Main:z from

Percentage Accurate: 91.6% → 99.8%

Time: 27.6s

Alternatives: 20

Speedup: 1.6×

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: 91.6% 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

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ t_3 := \frac{1}{\sqrt{z} + \sqrt{1 + z}}\\ \mathbf{if}\;t_2 - \sqrt{y} \leq 0.999995:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1} + \left(\frac{1}{t_2 + \sqrt{y}} + t_3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t_2 + \left(t_1 - \sqrt{y}\right)\right) - \sqrt{x}\right) + \left(t_3 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z))))))
   (if (<= (- t_2 (sqrt y)) 0.999995)
     (+ (/ 1.0 (+ (sqrt x) t_1)) (+ (/ 1.0 (+ t_2 (sqrt y))) t_3))
     (+
      (- (+ t_2 (- t_1 (sqrt y))) (sqrt x))
      (+ t_3 (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + y));
	double t_3 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	double tmp;
	if ((t_2 - sqrt(y)) <= 0.999995) {
		tmp = (1.0 / (sqrt(x) + t_1)) + ((1.0 / (t_2 + sqrt(y))) + t_3);
	} else {
		tmp = ((t_2 + (t_1 - sqrt(y))) - sqrt(x)) + (t_3 + (1.0 / (sqrt((1.0 + t)) + sqrt(t))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + y))
    t_3 = 1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))
    if ((t_2 - sqrt(y)) <= 0.999995d0) then
        tmp = (1.0d0 / (sqrt(x) + t_1)) + ((1.0d0 / (t_2 + sqrt(y))) + t_3)
    else
        tmp = ((t_2 + (t_1 - sqrt(y))) - sqrt(x)) + (t_3 + (1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + y));
	double t_3 = 1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)));
	double tmp;
	if ((t_2 - Math.sqrt(y)) <= 0.999995) {
		tmp = (1.0 / (Math.sqrt(x) + t_1)) + ((1.0 / (t_2 + Math.sqrt(y))) + t_3);
	} else {
		tmp = ((t_2 + (t_1 - Math.sqrt(y))) - Math.sqrt(x)) + (t_3 + (1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + y))
	t_3 = 1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))
	tmp = 0
	if (t_2 - math.sqrt(y)) <= 0.999995:
		tmp = (1.0 / (math.sqrt(x) + t_1)) + ((1.0 / (t_2 + math.sqrt(y))) + t_3)
	else:
		tmp = ((t_2 + (t_1 - math.sqrt(y))) - math.sqrt(x)) + (t_3 + (1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))
	tmp = 0.0
	if (Float64(t_2 - sqrt(y)) <= 0.999995)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_1)) + Float64(Float64(1.0 / Float64(t_2 + sqrt(y))) + t_3));
	else
		tmp = Float64(Float64(Float64(t_2 + Float64(t_1 - sqrt(y))) - sqrt(x)) + Float64(t_3 + Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((1.0 + y));
	t_3 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	tmp = 0.0;
	if ((t_2 - sqrt(y)) <= 0.999995)
		tmp = (1.0 / (sqrt(x) + t_1)) + ((1.0 / (t_2 + sqrt(y))) + t_3);
	else
		tmp = ((t_2 + (t_1 - sqrt(y))) - sqrt(x)) + (t_3 + (1.0 / (sqrt((1.0 + t)) + sqrt(t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 0.999995], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)) < 0.99999499999999997

   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. associate-+l+ 87.7%

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

      3. +-commutative 87.7%

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

      4. +-commutative 87.7%

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

      5. +-commutative 87.7%

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

   3. Simplified 87.7%

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

      1. flip-- 87.7%

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

      2. add-sqr-sqrt 69.3%

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

      3. add-sqr-sqrt 88.3%

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

   5. Applied egg-rr 88.3%

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

      1. +-commutative 88.3%

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

      2. associate--l+ 91.5%

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

      3. +-inverses 91.5%

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

      4. metadata-eval 91.5%

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

      5. +-commutative 91.5%

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

      6. +-commutative 91.5%

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

   7. Simplified 91.5%

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

      1. flip-- 91.6%

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

      2. add-sqr-sqrt 56.6%

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

      3. add-sqr-sqrt 91.7%

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

   9. Applied egg-rr 91.7%

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

       1. associate--l+ 94.7%

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

       2. +-inverses 94.7%

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

       3. metadata-eval 94.7%

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

   11. Simplified 94.7%

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

       1. flip-- 94.7%

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

       2. add-sqr-sqrt 74.9%

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

       3. add-sqr-sqrt 95.9%

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

   13. Applied egg-rr 95.9%

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

       1. associate--l+ 98.2%

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

       2. +-inverses 98.2%

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

       3. metadata-eval 98.2%

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

       4. +-commutative 98.2%

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

       5. +-commutative 98.2%

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

   15. Simplified 98.2%

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

   16. Taylor expanded in t around inf 54.2%

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

   if 0.99999499999999997 < (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))

