      11. associate--l+ 67.9%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      12. +-inverses 67.9%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      13. metadata-eval 67.9%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      14. sub-neg 67.9%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{1 + x} + \left(-\left(-\sqrt{x}\right)\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      15. remove-double-neg 67.9%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \color{blue}{\sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   8. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   9. Taylor expanded in t around inf 19.8%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 27.7%

          \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

       2. +-commutative 27.7%

          \[\leadsto \sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       3. +-commutative 27.7%

          \[\leadsto \sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

   11. Simplified 27.7%

       \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)} \]

   12. Taylor expanded in z around inf 17.6%

       \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}} \]
   13. Step-by-step derivation

       1. +-commutative 17.6%

          \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \sqrt{1 + y}\right)} - \sqrt{y} \]

       2. associate--l+ 25.5%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   14. Simplified 25.5%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-27}:\\ \;\;\;\;1 + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{y + 1} - \sqrt{y}\right) + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 3.3 \cdot 10^{+35}:\\ \;\;\;\;1 + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y} + t\_1} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

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 (+ y 1.0))))
   (if (<= z 3.3e+35)
     (+ 1.0 (+ (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))) (- t_1 (sqrt y))))
     (+ (/ 1.0 (+ (sqrt y) t_1)) (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double tmp;
	if (z <= 3.3e+35) {
		tmp = 1.0 + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (t_1 - sqrt(y)));
	} else {
		tmp = (1.0 / (sqrt(y) + t_1)) + (1.0 / (sqrt((1.0 + x)) + 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((y + 1.0d0))
    if (z <= 3.3d+35) then
        tmp = 1.0d0 + ((1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + (t_1 - sqrt(y)))
    else
        tmp = (1.0d0 / (sqrt(y) + t_1)) + (1.0d0 / (sqrt((1.0d0 + x)) + 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((y + 1.0));
	double tmp;
	if (z <= 3.3e+35) {
		tmp = 1.0 + ((1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + (t_1 - Math.sqrt(y)));
	} else {
		tmp = (1.0 / (Math.sqrt(y) + t_1)) + (1.0 / (Math.sqrt((1.0 + x)) + 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((y + 1.0))
	tmp = 0
	if z <= 3.3e+35:
		tmp = 1.0 + ((1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + (t_1 - math.sqrt(y)))
	else:
		tmp = (1.0 / (math.sqrt(y) + t_1)) + (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 3.3e+35)
		tmp = Float64(1.0 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + Float64(t_1 - sqrt(y))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) + Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + 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((y + 1.0));
	tmp = 0.0;
	if (z <= 3.3e+35)
		tmp = 1.0 + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (t_1 - sqrt(y)));
	else
		tmp = (1.0 / (sqrt(y) + t_1)) + (1.0 / (sqrt((1.0 + x)) + 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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 3.3e+35], N[(1.0 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 3.3000000000000002e35

   1. Initial program 93.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 93.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. associate-+l+ 93.8%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 93.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      4. add0 93.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + 0\right)}\right)\right) \]

      5. +-commutative 93.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + 0\right)\right)\right) \]

      6. add0 93.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)}\right)\right) \]

      7. +-commutative 93.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right)\right) \]

   3. Simplified 93.8%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 56.5%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}}\right) \]

      2. div-inv 56.5%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\left(\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}}\right) \]

      3. add-sqr-sqrt 56.1%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) \]

      4. add-sqr-sqrt 56.6%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(1 + z\right) - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) \]

   6. Applied egg-rr 94.1%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\color{blue}{\left(\left(1 + z\right) - z\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 56.6%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\left(\left(1 + z\right) - z\right) \cdot 1}{\sqrt{1 + z} + \sqrt{z}}}\right) \]

      2. *-rgt-identity 56.6%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{\left(1 + z\right) - z}}{\sqrt{1 + z} + \sqrt{z}}\right) \]

      3. remove-double-neg 56.6%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\left(-\left(-\sqrt{z}\right)\right)}}\right) \]

      4. sub-neg 56.6%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}}\right) \]

      5. div-sub 56.6%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\left(\frac{1 + z}{\sqrt{1 + z} - \left(-\sqrt{z}\right)} - \frac{z}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right)}\right) \]

      6. rem-square-sqrt 56.1%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1 + z}{\sqrt{1 + z} - \left(-\sqrt{z}\right)} - \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right)\right) \]

      7. sqr-neg 56.1%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1 + z}{\sqrt{1 + z} - \left(-\sqrt{z}\right)} - \frac{\color{blue}{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right)\right) \]

      8. div-sub 56.1%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\left(1 + z\right) - \left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}}\right) \]

      9. sqr-neg 56.1%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\left(1 + z\right) - \color{blue}{\sqrt{z} \cdot \sqrt{z}}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right) \]

      10. rem-square-sqrt 56.6%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\left(1 + z\right) - \color{blue}{z}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right) \]

      11. associate--l+ 58.6%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{1 + \left(z - z\right)}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right) \]

      12. +-inverses 58.6%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right) \]

      13. metadata-eval 58.6%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{1}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right) \]

      14. sub-neg 58.6%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{1 + z} + \left(-\left(-\sqrt{z}\right)\right)}}\right) \]

      15. remove-double-neg 58.6%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) \]

   8. Simplified 96.6%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   9. Taylor expanded in t around inf 57.6%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{0}\right)\right) \]

   10. Taylor expanded in x around 0 30.3%

       \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \sqrt{y}} \]
   11. Step-by-step derivation

       1. associate--l+ 46.5%

          \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \sqrt{y}\right)} \]

       2. +-commutative 46.5%

          \[\leadsto 1 + \left(\color{blue}{\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) \]

       3. +-commutative 46.5%

          \[\leadsto 1 + \left(\left(\frac{1}{\color{blue}{\sqrt{1 + z} + \sqrt{z}}} + \sqrt{1 + y}\right) - \sqrt{y}\right) \]

       4. associate-+r- 56.3%

          \[\leadsto 1 + \color{blue}{\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} \]

