Main:z from

Percentage Accurate: 91.7% → 99.8%

Time: 38.1s

Alternatives: 20

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := t_2 - \sqrt{z}\\ t_4 := \sqrt{1 + y}\\ \mathbf{if}\;t_3 \leq 0.002:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + t_1} + \frac{1}{t_4 + \sqrt{y}}\right) + \frac{1}{\sqrt{z} + t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - \sqrt{x}\right) + \left(\left(t_4 - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + t_3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- t_2 (sqrt z)))
        (t_4 (sqrt (+ 1.0 y))))
   (if (<= t_3 0.002)
     (+
      (+ (/ 1.0 (+ (sqrt x) t_1)) (/ 1.0 (+ t_4 (sqrt y))))
      (/ 1.0 (+ (sqrt z) t_2)))
     (+
      (- t_1 (sqrt x))
      (+ (- t_4 (sqrt y)) (+ (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t))) t_3))))))

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 = sqrt((1.0 + z));
	double t_3 = t_2 - sqrt(z);
	double t_4 = sqrt((1.0 + y));
	double tmp;
	if (t_3 <= 0.002) {
		tmp = ((1.0 / (sqrt(x) + t_1)) + (1.0 / (t_4 + sqrt(y)))) + (1.0 / (sqrt(z) + t_2));
	} else {
		tmp = (t_1 - sqrt(x)) + ((t_4 - sqrt(y)) + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + t_3));
	}
	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 = sqrt((1.0d0 + z))
    t_3 = t_2 - sqrt(z)
    t_4 = sqrt((1.0d0 + y))
    if (t_3 <= 0.002d0) then
        tmp = ((1.0d0 / (sqrt(x) + t_1)) + (1.0d0 / (t_4 + sqrt(y)))) + (1.0d0 / (sqrt(z) + t_2))
    else
        tmp = (t_1 - sqrt(x)) + ((t_4 - sqrt(y)) + ((1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t))) + t_3))
    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 = Math.sqrt((1.0 + z));
	double t_3 = t_2 - Math.sqrt(z);
	double t_4 = Math.sqrt((1.0 + y));
	double tmp;
	if (t_3 <= 0.002) {
		tmp = ((1.0 / (Math.sqrt(x) + t_1)) + (1.0 / (t_4 + Math.sqrt(y)))) + (1.0 / (Math.sqrt(z) + t_2));
	} else {
		tmp = (t_1 - Math.sqrt(x)) + ((t_4 - Math.sqrt(y)) + ((1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))) + t_3));
	}
	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 = math.sqrt((1.0 + z))
	t_3 = t_2 - math.sqrt(z)
	t_4 = math.sqrt((1.0 + y))
	tmp = 0
	if t_3 <= 0.002:
		tmp = ((1.0 / (math.sqrt(x) + t_1)) + (1.0 / (t_4 + math.sqrt(y)))) + (1.0 / (math.sqrt(z) + t_2))
	else:
		tmp = (t_1 - math.sqrt(x)) + ((t_4 - math.sqrt(y)) + ((1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))) + t_3))
	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 = sqrt(Float64(1.0 + z))
	t_3 = Float64(t_2 - sqrt(z))
	t_4 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (t_3 <= 0.002)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_1)) + Float64(1.0 / Float64(t_4 + sqrt(y)))) + Float64(1.0 / Float64(sqrt(z) + t_2)));
	else
		tmp = Float64(Float64(t_1 - sqrt(x)) + Float64(Float64(t_4 - sqrt(y)) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))) + t_3)));
	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 = sqrt((1.0 + z));
	t_3 = t_2 - sqrt(z);
	t_4 = sqrt((1.0 + y));
	tmp = 0.0;
	if (t_3 <= 0.002)
		tmp = ((1.0 / (sqrt(x) + t_1)) + (1.0 / (t_4 + sqrt(y)))) + (1.0 / (sqrt(z) + t_2));
	else
		tmp = (t_1 - sqrt(x)) + ((t_4 - sqrt(y)) + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + t_3));
	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[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.002], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$4 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z)) < 2e-3

   1. Initial program 84.7%

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

      1. associate-+l+ 84.7%

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

      2. associate-+l+ 84.7%

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

      3. +-commutative 84.7%

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

      4. +-commutative 84.7%

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

      5. +-commutative 84.7%

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

   3. Simplified 84.7%

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

      1. flip-- 84.9%

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

      2. add-sqr-sqrt 71.0%

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

      3. add-sqr-sqrt 85.3%

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

   5. Applied egg-rr 85.3%

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

      1. +-commutative 85.3%

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

      2. associate--l+ 87.6%

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

      3. +-inverses 87.6%

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

      4. metadata-eval 87.6%

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

      5. +-commutative 87.6%

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

      6. +-commutative 87.6%

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

   7. Simplified 87.6%

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

      1. flip-- 87.7%

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

      2. add-sqr-sqrt 73.8%

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

      3. add-sqr-sqrt 88.2%

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

   9. Applied egg-rr 88.2%

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

       1. associate--l+ 91.4%

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

       2. +-inverses 91.4%

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

       3. metadata-eval 91.4%

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

   11. Simplified 91.4%

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

       1. flip-- 91.6%

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

       2. add-sqr-sqrt 52.0%

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

       3. add-sqr-sqrt 91.9%

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

   13. Applied egg-rr 91.9%

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

       1. associate--l+ 96.3%

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

       2. +-inverses 96.3%

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

       3. metadata-eval 96.3%

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

       4. +-commutative 96.3%

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

       5. +-commutative 96.3%

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

   15. Simplified 96.3%

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

   16. Taylor expanded in t around inf 57.1%

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

       1. associate-+r+ 57.1%

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

       2. +-commutative 57.1%

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

   18. Simplified 57.1%

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

   if 2e-3 < (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))

   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. +-commutative 96.9%

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

      5. +-commutative 96.9%

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

   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. Step-by-step derivation

      1. flip-- 97.1%

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

      2. add-sqr-sqrt 72.6%

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

      3. add-sqr-sqrt 97.2%

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

   5. Applied egg-rr 97.2%

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

      1. associate--l+ 97.4%

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

      2. +-inverses 97.4%

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

      3. metadata-eval 97.4%

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

      4. +-commutative 97.4%

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

      5. +-commutative 97.4%

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

   7. Simplified 97.4%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 0.002:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := t_2 - \sqrt{z}\\ \mathbf{if}\;t_3 \leq 0.002:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + t_1} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{z} + t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - \sqrt{x}\right) + \left(\left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + t_3\right) + \left(\left(1 + y \cdot 0.5\right) - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))) (t_2 (sqrt (+ 1.0 z))) (t_3 (- t_2 (sqrt z))))
   (if (<= t_3 0.002)
     (+
      (+ (/ 1.0 (+ (sqrt x) t_1)) (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))
      (/ 1.0 (+ (sqrt z) t_2)))
     (+
      (- t_1 (sqrt x))
      (+
       (+ (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t))) t_3)
       (- (+ 1.0 (* y 0.5)) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + z));
	double t_3 = t_2 - sqrt(z);
	double tmp;
	if (t_3 <= 0.002) {
		tmp = ((1.0 / (sqrt(x) + t_1)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)))) + (1.0 / (sqrt(z) + t_2));
	} else {
		tmp = (t_1 - sqrt(x)) + (((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + t_3) + ((1.0 + (y * 0.5)) - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + z))
    t_3 = t_2 - sqrt(z)
    if (t_3 <= 0.002d0) then
        tmp = ((1.0d0 / (sqrt(x) + t_1)) + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y)))) + (1.0d0 / (sqrt(z) + t_2))
    else
        tmp = (t_1 - sqrt(x)) + (((1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t))) + t_3) + ((1.0d0 + (y * 0.5d0)) - sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = t_2 - Math.sqrt(z);
	double tmp;
	if (t_3 <= 0.002) {
		tmp = ((1.0 / (Math.sqrt(x) + t_1)) + (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt(z) + t_2));
	} else {
		tmp = (t_1 - Math.sqrt(x)) + (((1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))) + t_3) + ((1.0 + (y * 0.5)) - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + z))
	t_3 = t_2 - math.sqrt(z)
	tmp = 0
	if t_3 <= 0.002:
		tmp = ((1.0 / (math.sqrt(x) + t_1)) + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y)))) + (1.0 / (math.sqrt(z) + t_2))
	else:
		tmp = (t_1 - math.sqrt(x)) + (((1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))) + t_3) + ((1.0 + (y * 0.5)) - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(t_2 - sqrt(z))
	tmp = 0.0
	if (t_3 <= 0.002)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_1)) + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(z) + t_2)));
	else
		tmp = Float64(Float64(t_1 - sqrt(x)) + Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))) + t_3) + Float64(Float64(1.0 + Float64(y * 0.5)) - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((1.0 + z));
	t_3 = t_2 - sqrt(z);
	tmp = 0.0;
	if (t_3 <= 0.002)
		tmp = ((1.0 / (sqrt(x) + t_1)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)))) + (1.0 / (sqrt(z) + t_2));
	else
		tmp = (t_1 - sqrt(x)) + (((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + t_3) + ((1.0 + (y * 0.5)) - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.002], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + N[(N[(1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z)) < 2e-3

   1. Initial program 84.7%

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

      1. associate-+l+ 84.7%

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

      2. associate-+l+ 84.7%

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

      3. +-commutative 84.7%

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

      4. +-commutative 84.7%

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

      5. +-commutative 84.7%

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

   3. Simplified 84.7%

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

      1. flip-- 84.9%

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

      2. add-sqr-sqrt 71.0%

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

      3. add-sqr-sqrt 85.3%

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

   5. Applied egg-rr 85.3%

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

      1. +-commutative 85.3%

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

      2. associate--l+ 87.6%

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

      3. +-inverses 87.6%

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

      4. metadata-eval 87.6%

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

      5. +-commutative 87.6%

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

      6. +-commutative 87.6%

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

   7. Simplified 87.6%

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

      1. flip-- 87.7%

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

      2. add-sqr-sqrt 73.8%

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

      3. add-sqr-sqrt 88.2%

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

   9. Applied egg-rr 88.2%

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

       1. associate--l+ 91.4%

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

       2. +-inverses 91.4%

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

       3. metadata-eval 91.4%

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

   11. Simplified 91.4%

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

       1. flip-- 91.6%

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

       2. add-sqr-sqrt 52.0%

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

       3. add-sqr-sqrt 91.9%

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

   13. Applied egg-rr 91.9%

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

       1. associate--l+ 96.3%

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

       2. +-inverses 96.3%

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

       3. metadata-eval 96.3%

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

       4. +-commutative 96.3%

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

       5. +-commutative 96.3%

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

   15. Simplified 96.3%

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

   16. Taylor expanded in t around inf 57.1%

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

       1. associate-+r+ 57.1%

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

       2. +-commutative 57.1%

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

   18. Simplified 57.1%

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

   if 2e-3 < (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))