   1. Initial program 96.4%

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

      1. associate-+l+ 96.4%

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

      2. +-commutative 96.4%

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

      3. associate-+r- 60.1%

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

      4. associate-+l- 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 39.4%

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

      1. associate-+r- 39.4%

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

   5. Applied egg-rr 39.4%

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

      1. associate-+l- 39.8%

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

      2. associate--r- 43.7%

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

      3. associate-+r- 43.6%

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

      4. +-commutative 43.6%

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

      5. associate--l+ 43.7%

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

      6. +-commutative 43.7%

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

      7. associate-+l- 49.8%

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

      8. associate--r- 60.1%

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

   7. Simplified 60.1%

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

      1. +-commutative 60.1%

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

      2. flip-- 60.4%

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

      3. add-sqr-sqrt 47.4%

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

      4. add-sqr-sqrt 60.4%

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

   9. Applied egg-rr 60.4%

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

       1. associate--l+ 60.7%

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

       2. +-inverses 60.7%

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

       3. metadata-eval 60.7%

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

   11. Simplified 60.7%

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

       1. +-commutative 60.7%

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

       2. flip-- 60.7%

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

       3. add-sqr-sqrt 48.5%

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

       4. add-sqr-sqrt 60.8%

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

   13. Applied egg-rr 60.8%

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

       1. associate--l+ 61.5%

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

       2. +-inverses 61.5%

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

       3. metadata-eval 61.5%

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

   15. Simplified 61.5%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 0.999995:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + x} - \sqrt{y}\right)\right) - \sqrt{x}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := t_1 - \sqrt{z}\\ t_3 := \sqrt{1 + x}\\ t_4 := t_3 - \sqrt{x}\\ t_5 := \sqrt{1 + y}\\ t_6 := t_5 - \sqrt{y}\\ \mathbf{if}\;\left(t_4 + t_6\right) + t_2 \leq 2.1:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_3} + \left(\frac{1}{t_5 + \sqrt{y}} + \frac{1}{\sqrt{z} + t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 + \left(t_6 + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + t_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z)))
        (t_2 (- t_1 (sqrt z)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (- t_3 (sqrt x)))
        (t_5 (sqrt (+ 1.0 y)))
        (t_6 (- t_5 (sqrt y))))
   (if (<= (+ (+ t_4 t_6) t_2) 2.1)
     (+
      (/ 1.0 (+ (sqrt x) t_3))
      (+ (/ 1.0 (+ t_5 (sqrt y))) (/ 1.0 (+ (sqrt z) t_1))))
     (+ t_4 (+ t_6 (+ (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t))) 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));
	double t_2 = t_1 - sqrt(z);
	double t_3 = sqrt((1.0 + x));
	double t_4 = t_3 - sqrt(x);
	double t_5 = sqrt((1.0 + y));
	double t_6 = t_5 - sqrt(y);
	double tmp;
	if (((t_4 + t_6) + t_2) <= 2.1) {
		tmp = (1.0 / (sqrt(x) + t_3)) + ((1.0 / (t_5 + sqrt(y))) + (1.0 / (sqrt(z) + t_1)));
	} else {
		tmp = t_4 + (t_6 + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + t_2));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = t_1 - sqrt(z)
    t_3 = sqrt((1.0d0 + x))
    t_4 = t_3 - sqrt(x)
    t_5 = sqrt((1.0d0 + y))
    t_6 = t_5 - sqrt(y)
    if (((t_4 + t_6) + t_2) <= 2.1d0) then
        tmp = (1.0d0 / (sqrt(x) + t_3)) + ((1.0d0 / (t_5 + sqrt(y))) + (1.0d0 / (sqrt(z) + t_1)))
    else
        tmp = t_4 + (t_6 + ((1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t))) + 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));
	double t_2 = t_1 - Math.sqrt(z);
	double t_3 = Math.sqrt((1.0 + x));
	double t_4 = t_3 - Math.sqrt(x);
	double t_5 = Math.sqrt((1.0 + y));
	double t_6 = t_5 - Math.sqrt(y);
	double tmp;
	if (((t_4 + t_6) + t_2) <= 2.1) {
		tmp = (1.0 / (Math.sqrt(x) + t_3)) + ((1.0 / (t_5 + Math.sqrt(y))) + (1.0 / (Math.sqrt(z) + t_1)));
	} else {
		tmp = t_4 + (t_6 + ((1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))) + 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))
	t_2 = t_1 - math.sqrt(z)
	t_3 = math.sqrt((1.0 + x))
	t_4 = t_3 - math.sqrt(x)
	t_5 = math.sqrt((1.0 + y))
	t_6 = t_5 - math.sqrt(y)
	tmp = 0
	if ((t_4 + t_6) + t_2) <= 2.1:
		tmp = (1.0 / (math.sqrt(x) + t_3)) + ((1.0 / (t_5 + math.sqrt(y))) + (1.0 / (math.sqrt(z) + t_1)))
	else:
		tmp = t_4 + (t_6 + ((1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))) + t_2))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = Float64(t_1 - sqrt(z))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = Float64(t_3 - sqrt(x))
	t_5 = sqrt(Float64(1.0 + y))
	t_6 = Float64(t_5 - sqrt(y))
	tmp = 0.0
	if (Float64(Float64(t_4 + t_6) + t_2) <= 2.1)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_3)) + Float64(Float64(1.0 / Float64(t_5 + sqrt(y))) + Float64(1.0 / Float64(sqrt(z) + t_1))));
	else
		tmp = Float64(t_4 + Float64(t_6 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))) + 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));
	t_2 = t_1 - sqrt(z);
	t_3 = sqrt((1.0 + x));
	t_4 = t_3 - sqrt(x);
	t_5 = sqrt((1.0 + y));
	t_6 = t_5 - sqrt(y);
	tmp = 0.0;
	if (((t_4 + t_6) + t_2) <= 2.1)
		tmp = (1.0 / (sqrt(x) + t_3)) + ((1.0 / (t_5 + sqrt(y))) + (1.0 / (sqrt(z) + t_1)));
	else
		tmp = t_4 + (t_6 + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + t_2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$4 + t$95$6), $MachinePrecision] + t$95$2), $MachinePrecision], 2.1], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$5 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 + N[(t$95$6 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))) < 2.10000000000000009

   1. Initial program 91.4%

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

      1. associate-+l+ 91.4%

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

      2. associate-+l+ 91.4%

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

      3. +-commutative 91.4%

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

      4. +-commutative 91.4%

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

      5. +-commutative 91.4%

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

   3. Simplified 91.4%

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

      1. flip-- 91.4%

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

      2. add-sqr-sqrt 70.3%

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

      3. add-sqr-sqrt 92.2%

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

   5. Applied egg-rr 92.2%

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

      1. +-commutative 92.2%

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

      2. associate--l+ 94.2%

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

      3. +-inverses 94.2%

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

      4. metadata-eval 94.2%

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

      5. +-commutative 94.2%

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

      6. +-commutative 94.2%

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

   7. Simplified 94.2%

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

      1. flip-- 94.3%

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

      2. add-sqr-sqrt 74.9%

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

      3. add-sqr-sqrt 94.3%

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

   9. Applied egg-rr 94.3%

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

       1. associate--l+ 96.0%

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

       2. +-inverses 96.0%

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

       3. metadata-eval 96.0%

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

   11. Simplified 96.0%

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

       1. flip-- 96.4%

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

       2. add-sqr-sqrt 73.5%

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

       3. add-sqr-sqrt 97.2%

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

   13. Applied egg-rr 97.2%

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

       1. associate--l+ 98.6%

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

       2. +-inverses 98.6%

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

       3. metadata-eval 98.6%

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

       4. +-commutative 98.6%

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

       5. +-commutative 98.6%

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

   15. Simplified 98.6%

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

   16. Taylor expanded in t around inf 54.1%

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

   if 2.10000000000000009 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z)))

   1. Initial program 98.1%

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

      1. associate-+l+ 98.1%

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

      2. associate-+l+ 98.1%

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

      3. +-commutative 98.1%

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

      4. +-commutative 98.1%

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

      5. +-commutative 98.1%

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

   3. Simplified 98.1%

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

      1. flip-- 98.3%

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

      2. add-sqr-sqrt 70.1%

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

      3. add-sqr-sqrt 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. associate--l+ 99.8%

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

      2. +-inverses 99.8%

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

      3. metadata-eval 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2.1:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

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

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))) + ((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + (sqrt((1.0 + t)) - sqrt(t))));
}

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

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))) + ((1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))) + (math.sqrt((1.0 + t)) - math.sqrt(t))))

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(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z)))) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t)))))
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))) + ((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + (sqrt((1.0 + t)) - sqrt(t))));
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[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.2%