       5. +-commutative 56.3%

          \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)} \]

   12. Simplified 56.3%

       \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)} \]

   if 3.3000000000000002e35 < z

   1. Initial program 81.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 81.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. associate-+l+ 81.4%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 81.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      4. +-commutative 81.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

      5. associate-+l- 81.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

      6. +-commutative 81.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

      7. +-commutative 81.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

   3. Simplified 81.4%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 81.6%

         \[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      2. div-inv 81.6%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      3. add-sqr-sqrt 63.8%

         \[\leadsto \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      4. +-commutative 63.8%

         \[\leadsto \left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      5. add-sqr-sqrt 81.7%

         \[\leadsto \left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative 81.7%

         \[\leadsto \left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   6. Applied egg-rr 81.7%

      \[\leadsto \color{blue}{\left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 81.7%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + x\right) - x\right) \cdot 1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      2. *-rgt-identity 81.7%

         \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      3. remove-double-neg 81.7%

         \[\leadsto \frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \color{blue}{\left(-\left(-\sqrt{x}\right)\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      4. sub-neg 81.7%

         \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      5. div-sub 81.5%

         \[\leadsto \color{blue}{\left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      6. rem-square-sqrt 63.8%

         \[\leadsto \left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      7. sqr-neg 63.8%

         \[\leadsto \left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{\color{blue}{\left(-\sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      8. div-sub 63.8%

         \[\leadsto \color{blue}{\frac{\left(1 + x\right) - \left(-\sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      9. sqr-neg 63.8%

         \[\leadsto \frac{\left(1 + x\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      10. rem-square-sqrt 81.7%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      11. associate--l+ 88.2%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      12. +-inverses 88.2%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      13. metadata-eval 88.2%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      14. sub-neg 88.2%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{1 + x} + \left(-\left(-\sqrt{x}\right)\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      15. remove-double-neg 88.2%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \color{blue}{\sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   8. Simplified 88.2%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. flip-- 88.5%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      2. div-inv 88.5%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\left(\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      3. add-sqr-sqrt 73.7%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\color{blue}{\left(1 + y\right)} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      4. add-sqr-sqrt 88.8%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\left(1 + y\right) - \color{blue}{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   10. Applied egg-rr 88.8%

       \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\left(\left(1 + y\right) - y\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 88.8%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{\left(\left(1 + y\right) - y\right) \cdot 1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       2. *-rgt-identity 88.8%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{\color{blue}{\left(1 + y\right) - y}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       3. remove-double-neg 88.8%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       4. sub-neg 88.8%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       5. div-sub 88.2%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\left(\frac{1 + y}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} - \frac{y}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}\right)} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       6. rem-square-sqrt 73.4%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\frac{1 + y}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} - \frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       7. sqr-neg 73.4%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\frac{1 + y}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} - \frac{\color{blue}{\left(-\sqrt{y}\right) \cdot \left(-\sqrt{y}\right)}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       8. div-sub 73.7%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{\left(1 + y\right) - \left(-\sqrt{y}\right) \cdot \left(-\sqrt{y}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       9. sqr-neg 73.7%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{\left(1 + y\right) - \color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       10. rem-square-sqrt 88.8%

           \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{\left(1 + y\right) - \color{blue}{y}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       11. associate--l+ 93.0%

           \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{\color{blue}{1 + \left(y - y\right)}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       12. +-inverses 93.0%

           \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       13. metadata-eval 93.0%

           \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{\color{blue}{1}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       14. sub-neg 93.0%

           \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\color{blue}{\sqrt{1 + y} + \left(-\left(-\sqrt{y}\right)\right)}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       15. remove-double-neg 93.0%

           \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   12. Simplified 93.0%

       \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   13. Taylor expanded in t around inf 55.3%

       \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)}\right) \]

   14. Taylor expanded in z around inf 55.3%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{+35}:\\ \;\;\;\;1 + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 3 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\left(1 + t\_1\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y} + t\_1} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

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 (+ y 1.0))))
   (if (<= z 3e+30)
     (+ (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))) (- (+ 1.0 t_1) (sqrt y)))
     (+ (/ 1.0 (+ (sqrt y) t_1)) (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double tmp;
	if (z <= 3e+30) {
		tmp = (1.0 / (sqrt((1.0 + z)) + sqrt(z))) + ((1.0 + t_1) - sqrt(y));
	} else {
		tmp = (1.0 / (sqrt(y) + t_1)) + (1.0 / (sqrt((1.0 + x)) + 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((y + 1.0d0))
    if (z <= 3d+30) then
        tmp = (1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + ((1.0d0 + t_1) - sqrt(y))
    else
        tmp = (1.0d0 / (sqrt(y) + t_1)) + (1.0d0 / (sqrt((1.0d0 + x)) + 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((y + 1.0));
	double tmp;
	if (z <= 3e+30) {
		tmp = (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + ((1.0 + t_1) - Math.sqrt(y));
	} else {
		tmp = (1.0 / (Math.sqrt(y) + t_1)) + (1.0 / (Math.sqrt((1.0 + x)) + 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((y + 1.0))
	tmp = 0
	if z <= 3e+30:
		tmp = (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + ((1.0 + t_1) - math.sqrt(y))
	else:
		tmp = (1.0 / (math.sqrt(y) + t_1)) + (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 3e+30)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + Float64(Float64(1.0 + t_1) - sqrt(y)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) + Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + 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((y + 1.0));
	tmp = 0.0;
	if (z <= 3e+30)
		tmp = (1.0 / (sqrt((1.0 + z)) + sqrt(z))) + ((1.0 + t_1) - sqrt(y));
	else
		tmp = (1.0 / (sqrt(y) + t_1)) + (1.0 / (sqrt((1.0 + x)) + 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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 3e+30], N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + t$95$1), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 2.99999999999999978e30

   1. Initial program 93.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 93.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. associate-+l+ 93.8%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 93.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      4. add0 93.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + 0\right)}\right)\right) \]

      5. +-commutative 93.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + 0\right)\right)\right) \]