   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. +-commutative 96.9%

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

      5. +-commutative 96.9%

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

   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. Step-by-step derivation

      1. flip-- 97.1%

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

      2. add-sqr-sqrt 72.6%

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

      3. add-sqr-sqrt 97.2%

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

   5. Applied egg-rr 97.2%

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

      1. associate--l+ 97.4%

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

      2. +-inverses 97.4%

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

      3. metadata-eval 97.4%

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

      4. +-commutative 97.4%

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

      5. +-commutative 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in y around 0 50.1%

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

      1. *-commutative 50.1%

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

   10. Simplified 50.1%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 0.002:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\left(1 + y \cdot 0.5\right) - \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

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

C

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

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(x) + sqrt((1.0d0 + x)))) + ((1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))) + ((sqrt((1.0d0 + t)) - sqrt(t)) + (1.0d0 / (sqrt(z) + sqrt((1.0d0 + z))))))
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(x) + Math.sqrt((1.0 + x)))) + ((1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z))))));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + ((1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) + ((math.sqrt((1.0 + t)) - math.sqrt(t)) + (1.0 / (math.sqrt(z) + math.sqrt((1.0 + z))))))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))) + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z)))))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))));
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[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.9%

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

   1. associate-+l+ 90.9%

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

   2. associate-+l+ 90.9%

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

   3. +-commutative 90.9%

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

   4. +-commutative 90.9%

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

   5. +-commutative 90.9%

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

3. Simplified 90.9%

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

   1. flip-- 91.1%

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

   2. add-sqr-sqrt 72.6%

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

   3. add-sqr-sqrt 91.5%

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

5. Applied egg-rr 91.5%

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

   1. +-commutative 91.5%

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

   2. associate--l+ 92.9%

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

   3. +-inverses 92.9%

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

   4. metadata-eval 92.9%

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

   5. +-commutative 92.9%

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

   6. +-commutative 92.9%

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

7. Simplified 92.9%

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

   1. flip-- 93.1%

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

   2. add-sqr-sqrt 75.6%

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

   3. add-sqr-sqrt 93.5%

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

9. Applied egg-rr 93.5%

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

    1. associate--l+ 95.5%

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

    2. +-inverses 95.5%

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

    3. metadata-eval 95.5%

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

11. Simplified 95.5%

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

    1. flip-- 95.5%

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

    2. add-sqr-sqrt 76.0%

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

    3. add-sqr-sqrt 95.7%

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

13. Applied egg-rr 95.7%

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

    1. associate--l+ 97.9%

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

    2. +-inverses 97.9%

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

    3. metadata-eval 97.9%

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

    4. +-commutative 97.9%

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

    5. +-commutative 97.9%

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

15. Simplified 97.9%

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

16. Final simplification 97.9%

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

Alternative 4: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;z \leq 3.6 \cdot 10^{-10}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(1 + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(t_1 - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + t_2} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{z} + t_1}\\ \end{array} \end{array} \]

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 z))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= z 3.6e-10)
     (+
      (- t_2 (sqrt x))
      (+ 1.0 (+ (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t))) (- t_1 (sqrt z)))))
     (+
      (+ (/ 1.0 (+ (sqrt x) t_2)) (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))
      (/ 1.0 (+ (sqrt z) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (z <= 3.6e-10) {
		tmp = (t_2 - sqrt(x)) + (1.0 + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + (t_1 - sqrt(z))));
	} else {
		tmp = ((1.0 / (sqrt(x) + t_2)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)))) + (1.0 / (sqrt(z) + t_1));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + x))
    if (z <= 3.6d-10) then
        tmp = (t_2 - sqrt(x)) + (1.0d0 + ((1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t))) + (t_1 - sqrt(z))))
    else
        tmp = ((1.0d0 / (sqrt(x) + t_2)) + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y)))) + (1.0d0 / (sqrt(z) + t_1))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (z <= 3.6e-10) {
		tmp = (t_2 - Math.sqrt(x)) + (1.0 + ((1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))) + (t_1 - Math.sqrt(z))));
	} else {
		tmp = ((1.0 / (Math.sqrt(x) + t_2)) + (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt(z) + t_1));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if z <= 3.6e-10:
		tmp = (t_2 - math.sqrt(x)) + (1.0 + ((1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))) + (t_1 - math.sqrt(z))))
	else:
		tmp = ((1.0 / (math.sqrt(x) + t_2)) + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y)))) + (1.0 / (math.sqrt(z) + t_1))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (z <= 3.6e-10)
		tmp = Float64(Float64(t_2 - sqrt(x)) + Float64(1.0 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))) + Float64(t_1 - sqrt(z)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_2)) + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(z) + t_1)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (z <= 3.6e-10)
		tmp = (t_2 - sqrt(x)) + (1.0 + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + (t_1 - sqrt(z))));
	else
		tmp = ((1.0 / (sqrt(x) + t_2)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)))) + (1.0 / (sqrt(z) + t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 3.6e-10], N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 3.6e-10

   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. +-commutative 96.9%

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

      5. +-commutative 96.9%

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

   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. Step-by-step derivation

      1. flip-- 97.1%

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

      2. add-sqr-sqrt 72.7%

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

      3. add-sqr-sqrt 97.3%

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

   5. Applied egg-rr 97.3%

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

      1. associate--l+ 97.4%

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

      2. +-inverses 97.4%

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

      3. metadata-eval 97.4%

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

      4. +-commutative 97.4%

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

      5. +-commutative 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in y around 0 54.3%

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

   if 3.6e-10 < z

   1. Initial program 85.2%

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

      1. associate-+l+ 85.2%

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

      2. associate-+l+ 85.2%

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

      3. +-commutative 85.2%

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

      4. +-commutative 85.2%

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

      5. +-commutative 85.2%

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

   3. Simplified 85.2%

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

      1. flip-- 85.6%

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

      2. add-sqr-sqrt 71.6%

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

      3. add-sqr-sqrt 85.9%

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

   5. Applied egg-rr 85.9%

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

      1. +-commutative 85.9%

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

      2. associate--l+ 88.1%

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

      3. +-inverses 88.1%

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

      4. metadata-eval 88.1%

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

      5. +-commutative 88.1%

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

      6. +-commutative 88.1%

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

   7. Simplified 88.1%

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

      1. flip-- 88.2%

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

      2. add-sqr-sqrt 75.0%

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

      3. add-sqr-sqrt 88.7%

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

   9. Applied egg-rr 88.7%

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

       1. associate--l+ 91.8%

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

       2. +-inverses 91.8%

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

       3. metadata-eval 91.8%

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

   11. Simplified 91.8%

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

       1. flip-- 92.0%

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

       2. add-sqr-sqrt 54.2%

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

       3. add-sqr-sqrt 92.3%

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

   13. Applied egg-rr 92.3%

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

       1. associate--l+ 96.5%

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

       2. +-inverses 96.5%

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

       3. metadata-eval 96.5%

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

       4. +-commutative 96.5%

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

       5. +-commutative 96.5%

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

   15. Simplified 96.5%

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

   16. Taylor expanded in t around inf 57.9%

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

       1. associate-+r+ 57.9%

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

       2. +-commutative 57.9%

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

   18. Simplified 57.9%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{-10}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(1 + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\\ \end{array} \]

Alternative 5: 97.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 3.2 \cdot 10^{-34}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(1 + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + t_1\right)\right)\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+55}:\\ \;\;\;\;t_2 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(t_1 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 z)) (sqrt z))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= y 3.2e-34)
     (+ (- t_2 (sqrt x)) (+ 1.0 (+ (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t))) t_1)))
     (if (<= y 1.1e+55)
       (+ t_2 (+ (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))) (- t_1 (sqrt x))))
       (/ 1.0 (+ (sqrt x) t_2))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z)) - sqrt(z);
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (y <= 3.2e-34) {
		tmp = (t_2 - sqrt(x)) + (1.0 + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + t_1));
	} else if (y <= 1.1e+55) {
		tmp = t_2 + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (t_1 - sqrt(x)));
	} else {
		tmp = 1.0 / (sqrt(x) + t_2);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z)) - sqrt(z)
    t_2 = sqrt((1.0d0 + x))
    if (y <= 3.2d-34) then
        tmp = (t_2 - sqrt(x)) + (1.0d0 + ((1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t))) + t_1))
    else if (y <= 1.1d+55) then
        tmp = t_2 + ((1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))) + (t_1 - sqrt(x)))
    else
        tmp = 1.0d0 / (sqrt(x) + t_2)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 3.2e-34) {
		tmp = (t_2 - Math.sqrt(x)) + (1.0 + ((1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))) + t_1));
	} else if (y <= 1.1e+55) {
		tmp = t_2 + ((1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) + (t_1 - Math.sqrt(x)));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_2);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z)) - math.sqrt(z)
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 3.2e-34:
		tmp = (t_2 - math.sqrt(x)) + (1.0 + ((1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))) + t_1))
	elif y <= 1.1e+55:
		tmp = t_2 + ((1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) + (t_1 - math.sqrt(x)))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_2)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 3.2e-34)
		tmp = Float64(Float64(t_2 - sqrt(x)) + Float64(1.0 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))) + t_1)));
	elseif (y <= 1.1e+55)
		tmp = Float64(t_2 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))) + Float64(t_1 - sqrt(x))));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_2));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z)) - sqrt(z);
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 3.2e-34)
		tmp = (t_2 - sqrt(x)) + (1.0 + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + t_1));
	elseif (y <= 1.1e+55)
		tmp = t_2 + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (t_1 - sqrt(x)));
	else
		tmp = 1.0 / (sqrt(x) + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 3.2e-34], N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+55], N[(t$95$2 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.20000000000000003e-34