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

   1. associate-+l+ 92.2%

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

   2. associate-+l+ 92.2%

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

   3. +-commutative 92.2%

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

   4. +-commutative 92.2%

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

   5. +-commutative 92.2%

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

3. Simplified 92.2%

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

   1. flip-- 92.2%

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

   2. add-sqr-sqrt 73.9%

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

   3. add-sqr-sqrt 93.0%

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

5. Applied egg-rr 93.0%

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

   1. +-commutative 93.0%

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

   2. associate--l+ 94.7%

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

   3. +-inverses 94.7%

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

   4. metadata-eval 94.7%

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

   5. +-commutative 94.7%

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

   6. +-commutative 94.7%

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

7. Simplified 94.7%

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

   1. flip-- 94.8%

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

   2. add-sqr-sqrt 77.9%

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

   3. add-sqr-sqrt 94.8%

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

9. Applied egg-rr 94.8%

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

    1. associate--l+ 96.2%

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

    2. +-inverses 96.2%

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

    3. metadata-eval 96.2%

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

11. Simplified 96.2%

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

    1. flip-- 96.6%

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

    2. add-sqr-sqrt 76.7%

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

    3. add-sqr-sqrt 97.3%

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

13. Applied egg-rr 97.3%

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

    1. associate--l+ 98.5%

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

    2. +-inverses 98.5%

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

    3. metadata-eval 98.5%

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

    4. +-commutative 98.5%

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

    5. +-commutative 98.5%

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

15. Simplified 98.5%

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

16. Final simplification 98.5%

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

Alternative 4: 97.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + x}\\ t_4 := t_3 - \sqrt{x}\\ \mathbf{if}\;y \leq 9 \cdot 10^{-86}:\\ \;\;\;\;t_4 + \left(1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(t_2 - \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 10^{+33}:\\ \;\;\;\;t_4 + \left(\frac{1}{t_1 + \sqrt{y}} + \frac{1}{\sqrt{z} + t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_3} + \left(\left(t_1 - \sqrt{y}\right) - \left(\sqrt{z} - t_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (- t_3 (sqrt x))))
   (if (<= y 9e-86)
     (+ t_4 (+ 1.0 (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (- t_2 (sqrt z)))))
     (if (<= y 1e+33)
       (+ t_4 (+ (/ 1.0 (+ t_1 (sqrt y))) (/ 1.0 (+ (sqrt z) t_2))))
       (+ (/ 1.0 (+ (sqrt x) t_3)) (- (- t_1 (sqrt y)) (- (sqrt z) t_2)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + x));
	double t_4 = t_3 - sqrt(x);
	double tmp;
	if (y <= 9e-86) {
		tmp = t_4 + (1.0 + ((sqrt((1.0 + t)) - sqrt(t)) + (t_2 - sqrt(z))));
	} else if (y <= 1e+33) {
		tmp = t_4 + ((1.0 / (t_1 + sqrt(y))) + (1.0 / (sqrt(z) + t_2)));
	} else {
		tmp = (1.0 / (sqrt(x) + t_3)) + ((t_1 - sqrt(y)) - (sqrt(z) - t_2));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + z))
    t_3 = sqrt((1.0d0 + x))
    t_4 = t_3 - sqrt(x)
    if (y <= 9d-86) then
        tmp = t_4 + (1.0d0 + ((sqrt((1.0d0 + t)) - sqrt(t)) + (t_2 - sqrt(z))))
    else if (y <= 1d+33) then
        tmp = t_4 + ((1.0d0 / (t_1 + sqrt(y))) + (1.0d0 / (sqrt(z) + t_2)))
    else
        tmp = (1.0d0 / (sqrt(x) + t_3)) + ((t_1 - sqrt(y)) - (sqrt(z) - t_2))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = Math.sqrt((1.0 + x));
	double t_4 = t_3 - Math.sqrt(x);
	double tmp;
	if (y <= 9e-86) {
		tmp = t_4 + (1.0 + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (t_2 - Math.sqrt(z))));
	} else if (y <= 1e+33) {
		tmp = t_4 + ((1.0 / (t_1 + Math.sqrt(y))) + (1.0 / (Math.sqrt(z) + t_2)));
	} else {
		tmp = (1.0 / (Math.sqrt(x) + t_3)) + ((t_1 - Math.sqrt(y)) - (Math.sqrt(z) - t_2));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((1.0 + z))
	t_3 = math.sqrt((1.0 + x))
	t_4 = t_3 - math.sqrt(x)
	tmp = 0
	if y <= 9e-86:
		tmp = t_4 + (1.0 + ((math.sqrt((1.0 + t)) - math.sqrt(t)) + (t_2 - math.sqrt(z))))
	elif y <= 1e+33:
		tmp = t_4 + ((1.0 / (t_1 + math.sqrt(y))) + (1.0 / (math.sqrt(z) + t_2)))
	else:
		tmp = (1.0 / (math.sqrt(x) + t_3)) + ((t_1 - math.sqrt(y)) - (math.sqrt(z) - t_2))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = Float64(t_3 - sqrt(x))
	tmp = 0.0
	if (y <= 9e-86)
		tmp = Float64(t_4 + Float64(1.0 + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(t_2 - sqrt(z)))));
	elseif (y <= 1e+33)
		tmp = Float64(t_4 + Float64(Float64(1.0 / Float64(t_1 + sqrt(y))) + Float64(1.0 / Float64(sqrt(z) + t_2))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_3)) + Float64(Float64(t_1 - sqrt(y)) - Float64(sqrt(z) - t_2)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((1.0 + z));
	t_3 = sqrt((1.0 + x));
	t_4 = t_3 - sqrt(x);
	tmp = 0.0;
	if (y <= 9e-86)
		tmp = t_4 + (1.0 + ((sqrt((1.0 + t)) - sqrt(t)) + (t_2 - sqrt(z))));
	elseif (y <= 1e+33)
		tmp = t_4 + ((1.0 / (t_1 + sqrt(y))) + (1.0 / (sqrt(z) + t_2)));
	else
		tmp = (1.0 / (sqrt(x) + t_3)) + ((t_1 - sqrt(y)) - (sqrt(z) - t_2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 9e-86], N[(t$95$4 + N[(1.0 + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+33], N[(t$95$4 + N[(N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.9999999999999995e-86

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. associate-+l+ 96.2%

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

      3. +-commutative 96.2%

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

      4. +-commutative 96.2%

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

      5. +-commutative 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in y around 0 96.2%