      6. add0 93.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)}\right)\right) \]

      7. +-commutative 93.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right)\right) \]

   3. Simplified 93.8%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 56.5%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}}\right) \]

      2. div-inv 56.5%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\left(\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}}\right) \]

      3. add-sqr-sqrt 56.1%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) \]

      4. add-sqr-sqrt 56.6%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(1 + z\right) - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) \]

   6. Applied egg-rr 94.1%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\color{blue}{\left(\left(1 + z\right) - z\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 56.6%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\left(\left(1 + z\right) - z\right) \cdot 1}{\sqrt{1 + z} + \sqrt{z}}}\right) \]

      2. *-rgt-identity 56.6%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{\left(1 + z\right) - z}}{\sqrt{1 + z} + \sqrt{z}}\right) \]

      3. remove-double-neg 56.6%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\left(-\left(-\sqrt{z}\right)\right)}}\right) \]

      4. sub-neg 56.6%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}}\right) \]

      5. div-sub 56.6%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\left(\frac{1 + z}{\sqrt{1 + z} - \left(-\sqrt{z}\right)} - \frac{z}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right)}\right) \]

      6. rem-square-sqrt 56.1%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1 + z}{\sqrt{1 + z} - \left(-\sqrt{z}\right)} - \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right)\right) \]

      7. sqr-neg 56.1%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1 + z}{\sqrt{1 + z} - \left(-\sqrt{z}\right)} - \frac{\color{blue}{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right)\right) \]

      8. div-sub 56.1%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\left(1 + z\right) - \left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}}\right) \]

      9. sqr-neg 56.1%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\left(1 + z\right) - \color{blue}{\sqrt{z} \cdot \sqrt{z}}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right) \]

      10. rem-square-sqrt 56.6%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\left(1 + z\right) - \color{blue}{z}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right) \]

      11. associate--l+ 58.6%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{1 + \left(z - z\right)}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right) \]

      12. +-inverses 58.6%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right) \]

      13. metadata-eval 58.6%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{1}}{\sqrt{1 + z} - \left(-\sqrt{z}\right)}\right) \]

      14. sub-neg 58.6%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{1 + z} + \left(-\left(-\sqrt{z}\right)\right)}}\right) \]

      15. remove-double-neg 58.6%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) \]

   8. Simplified 96.6%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   9. Taylor expanded in t around inf 57.6%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{0}\right)\right) \]
   10. Step-by-step derivation

       1. *-un-lft-identity 57.6%

          \[\leadsto \color{blue}{1 \cdot \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + 0\right)\right)\right)} \]

       2. add0 57.6%

          \[\leadsto 1 \cdot \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}}\right)\right) \]

       3. associate-+r+ 57.6%

          \[\leadsto 1 \cdot \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)} \]

       4. +-commutative 57.6%

          \[\leadsto 1 \cdot \left(\left(\left(\sqrt{\color{blue}{1 + x}} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) \]

   11. Applied egg-rr 57.6%

       \[\leadsto \color{blue}{1 \cdot \left(\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)} \]

   12. Taylor expanded in x around 0 46.2%

       \[\leadsto 1 \cdot \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right)} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) \]

   if 2.99999999999999978e30 < z

   1. Initial program 81.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 81.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. associate-+l+ 81.4%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 81.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      4. +-commutative 81.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

      5. associate-+l- 81.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

      6. +-commutative 81.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

      7. +-commutative 81.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

   3. Simplified 81.4%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 81.6%

         \[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      2. div-inv 81.6%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      3. add-sqr-sqrt 63.8%

         \[\leadsto \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      4. +-commutative 63.8%

         \[\leadsto \left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      5. add-sqr-sqrt 81.7%

         \[\leadsto \left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative 81.7%

         \[\leadsto \left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   6. Applied egg-rr 81.7%

      \[\leadsto \color{blue}{\left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 81.7%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + x\right) - x\right) \cdot 1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      2. *-rgt-identity 81.7%

         \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      3. remove-double-neg 81.7%

         \[\leadsto \frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \color{blue}{\left(-\left(-\sqrt{x}\right)\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      4. sub-neg 81.7%

         \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      5. div-sub 81.5%

         \[\leadsto \color{blue}{\left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      6. rem-square-sqrt 63.8%

         \[\leadsto \left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      7. sqr-neg 63.8%

         \[\leadsto \left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{\color{blue}{\left(-\sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      8. div-sub 63.8%

         \[\leadsto \color{blue}{\frac{\left(1 + x\right) - \left(-\sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      9. sqr-neg 63.8%

         \[\leadsto \frac{\left(1 + x\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      10. rem-square-sqrt 81.7%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      11. associate--l+ 88.2%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      12. +-inverses 88.2%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      13. metadata-eval 88.2%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      14. sub-neg 88.2%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{1 + x} + \left(-\left(-\sqrt{x}\right)\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      15. remove-double-neg 88.2%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \color{blue}{\sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   8. Simplified 88.2%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. flip-- 88.5%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      2. div-inv 88.5%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\left(\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      3. add-sqr-sqrt 73.7%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\color{blue}{\left(1 + y\right)} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      4. add-sqr-sqrt 88.8%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\left(1 + y\right) - \color{blue}{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   10. Applied egg-rr 88.8%

       \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\left(\left(1 + y\right) - y\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 88.8%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{\left(\left(1 + y\right) - y\right) \cdot 1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       2. *-rgt-identity 88.8%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{\color{blue}{\left(1 + y\right) - y}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       3. remove-double-neg 88.8%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       4. sub-neg 88.8%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       5. div-sub 88.2%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\left(\frac{1 + y}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} - \frac{y}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}\right)} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       6. rem-square-sqrt 73.4%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\frac{1 + y}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} - \frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       7. sqr-neg 73.4%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\frac{1 + y}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} - \frac{\color{blue}{\left(-\sqrt{y}\right) \cdot \left(-\sqrt{y}\right)}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       8. div-sub 73.7%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{\left(1 + y\right) - \left(-\sqrt{y}\right) \cdot \left(-\sqrt{y}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       9. sqr-neg 73.7%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{\left(1 + y\right) - \color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       10. rem-square-sqrt 88.8%