   1. Initial program 97.0%

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

      1. associate-+l+ 97.0%

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

      2. associate-+l+ 97.0%

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

      3. +-commutative 97.0%

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

      4. +-commutative 97.0%

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

      5. +-commutative 97.0%

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

   3. Simplified 97.0%

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

      1. flip-- 97.0%

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

      2. add-sqr-sqrt 74.7%

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

      3. add-sqr-sqrt 97.6%

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

   5. Applied egg-rr 97.6%

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

      1. associate--l+ 97.6%

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

      2. +-inverses 97.6%

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

      3. metadata-eval 97.6%

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

      4. +-commutative 97.6%

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

      5. +-commutative 97.6%

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

   7. Simplified 97.6%

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

   8. Taylor expanded in y around 0 97.6%

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

   if 3.20000000000000003e-34 < y < 1.10000000000000005e55

   1. Initial program 89.9%

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

      1. associate-+l+ 89.9%

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

      2. +-commutative 89.9%

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

      3. associate-+r- 69.5%

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

      4. associate-+l- 52.7%

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

      5. +-commutative 52.7%

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

      6. +-commutative 52.7%

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

      7. associate--l+ 52.7%

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

   3. Simplified 33.0%

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

   4. Taylor expanded in t around inf 32.1%

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

      1. associate--l+ 33.5%

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

      2. +-commutative 33.5%

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

   6. Simplified 33.5%

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

      1. flip-- 91.3%

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

      2. add-sqr-sqrt 72.7%

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

      3. add-sqr-sqrt 93.8%

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

   8. Applied egg-rr 35.0%

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

      1. associate--l+ 98.8%

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

      2. +-inverses 98.8%

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

      3. metadata-eval 98.8%

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

   10. Simplified 35.5%

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

   if 1.10000000000000005e55 < y

   1. Initial program 84.2%

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

      1. associate-+l+ 84.2%

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

      2. +-commutative 84.2%

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

      3. associate-+r- 84.2%

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

      4. associate-+l- 54.4%

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

      5. +-commutative 54.4%

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

      6. +-commutative 54.4%

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

      7. associate--l+ 54.4%

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

   3. Simplified 32.1%

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

   4. Taylor expanded in t around inf 37.2%

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

      1. associate--l+ 36.9%

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

      2. +-commutative 36.9%

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

   6. Simplified 36.9%

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

   7. Taylor expanded in z around inf 23.2%

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

      1. +-commutative 23.2%

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

   9. Simplified 23.2%

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

   10. Taylor expanded in y around inf 22.4%

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

       1. flip-- 22.4%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 22.7%

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

       3. add-sqr-sqrt 22.4%

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

       4. +-commutative 22.4%

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

   12. Applied egg-rr 22.4%

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

       1. associate--l+ 26.2%

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

       2. +-inverses 26.2%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 26.2%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{x} + \sqrt{1 + x}} \]

   14. Simplified 26.2%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-34}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(1 + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{1 + x} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 6: 92.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+15}:\\ \;\;\;\;\left(\mathsf{hypot}\left(1, \sqrt{t}\right) + \mathsf{hypot}\left(1, \sqrt{y}\right)\right) + \left(2 - \left(\sqrt{y} + \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \frac{-1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 3.5e+15)
   (+
    (+ (hypot 1.0 (sqrt t)) (hypot 1.0 (sqrt y)))
    (- 2.0 (+ (sqrt y) (sqrt t))))
   (+
    (sqrt (+ 1.0 x))
    (-
     (- (sqrt (+ 1.0 y)) (sqrt y))
     (+ (sqrt x) (/ -1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.5e+15) {
		tmp = (hypot(1.0, sqrt(t)) + hypot(1.0, sqrt(y))) + (2.0 - (sqrt(y) + sqrt(t)));
	} else {
		tmp = sqrt((1.0 + x)) + ((sqrt((1.0 + y)) - sqrt(y)) - (sqrt(x) + (-1.0 / (sqrt(z) + sqrt((1.0 + z))))));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.5e+15) {
		tmp = (Math.hypot(1.0, Math.sqrt(t)) + Math.hypot(1.0, Math.sqrt(y))) + (2.0 - (Math.sqrt(y) + Math.sqrt(t)));
	} else {
		tmp = Math.sqrt((1.0 + x)) + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) - (Math.sqrt(x) + (-1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z))))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 3.5e+15:
		tmp = (math.hypot(1.0, math.sqrt(t)) + math.hypot(1.0, math.sqrt(y))) + (2.0 - (math.sqrt(y) + math.sqrt(t)))
	else:
		tmp = math.sqrt((1.0 + x)) + ((math.sqrt((1.0 + y)) - math.sqrt(y)) - (math.sqrt(x) + (-1.0 / (math.sqrt(z) + math.sqrt((1.0 + z))))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3.5e+15)
		tmp = Float64(Float64(hypot(1.0, sqrt(t)) + hypot(1.0, sqrt(y))) + Float64(2.0 - Float64(sqrt(y) + sqrt(t))));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) - Float64(sqrt(x) + Float64(-1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z)))))));
	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 (t <= 3.5e+15)
		tmp = (hypot(1.0, sqrt(t)) + hypot(1.0, sqrt(y))) + (2.0 - (sqrt(y) + sqrt(t)));
	else
		tmp = sqrt((1.0 + x)) + ((sqrt((1.0 + y)) - sqrt(y)) - (sqrt(x) + (-1.0 / (sqrt(z) + sqrt((1.0 + z))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 3.5e+15], N[(N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[t], $MachinePrecision] ^ 2], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + N[Sqrt[y], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + N[(2.0 - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(-1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.5e15

   1. Initial program 96.0%

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

      1. associate-+l+ 96.0%

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

      2. associate-+l+ 96.0%

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

      3. +-commutative 96.0%

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

      4. +-commutative 96.0%

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

      5. +-commutative 96.0%

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

   3. Simplified 96.0%

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

      1. flip-- 96.0%

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

      2. add-sqr-sqrt 74.5%

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

      3. add-sqr-sqrt 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. +-commutative 96.5%

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

      2. associate--l+ 97.0%

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

      3. +-inverses 97.0%

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

      4. metadata-eval 97.0%

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

      5. +-commutative 97.0%

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

      6. +-commutative 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in x around 0 22.4%

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

      1. associate--l+ 40.6%

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

      2. associate--r+ 40.6%

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

      3. associate-+r+ 40.6%

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

      4. +-commutative 40.6%

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

   10. Simplified 40.6%

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

   11. Taylor expanded in z around 0 24.0%

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

       1. +-commutative 24.0%

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

       2. associate--l+ 24.0%

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

       3. rem-square-sqrt 24.0%

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

       4. hypot-1-def 24.0%

          \[\leadsto \left(\color{blue}{\mathsf{hypot}\left(1, \sqrt{t}\right)} + \sqrt{1 + y}\right) + \left(2 - \left(\sqrt{t} + \sqrt{y}\right)\right) \]

       5. rem-square-sqrt 24.0%

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

       6. hypot-1-def 24.0%

          \[\leadsto \left(\mathsf{hypot}\left(1, \sqrt{t}\right) + \color{blue}{\mathsf{hypot}\left(1, \sqrt{y}\right)}\right) + \left(2 - \left(\sqrt{t} + \sqrt{y}\right)\right) \]

       7. +-commutative 24.0%

          \[\leadsto \left(\mathsf{hypot}\left(1, \sqrt{t}\right) + \mathsf{hypot}\left(1, \sqrt{y}\right)\right) + \left(2 - \color{blue}{\left(\sqrt{y} + \sqrt{t}\right)}\right) \]

   13. Simplified 24.0%

       \[\leadsto \color{blue}{\left(\mathsf{hypot}\left(1, \sqrt{t}\right) + \mathsf{hypot}\left(1, \sqrt{y}\right)\right) + \left(2 - \left(\sqrt{y} + \sqrt{t}\right)\right)} \]

   if 3.5e15 < t

   1. Initial program 86.1%

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

      1. associate-+l+ 86.1%

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

      2. +-commutative 86.1%

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

      3. associate-+r- 70.3%

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

      4. associate-+l- 59.2%

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

      5. +-commutative 59.2%

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

      6. +-commutative 59.2%

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

      7. associate--l+ 59.2%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in t around inf 57.2%

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

      1. associate--l+ 59.2%

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

      2. +-commutative 59.2%

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

   6. Simplified 59.2%

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

      1. flip-- 59.2%

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

      2. add-sqr-sqrt 46.6%

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

      3. add-sqr-sqrt 59.2%

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

      4. +-commutative 59.2%

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

      5. +-commutative 59.2%

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

   8. Applied egg-rr 59.2%

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

      1. +-commutative 59.2%

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

   10. Simplified 59.2%

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

   11. Taylor expanded in z around 0 59.5%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+15}:\\ \;\;\;\;\left(\mathsf{hypot}\left(1, \sqrt{t}\right) + \mathsf{hypot}\left(1, \sqrt{y}\right)\right) + \left(2 - \left(\sqrt{y} + \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \frac{-1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\\ \end{array} \]

Alternative 7: 92.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+15}:\\ \;\;\;\;\left(\mathsf{hypot}\left(1, \sqrt{t}\right) + \mathsf{hypot}\left(1, \sqrt{y}\right)\right) + \left(2 - \left(\sqrt{y} + \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 3.5e+15)
   (+
    (+ (hypot 1.0 (sqrt t)) (hypot 1.0 (sqrt y)))
    (- 2.0 (+ (sqrt y) (sqrt t))))
   (+
    (sqrt (+ 1.0 x))
    (+
     (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))
     (- (- (sqrt (+ 1.0 z)) (sqrt z)) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.5e+15) {
		tmp = (hypot(1.0, sqrt(t)) + hypot(1.0, sqrt(y))) + (2.0 - (sqrt(y) + sqrt(t)));
	} else {
		tmp = sqrt((1.0 + x)) + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + ((sqrt((1.0 + z)) - sqrt(z)) - sqrt(x)));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.5e+15) {
		tmp = (Math.hypot(1.0, Math.sqrt(t)) + Math.hypot(1.0, Math.sqrt(y))) + (2.0 - (Math.sqrt(y) + Math.sqrt(t)));
	} else {
		tmp = Math.sqrt((1.0 + x)) + ((1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) + ((Math.sqrt((1.0 + z)) - Math.sqrt(z)) - Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 3.5e+15:
		tmp = (math.hypot(1.0, math.sqrt(t)) + math.hypot(1.0, math.sqrt(y))) + (2.0 - (math.sqrt(y) + math.sqrt(t)))
	else:
		tmp = math.sqrt((1.0 + x)) + ((1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) + ((math.sqrt((1.0 + z)) - math.sqrt(z)) - 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 (t <= 3.5e+15)
		tmp = Float64(Float64(hypot(1.0, sqrt(t)) + hypot(1.0, sqrt(y))) + Float64(2.0 - Float64(sqrt(y) + sqrt(t))));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))) + Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) - 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 (t <= 3.5e+15)
		tmp = (hypot(1.0, sqrt(t)) + hypot(1.0, sqrt(y))) + (2.0 - (sqrt(y) + sqrt(t)));
	else
		tmp = sqrt((1.0 + x)) + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + ((sqrt((1.0 + z)) - sqrt(z)) - 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[t, 3.5e+15], N[(N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[t], $MachinePrecision] ^ 2], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + N[Sqrt[y], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + N[(2.0 - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.5e15

   1. Initial program 96.0%

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

      1. associate-+l+ 96.0%

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

      2. associate-+l+ 96.0%

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

      3. +-commutative 96.0%

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

      4. +-commutative 96.0%

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

      5. +-commutative 96.0%

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

   3. Simplified 96.0%

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

      1. flip-- 96.0%

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

      2. add-sqr-sqrt 74.5%

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

      3. add-sqr-sqrt 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. +-commutative 96.5%

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

      2. associate--l+ 97.0%

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

      3. +-inverses 97.0%

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

      4. metadata-eval 97.0%

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

      5. +-commutative 97.0%

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

      6. +-commutative 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in x around 0 22.4%