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

   if 8.9999999999999995e-86 < y < 9.9999999999999995e32

   1. Initial program 93.3%

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

      1. associate-+l+ 93.3%

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

      2. associate-+l+ 93.3%

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

      3. +-commutative 93.3%

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

      4. +-commutative 93.3%

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

      5. +-commutative 93.3%

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

   3. Simplified 93.3%

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

      1. flip-- 93.8%

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

      2. add-sqr-sqrt 88.5%

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

      3. add-sqr-sqrt 93.8%

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

   5. Applied egg-rr 93.6%

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

      1. associate--l+ 95.6%

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

      2. +-inverses 95.6%

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

      3. metadata-eval 95.6%

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

   7. Simplified 95.0%

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

   8. Taylor expanded in t around inf 55.8%

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

      1. flip-- 95.6%

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

      2. add-sqr-sqrt 77.5%

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

      3. add-sqr-sqrt 97.8%

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

   10. Applied egg-rr 57.3%

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

       1. associate--l+ 97.8%

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

       2. +-inverses 97.8%

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

       3. metadata-eval 97.8%

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

       4. +-commutative 97.8%

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

       5. +-commutative 97.8%

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

   12. Simplified 57.3%

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

   if 9.9999999999999995e32 < y

   1. Initial program 88.4%

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

      1. associate-+l+ 88.4%

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

      2. associate-+l+ 88.4%

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

      3. +-commutative 88.4%

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

      4. +-commutative 88.4%

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

      5. +-commutative 88.4%

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

   3. Simplified 88.4%

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

      1. flip-- 88.3%

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

      2. add-sqr-sqrt 69.7%

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

      3. add-sqr-sqrt 89.0%

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

   5. Applied egg-rr 89.0%

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

      1. +-commutative 89.0%

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

      2. associate--l+ 92.5%

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

      3. +-inverses 92.5%

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

      4. metadata-eval 92.5%

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

      5. +-commutative 92.5%

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

      6. +-commutative 92.5%

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

   7. Simplified 92.5%

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

   8. Taylor expanded in t around inf 48.7%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-86}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 10^{+33}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\\ \end{array} \]

Alternative 5: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 8.5 \cdot 10^{+15}:\\ \;\;\;\;2 + \left(t_1 + \left(\mathsf{hypot}\left(1, \sqrt{t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{z} + t_1}\right)\\ \end{array} \end{array} \]

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 8.5e+15)
     (+ 2.0 (+ t_1 (- (hypot 1.0 (sqrt t)) (+ (sqrt z) (sqrt t)))))
     (+
      (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))
      (+ (/ 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 tmp;
	if (t <= 8.5e+15) {
		tmp = 2.0 + (t_1 + (hypot(1.0, sqrt(t)) - (sqrt(z) + sqrt(t))));
	} else {
		tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (1.0 / (sqrt(z) + t_1)));
	}
	return tmp;
}

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 <= 8.5e+15) {
		tmp = 2.0 + (t_1 + (Math.hypot(1.0, Math.sqrt(t)) - (Math.sqrt(z) + Math.sqrt(t))));
	} else {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + ((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))
	tmp = 0
	if t <= 8.5e+15:
		tmp = 2.0 + (t_1 + (math.hypot(1.0, math.sqrt(t)) - (math.sqrt(z) + math.sqrt(t))))
	else:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + ((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))
	tmp = 0.0
	if (t <= 8.5e+15)
		tmp = Float64(2.0 + Float64(t_1 + Float64(hypot(1.0, sqrt(t)) - Float64(sqrt(z) + sqrt(t)))));
	else
		tmp = 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(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));
	tmp = 0.0;
	if (t <= 8.5e+15)
		tmp = 2.0 + (t_1 + (hypot(1.0, sqrt(t)) - (sqrt(z) + sqrt(t))));
	else
		tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) + ((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]}, If[LessEqual[t, 8.5e+15], N[(2.0 + N[(t$95$1 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[t], $MachinePrecision] ^ 2], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 8.5e15

   1. Initial program 97.4%

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

      1. associate-+l+ 97.4%

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

      2. +-commutative 97.4%

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

      3. associate-+r- 79.6%

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

      4. associate-+l- 49.8%

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

      5. +-commutative 49.8%

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

      6. +-commutative 49.8%

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

      7. associate--l+ 49.8%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in y around 0 18.3%

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

      1. associate--l+ 38.6%

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

      2. +-commutative 38.6%

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

      3. associate--l+ 38.6%

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

      4. +-commutative 38.6%

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

      5. +-commutative 38.6%

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

      6. +-commutative 38.6%

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

      7. +-commutative 38.6%

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

   6. Simplified 38.6%

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

      1. expm1-log1p-u 18.3%

         \[\leadsto 1 + \left(\left(\sqrt{z + 1} + \sqrt{1 + x}\right) + \left(\sqrt{t + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{t} + \left(\sqrt{z} + \sqrt{x}\right)\right)\right)}\right)\right) \]

   8. Applied egg-rr 18.3%

      \[\leadsto 1 + \left(\left(\sqrt{z + 1} + \sqrt{1 + x}\right) + \left(\sqrt{t + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{t} + \left(\sqrt{z} + \sqrt{x}\right)\right)\right)}\right)\right) \]

   9. Taylor expanded in x around 0 21.1%

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

       1. associate--l+ 43.9%

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

       2. +-commutative 43.9%

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

       3. associate--l+ 43.9%

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

       4. rem-square-sqrt 43.9%

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

       5. hypot-1-def 43.9%

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

   11. Simplified 43.9%

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

   if 8.5e15 < 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. associate-+l+ 86.1%

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

      3. +-commutative 86.1%

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

      4. +-commutative 86.1%

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

      5. +-commutative 86.1%

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

   3. Simplified 86.1%

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

      1. flip-- 86.1%

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

      2. add-sqr-sqrt 70.5%

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

      3. add-sqr-sqrt 86.7%

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

   5. Applied egg-rr 86.7%

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

      1. +-commutative 86.7%

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

      2. associate--l+ 90.1%

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

      3. +-inverses 90.1%

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

      4. metadata-eval 90.1%

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

      5. +-commutative 90.1%

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

      6. +-commutative 90.1%

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

   7. Simplified 90.1%

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

      1. flip-- 90.2%

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

      2. add-sqr-sqrt 74.8%

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

      3. add-sqr-sqrt 90.2%

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

   9. Applied egg-rr 90.2%

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

       1. associate--l+ 93.0%

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

       2. +-inverses 93.0%

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

       3. metadata-eval 93.0%

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

   11. Simplified 93.0%

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

       1. flip-- 93.4%

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

       2. add-sqr-sqrt 72.1%

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

       3. add-sqr-sqrt 94.6%

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

   13. Applied egg-rr 94.6%

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

       1. associate--l+ 97.2%

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

       2. +-inverses 97.2%

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

       3. metadata-eval 97.2%

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

       4. +-commutative 97.2%

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

       5. +-commutative 97.2%

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

   15. Simplified 97.2%

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

   16. Taylor expanded in t around inf 97.2%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{+15}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} + \left(\mathsf{hypot}\left(1, \sqrt{t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \end{array} \]