           \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{\left(1 + y\right) - \color{blue}{y}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       11. associate--l+ 93.0%

           \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{\color{blue}{1 + \left(y - y\right)}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       12. +-inverses 93.0%

           \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       13. metadata-eval 93.0%

           \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{\color{blue}{1}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       14. sub-neg 93.0%

           \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\color{blue}{\sqrt{1 + y} + \left(-\left(-\sqrt{y}\right)\right)}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

       15. remove-double-neg 93.0%

           \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   12. Simplified 93.0%

       \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   13. Taylor expanded in t around inf 55.3%

       \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)}\right) \]

   14. Taylor expanded in z around inf 55.3%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\left(1 + \sqrt{y + 1}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 4.7 \cdot 10^{-39}:\\ \;\;\;\;2 + \left(\left(t\_1 + \sqrt{1 + t}\right) - \left(\sqrt{x} + \sqrt{t}\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+18}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1 + \sqrt{x}}\\ \end{array} \end{array} \]

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 (+ 1.0 x))))
   (if (<= y 4.7e-39)
     (+ 2.0 (- (+ t_1 (sqrt (+ 1.0 t))) (+ (sqrt x) (sqrt t))))
     (if (<= y 5e+18)
       (+ 1.0 (- (sqrt (+ y 1.0)) (sqrt y)))
       (/ 1.0 (+ t_1 (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 + x));
	double tmp;
	if (y <= 4.7e-39) {
		tmp = 2.0 + ((t_1 + sqrt((1.0 + t))) - (sqrt(x) + sqrt(t)));
	} else if (y <= 5e+18) {
		tmp = 1.0 + (sqrt((y + 1.0)) - sqrt(y));
	} else {
		tmp = 1.0 / (t_1 + 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 + x))
    if (y <= 4.7d-39) then
        tmp = 2.0d0 + ((t_1 + sqrt((1.0d0 + t))) - (sqrt(x) + sqrt(t)))
    else if (y <= 5d+18) then
        tmp = 1.0d0 + (sqrt((y + 1.0d0)) - sqrt(y))
    else
        tmp = 1.0d0 / (t_1 + 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 + x));
	double tmp;
	if (y <= 4.7e-39) {
		tmp = 2.0 + ((t_1 + Math.sqrt((1.0 + t))) - (Math.sqrt(x) + Math.sqrt(t)));
	} else if (y <= 5e+18) {
		tmp = 1.0 + (Math.sqrt((y + 1.0)) - Math.sqrt(y));
	} else {
		tmp = 1.0 / (t_1 + 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 + x))
	tmp = 0
	if y <= 4.7e-39:
		tmp = 2.0 + ((t_1 + math.sqrt((1.0 + t))) - (math.sqrt(x) + math.sqrt(t)))
	elif y <= 5e+18:
		tmp = 1.0 + (math.sqrt((y + 1.0)) - math.sqrt(y))
	else:
		tmp = 1.0 / (t_1 + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 4.7e-39)
		tmp = Float64(2.0 + Float64(Float64(t_1 + sqrt(Float64(1.0 + t))) - Float64(sqrt(x) + sqrt(t))));
	elseif (y <= 5e+18)
		tmp = Float64(1.0 + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)));
	else
		tmp = Float64(1.0 / Float64(t_1 + 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 + x));
	tmp = 0.0;
	if (y <= 4.7e-39)
		tmp = 2.0 + ((t_1 + sqrt((1.0 + t))) - (sqrt(x) + sqrt(t)));
	elseif (y <= 5e+18)
		tmp = 1.0 + (sqrt((y + 1.0)) - sqrt(y));
	else
		tmp = 1.0 / (t_1 + 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[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.7e-39], N[(2.0 + N[(N[(t$95$1 + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+18], N[(1.0 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.7000000000000002e-39

   1. Initial program 96.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 96.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. associate-+l+ 96.6%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 96.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      4. +-commutative 96.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

      5. associate-+l- 70.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

      6. +-commutative 70.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

      7. +-commutative 70.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

   3. Simplified 70.6%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 52.1%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\left(1 + \sqrt{1 + t}\right) - \sqrt{t}\right)}\right) \]
   6. Step-by-step derivation

      1. associate--l+ 62.2%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(1 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)}\right) \]

   7. Simplified 62.2%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(1 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)}\right) \]

   8. Taylor expanded in y around 0 17.8%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \sqrt{1 + x}\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)} \]
   9. Step-by-step derivation

      1. associate--l+ 58.9%

         \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right)\right)} \]

   10. Simplified 58.9%

       \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right)\right)} \]

   if 4.7000000000000002e-39 < y < 5e18

   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. associate-+l+ 88.7%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 88.7%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      4. +-commutative 88.7%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

      5. associate-+l- 76.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

      6. +-commutative 76.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

      7. +-commutative 76.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

   3. Simplified 76.9%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 48.6%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{1 + t} - \sqrt{t}\right)}\right) \]

   6. Taylor expanded in t around inf 18.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 18.3%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 18.3%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   8. Simplified 18.3%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   9. Taylor expanded in x around 0 48.3%

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   10. Step-by-step derivation

       1. associate--l+ 48.3%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   11. Simplified 48.3%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   if 5e18 < y

   1. Initial program 83.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 83.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. associate-+l+ 83.7%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 83.7%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      4. +-commutative 83.7%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

      5. associate-+l- 65.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

      6. +-commutative 65.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

      7. +-commutative 65.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

   3. Simplified 65.4%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 40.6%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{1 + t} - \sqrt{t}\right)}\right) \]

   6. Taylor expanded in t around inf 3.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 18.6%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 18.6%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   8. Simplified 18.6%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   9. Taylor expanded in y around inf 19.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   10. Step-by-step derivation

       1. flip-- 19.2%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 19.6%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. add-sqr-sqrt 19.2%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \]