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

      1. associate--l+ 40.6%

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

      2. associate--r+ 40.6%

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

      3. associate-+r+ 40.6%

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

      4. +-commutative 40.6%

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

   10. Simplified 40.6%

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

   11. Taylor expanded in z around 0 24.0%

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

       1. +-commutative 24.0%

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

       2. associate--l+ 24.0%

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

       3. rem-square-sqrt 24.0%

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

       4. hypot-1-def 24.0%

          \[\leadsto \left(\color{blue}{\mathsf{hypot}\left(1, \sqrt{t}\right)} + \sqrt{1 + y}\right) + \left(2 - \left(\sqrt{t} + \sqrt{y}\right)\right) \]

       5. rem-square-sqrt 24.0%

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

       6. hypot-1-def 24.0%

          \[\leadsto \left(\mathsf{hypot}\left(1, \sqrt{t}\right) + \color{blue}{\mathsf{hypot}\left(1, \sqrt{y}\right)}\right) + \left(2 - \left(\sqrt{t} + \sqrt{y}\right)\right) \]

       7. +-commutative 24.0%

          \[\leadsto \left(\mathsf{hypot}\left(1, \sqrt{t}\right) + \mathsf{hypot}\left(1, \sqrt{y}\right)\right) + \left(2 - \color{blue}{\left(\sqrt{y} + \sqrt{t}\right)}\right) \]

   13. Simplified 24.0%

       \[\leadsto \color{blue}{\left(\mathsf{hypot}\left(1, \sqrt{t}\right) + \mathsf{hypot}\left(1, \sqrt{y}\right)\right) + \left(2 - \left(\sqrt{y} + \sqrt{t}\right)\right)} \]

   if 3.5e15 < t

   1. Initial program 86.1%

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

      1. associate-+l+ 86.1%

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

      2. +-commutative 86.1%

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

      3. associate-+r- 70.3%

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

      4. associate-+l- 59.2%

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

      5. +-commutative 59.2%

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

      6. +-commutative 59.2%

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

      7. associate--l+ 59.2%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in t around inf 57.2%

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

      1. associate--l+ 59.2%

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

      2. +-commutative 59.2%

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

   6. Simplified 59.2%

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

      1. flip-- 89.3%

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

      2. add-sqr-sqrt 73.1%

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

      3. add-sqr-sqrt 90.1%

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

   8. Applied egg-rr 59.7%

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

      1. associate--l+ 93.3%

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

      2. +-inverses 93.3%

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

      3. metadata-eval 93.3%

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

   10. Simplified 59.8%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+15}:\\ \;\;\;\;\left(\mathsf{hypot}\left(1, \sqrt{t}\right) + \mathsf{hypot}\left(1, \sqrt{y}\right)\right) + \left(2 - \left(\sqrt{y} + \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 8: 91.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+15}:\\ \;\;\;\;\left(\mathsf{hypot}\left(1, \sqrt{t}\right) + \mathsf{hypot}\left(1, \sqrt{y}\right)\right) + \left(2 - \left(\sqrt{y} + \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 3.5e+15)
   (+
    (+ (hypot 1.0 (sqrt t)) (hypot 1.0 (sqrt y)))
    (- 2.0 (+ (sqrt y) (sqrt t))))
   (+
    1.0
    (+ (sqrt (+ 1.0 y)) (- (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.5e+15) {
		tmp = (hypot(1.0, sqrt(t)) + hypot(1.0, sqrt(y))) + (2.0 - (sqrt(y) + sqrt(t)));
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) + ((1.0 / (sqrt(z) + sqrt((1.0 + z)))) - sqrt(y)));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.5e+15) {
		tmp = (Math.hypot(1.0, Math.sqrt(t)) + Math.hypot(1.0, Math.sqrt(y))) + (2.0 - (Math.sqrt(y) + Math.sqrt(t)));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) + ((1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)))) - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 3.5e+15:
		tmp = (math.hypot(1.0, math.sqrt(t)) + math.hypot(1.0, math.sqrt(y))) + (2.0 - (math.sqrt(y) + math.sqrt(t)))
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) + ((1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))) - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3.5e+15)
		tmp = Float64(Float64(hypot(1.0, sqrt(t)) + hypot(1.0, sqrt(y))) + Float64(2.0 - Float64(sqrt(y) + sqrt(t))));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) + Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z)))) - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 3.5e+15)
		tmp = (hypot(1.0, sqrt(t)) + hypot(1.0, sqrt(y))) + (2.0 - (sqrt(y) + sqrt(t)));
	else
		tmp = 1.0 + (sqrt((1.0 + y)) + ((1.0 / (sqrt(z) + sqrt((1.0 + z)))) - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 3.5e+15], N[(N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[t], $MachinePrecision] ^ 2], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + N[Sqrt[y], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + N[(2.0 - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.5e15

   1. Initial program 96.0%

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

      1. associate-+l+ 96.0%

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

      2. associate-+l+ 96.0%

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

      3. +-commutative 96.0%

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

      4. +-commutative 96.0%

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

      5. +-commutative 96.0%

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

   3. Simplified 96.0%

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

      1. flip-- 96.0%

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

      2. add-sqr-sqrt 74.5%

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

      3. add-sqr-sqrt 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. +-commutative 96.5%

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

      2. associate--l+ 97.0%

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

      3. +-inverses 97.0%

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

      4. metadata-eval 97.0%

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

      5. +-commutative 97.0%

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

      6. +-commutative 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in x around 0 22.4%

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

      1. associate--l+ 40.6%

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

      2. associate--r+ 40.6%

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

      3. associate-+r+ 40.6%

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

      4. +-commutative 40.6%

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

   10. Simplified 40.6%

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

   11. Taylor expanded in z around 0 24.0%

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

       1. +-commutative 24.0%

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

       2. associate--l+ 24.0%

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

       3. rem-square-sqrt 24.0%

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

       4. hypot-1-def 24.0%

          \[\leadsto \left(\color{blue}{\mathsf{hypot}\left(1, \sqrt{t}\right)} + \sqrt{1 + y}\right) + \left(2 - \left(\sqrt{t} + \sqrt{y}\right)\right) \]

       5. rem-square-sqrt 24.0%

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

       6. hypot-1-def 24.0%

          \[\leadsto \left(\mathsf{hypot}\left(1, \sqrt{t}\right) + \color{blue}{\mathsf{hypot}\left(1, \sqrt{y}\right)}\right) + \left(2 - \left(\sqrt{t} + \sqrt{y}\right)\right) \]

       7. +-commutative 24.0%

          \[\leadsto \left(\mathsf{hypot}\left(1, \sqrt{t}\right) + \mathsf{hypot}\left(1, \sqrt{y}\right)\right) + \left(2 - \color{blue}{\left(\sqrt{y} + \sqrt{t}\right)}\right) \]

   13. Simplified 24.0%

       \[\leadsto \color{blue}{\left(\mathsf{hypot}\left(1, \sqrt{t}\right) + \mathsf{hypot}\left(1, \sqrt{y}\right)\right) + \left(2 - \left(\sqrt{y} + \sqrt{t}\right)\right)} \]

   if 3.5e15 < t

   1. Initial program 86.1%

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

      1. associate-+l+ 86.1%

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

      2. +-commutative 86.1%

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

      3. associate-+r- 70.3%

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

      4. associate-+l- 59.2%

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

      5. +-commutative 59.2%

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

      6. +-commutative 59.2%

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

      7. associate--l+ 59.2%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in t around inf 57.2%

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

      1. associate--l+ 59.2%

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

      2. +-commutative 59.2%

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

   6. Simplified 59.2%

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

      1. flip-- 59.2%

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

      2. add-sqr-sqrt 46.6%

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

      3. add-sqr-sqrt 59.2%

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

      4. +-commutative 59.2%

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

      5. +-commutative 59.2%

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

   8. Applied egg-rr 59.2%

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

      1. +-commutative 59.2%

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

   10. Simplified 59.2%

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

   11. Taylor expanded in x around 0 35.8%

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

       1. associate--l+ 58.8%

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

       2. associate--l+ 58.8%

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

       3. +-commutative 58.8%

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

   13. Simplified 58.8%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+15}:\\ \;\;\;\;\left(\mathsf{hypot}\left(1, \sqrt{t}\right) + \mathsf{hypot}\left(1, \sqrt{y}\right)\right) + \left(2 - \left(\sqrt{y} + \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} - \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 9: 91.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 3.5 \cdot 10^{+15}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(t_1 - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(\frac{1}{\sqrt{z} + t_1} - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))))
   (if (<= t 3.5e+15)
     (+ 2.0 (+ (sqrt (+ 1.0 t)) (- t_1 (+ (sqrt z) (sqrt t)))))
     (+ 1.0 (+ (sqrt (+ 1.0 y)) (- (/ 1.0 (+ (sqrt z) t_1)) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double tmp;
	if (t <= 3.5e+15) {
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) + ((1.0 / (sqrt(z) + t_1)) - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if (t <= 3.5d+15) then
        tmp = 2.0d0 + (sqrt((1.0d0 + t)) + (t_1 - (sqrt(z) + sqrt(t))))
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) + ((1.0d0 / (sqrt(z) + t_1)) - sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double tmp;
	if (t <= 3.5e+15) {
		tmp = 2.0 + (Math.sqrt((1.0 + t)) + (t_1 - (Math.sqrt(z) + Math.sqrt(t))));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) + ((1.0 / (Math.sqrt(z) + t_1)) - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	tmp = 0
	if t <= 3.5e+15:
		tmp = 2.0 + (math.sqrt((1.0 + t)) + (t_1 - (math.sqrt(z) + math.sqrt(t))))
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) + ((1.0 / (math.sqrt(z) + t_1)) - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (t <= 3.5e+15)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + t)) + Float64(t_1 - Float64(sqrt(z) + sqrt(t)))));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) + Float64(Float64(1.0 / Float64(sqrt(z) + t_1)) - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	tmp = 0.0;
	if (t <= 3.5e+15)
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	else
		tmp = 1.0 + (sqrt((1.0 + y)) + ((1.0 / (sqrt(z) + t_1)) - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 3.5e+15], N[(2.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 3.5e15

   1. Initial program 96.0%

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

      1. associate-+l+ 96.0%

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

      2. associate-+l+ 96.0%

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

      3. +-commutative 96.0%

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

      4. +-commutative 96.0%

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

      5. +-commutative 96.0%

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

   3. Simplified 96.0%

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

      1. flip-- 96.0%

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

      2. add-sqr-sqrt 74.5%

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

      3. add-sqr-sqrt 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. +-commutative 96.5%

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

      2. associate--l+ 97.0%

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

      3. +-inverses 97.0%

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

      4. metadata-eval 97.0%

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

      5. +-commutative 97.0%

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

      6. +-commutative 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in x around 0 22.4%