Alternative 6: 93.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 1.06 \cdot 10^{-43}:\\ \;\;\;\;2 + \left(t_2 + \left(\mathsf{hypot}\left(1, \sqrt{t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+29}:\\ \;\;\;\;t_1 + \left(\left(t_3 - \sqrt{y}\right) - \left(\sqrt{x} + \frac{-1}{\sqrt{z} + t_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_3 + \sqrt{y}} + \left(t_1 - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))) (t_2 (sqrt (+ 1.0 z))) (t_3 (sqrt (+ 1.0 y))))
   (if (<= z 1.06e-43)
     (+ 2.0 (+ t_2 (- (hypot 1.0 (sqrt t)) (+ (sqrt z) (sqrt t)))))
     (if (<= z 1.25e+29)
       (+ t_1 (- (- t_3 (sqrt y)) (+ (sqrt x) (/ -1.0 (+ (sqrt z) t_2)))))
       (+ (/ 1.0 (+ t_3 (sqrt y))) (- t_1 (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + y));
	double tmp;
	if (z <= 1.06e-43) {
		tmp = 2.0 + (t_2 + (hypot(1.0, sqrt(t)) - (sqrt(z) + sqrt(t))));
	} else if (z <= 1.25e+29) {
		tmp = t_1 + ((t_3 - sqrt(y)) - (sqrt(x) + (-1.0 / (sqrt(z) + t_2))));
	} else {
		tmp = (1.0 / (t_3 + sqrt(y))) + (t_1 - 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 t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 1.06e-43) {
		tmp = 2.0 + (t_2 + (Math.hypot(1.0, Math.sqrt(t)) - (Math.sqrt(z) + Math.sqrt(t))));
	} else if (z <= 1.25e+29) {
		tmp = t_1 + ((t_3 - Math.sqrt(y)) - (Math.sqrt(x) + (-1.0 / (Math.sqrt(z) + t_2))));
	} else {
		tmp = (1.0 / (t_3 + Math.sqrt(y))) + (t_1 - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + z))
	t_3 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 1.06e-43:
		tmp = 2.0 + (t_2 + (math.hypot(1.0, math.sqrt(t)) - (math.sqrt(z) + math.sqrt(t))))
	elif z <= 1.25e+29:
		tmp = t_1 + ((t_3 - math.sqrt(y)) - (math.sqrt(x) + (-1.0 / (math.sqrt(z) + t_2))))
	else:
		tmp = (1.0 / (t_3 + math.sqrt(y))) + (t_1 - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 1.06e-43)
		tmp = Float64(2.0 + Float64(t_2 + Float64(hypot(1.0, sqrt(t)) - Float64(sqrt(z) + sqrt(t)))));
	elseif (z <= 1.25e+29)
		tmp = Float64(t_1 + Float64(Float64(t_3 - sqrt(y)) - Float64(sqrt(x) + Float64(-1.0 / Float64(sqrt(z) + t_2)))));
	else
		tmp = Float64(Float64(1.0 / Float64(t_3 + sqrt(y))) + Float64(t_1 - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((1.0 + z));
	t_3 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 1.06e-43)
		tmp = 2.0 + (t_2 + (hypot(1.0, sqrt(t)) - (sqrt(z) + sqrt(t))));
	elseif (z <= 1.25e+29)
		tmp = t_1 + ((t_3 - sqrt(y)) - (sqrt(x) + (-1.0 / (sqrt(z) + t_2))));
	else
		tmp = (1.0 / (t_3 + sqrt(y))) + (t_1 - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1.06e-43], N[(2.0 + N[(t$95$2 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[t], $MachinePrecision] ^ 2], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.25e+29], N[(t$95$1 + N[(N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(-1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(t$95$3 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.05999999999999994e-43

   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- 77.0%

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

      4. associate-+l- 46.5%

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

      5. +-commutative 46.5%

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

      6. +-commutative 46.5%

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

      7. associate--l+ 46.5%

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

   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 y around 0 19.2%

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

      1. associate--l+ 39.3%

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

      2. +-commutative 39.3%

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

      3. associate--l+ 42.3%

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

      4. +-commutative 42.3%

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

      5. +-commutative 42.3%

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

      6. +-commutative 42.3%

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

      7. +-commutative 42.3%

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

   6. Simplified 42.3%

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

      1. expm1-log1p-u 18.5%

         \[\leadsto 1 + \left(\left(\sqrt{z + 1} + \sqrt{1 + x}\right) + \left(\sqrt{t + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{t} + \left(\sqrt{z} + \sqrt{x}\right)\right)\right)}\right)\right) \]

   8. Applied egg-rr 18.5%

      \[\leadsto 1 + \left(\left(\sqrt{z + 1} + \sqrt{1 + x}\right) + \left(\sqrt{t + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{t} + \left(\sqrt{z} + \sqrt{x}\right)\right)\right)}\right)\right) \]

   9. Taylor expanded in x around 0 22.6%

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

       1. associate--l+ 46.8%

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

       2. +-commutative 46.8%

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

       3. associate--l+ 37.3%

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

       4. rem-square-sqrt 37.3%

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

       5. hypot-1-def 37.3%

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

   11. Simplified 37.3%

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

   if 1.05999999999999994e-43 < z < 1.25e29

   1. Initial program 80.9%

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

      1. associate-+l+ 80.9%

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

      2. +-commutative 80.9%

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

      3. associate-+r- 63.8%

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

      4. associate-+l- 41.5%

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

      5. +-commutative 41.5%

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

      6. +-commutative 41.5%

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

      7. associate--l+ 41.5%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in t around inf 23.5%

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

      1. associate--l+ 25.1%

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

      2. +-commutative 25.1%

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

   6. Simplified 25.1%

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

      1. flip-- 22.7%

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

      2. add-sqr-sqrt 22.7%

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

      3. add-sqr-sqrt 22.7%

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

      4. +-commutative 22.7%

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

      5. +-commutative 22.7%

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

   8. Applied egg-rr 26.0%

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

      1. +-commutative 22.7%

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

      2. associate--r+ 22.7%

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

      3. +-inverses 22.7%

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

      4. metadata-eval 22.7%

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

      5. +-commutative 22.7%

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

   10. Simplified 26.0%

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

   if 1.25e29 < z

   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. associate-+l+ 89.9%

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

      3. +-commutative 89.9%

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

      4. +-commutative 89.9%

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

      5. +-commutative 89.9%

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

   3. Simplified 89.9%

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

      1. flip-- 93.4%

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

      2. add-sqr-sqrt 79.8%

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

      3. add-sqr-sqrt 93.4%

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

   5. Applied egg-rr 90.1%

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

      1. associate--l+ 96.5%

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

      2. +-inverses 96.5%

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

      3. metadata-eval 96.5%

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

   7. Simplified 93.2%

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

   8. Taylor expanded in t around inf 44.0%

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

   9. Taylor expanded in z around inf 29.4%

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

       1. +-commutative 29.4%

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

       2. +-commutative 29.4%

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

       3. associate--l+ 44.0%

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

       4. +-commutative 44.0%

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

       5. +-commutative 44.0%

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

   11. Simplified 44.0%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.06 \cdot 10^{-43}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} + \left(\mathsf{hypot}\left(1, \sqrt{t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \frac{-1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \end{array} \]