   11. Applied egg-rr 19.2%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} \]
   12. Step-by-step derivation

       1. associate--l+ 25.2%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} + \sqrt{x}} \]

       2. +-inverses 25.2%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. metadata-eval 25.2%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 25.2%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   13. Simplified 25.2%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-39}:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + x} + \sqrt{1 + t}\right) - \left(\sqrt{x} + \sqrt{t}\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+18}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 88.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ \mathbf{if}\;y \leq 4.7 \cdot 10^{-39}:\\ \;\;\;\;t\_1 + \left(1 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;1 + \left(t\_1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

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 (+ y 1.0))))
   (if (<= y 4.7e-39)
     (+ t_1 (+ 1.0 (- (sqrt (+ 1.0 z)) (+ (sqrt y) (sqrt z)))))
     (if (<= y 1.5e+19)
       (+ 1.0 (- t_1 (sqrt y)))
       (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double tmp;
	if (y <= 4.7e-39) {
		tmp = t_1 + (1.0 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z))));
	} else if (y <= 1.5e+19) {
		tmp = 1.0 + (t_1 - sqrt(y));
	} else {
		tmp = 1.0 / (sqrt((1.0 + x)) + 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((y + 1.0d0))
    if (y <= 4.7d-39) then
        tmp = t_1 + (1.0d0 + (sqrt((1.0d0 + z)) - (sqrt(y) + sqrt(z))))
    else if (y <= 1.5d+19) then
        tmp = 1.0d0 + (t_1 - sqrt(y))
    else
        tmp = 1.0d0 / (sqrt((1.0d0 + x)) + 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((y + 1.0));
	double tmp;
	if (y <= 4.7e-39) {
		tmp = t_1 + (1.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(y) + Math.sqrt(z))));
	} else if (y <= 1.5e+19) {
		tmp = 1.0 + (t_1 - Math.sqrt(y));
	} else {
		tmp = 1.0 / (Math.sqrt((1.0 + x)) + 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((y + 1.0))
	tmp = 0
	if y <= 4.7e-39:
		tmp = t_1 + (1.0 + (math.sqrt((1.0 + z)) - (math.sqrt(y) + math.sqrt(z))))
	elif y <= 1.5e+19:
		tmp = 1.0 + (t_1 - math.sqrt(y))
	else:
		tmp = 1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (y <= 4.7e-39)
		tmp = Float64(t_1 + Float64(1.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(y) + sqrt(z)))));
	elseif (y <= 1.5e+19)
		tmp = Float64(1.0 + Float64(t_1 - sqrt(y)));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + 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((y + 1.0));
	tmp = 0.0;
	if (y <= 4.7e-39)
		tmp = t_1 + (1.0 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z))));
	elseif (y <= 1.5e+19)
		tmp = 1.0 + (t_1 - sqrt(y));
	else
		tmp = 1.0 / (sqrt((1.0 + x)) + 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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.7e-39], N[(t$95$1 + N[(1.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+19], N[(1.0 + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.7000000000000002e-39

   1. Initial program 96.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 96.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. associate-+l+ 96.6%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 96.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      4. +-commutative 96.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

      5. associate-+l- 70.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

      6. +-commutative 70.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

      7. +-commutative 70.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

   3. Simplified 70.6%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 70.9%

         \[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      2. div-inv 70.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      3. add-sqr-sqrt 55.9%

         \[\leadsto \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      4. +-commutative 55.9%

         \[\leadsto \left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      5. add-sqr-sqrt 70.9%

         \[\leadsto \left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative 70.9%

         \[\leadsto \left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   6. Applied egg-rr 70.9%

      \[\leadsto \color{blue}{\left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 70.9%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + x\right) - x\right) \cdot 1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      2. *-rgt-identity 70.9%

         \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      3. remove-double-neg 70.9%

         \[\leadsto \frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \color{blue}{\left(-\left(-\sqrt{x}\right)\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      4. sub-neg 70.9%

         \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      5. div-sub 70.7%

         \[\leadsto \color{blue}{\left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      6. rem-square-sqrt 55.9%

         \[\leadsto \left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      7. sqr-neg 55.9%

         \[\leadsto \left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{\color{blue}{\left(-\sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      8. div-sub 55.9%

         \[\leadsto \color{blue}{\frac{\left(1 + x\right) - \left(-\sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      9. sqr-neg 55.9%

         \[\leadsto \frac{\left(1 + x\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      10. rem-square-sqrt 70.9%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      11. associate--l+ 71.3%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      12. +-inverses 71.3%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      13. metadata-eval 71.3%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      14. sub-neg 71.3%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{1 + x} + \left(-\left(-\sqrt{x}\right)\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      15. remove-double-neg 71.3%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \color{blue}{\sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   8. Simplified 71.3%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   9. Taylor expanded in t around inf 42.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 52.7%

          \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

       2. +-commutative 52.7%

          \[\leadsto \sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       3. +-commutative 52.7%

          \[\leadsto \sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

   11. Simplified 52.7%

       \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)} \]

   12. Taylor expanded in x around 0 49.9%

       \[\leadsto \sqrt{1 + y} + \color{blue}{\left(\left(1 + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   13. Step-by-step derivation

       1. associate--l+ 62.2%

          \[\leadsto \sqrt{1 + y} + \color{blue}{\left(1 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   14. Simplified 62.2%

       \[\leadsto \sqrt{1 + y} + \color{blue}{\left(1 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   if 4.7000000000000002e-39 < y < 1.5e19

   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. associate-+l+ 88.7%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 88.7%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      4. +-commutative 88.7%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

      5. associate-+l- 76.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

      6. +-commutative 76.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

      7. +-commutative 76.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

   3. Simplified 76.9%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 48.6%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{1 + t} - \sqrt{t}\right)}\right) \]

   6. Taylor expanded in t around inf 18.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 18.3%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 18.3%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   8. Simplified 18.3%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   9. Taylor expanded in x around 0 48.3%

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   10. Step-by-step derivation

       1. associate--l+ 48.3%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   11. Simplified 48.3%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   if 1.5e19 < y