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

      1. associate--l+ 40.6%

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

      2. associate--r+ 40.6%

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

      3. associate-+r+ 40.6%

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

      4. +-commutative 40.6%

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

   10. Simplified 40.6%

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

   11. Taylor expanded in y around 0 25.0%

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

       1. associate--l+ 47.6%

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

       2. associate--l+ 40.0%

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

       3. +-commutative 40.0%

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

   13. Simplified 40.0%

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

   if 3.5e15 < t

   1. Initial program 86.1%

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

      1. associate-+l+ 86.1%

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

      2. +-commutative 86.1%

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

      3. associate-+r- 70.3%

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

      4. associate-+l- 59.2%

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

      5. +-commutative 59.2%

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

      6. +-commutative 59.2%

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

      7. associate--l+ 59.2%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in t around inf 57.2%

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

      1. associate--l+ 59.2%

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

      2. +-commutative 59.2%

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

   6. Simplified 59.2%

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

      1. flip-- 59.2%

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

      2. add-sqr-sqrt 46.6%

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

      3. add-sqr-sqrt 59.2%

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

      4. +-commutative 59.2%

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

      5. +-commutative 59.2%

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

   8. Applied egg-rr 59.2%

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

      1. +-commutative 59.2%

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

   10. Simplified 59.2%

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

   11. Taylor expanded in x around 0 35.8%

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

       1. associate--l+ 58.8%

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

       2. associate--l+ 58.8%

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

       3. +-commutative 58.8%

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

   13. Simplified 58.8%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+15}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + z} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} - \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 10: 84.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+223}:\\ \;\;\;\;t_1 + \left(t_2 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+253}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_2 - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))) (t_2 (sqrt (+ 1.0 y))))
   (if (<= z 1.3e+15)
     (- (+ (sqrt (+ 1.0 z)) 2.0) (+ (sqrt y) (sqrt z)))
     (if (<= z 1.05e+223)
       (+ t_1 (- t_2 (+ (sqrt x) (sqrt y))))
       (if (<= z 1.75e+253)
         (/ 1.0 (+ (sqrt x) t_1))
         (+ 1.0 (- t_2 (sqrt y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + y));
	double tmp;
	if (z <= 1.3e+15) {
		tmp = (sqrt((1.0 + z)) + 2.0) - (sqrt(y) + sqrt(z));
	} else if (z <= 1.05e+223) {
		tmp = t_1 + (t_2 - (sqrt(x) + sqrt(y)));
	} else if (z <= 1.75e+253) {
		tmp = 1.0 / (sqrt(x) + t_1);
	} else {
		tmp = 1.0 + (t_2 - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + y))
    if (z <= 1.3d+15) then
        tmp = (sqrt((1.0d0 + z)) + 2.0d0) - (sqrt(y) + sqrt(z))
    else if (z <= 1.05d+223) then
        tmp = t_1 + (t_2 - (sqrt(x) + sqrt(y)))
    else if (z <= 1.75d+253) then
        tmp = 1.0d0 / (sqrt(x) + t_1)
    else
        tmp = 1.0d0 + (t_2 - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 1.3e+15) {
		tmp = (Math.sqrt((1.0 + z)) + 2.0) - (Math.sqrt(y) + Math.sqrt(z));
	} else if (z <= 1.05e+223) {
		tmp = t_1 + (t_2 - (Math.sqrt(x) + Math.sqrt(y)));
	} else if (z <= 1.75e+253) {
		tmp = 1.0 / (Math.sqrt(x) + t_1);
	} else {
		tmp = 1.0 + (t_2 - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 1.3e+15:
		tmp = (math.sqrt((1.0 + z)) + 2.0) - (math.sqrt(y) + math.sqrt(z))
	elif z <= 1.05e+223:
		tmp = t_1 + (t_2 - (math.sqrt(x) + math.sqrt(y)))
	elif z <= 1.75e+253:
		tmp = 1.0 / (math.sqrt(x) + t_1)
	else:
		tmp = 1.0 + (t_2 - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 1.3e+15)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) + 2.0) - Float64(sqrt(y) + sqrt(z)));
	elseif (z <= 1.05e+223)
		tmp = Float64(t_1 + Float64(t_2 - Float64(sqrt(x) + sqrt(y))));
	elseif (z <= 1.75e+253)
		tmp = Float64(1.0 / Float64(sqrt(x) + t_1));
	else
		tmp = Float64(1.0 + Float64(t_2 - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 1.3e+15)
		tmp = (sqrt((1.0 + z)) + 2.0) - (sqrt(y) + sqrt(z));
	elseif (z <= 1.05e+223)
		tmp = t_1 + (t_2 - (sqrt(x) + sqrt(y)));
	elseif (z <= 1.75e+253)
		tmp = 1.0 / (sqrt(x) + t_1);
	else
		tmp = 1.0 + (t_2 - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1.3e+15], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e+223], N[(t$95$1 + N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.75e+253], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < 1.3e15

   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. +-commutative 96.5%

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

      3. associate-+r- 82.3%

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

      4. associate-+l- 60.6%

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

      5. +-commutative 60.6%

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

      6. +-commutative 60.6%

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

      7. associate--l+ 60.6%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in t around inf 37.7%

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

      1. associate--l+ 37.7%

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

      2. +-commutative 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in y around 0 18.7%

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

   8. Taylor expanded in x around 0 26.6%

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

   if 1.3e15 < z < 1.04999999999999995e223

   1. Initial program 84.9%

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

      1. associate-+l+ 84.9%

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

      2. +-commutative 84.9%

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

      3. associate-+r- 66.8%

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

      4. associate-+l- 56.3%

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

      5. +-commutative 56.3%

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

      6. +-commutative 56.3%

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

      7. associate--l+ 56.3%

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

   3. Simplified 39.6%

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

   4. Taylor expanded in t around inf 33.2%

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

      1. associate--l+ 35.9%

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

      2. +-commutative 35.9%

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

   6. Simplified 35.9%

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

   7. Taylor expanded in z around inf 35.8%

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

      1. +-commutative 35.8%

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

   9. Simplified 35.8%

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

   if 1.04999999999999995e223 < z < 1.74999999999999989e253

   1. Initial program 80.7%

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

      1. associate-+l+ 80.7%

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

      2. +-commutative 80.7%

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

      3. associate-+r- 68.8%

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

      4. associate-+l- 61.3%

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

      5. +-commutative 61.3%

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

      6. +-commutative 61.3%

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

      7. associate--l+ 61.3%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in t around inf 41.0%

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

      1. associate--l+ 39.9%

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

      2. +-commutative 39.9%

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

   6. Simplified 39.9%

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

   7. Taylor expanded in z around inf 40.4%

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

      1. +-commutative 40.4%

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

   9. Simplified 40.4%

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

   10. Taylor expanded in y around inf 28.7%

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

       1. flip-- 28.7%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 29.0%

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

       3. add-sqr-sqrt 28.7%

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

       4. +-commutative 28.7%

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

   12. Applied egg-rr 28.7%

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

       1. associate--l+ 36.3%

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

       2. +-inverses 36.3%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 36.3%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{x} + \sqrt{1 + x}} \]

   14. Simplified 36.3%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]

   if 1.74999999999999989e253 < z

   1. Initial program 86.8%

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

      1. associate-+l+ 86.8%

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

      2. +-commutative 86.8%

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

      3. associate-+r- 66.8%

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

      4. associate-+l- 55.9%

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

      5. +-commutative 55.9%

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

      6. +-commutative 55.9%

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

      7. associate--l+ 55.9%

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

   3. Simplified 28.7%

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

   4. Taylor expanded in t around inf 35.5%

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

      1. associate--l+ 36.6%

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

      2. +-commutative 36.6%

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

   6. Simplified 36.6%

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

   7. Taylor expanded in z around inf 36.6%

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

      1. +-commutative 36.6%

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

   9. Simplified 36.6%

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

   10. Taylor expanded in x around 0 41.3%

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

       1. associate--l+ 59.0%

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

   12. Simplified 59.0%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+223}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+253}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 11: 89.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;y \leq 3.45 \cdot 10^{-34}:\\ \;\;\;\;1 + \left(t_2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+54}:\\ \;\;\;\;t_1 + \left(\left(t_2 - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

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))) (t_2 (sqrt (+ 1.0 y))))
   (if (<= y 3.45e-34)
     (+ 1.0 (+ t_2 (- (sqrt (+ 1.0 z)) (+ (sqrt y) (sqrt z)))))
     (if (<= y 7.8e+54)
       (+ t_1 (- (- t_2 (sqrt y)) (sqrt x)))
       (/ 1.0 (+ (sqrt x) 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));
	double t_2 = sqrt((1.0 + y));
	double tmp;
	if (y <= 3.45e-34) {
		tmp = 1.0 + (t_2 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z))));
	} else if (y <= 7.8e+54) {
		tmp = t_1 + ((t_2 - sqrt(y)) - sqrt(x));
	} else {
		tmp = 1.0 / (sqrt(x) + 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) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + y))
    if (y <= 3.45d-34) then
        tmp = 1.0d0 + (t_2 + (sqrt((1.0d0 + z)) - (sqrt(y) + sqrt(z))))
    else if (y <= 7.8d+54) then
        tmp = t_1 + ((t_2 - sqrt(y)) - sqrt(x))
    else
        tmp = 1.0d0 / (sqrt(x) + 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));
	double t_2 = Math.sqrt((1.0 + y));
	double tmp;
	if (y <= 3.45e-34) {
		tmp = 1.0 + (t_2 + (Math.sqrt((1.0 + z)) - (Math.sqrt(y) + Math.sqrt(z))));
	} else if (y <= 7.8e+54) {
		tmp = t_1 + ((t_2 - Math.sqrt(y)) - Math.sqrt(x));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + 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))
	t_2 = math.sqrt((1.0 + y))
	tmp = 0
	if y <= 3.45e-34:
		tmp = 1.0 + (t_2 + (math.sqrt((1.0 + z)) - (math.sqrt(y) + math.sqrt(z))))
	elif y <= 7.8e+54:
		tmp = t_1 + ((t_2 - math.sqrt(y)) - math.sqrt(x))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_1)
	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 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (y <= 3.45e-34)
		tmp = Float64(1.0 + Float64(t_2 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(y) + sqrt(z)))));
	elseif (y <= 7.8e+54)
		tmp = Float64(t_1 + Float64(Float64(t_2 - sqrt(y)) - sqrt(x)));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + 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));
	t_2 = sqrt((1.0 + y));
	tmp = 0.0;
	if (y <= 3.45e-34)
		tmp = 1.0 + (t_2 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z))));
	elseif (y <= 7.8e+54)
		tmp = t_1 + ((t_2 - sqrt(y)) - sqrt(x));
	else
		tmp = 1.0 / (sqrt(x) + 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[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 3.45e-34], N[(1.0 + N[(t$95$2 + 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, 7.8e+54], N[(t$95$1 + N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.45e-34