Alternative 7: 96.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + x}\\ t_3 := t_2 - \sqrt{x}\\ \mathbf{if}\;y \leq 1.1 \cdot 10^{-85}:\\ \;\;\;\;t_3 + \left(1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + t_1\right)\right)\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+35}:\\ \;\;\;\;t_3 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \end{array} \]

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)))
        (t_3 (- t_2 (sqrt x))))
   (if (<= y 1.1e-85)
     (+ t_3 (+ 1.0 (+ (- (sqrt (+ 1.0 t)) (sqrt t)) t_1)))
     (if (<= y 1.95e+35)
       (+ t_3 (+ (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))) t_1))
       (/ 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 t_3 = t_2 - sqrt(x);
	double tmp;
	if (y <= 1.1e-85) {
		tmp = t_3 + (1.0 + ((sqrt((1.0 + t)) - sqrt(t)) + t_1));
	} else if (y <= 1.95e+35) {
		tmp = t_3 + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + t_1);
	} 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) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z)) - sqrt(z)
    t_2 = sqrt((1.0d0 + x))
    t_3 = t_2 - sqrt(x)
    if (y <= 1.1d-85) then
        tmp = t_3 + (1.0d0 + ((sqrt((1.0d0 + t)) - sqrt(t)) + t_1))
    else if (y <= 1.95d+35) then
        tmp = t_3 + ((1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))) + t_1)
    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 t_3 = t_2 - Math.sqrt(x);
	double tmp;
	if (y <= 1.1e-85) {
		tmp = t_3 + (1.0 + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + t_1));
	} else if (y <= 1.95e+35) {
		tmp = t_3 + ((1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) + t_1);
	} 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))
	t_3 = t_2 - math.sqrt(x)
	tmp = 0
	if y <= 1.1e-85:
		tmp = t_3 + (1.0 + ((math.sqrt((1.0 + t)) - math.sqrt(t)) + t_1))
	elif y <= 1.95e+35:
		tmp = t_3 + ((1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) + t_1)
	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))
	t_3 = Float64(t_2 - sqrt(x))
	tmp = 0.0
	if (y <= 1.1e-85)
		tmp = Float64(t_3 + Float64(1.0 + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + t_1)));
	elseif (y <= 1.95e+35)
		tmp = Float64(t_3 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))) + t_1));
	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));
	t_3 = t_2 - sqrt(x);
	tmp = 0.0;
	if (y <= 1.1e-85)
		tmp = t_3 + (1.0 + ((sqrt((1.0 + t)) - sqrt(t)) + t_1));
	elseif (y <= 1.95e+35)
		tmp = t_3 + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + t_1);
	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]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.1e-85], N[(t$95$3 + N[(1.0 + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.95e+35], N[(t$95$3 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $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 < 1.1e-85

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. associate-+l+ 96.2%

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

      3. +-commutative 96.2%

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

      4. +-commutative 96.2%

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

      5. +-commutative 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in y around 0 96.2%

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

   if 1.1e-85 < y < 1.95e35

   1. Initial program 93.2%

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

      1. associate-+l+ 93.2%

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

      2. associate-+l+ 93.2%

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

      3. +-commutative 93.2%

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

      4. +-commutative 93.2%

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

      5. +-commutative 93.2%

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

   3. Simplified 93.2%

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

      1. flip-- 94.2%

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

      2. add-sqr-sqrt 87.4%

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

      3. add-sqr-sqrt 94.2%

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

   5. Applied egg-rr 93.5%

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

      1. associate--l+ 95.8%

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

      2. +-inverses 95.8%

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

      3. metadata-eval 95.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in t around inf 54.9%

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

   if 1.95e35 < y

   1. Initial program 88.3%

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

      1. associate-+l+ 88.3%

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

      2. +-commutative 88.3%

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

      3. associate-+r- 88.3%

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

      4. associate-+l- 45.4%

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

      5. +-commutative 45.4%

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

      6. +-commutative 45.4%

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

      7. associate--l+ 45.4%

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

   3. Simplified 27.4%

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

   4. Taylor expanded in t around inf 28.7%

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

      1. associate--l+ 29.6%

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

      2. +-commutative 29.6%

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

   6. Simplified 29.6%

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

   7. Taylor expanded in z around inf 17.0%

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

      1. +-commutative 17.0%

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

   9. Simplified 17.0%

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

   10. Taylor expanded in y around inf 16.9%

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

       1. flip-- 16.9%

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

       2. add-sqr-sqrt 17.4%

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

       3. add-sqr-sqrt 17.0%

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

   12. Applied egg-rr 17.0%

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

       1. associate--l+ 21.6%

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

       2. +-inverses 21.6%

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

       3. metadata-eval 21.6%

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

   14. Simplified 21.6%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-85}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+35}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 8: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 8.5 \cdot 10^{+15}:\\ \;\;\;\;2 + \left(t_1 + \left(\mathsf{hypot}\left(1, \sqrt{t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(t_1 - \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))))
   (if (<= t 8.5e+15)
     (+ 2.0 (+ t_1 (- (hypot 1.0 (sqrt t)) (+ (sqrt z) (sqrt t)))))
     (+
      (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))
      (+ (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))) (- t_1 (sqrt z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double tmp;
	if (t <= 8.5e+15) {
		tmp = 2.0 + (t_1 + (hypot(1.0, sqrt(t)) - (sqrt(z) + sqrt(t))));
	} else {
		tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (t_1 - sqrt(z)));
	}
	return tmp;
}

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 <= 8.5e+15) {
		tmp = 2.0 + (t_1 + (Math.hypot(1.0, Math.sqrt(t)) - (Math.sqrt(z) + Math.sqrt(t))));
	} else {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + ((1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) + (t_1 - Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	tmp = 0
	if t <= 8.5e+15:
		tmp = 2.0 + (t_1 + (math.hypot(1.0, math.sqrt(t)) - (math.sqrt(z) + math.sqrt(t))))
	else:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + ((1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) + (t_1 - math.sqrt(z)))
	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 <= 8.5e+15)
		tmp = Float64(2.0 + Float64(t_1 + Float64(hypot(1.0, sqrt(t)) - Float64(sqrt(z) + sqrt(t)))));
	else
		tmp = 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(t_1 - sqrt(z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	tmp = 0.0;
	if (t <= 8.5e+15)
		tmp = 2.0 + (t_1 + (hypot(1.0, sqrt(t)) - (sqrt(z) + sqrt(t))));
	else
		tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (t_1 - sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 8.5e+15], N[(2.0 + N[(t$95$1 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[t], $MachinePrecision] ^ 2], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 8.5e15