   1. Initial program 83.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 83.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. associate-+l+ 83.7%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 83.7%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      4. +-commutative 83.7%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

      5. associate-+l- 65.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

      6. +-commutative 65.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

      7. +-commutative 65.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

   3. Simplified 65.4%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 40.6%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{1 + t} - \sqrt{t}\right)}\right) \]

   6. Taylor expanded in t around inf 3.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 18.6%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 18.6%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   8. Simplified 18.6%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   9. Taylor expanded in y around inf 19.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   10. Step-by-step derivation

       1. flip-- 19.2%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 19.6%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. add-sqr-sqrt 19.2%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \]

   11. Applied egg-rr 19.2%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} \]
   12. Step-by-step derivation

       1. associate--l+ 25.2%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} + \sqrt{x}} \]

       2. +-inverses 25.2%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. metadata-eval 25.2%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 25.2%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   13. Simplified 25.2%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{y + 1} + \left(1 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 2.55 \cdot 10^{+15}:\\ \;\;\;\;t\_1 + \left(1 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \sqrt{y}\right) + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

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 (+ y 1.0))))
   (if (<= z 2.55e+15)
     (+ t_1 (+ 1.0 (- (sqrt (+ 1.0 z)) (+ (sqrt y) (sqrt z)))))
     (+ (- t_1 (sqrt y)) (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double tmp;
	if (z <= 2.55e+15) {
		tmp = t_1 + (1.0 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z))));
	} else {
		tmp = (t_1 - sqrt(y)) + (1.0 / (sqrt((1.0 + x)) + 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((y + 1.0d0))
    if (z <= 2.55d+15) then
        tmp = t_1 + (1.0d0 + (sqrt((1.0d0 + z)) - (sqrt(y) + sqrt(z))))
    else
        tmp = (t_1 - sqrt(y)) + (1.0d0 / (sqrt((1.0d0 + x)) + 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((y + 1.0));
	double tmp;
	if (z <= 2.55e+15) {
		tmp = t_1 + (1.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(y) + Math.sqrt(z))));
	} else {
		tmp = (t_1 - Math.sqrt(y)) + (1.0 / (Math.sqrt((1.0 + x)) + 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((y + 1.0))
	tmp = 0
	if z <= 2.55e+15:
		tmp = t_1 + (1.0 + (math.sqrt((1.0 + z)) - (math.sqrt(y) + math.sqrt(z))))
	else:
		tmp = (t_1 - math.sqrt(y)) + (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 2.55e+15)
		tmp = Float64(t_1 + Float64(1.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(y) + sqrt(z)))));
	else
		tmp = Float64(Float64(t_1 - sqrt(y)) + Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + 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((y + 1.0));
	tmp = 0.0;
	if (z <= 2.55e+15)
		tmp = t_1 + (1.0 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z))));
	else
		tmp = (t_1 - sqrt(y)) + (1.0 / (sqrt((1.0 + x)) + 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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 2.55e+15], N[(t$95$1 + N[(1.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 2.55e15

   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. associate-+l+ 96.5%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 96.5%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      4. +-commutative 96.5%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

      5. associate-+l- 59.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

      6. +-commutative 59.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

      7. +-commutative 59.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

   3. Simplified 59.8%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 60.1%

         \[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      2. div-inv 60.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      3. add-sqr-sqrt 51.0%

         \[\leadsto \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      4. +-commutative 51.0%

         \[\leadsto \left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      5. add-sqr-sqrt 60.1%

         \[\leadsto \left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative 60.1%

         \[\leadsto \left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   6. Applied egg-rr 60.1%

      \[\leadsto \color{blue}{\left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 60.1%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + x\right) - x\right) \cdot 1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      2. *-rgt-identity 60.1%

         \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      3. remove-double-neg 60.1%

         \[\leadsto \frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \color{blue}{\left(-\left(-\sqrt{x}\right)\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      4. sub-neg 60.1%

         \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      5. div-sub 59.8%

         \[\leadsto \color{blue}{\left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      6. rem-square-sqrt 50.7%

         \[\leadsto \left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      7. sqr-neg 50.7%

         \[\leadsto \left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{\color{blue}{\left(-\sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      8. div-sub 51.0%

         \[\leadsto \color{blue}{\frac{\left(1 + x\right) - \left(-\sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      9. sqr-neg 51.0%

         \[\leadsto \frac{\left(1 + x\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      10. rem-square-sqrt 60.1%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      11. associate--l+ 60.6%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      12. +-inverses 60.6%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      13. metadata-eval 60.6%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      14. sub-neg 60.6%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{1 + x} + \left(-\left(-\sqrt{x}\right)\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      15. remove-double-neg 60.6%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \color{blue}{\sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   8. Simplified 60.6%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   9. Taylor expanded in t around inf 32.4%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 32.4%

          \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

       2. +-commutative 32.4%

          \[\leadsto \sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       3. +-commutative 32.4%

          \[\leadsto \sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

   11. Simplified 32.4%

       \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)} \]

   12. Taylor expanded in x around 0 29.6%

       \[\leadsto \sqrt{1 + y} + \color{blue}{\left(\left(1 + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   13. Step-by-step derivation

       1. associate--l+ 29.6%

          \[\leadsto \sqrt{1 + y} + \color{blue}{\left(1 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   14. Simplified 29.6%

       \[\leadsto \sqrt{1 + y} + \color{blue}{\left(1 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   if 2.55e15 < z

   1. Initial program 79.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 79.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. associate-+l+ 79.6%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 79.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      4. +-commutative 79.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

      5. associate-+l- 79.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

      6. +-commutative 79.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

      7. +-commutative 79.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

   3. Simplified 79.6%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 79.8%

         \[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      2. div-inv 79.8%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      3. add-sqr-sqrt 60.8%

         \[\leadsto \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      4. +-commutative 60.8%

         \[\leadsto \left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      5. add-sqr-sqrt 79.8%

         \[\leadsto \left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative 79.8%

         \[\leadsto \left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   6. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{\left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 79.8%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + x\right) - x\right) \cdot 1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      2. *-rgt-identity 79.8%