   1. Initial program 97.0%

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

      1. associate-+l+ 97.0%

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

      2. +-commutative 97.0%

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

      3. associate-+r- 69.1%

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

      4. associate-+l- 65.0%

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

      5. +-commutative 65.0%

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

      6. +-commutative 65.0%

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

      7. associate--l+ 65.0%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in t around inf 37.2%

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

      1. associate--l+ 38.8%

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

      2. +-commutative 38.8%

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

   6. Simplified 38.8%

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

   7. Taylor expanded in x around 0 28.2%

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

      1. associate--l+ 42.9%

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

      2. associate--l+ 56.5%

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

   9. Simplified 56.5%

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

   if 3.45e-34 < y < 7.8000000000000005e54

   1. Initial program 89.9%

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

      1. associate-+l+ 89.9%

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

      2. +-commutative 89.9%

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

      3. associate-+r- 69.5%

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

      4. associate-+l- 52.7%

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

      5. +-commutative 52.7%

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

      6. +-commutative 52.7%

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

      7. associate--l+ 52.7%

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

   3. Simplified 33.0%

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

   4. Taylor expanded in t around inf 32.1%

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

      1. associate--l+ 33.5%

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

      2. +-commutative 33.5%

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

   6. Simplified 33.5%

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

   7. Taylor expanded in z around inf 19.1%

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

      1. +-commutative 19.1%

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

   9. Simplified 19.1%

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

       1. sub-neg 19.1%

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

       2. +-commutative 19.1%

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

   11. Applied egg-rr 19.1%

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

       1. sub-neg 19.1%

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

       2. +-commutative 19.1%

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

       3. associate--r+ 19.5%

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

   13. Simplified 19.5%

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

   if 7.8000000000000005e54 < y

   1. Initial program 84.2%

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

      1. associate-+l+ 84.2%

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

      2. +-commutative 84.2%

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

      3. associate-+r- 84.2%

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

      4. associate-+l- 54.4%

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

      5. +-commutative 54.4%

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

      6. +-commutative 54.4%

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

      7. associate--l+ 54.4%

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

   3. Simplified 32.1%

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

   4. Taylor expanded in t around inf 37.2%

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

      1. associate--l+ 36.9%

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

      2. +-commutative 36.9%

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

   6. Simplified 36.9%

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

   7. Taylor expanded in z around inf 23.2%

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

      1. +-commutative 23.2%

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

   9. Simplified 23.2%

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

   10. Taylor expanded in y around inf 22.4%

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

       1. flip-- 22.4%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 22.7%

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

       3. add-sqr-sqrt 22.4%

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

       4. +-commutative 22.4%

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

   12. Applied egg-rr 22.4%

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

       1. associate--l+ 26.2%

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

       2. +-inverses 26.2%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 26.2%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{x} + \sqrt{1 + x}} \]

   14. Simplified 26.2%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.45 \cdot 10^{-34}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 12: 89.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 8.8 \cdot 10^{-7}:\\ \;\;\;\;t_1 + \left(\left(1 - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 10^{+55}:\\ \;\;\;\;t_1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

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))))
   (if (<= y 8.8e-7)
     (+ t_1 (+ (- 1.0 (sqrt y)) (- (sqrt (+ 1.0 z)) (sqrt z))))
     (if (<= y 1e+55)
       (+ t_1 (- (- (sqrt (+ 1.0 y)) (sqrt y)) (sqrt x)))
       (/ 1.0 (+ (sqrt x) 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));
	double tmp;
	if (y <= 8.8e-7) {
		tmp = t_1 + ((1.0 - sqrt(y)) + (sqrt((1.0 + z)) - sqrt(z)));
	} else if (y <= 1e+55) {
		tmp = t_1 + ((sqrt((1.0 + y)) - sqrt(y)) - sqrt(x));
	} else {
		tmp = 1.0 / (sqrt(x) + 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))
    if (y <= 8.8d-7) then
        tmp = t_1 + ((1.0d0 - sqrt(y)) + (sqrt((1.0d0 + z)) - sqrt(z)))
    else if (y <= 1d+55) then
        tmp = t_1 + ((sqrt((1.0d0 + y)) - sqrt(y)) - sqrt(x))
    else
        tmp = 1.0d0 / (sqrt(x) + 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));
	double tmp;
	if (y <= 8.8e-7) {
		tmp = t_1 + ((1.0 - Math.sqrt(y)) + (Math.sqrt((1.0 + z)) - Math.sqrt(z)));
	} else if (y <= 1e+55) {
		tmp = t_1 + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) - Math.sqrt(x));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + 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))
	tmp = 0
	if y <= 8.8e-7:
		tmp = t_1 + ((1.0 - math.sqrt(y)) + (math.sqrt((1.0 + z)) - math.sqrt(z)))
	elif y <= 1e+55:
		tmp = t_1 + ((math.sqrt((1.0 + y)) - math.sqrt(y)) - math.sqrt(x))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_1)
	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 <= 8.8e-7)
		tmp = Float64(t_1 + Float64(Float64(1.0 - sqrt(y)) + Float64(sqrt(Float64(1.0 + z)) - sqrt(z))));
	elseif (y <= 1e+55)
		tmp = Float64(t_1 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) - sqrt(x)));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + 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));
	tmp = 0.0;
	if (y <= 8.8e-7)
		tmp = t_1 + ((1.0 - sqrt(y)) + (sqrt((1.0 + z)) - sqrt(z)));
	elseif (y <= 1e+55)
		tmp = t_1 + ((sqrt((1.0 + y)) - sqrt(y)) - sqrt(x));
	else
		tmp = 1.0 / (sqrt(x) + 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[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 8.8e-7], N[(t$95$1 + N[(N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+55], N[(t$95$1 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.8000000000000004e-7

   1. Initial program 97.0%

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

      1. associate-+l+ 97.0%

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

      2. +-commutative 97.0%

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

      3. associate-+r- 67.6%

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

      4. associate-+l- 63.3%

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

      5. +-commutative 63.3%

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

      6. +-commutative 63.3%

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

      7. associate--l+ 63.3%

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

   3. Simplified 42.3%

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

   4. Taylor expanded in t around inf 36.2%

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

      1. associate--l+ 37.6%

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

      2. +-commutative 37.6%

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

   6. Simplified 37.6%

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

   7. Taylor expanded in y around 0 37.6%

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

   8. Taylor expanded in x around 0 36.0%

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

   if 8.8000000000000004e-7 < y < 1.00000000000000001e55

   1. Initial program 87.7%

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

      1. associate-+l+ 87.7%

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

      2. +-commutative 87.7%

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

      3. associate-+r- 75.2%

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

      4. associate-+l- 54.8%

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

      5. +-commutative 54.8%

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

      6. +-commutative 54.8%

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

      7. associate--l+ 54.8%

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

   3. Simplified 33.4%

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

   4. Taylor expanded in t around inf 34.1%

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

      1. associate--l+ 36.1%

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

      2. +-commutative 36.1%

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

   6. Simplified 36.1%

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

   7. Taylor expanded in z around inf 21.8%

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

      1. +-commutative 21.8%

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

   9. Simplified 21.8%

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

       1. sub-neg 21.8%

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

       2. +-commutative 21.8%

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

   11. Applied egg-rr 21.8%

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

       1. sub-neg 21.8%

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

       2. +-commutative 21.8%

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

       3. associate--r+ 22.3%

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

   13. Simplified 22.3%

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

   if 1.00000000000000001e55 < y

   1. Initial program 84.2%

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

      1. associate-+l+ 84.2%

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

      2. +-commutative 84.2%

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

      3. associate-+r- 84.2%

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

      4. associate-+l- 54.4%

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

      5. +-commutative 54.4%

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

      6. +-commutative 54.4%

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

      7. associate--l+ 54.4%

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

   3. Simplified 32.1%

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

   4. Taylor expanded in t around inf 37.2%

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

      1. associate--l+ 36.9%

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

      2. +-commutative 36.9%

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

   6. Simplified 36.9%

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

   7. Taylor expanded in z around inf 23.2%

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

      1. +-commutative 23.2%

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

   9. Simplified 23.2%

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

   10. Taylor expanded in y around inf 22.4%

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

       1. flip-- 22.4%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 22.7%

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

       3. add-sqr-sqrt 22.4%

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

       4. +-commutative 22.4%

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

   12. Applied egg-rr 22.4%

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

       1. associate--l+ 26.2%

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

       2. +-inverses 26.2%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 26.2%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{x} + \sqrt{1 + x}} \]

   14. Simplified 26.2%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.8 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(1 - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 10^{+55}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 13: 86.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.2e+15)
   (- (+ (sqrt (+ 1.0 z)) 2.0) (+ (sqrt y) (sqrt z)))
   (+ (sqrt (+ 1.0 x)) (- (- (sqrt (+ 1.0 y)) (sqrt y)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.2e+15) {
		tmp = (sqrt((1.0 + z)) + 2.0) - (sqrt(y) + sqrt(z));
	} else {
		tmp = sqrt((1.0 + x)) + ((sqrt((1.0 + y)) - sqrt(y)) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.2d+15) then
        tmp = (sqrt((1.0d0 + z)) + 2.0d0) - (sqrt(y) + sqrt(z))
    else
        tmp = sqrt((1.0d0 + x)) + ((sqrt((1.0d0 + y)) - sqrt(y)) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.2e+15) {
		tmp = (Math.sqrt((1.0 + z)) + 2.0) - (Math.sqrt(y) + Math.sqrt(z));
	} else {
		tmp = Math.sqrt((1.0 + x)) + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.2e+15:
		tmp = (math.sqrt((1.0 + z)) + 2.0) - (math.sqrt(y) + math.sqrt(z))
	else:
		tmp = math.sqrt((1.0 + x)) + ((math.sqrt((1.0 + y)) - math.sqrt(y)) - 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 (z <= 1.2e+15)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) + 2.0) - Float64(sqrt(y) + sqrt(z)));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.2e+15)
		tmp = (sqrt((1.0 + z)) + 2.0) - (sqrt(y) + sqrt(z));
	else
		tmp = sqrt((1.0 + x)) + ((sqrt((1.0 + y)) - sqrt(y)) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 1.2e+15], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.2e15

   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. +-commutative 96.5%

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

      3. associate-+r- 82.3%

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

      4. associate-+l- 60.6%

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

      5. +-commutative 60.6%

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

      6. +-commutative 60.6%

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

      7. associate--l+ 60.6%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in t around inf 37.7%

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

      1. associate--l+ 37.7%

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

      2. +-commutative 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in y around 0 18.7%

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

   8. Taylor expanded in x around 0 26.6%

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

   if 1.2e15 < z

   1. Initial program 84.8%

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

      1. associate-+l+ 84.8%

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

      2. +-commutative 84.8%

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

      3. associate-+r- 67.0%

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

      4. associate-+l- 56.8%

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

      5. +-commutative 56.8%

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

      6. +-commutative 56.8%

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

      7. associate--l+ 56.8%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in t around inf 34.7%