   1. Initial program 97.4%

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

      1. associate-+l+ 97.4%

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

      2. +-commutative 97.4%

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

      3. associate-+r- 79.6%

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

      4. associate-+l- 49.8%

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

      5. +-commutative 49.8%

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

      6. +-commutative 49.8%

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

      7. associate--l+ 49.8%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in y around 0 18.3%

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

      1. associate--l+ 38.6%

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

      2. +-commutative 38.6%

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

      3. associate--l+ 38.6%

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

      4. +-commutative 38.6%

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

      5. +-commutative 38.6%

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

      6. +-commutative 38.6%

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

      7. +-commutative 38.6%

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

   6. Simplified 38.6%

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

      1. expm1-log1p-u 18.3%

         \[\leadsto 1 + \left(\left(\sqrt{z + 1} + \sqrt{1 + x}\right) + \left(\sqrt{t + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{t} + \left(\sqrt{z} + \sqrt{x}\right)\right)\right)}\right)\right) \]

   8. Applied egg-rr 18.3%

      \[\leadsto 1 + \left(\left(\sqrt{z + 1} + \sqrt{1 + x}\right) + \left(\sqrt{t + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{t} + \left(\sqrt{z} + \sqrt{x}\right)\right)\right)}\right)\right) \]

   9. Taylor expanded in x around 0 21.1%

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

       1. associate--l+ 43.9%

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

       2. +-commutative 43.9%

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

       3. associate--l+ 43.9%

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

       4. rem-square-sqrt 43.9%

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

       5. hypot-1-def 43.9%

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

   11. Simplified 43.9%

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

   if 8.5e15 < 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. associate-+l+ 86.1%

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

      3. +-commutative 86.1%

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

      4. +-commutative 86.1%

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

      5. +-commutative 86.1%

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

   3. Simplified 86.1%

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

      1. flip-- 86.1%

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

      2. add-sqr-sqrt 70.5%

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

      3. add-sqr-sqrt 86.7%

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

   5. Applied egg-rr 86.7%

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

      1. +-commutative 86.7%

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

      2. associate--l+ 90.1%

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

      3. +-inverses 90.1%

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

      4. metadata-eval 90.1%

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

      5. +-commutative 90.1%

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

      6. +-commutative 90.1%

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

   7. Simplified 90.1%

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

      1. flip-- 90.2%

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

      2. add-sqr-sqrt 74.8%

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

      3. add-sqr-sqrt 90.2%

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

   9. Applied egg-rr 90.2%

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

       1. associate--l+ 93.0%

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

       2. +-inverses 93.0%

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

       3. metadata-eval 93.0%

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

   11. Simplified 93.0%

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

   12. Taylor expanded in t around inf 93.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{+15}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} + \left(\mathsf{hypot}\left(1, \sqrt{t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \end{array} \]

Alternative 9: 96.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := t_1 - \sqrt{x}\\ \mathbf{if}\;y \leq 7.2 \cdot 10^{-32}:\\ \;\;\;\;t_2 + \left(1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\sqrt{1 + y} + \sqrt{y}} + t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

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 (- t_1 (sqrt x))))
   (if (<= y 7.2e-32)
     (+
      t_2
      (+ 1.0 (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (- (sqrt (+ 1.0 z)) (sqrt z)))))
     (if (<= y 2e+30)
       (+ (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))) t_2)
       (/ 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 = t_1 - sqrt(x);
	double tmp;
	if (y <= 7.2e-32) {
		tmp = t_2 + (1.0 + ((sqrt((1.0 + t)) - sqrt(t)) + (sqrt((1.0 + z)) - sqrt(z))));
	} else if (y <= 2e+30) {
		tmp = (1.0 / (sqrt((1.0 + y)) + sqrt(y))) + t_2;
	} 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 = t_1 - sqrt(x)
    if (y <= 7.2d-32) then
        tmp = t_2 + (1.0d0 + ((sqrt((1.0d0 + t)) - sqrt(t)) + (sqrt((1.0d0 + z)) - sqrt(z))))
    else if (y <= 2d+30) then
        tmp = (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))) + t_2
    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 = t_1 - Math.sqrt(x);
	double tmp;
	if (y <= 7.2e-32) {
		tmp = t_2 + (1.0 + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (Math.sqrt((1.0 + z)) - Math.sqrt(z))));
	} else if (y <= 2e+30) {
		tmp = (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) + t_2;
	} 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 = t_1 - math.sqrt(x)
	tmp = 0
	if y <= 7.2e-32:
		tmp = t_2 + (1.0 + ((math.sqrt((1.0 + t)) - math.sqrt(t)) + (math.sqrt((1.0 + z)) - math.sqrt(z))))
	elif y <= 2e+30:
		tmp = (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) + t_2
	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 = Float64(t_1 - sqrt(x))
	tmp = 0.0
	if (y <= 7.2e-32)
		tmp = Float64(t_2 + Float64(1.0 + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(sqrt(Float64(1.0 + z)) - sqrt(z)))));
	elseif (y <= 2e+30)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))) + t_2);
	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 = t_1 - sqrt(x);
	tmp = 0.0;
	if (y <= 7.2e-32)
		tmp = t_2 + (1.0 + ((sqrt((1.0 + t)) - sqrt(t)) + (sqrt((1.0 + z)) - sqrt(z))));
	elseif (y <= 2e+30)
		tmp = (1.0 / (sqrt((1.0 + y)) + sqrt(y))) + t_2;
	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[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.2e-32], N[(t$95$2 + N[(1.0 + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+30], N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.19999999999999986e-32