         \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      3. remove-double-neg 79.8%

         \[\leadsto \frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \color{blue}{\left(-\left(-\sqrt{x}\right)\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      4. sub-neg 79.8%

         \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      5. div-sub 79.7%

         \[\leadsto \color{blue}{\left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      6. rem-square-sqrt 60.8%

         \[\leadsto \left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      7. sqr-neg 60.8%

         \[\leadsto \left(\frac{1 + x}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} - \frac{\color{blue}{\left(-\sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      8. div-sub 60.8%

         \[\leadsto \color{blue}{\frac{\left(1 + x\right) - \left(-\sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}{\sqrt{1 + x} - \left(-\sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      9. sqr-neg 60.8%

         \[\leadsto \frac{\left(1 + x\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      10. rem-square-sqrt 79.8%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      11. associate--l+ 85.9%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      12. +-inverses 85.9%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      13. metadata-eval 85.9%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} - \left(-\sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      14. sub-neg 85.9%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{1 + x} + \left(-\left(-\sqrt{x}\right)\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

      15. remove-double-neg 85.9%

          \[\leadsto \frac{1}{\sqrt{1 + x} + \color{blue}{\sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   8. Simplified 85.9%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right) \]

   9. Taylor expanded in t around inf 5.8%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 18.2%

          \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

       2. +-commutative 18.2%

          \[\leadsto \sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) \]

       3. +-commutative 18.2%

          \[\leadsto \sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

   11. Simplified 18.2%

       \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)} \]

   12. Taylor expanded in z around inf 28.7%

       \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}} \]
   13. Step-by-step derivation

       1. +-commutative 28.7%

          \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \sqrt{1 + y}\right)} - \sqrt{y} \]

       2. associate--l+ 48.7%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   14. Simplified 48.7%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.55 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{y + 1} + \left(1 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{y + 1} - \sqrt{y}\right) + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;y \leq 135:\\ \;\;\;\;1 + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 x)) (sqrt x))))
   (if (<= y 135.0) (+ 1.0 t_1) t_1)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (y <= 135.0) {
		tmp = 1.0 + t_1;
	} else {
		tmp = 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((1.0d0 + x)) - sqrt(x)
    if (y <= 135.0d0) then
        tmp = 1.0d0 + t_1
    else
        tmp = 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((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (y <= 135.0) {
		tmp = 1.0 + t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if y <= 135.0:
		tmp = 1.0 + t_1
	else:
		tmp = t_1
	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 + x)) - sqrt(x))
	tmp = 0.0
	if (y <= 135.0)
		tmp = Float64(1.0 + t_1);
	else
		tmp = 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((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (y <= 135.0)
		tmp = 1.0 + t_1;
	else
		tmp = 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[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 135.0], N[(1.0 + t$95$1), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if y < 135

   1. Initial program 96.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 96.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. associate-+l+ 96.6%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 96.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      4. +-commutative 96.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

      5. associate-+l- 72.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

      6. +-commutative 72.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

      7. +-commutative 72.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

   3. Simplified 72.9%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 52.0%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{1 + t} - \sqrt{t}\right)}\right) \]

   6. Taylor expanded in t around inf 24.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 24.1%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 24.1%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   8. Simplified 24.1%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   9. Taylor expanded in y around 0 23.5%

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + x}\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. associate--l+ 35.4%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

   11. Simplified 35.4%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

   if 135 < y

   1. Initial program 83.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 83.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. associate-+l+ 83.2%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 83.2%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      4. +-commutative 83.2%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

      5. associate-+l- 65.2%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

      6. +-commutative 65.2%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

      7. +-commutative 65.2%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

   3. Simplified 65.2%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 40.0%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{1 + t} - \sqrt{t}\right)}\right) \]

   6. Taylor expanded in t around inf 4.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 18.9%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 18.9%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   8. Simplified 18.9%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   9. Taylor expanded in y around inf 19.1%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 135:\\ \;\;\;\;1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-27}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

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 (<= x 3.2e-27)
   (+ 1.0 (- (sqrt (+ y 1.0)) (sqrt y)))
   (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 3.2e-27) {
		tmp = 1.0 + (sqrt((y + 1.0)) - sqrt(y));
	} else {
		tmp = 1.0 / (sqrt((1.0 + x)) + 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 (x <= 3.2d-27) then
        tmp = 1.0d0 + (sqrt((y + 1.0d0)) - sqrt(y))
    else
        tmp = 1.0d0 / (sqrt((1.0d0 + x)) + 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 (x <= 3.2e-27) {
		tmp = 1.0 + (Math.sqrt((y + 1.0)) - Math.sqrt(y));
	} else {
		tmp = 1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 3.2e-27:
		tmp = 1.0 + (math.sqrt((y + 1.0)) - math.sqrt(y))
	else:
		tmp = 1.0 / (math.sqrt((1.0 + x)) + 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 (x <= 3.2e-27)
		tmp = Float64(1.0 + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + 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 (x <= 3.2e-27)
		tmp = 1.0 + (sqrt((y + 1.0)) - sqrt(y));
	else
		tmp = 1.0 / (sqrt((1.0 + x)) + 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[x, 3.2e-27], N[(1.0 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.19999999999999991e-27

   1. Initial program 96.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 96.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. associate-+l+ 96.6%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 96.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      4. +-commutative 96.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

      5. associate-+l- 76.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

      6. +-commutative 76.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

      7. +-commutative 76.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

   3. Simplified 76.1%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 51.0%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{1 + t} - \sqrt{t}\right)}\right) \]

   6. Taylor expanded in t around inf 20.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 35.7%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 35.7%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   8. Simplified 35.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   9. Taylor expanded in x around 0 20.2%

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   10. Step-by-step derivation

       1. associate--l+ 35.7%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   11. Simplified 35.7%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   if 3.19999999999999991e-27 < x

   1. Initial program 82.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 82.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. associate-+l+ 82.3%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. +-commutative 82.3%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      4. +-commutative 82.3%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