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

      1. associate--l+ 36.6%

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

      2. +-commutative 36.6%

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

   6. Simplified 36.6%

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

   7. Taylor expanded in z around inf 36.5%

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

      1. +-commutative 36.5%

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

   9. Simplified 36.5%

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

       1. sub-neg 36.5%

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

       2. +-commutative 36.5%

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

   11. Applied egg-rr 36.5%

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

       1. sub-neg 36.5%

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

       2. +-commutative 36.5%

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

       3. associate--r+ 36.6%

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

   13. Simplified 36.6%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 14: 84.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + z} + \left(2 - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+223}:\\ \;\;\;\;1 + \left(\mathsf{hypot}\left(1, \sqrt{y}\right) - \sqrt{y}\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+253}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.7e+15)
   (+ (sqrt (+ 1.0 z)) (- 2.0 (+ (sqrt y) (sqrt z))))
   (if (<= z 1.05e+223)
     (+ 1.0 (- (hypot 1.0 (sqrt y)) (sqrt y)))
     (if (<= z 1.75e+253)
       (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))
       (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.7e+15) {
		tmp = sqrt((1.0 + z)) + (2.0 - (sqrt(y) + sqrt(z)));
	} else if (z <= 1.05e+223) {
		tmp = 1.0 + (hypot(1.0, sqrt(y)) - sqrt(y));
	} else if (z <= 1.75e+253) {
		tmp = 1.0 / (sqrt(x) + sqrt((1.0 + x)));
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.7e+15) {
		tmp = Math.sqrt((1.0 + z)) + (2.0 - (Math.sqrt(y) + Math.sqrt(z)));
	} else if (z <= 1.05e+223) {
		tmp = 1.0 + (Math.hypot(1.0, Math.sqrt(y)) - Math.sqrt(y));
	} else if (z <= 1.75e+253) {
		tmp = 1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.7e+15:
		tmp = math.sqrt((1.0 + z)) + (2.0 - (math.sqrt(y) + math.sqrt(z)))
	elif z <= 1.05e+223:
		tmp = 1.0 + (math.hypot(1.0, math.sqrt(y)) - math.sqrt(y))
	elif z <= 1.75e+253:
		tmp = 1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.7e+15)
		tmp = Float64(sqrt(Float64(1.0 + z)) + Float64(2.0 - Float64(sqrt(y) + sqrt(z))));
	elseif (z <= 1.05e+223)
		tmp = Float64(1.0 + Float64(hypot(1.0, sqrt(y)) - sqrt(y)));
	elseif (z <= 1.75e+253)
		tmp = Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x))));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.7e+15)
		tmp = sqrt((1.0 + z)) + (2.0 - (sqrt(y) + sqrt(z)));
	elseif (z <= 1.05e+223)
		tmp = 1.0 + (hypot(1.0, sqrt(y)) - sqrt(y));
	elseif (z <= 1.75e+253)
		tmp = 1.0 / (sqrt(x) + sqrt((1.0 + x)));
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 1.7e+15], N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(2.0 - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e+223], N[(1.0 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[y], $MachinePrecision] ^ 2], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.75e+253], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < 1.7e15

   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. +-commutative 96.5%

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

      3. associate-+r- 82.3%

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

      4. associate-+l- 60.6%

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

      5. +-commutative 60.6%

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

      6. +-commutative 60.6%

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

      7. associate--l+ 60.6%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in t around inf 37.7%

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

      1. associate--l+ 37.7%

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

      2. +-commutative 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in y around 0 18.7%

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

   8. Taylor expanded in x around 0 26.6%

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

      1. +-commutative 26.6%

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

      2. associate--l+ 26.6%

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

   10. Simplified 26.6%

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

   if 1.7e15 < z < 1.04999999999999995e223

   1. Initial program 84.9%

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

      1. associate-+l+ 84.9%

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

      2. +-commutative 84.9%

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

      3. associate-+r- 66.8%

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

      4. associate-+l- 56.3%

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

      5. +-commutative 56.3%

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

      6. +-commutative 56.3%

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

      7. associate--l+ 56.3%

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

   3. Simplified 39.6%

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

   4. Taylor expanded in t around inf 33.2%

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

      1. associate--l+ 35.9%

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

      2. +-commutative 35.9%

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

   6. Simplified 35.9%

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

      1. add-cbrt-cube 35.9%

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

      2. pow1 35.9%

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

      3. pow1 35.9%

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

      4. pow-prod-up 35.9%

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

      5. metadata-eval 35.9%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt[3]{{\left(\sqrt{1 + y} - \sqrt{y}\right)}^{\color{blue}{2}} \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

   8. Applied egg-rr 35.9%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\sqrt[3]{{\left(\sqrt{1 + y} - \sqrt{y}\right)}^{2} \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)}} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

   9. Taylor expanded in z around inf 22.7%

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

       1. +-commutative 22.7%

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

       2. +-commutative 22.7%

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

   11. Simplified 22.7%

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

   12. Taylor expanded in x around 0 33.6%

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

       1. associate--l+ 56.3%

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

       2. rem-square-sqrt 56.3%

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

       3. hypot-1-def 56.3%

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

   14. Simplified 56.3%

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

   if 1.04999999999999995e223 < z < 1.74999999999999989e253

   1. Initial program 80.7%

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

      1. associate-+l+ 80.7%

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

      2. +-commutative 80.7%

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

      3. associate-+r- 68.8%

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

      4. associate-+l- 61.3%

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

      5. +-commutative 61.3%

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

      6. +-commutative 61.3%

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

      7. associate--l+ 61.3%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in t around inf 41.0%

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

      1. associate--l+ 39.9%

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

      2. +-commutative 39.9%

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

   6. Simplified 39.9%

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

   7. Taylor expanded in z around inf 40.4%

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

      1. +-commutative 40.4%

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

   9. Simplified 40.4%

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

   10. Taylor expanded in y around inf 28.7%

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

       1. flip-- 28.7%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 29.0%

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

       3. add-sqr-sqrt 28.7%

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

       4. +-commutative 28.7%

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

   12. Applied egg-rr 28.7%

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

       1. associate--l+ 36.3%

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

       2. +-inverses 36.3%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 36.3%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{x} + \sqrt{1 + x}} \]

   14. Simplified 36.3%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]

   if 1.74999999999999989e253 < z

   1. Initial program 86.8%

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

      1. associate-+l+ 86.8%

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

      2. +-commutative 86.8%

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

      3. associate-+r- 66.8%

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

      4. associate-+l- 55.9%

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

      5. +-commutative 55.9%

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

      6. +-commutative 55.9%

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

      7. associate--l+ 55.9%

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

   3. Simplified 28.7%

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

   4. Taylor expanded in t around inf 35.5%

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

      1. associate--l+ 36.6%

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

      2. +-commutative 36.6%

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

   6. Simplified 36.6%

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

   7. Taylor expanded in z around inf 36.6%

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

      1. +-commutative 36.6%

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

   9. Simplified 36.6%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in x around 0 41.3%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   11. Step-by-step derivation

       1. associate--l+ 59.0%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   12. Simplified 59.0%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + z} + \left(2 - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+223}:\\ \;\;\;\;1 + \left(\mathsf{hypot}\left(1, \sqrt{y}\right) - \sqrt{y}\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+253}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 15: 84.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+223}:\\ \;\;\;\;1 + \left(\mathsf{hypot}\left(1, \sqrt{y}\right) - \sqrt{y}\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+253}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.2e+15)
   (- (+ (sqrt (+ 1.0 z)) 2.0) (+ (sqrt y) (sqrt z)))
   (if (<= z 1.05e+223)
     (+ 1.0 (- (hypot 1.0 (sqrt y)) (sqrt y)))
     (if (<= z 1.75e+253)
       (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))
       (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.2e+15) {
		tmp = (sqrt((1.0 + z)) + 2.0) - (sqrt(y) + sqrt(z));
	} else if (z <= 1.05e+223) {
		tmp = 1.0 + (hypot(1.0, sqrt(y)) - sqrt(y));
	} else if (z <= 1.75e+253) {
		tmp = 1.0 / (sqrt(x) + sqrt((1.0 + x)));
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.2e+15) {
		tmp = (Math.sqrt((1.0 + z)) + 2.0) - (Math.sqrt(y) + Math.sqrt(z));
	} else if (z <= 1.05e+223) {
		tmp = 1.0 + (Math.hypot(1.0, Math.sqrt(y)) - Math.sqrt(y));
	} else if (z <= 1.75e+253) {
		tmp = 1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.2e+15:
		tmp = (math.sqrt((1.0 + z)) + 2.0) - (math.sqrt(y) + math.sqrt(z))
	elif z <= 1.05e+223:
		tmp = 1.0 + (math.hypot(1.0, math.sqrt(y)) - math.sqrt(y))
	elif z <= 1.75e+253:
		tmp = 1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.2e+15)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) + 2.0) - Float64(sqrt(y) + sqrt(z)));
	elseif (z <= 1.05e+223)
		tmp = Float64(1.0 + Float64(hypot(1.0, sqrt(y)) - sqrt(y)));
	elseif (z <= 1.75e+253)
		tmp = Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x))));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.2e+15)
		tmp = (sqrt((1.0 + z)) + 2.0) - (sqrt(y) + sqrt(z));
	elseif (z <= 1.05e+223)
		tmp = 1.0 + (hypot(1.0, sqrt(y)) - sqrt(y));
	elseif (z <= 1.75e+253)
		tmp = 1.0 / (sqrt(x) + sqrt((1.0 + x)));
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 1.2e+15], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e+223], N[(1.0 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[y], $MachinePrecision] ^ 2], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.75e+253], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < 1.2e15

   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. +-commutative 96.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 82.3%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 60.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 60.6%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 60.6%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 60.6%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 37.9%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 37.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 37.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 37.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 37.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in y around 0 18.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\color{blue}{1} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

   8. Taylor expanded in x around 0 26.6%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   if 1.2e15 < z < 1.04999999999999995e223

   1. Initial program 84.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 84.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 84.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 66.8%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 56.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 56.3%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 56.3%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 56.3%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 33.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 35.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 35.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 35.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. add-cbrt-cube 35.9%

         \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\sqrt[3]{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)}} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

      2. pow1 35.9%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt[3]{\left(\color{blue}{{\left(\sqrt{1 + y} - \sqrt{y}\right)}^{1}} \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

      3. pow1 35.9%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt[3]{\left({\left(\sqrt{1 + y} - \sqrt{y}\right)}^{1} \cdot \color{blue}{{\left(\sqrt{1 + y} - \sqrt{y}\right)}^{1}}\right) \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

      4. pow-prod-up 35.9%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt[3]{\color{blue}{{\left(\sqrt{1 + y} - \sqrt{y}\right)}^{\left(1 + 1\right)}} \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

      5. metadata-eval 35.9%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt[3]{{\left(\sqrt{1 + y} - \sqrt{y}\right)}^{\color{blue}{2}} \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

   8. Applied egg-rr 35.9%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\sqrt[3]{{\left(\sqrt{1 + y} - \sqrt{y}\right)}^{2} \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)}} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

   9. Taylor expanded in z around inf 22.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   10. Step-by-step derivation

       1. +-commutative 22.7%

          \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

       2. +-commutative 22.7%

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

   11. Simplified 22.7%

       \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)} \]

   12. Taylor expanded in x around 0 33.6%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   13. Step-by-step derivation

       1. associate--l+ 56.3%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

       2. rem-square-sqrt 56.3%

          \[\leadsto 1 + \left(\sqrt{1 + \color{blue}{\sqrt{y} \cdot \sqrt{y}}} - \sqrt{y}\right) \]

       3. hypot-1-def 56.3%

          \[\leadsto 1 + \left(\color{blue}{\mathsf{hypot}\left(1, \sqrt{y}\right)} - \sqrt{y}\right) \]

   14. Simplified 56.3%

       \[\leadsto \color{blue}{1 + \left(\mathsf{hypot}\left(1, \sqrt{y}\right) - \sqrt{y}\right)} \]

   if 1.04999999999999995e223 < z < 1.74999999999999989e253

   1. Initial program 80.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 80.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 80.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 68.8%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 61.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 61.3%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 61.3%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 61.3%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 41.0%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 39.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 39.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 39.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 40.4%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 40.4%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 40.4%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 28.7%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 28.7%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 29.0%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. add-sqr-sqrt 28.7%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 28.7%

          \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   12. Applied egg-rr 28.7%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{1 + x}}} \]
   13. Step-by-step derivation

       1. associate--l+ 36.3%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{x} + \sqrt{1 + x}} \]

       2. +-inverses 36.3%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 36.3%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{x} + \sqrt{1 + x}} \]

   14. Simplified 36.3%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]

   if 1.74999999999999989e253 < z

   1. Initial program 86.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 86.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 86.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 66.8%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 55.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 55.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 55.9%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 55.9%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 28.7%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 35.5%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 36.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 36.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 36.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 36.6%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 36.6%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 36.6%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in x around 0 41.3%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   11. Step-by-step derivation

       1. associate--l+ 59.0%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   12. Simplified 59.0%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+223}:\\ \;\;\;\;1 + \left(\mathsf{hypot}\left(1, \sqrt{y}\right) - \sqrt{y}\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+253}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 16: 70.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;1 + \left(\mathsf{hypot}\left(1, \sqrt{y}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.5e+16)
   (+ 1.0 (- (hypot 1.0 (sqrt y)) (sqrt y)))
   (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.5e+16) {
		tmp = 1.0 + (hypot(1.0, sqrt(y)) - sqrt(y));
	} else {
		tmp = 1.0 / (sqrt(x) + sqrt((1.0 + x)));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.5e+16) {
		tmp = 1.0 + (Math.hypot(1.0, Math.sqrt(y)) - Math.sqrt(y));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 1.5e+16:
		tmp = 1.0 + (math.hypot(1.0, math.sqrt(y)) - math.sqrt(y))
	else:
		tmp = 1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.5e+16)
		tmp = Float64(1.0 + Float64(hypot(1.0, sqrt(y)) - sqrt(y)));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.5e+16)
		tmp = 1.0 + (hypot(1.0, sqrt(y)) - sqrt(y));
	else
		tmp = 1.0 / (sqrt(x) + sqrt((1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 1.5e+16], N[(1.0 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[y], $MachinePrecision] ^ 2], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5e16

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 96.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 96.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 67.0%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 62.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 62.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 62.1%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 62.1%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 39.8%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 36.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 38.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 38.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 38.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. add-cbrt-cube 38.2%

         \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\sqrt[3]{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)}} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

      2. pow1 38.2%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt[3]{\left(\color{blue}{{\left(\sqrt{1 + y} - \sqrt{y}\right)}^{1}} \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

      3. pow1 38.2%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt[3]{\left({\left(\sqrt{1 + y} - \sqrt{y}\right)}^{1} \cdot \color{blue}{{\left(\sqrt{1 + y} - \sqrt{y}\right)}^{1}}\right) \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

      4. pow-prod-up 38.2%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt[3]{\color{blue}{{\left(\sqrt{1 + y} - \sqrt{y}\right)}^{\left(1 + 1\right)}} \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

      5. metadata-eval 38.2%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt[3]{{\left(\sqrt{1 + y} - \sqrt{y}\right)}^{\color{blue}{2}} \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

   8. Applied egg-rr 38.2%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\sqrt[3]{{\left(\sqrt{1 + y} - \sqrt{y}\right)}^{2} \cdot \left(\sqrt{1 + y} - \sqrt{y}\right)}} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

   9. Taylor expanded in z around inf 25.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   10. Step-by-step derivation

       1. +-commutative 25.2%

          \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

       2. +-commutative 25.2%

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

   11. Simplified 25.2%

       \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)} \]

   12. Taylor expanded in x around 0 44.8%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   13. Step-by-step derivation

       1. associate--l+ 44.8%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

       2. rem-square-sqrt 44.8%

          \[\leadsto 1 + \left(\sqrt{1 + \color{blue}{\sqrt{y} \cdot \sqrt{y}}} - \sqrt{y}\right) \]

       3. hypot-1-def 44.8%

          \[\leadsto 1 + \left(\color{blue}{\mathsf{hypot}\left(1, \sqrt{y}\right)} - \sqrt{y}\right) \]

   14. Simplified 44.8%

       \[\leadsto \color{blue}{1 + \left(\mathsf{hypot}\left(1, \sqrt{y}\right) - \sqrt{y}\right)} \]

   if 1.5e16 < y

   1. Initial program 84.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 84.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 84.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 84.5%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 54.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 54.8%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 54.8%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 54.8%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 34.1%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 35.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 35.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 35.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 35.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 23.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 23.2%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 23.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 22.6%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 22.6%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 23.1%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. add-sqr-sqrt 22.6%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 22.6%

          \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   12. Applied egg-rr 22.6%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{1 + x}}} \]
   13. Step-by-step derivation

       1. associate--l+ 26.1%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{x} + \sqrt{1 + x}} \]

       2. +-inverses 26.1%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 26.1%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{x} + \sqrt{1 + x}} \]

   14. Simplified 26.1%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;1 + \left(\mathsf{hypot}\left(1, \sqrt{y}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 17: 62.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;y \leq 1.15:\\ \;\;\;\;1 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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 1.15) (+ 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 <= 1.15) {
		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 <= 1.15d0) 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 <= 1.15) {
		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 <= 1.15:
		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 <= 1.15)
		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 <= 1.15)
		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, 1.15], N[(1.0 + t$95$1), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1499999999999999

   1. Initial program 97.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 97.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 67.2%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 62.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 62.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 62.9%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 62.9%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 41.9%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 36.1%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 37.4%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 37.4%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 37.4%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 25.4%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 25.4%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 25.4%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around 0 25.4%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + x}\right) - \sqrt{x}} \]
   11. Step-by-step derivation

       1. associate--l+ 37.5%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

   12. Simplified 37.5%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

   if 1.1499999999999999 < y

   1. Initial program 84.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 84.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 84.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 82.7%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 54.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 54.7%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 54.7%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 54.7%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 32.5%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 36.5%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 36.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 36.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 36.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 23.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 23.2%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 23.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 22.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15:\\ \;\;\;\;1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 18: 70.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.5e+15)
   (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))
   (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.5e+15) {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	} else {
		tmp = 1.0 / (sqrt(x) + sqrt((1.0 + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 4.5d+15) then
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    else
        tmp = 1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.5e+15) {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 4.5e+15:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	else:
		tmp = 1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.5e+15)
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 4.5e+15)
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	else
		tmp = 1.0 / (sqrt(x) + sqrt((1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 4.5e+15], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5e15

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 96.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 96.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 67.0%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 62.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 62.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 62.1%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 62.1%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 39.8%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 36.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 38.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 38.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 38.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 25.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 25.2%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 25.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in x around 0 44.8%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   11. Step-by-step derivation

       1. associate--l+ 44.8%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   12. Simplified 44.8%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   if 4.5e15 < y

   1. Initial program 84.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 84.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 84.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 84.5%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 54.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 54.8%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 54.8%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 54.8%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 34.1%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 35.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 35.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 35.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 35.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 23.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 23.2%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 23.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 22.6%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 22.6%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 23.1%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. add-sqr-sqrt 22.6%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 22.6%

          \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   12. Applied egg-rr 22.6%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{1 + x}}} \]
   13. Step-by-step derivation

       1. associate--l+ 26.1%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{x} + \sqrt{1 + x}} \]

       2. +-inverses 26.1%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 26.1%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{x} + \sqrt{1 + x}} \]

   14. Simplified 26.1%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 19: 64.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + \left(\sqrt{1 + y} - \sqrt{y}\right) \end{array} \]

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 (+ 1.0 y)) (sqrt y))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + (sqrt((1.0 + y)) - 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((1.0d0 + y)) - 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((1.0 + y)) - Math.sqrt(y));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 + (math.sqrt((1.0 + y)) - 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(1.0 + y)) - 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((1.0 + y)) - 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[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.9%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 90.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. +-commutative 90.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. associate-+r- 75.0%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 58.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. +-commutative 58.8%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

   6. +-commutative 58.8%

      \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

   7. associate--l+ 58.8%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

3. Simplified 37.2%

   \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

4. Taylor expanded in t around inf 36.3%

   \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
5. Step-by-step derivation

   1. associate--l+ 37.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

   2. +-commutative 37.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

6. Simplified 37.2%

   \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

7. Taylor expanded in z around inf 24.3%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative 24.3%

      \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

9. Simplified 24.3%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

10. Taylor expanded in x around 0 27.1%

    \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
11. Step-by-step derivation

    1. associate--l+ 45.4%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

12. Simplified 45.4%

    \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

13. Final simplification 45.4%

    \[\leadsto 1 + \left(\sqrt{1 + y} - \sqrt{y}\right) \]

Alternative 20: 35.7% 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 90.9%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 90.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. +-commutative 90.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. associate-+r- 75.0%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 58.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. +-commutative 58.8%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

   6. +-commutative 58.8%

      \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

   7. associate--l+ 58.8%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

3. Simplified 37.2%

   \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

4. Taylor expanded in t around inf 36.3%

   \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
5. Step-by-step derivation

   1. associate--l+ 37.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

   2. +-commutative 37.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

6. Simplified 37.2%

   \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

7. Taylor expanded in z around inf 24.3%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative 24.3%

      \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

9. Simplified 24.3%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

10. Taylor expanded in y around inf 16.8%

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

11. Final simplification 16.8%

    \[\leadsto \sqrt{1 + x} - \sqrt{x} \]

Developer target: 99.5% 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 2023275 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :herbie-target
  (+ (+ (+ (/ 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))))