      5. associate-+l- 61.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

      6. +-commutative 61.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

      7. +-commutative 61.8%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

   3. Simplified 61.8%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 40.1%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{1 + t} - \sqrt{t}\right)}\right) \]

   6. Taylor expanded in t around inf 6.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 8.2%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 8.2%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   8. Simplified 8.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   9. Taylor expanded in y around inf 6.0%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   10. Step-by-step derivation

       1. flip-- 6.1%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 6.7%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. add-sqr-sqrt 6.1%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \]

   11. Applied egg-rr 6.1%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} \]
   12. Step-by-step derivation

       1. associate--l+ 13.3%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} + \sqrt{x}} \]

       2. +-inverses 13.3%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. metadata-eval 13.3%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 13.3%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   13. Simplified 13.3%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-27}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + \left(\sqrt{y + 1} - \sqrt{y}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (+ 1.0 (- (sqrt (+ y 1.0)) (sqrt y))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + (sqrt((y + 1.0)) - 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 = 1.0d0 + (sqrt((y + 1.0d0)) - 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 1.0 + (Math.sqrt((y + 1.0)) - Math.sqrt(y));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 + (math.sqrt((y + 1.0)) - math.sqrt(y))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 + (sqrt((y + 1.0)) - 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[(1.0 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. associate-+l+ 89.0%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   3. +-commutative 89.0%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

   4. +-commutative 89.0%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

   5. associate-+l- 68.5%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

   6. +-commutative 68.5%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

   7. +-commutative 68.5%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

3. Simplified 68.5%

   \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 45.3%

   \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{1 + t} - \sqrt{t}\right)}\right) \]

6. Taylor expanded in t around inf 13.1%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
7. Step-by-step derivation

   1. associate--l+ 21.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   2. +-commutative 21.2%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

8. Simplified 21.2%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

9. Taylor expanded in x around 0 25.2%

   \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
10. Step-by-step derivation

    1. associate--l+ 44.0%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

11. Simplified 44.0%

    \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

12. Final simplification 44.0%

    \[\leadsto 1 + \left(\sqrt{y + 1} - \sqrt{y}\right) \]
13. Add Preprocessing

Alternative 17: 35.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{1 + x} - \sqrt{x} \end{array} \]

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 (+ 1.0 x)) (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt((1.0 + x)) - sqrt(x);
}

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((1.0d0 + x)) - sqrt(x)
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((1.0 + x)) - Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt((1.0 + x)) - math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt((1.0 + x)) - sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

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. associate-+l+ 89.0%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   3. +-commutative 89.0%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

   4. +-commutative 89.0%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

   5. associate-+l- 68.5%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

   6. +-commutative 68.5%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

   7. +-commutative 68.5%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

3. Simplified 68.5%

   \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 45.3%

   \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{1 + t} - \sqrt{t}\right)}\right) \]

6. Taylor expanded in t around inf 13.1%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
7. Step-by-step derivation

   1. associate--l+ 21.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   2. +-commutative 21.2%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

8. Simplified 21.2%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

9. Taylor expanded in y around inf 15.5%

   \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

10. Final simplification 15.5%

    \[\leadsto \sqrt{1 + x} - \sqrt{x} \]
11. Add Preprocessing

Alternative 18: 34.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(1 + x \cdot 0.5\right) - \sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- (+ 1.0 (* x 0.5)) (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (1.0 + (x * 0.5)) - sqrt(x);
}

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 = (1.0d0 + (x * 0.5d0)) - sqrt(x)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return (1.0 + (x * 0.5)) - Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (1.0 + (x * 0.5)) - math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(1.0 + Float64(x * 0.5)) - sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (1.0 + (x * 0.5)) - sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

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. associate-+l+ 89.0%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   3. +-commutative 89.0%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

   4. +-commutative 89.0%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

   5. associate-+l- 68.5%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

   6. +-commutative 68.5%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

   7. +-commutative 68.5%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

3. Simplified 68.5%

   \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 45.3%

   \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{1 + t} - \sqrt{t}\right)}\right) \]

6. Taylor expanded in t around inf 13.1%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
7. Step-by-step derivation

   1. associate--l+ 21.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   2. +-commutative 21.2%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

8. Simplified 21.2%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

9. Taylor expanded in y around inf 15.5%

   \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

10. Taylor expanded in x around 0 15.4%

    \[\leadsto \color{blue}{\left(1 + 0.5 \cdot x\right)} - \sqrt{x} \]
11. Step-by-step derivation

    1. *-commutative 15.4%

       \[\leadsto \left(1 + \color{blue}{x \cdot 0.5}\right) - \sqrt{x} \]

12. Simplified 15.4%

    \[\leadsto \color{blue}{\left(1 + x \cdot 0.5\right)} - \sqrt{x} \]

13. Final simplification 15.4%

    \[\leadsto \left(1 + x \cdot 0.5\right) - \sqrt{x} \]
14. Add Preprocessing

Alternative 19: 34.4% accurate, 823.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 1.0)

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0;
}

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 = 1.0d0
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return 1.0
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := 1.0

Derivation

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. associate-+l+ 89.0%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   3. +-commutative 89.0%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

   4. +-commutative 89.0%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) \]

   5. associate-+l- 68.5%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)}\right) \]

   6. +-commutative 68.5%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + t}} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right) \]

   7. +-commutative 68.5%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right)\right)\right) \]

3. Simplified 68.5%

   \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 45.3%

   \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{1 + t} - \sqrt{t}\right)}\right) \]

6. Taylor expanded in t around inf 13.1%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
7. Step-by-step derivation

   1. associate--l+ 21.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   2. +-commutative 21.2%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

8. Simplified 21.2%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

9. Taylor expanded in y around inf 15.5%

   \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

10. Taylor expanded in x around 0 32.7%

    \[\leadsto \color{blue}{1} \]

11. Final simplification 32.7%

    \[\leadsto 1 \]
12. Add Preprocessing

Developer target: 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 2024046 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (+ (+ (+ (/ 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)))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))