   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. associate-+l+ 96.1%

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

      3. +-commutative 96.1%

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

      4. +-commutative 96.1%

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

      5. +-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in y around 0 96.1%

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

   if 7.19999999999999986e-32 < y < 2e30

   1. Initial program 90.7%

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

      1. associate-+l+ 90.7%

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

      2. associate-+l+ 90.7%

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

      3. +-commutative 90.7%

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

      4. +-commutative 90.7%

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

      5. +-commutative 90.7%

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

   3. Simplified 90.7%

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

      1. flip-- 92.0%

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

      2. add-sqr-sqrt 85.4%

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

      3. add-sqr-sqrt 92.0%

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

   5. Applied egg-rr 91.5%

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

      1. associate--l+ 96.1%

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

      2. +-inverses 96.1%

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

      3. metadata-eval 96.1%

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

   7. Simplified 94.6%

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

   8. Taylor expanded in t around inf 41.3%

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

   9. Taylor expanded in z around inf 17.2%

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

       1. +-commutative 17.2%

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

       2. +-commutative 17.2%

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

       3. associate--l+ 25.0%

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

       4. +-commutative 25.0%

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

       5. +-commutative 25.0%

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

   11. Simplified 25.0%

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

   if 2e30 < y

   1. Initial program 88.3%

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

      1. associate-+l+ 88.3%

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

      2. +-commutative 88.3%

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

      3. associate-+r- 88.3%

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

      4. associate-+l- 45.6%

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

      5. +-commutative 45.6%

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

      6. +-commutative 45.6%

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

      7. associate--l+ 45.6%

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

   3. Simplified 27.3%

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

   4. Taylor expanded in t around inf 28.8%

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

      1. associate--l+ 29.8%

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

      2. +-commutative 29.8%

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

   6. Simplified 29.8%

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

   7. Taylor expanded in z around inf 16.8%

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

      1. +-commutative 16.8%

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

   9. Simplified 16.8%

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

   10. Taylor expanded in y around inf 17.0%

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

       1. flip-- 17.0%

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

       2. add-sqr-sqrt 17.4%

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

       3. add-sqr-sqrt 17.1%

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

   12. Applied egg-rr 17.1%

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

       1. associate--l+ 21.6%

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

       2. +-inverses 21.6%

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

       3. metadata-eval 21.6%

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

   14. Simplified 21.6%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-32}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 10: 95.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 8.5 \cdot 10^{+15}:\\ \;\;\;\;2 + \left(t_1 + \left(\mathsf{hypot}\left(1, \sqrt{t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{z} - t_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))))
   (if (<= t 8.5e+15)
     (+ 2.0 (+ t_1 (- (hypot 1.0 (sqrt t)) (+ (sqrt z) (sqrt t)))))
     (+
      (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))
      (- (- (sqrt (+ 1.0 y)) (sqrt y)) (- (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 tmp;
	if (t <= 8.5e+15) {
		tmp = 2.0 + (t_1 + (hypot(1.0, sqrt(t)) - (sqrt(z) + sqrt(t))));
	} else {
		tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) + ((sqrt((1.0 + y)) - sqrt(y)) - (sqrt(z) - t_1));
	}
	return tmp;
}

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 <= 8.5e+15) {
		tmp = 2.0 + (t_1 + (Math.hypot(1.0, Math.sqrt(t)) - (Math.sqrt(z) + Math.sqrt(t))));
	} else {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) - (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))
	tmp = 0
	if t <= 8.5e+15:
		tmp = 2.0 + (t_1 + (math.hypot(1.0, math.sqrt(t)) - (math.sqrt(z) + math.sqrt(t))))
	else:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + ((math.sqrt((1.0 + y)) - math.sqrt(y)) - (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))
	tmp = 0.0
	if (t <= 8.5e+15)
		tmp = Float64(2.0 + Float64(t_1 + Float64(hypot(1.0, sqrt(t)) - Float64(sqrt(z) + sqrt(t)))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) - 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));
	tmp = 0.0;
	if (t <= 8.5e+15)
		tmp = 2.0 + (t_1 + (hypot(1.0, sqrt(t)) - (sqrt(z) + sqrt(t))));
	else
		tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) + ((sqrt((1.0 + y)) - sqrt(y)) - (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]}, If[LessEqual[t, 8.5e+15], N[(2.0 + N[(t$95$1 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[t], $MachinePrecision] ^ 2], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 8.5e15

   1. Initial program 97.4%

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

      1. associate-+l+ 97.4%

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

      2. +-commutative 97.4%

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

      3. associate-+r- 79.6%

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

      4. associate-+l- 49.8%

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

      5. +-commutative 49.8%

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

      6. +-commutative 49.8%

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

      7. associate--l+ 49.8%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in y around 0 18.3%

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

      1. associate--l+ 38.6%

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

      2. +-commutative 38.6%

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

      3. associate--l+ 38.6%

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

      4. +-commutative 38.6%

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

      5. +-commutative 38.6%

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

      6. +-commutative 38.6%

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

      7. +-commutative 38.6%

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

   6. Simplified 38.6%

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

      1. expm1-log1p-u 18.3%

         \[\leadsto 1 + \left(\left(\sqrt{z + 1} + \sqrt{1 + x}\right) + \left(\sqrt{t + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{t} + \left(\sqrt{z} + \sqrt{x}\right)\right)\right)}\right)\right) \]

   8. Applied egg-rr 18.3%

      \[\leadsto 1 + \left(\left(\sqrt{z + 1} + \sqrt{1 + x}\right) + \left(\sqrt{t + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{t} + \left(\sqrt{z} + \sqrt{x}\right)\right)\right)}\right)\right) \]

   9. Taylor expanded in x around 0 21.1%

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

       1. associate--l+ 43.9%

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

       2. +-commutative 43.9%

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

       3. associate--l+ 43.9%

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

       4. rem-square-sqrt 43.9%

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

       5. hypot-1-def 43.9%

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

   11. Simplified 43.9%

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

   if 8.5e15 < 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. associate-+l+ 86.1%

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

      3. +-commutative 86.1%

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

      4. +-commutative 86.1%

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

      5. +-commutative 86.1%

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

   3. Simplified 86.1%

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

      1. flip-- 86.1%

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

      2. add-sqr-sqrt 70.5%

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

      3. add-sqr-sqrt 86.7%

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

   5. Applied egg-rr 86.7%

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

      1. +-commutative 86.7%

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

      2. associate--l+ 90.1%

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

      3. +-inverses 90.1%

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

      4. metadata-eval 90.1%

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

      5. +-commutative 90.1%

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

      6. +-commutative 90.1%

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

   7. Simplified 90.1%

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

   8. Taylor expanded in t around inf 90.1%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{+15}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} + \left(\mathsf{hypot}\left(1, \sqrt{t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\\ \end{array} \]

Alternative 11: 92.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ \mathbf{if}\;z \leq 10^{-43}:\\ \;\;\;\;2 + \left(t_2 + \left(\mathsf{hypot}\left(1, \sqrt{t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+19}:\\ \;\;\;\;t_1 + \left(1 - \left(\sqrt{x} + \frac{-1}{\sqrt{z} + t_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(t_1 - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))) (t_2 (sqrt (+ 1.0 z))))
   (if (<= z 1e-43)
     (+ 2.0 (+ t_2 (- (hypot 1.0 (sqrt t)) (+ (sqrt z) (sqrt t)))))
     (if (<= z 6.8e+19)
       (+ t_1 (- 1.0 (+ (sqrt x) (/ -1.0 (+ (sqrt z) t_2)))))
       (+ (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))) (- t_1 (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + z));
	double tmp;
	if (z <= 1e-43) {
		tmp = 2.0 + (t_2 + (hypot(1.0, sqrt(t)) - (sqrt(z) + sqrt(t))));
	} else if (z <= 6.8e+19) {
		tmp = t_1 + (1.0 - (sqrt(x) + (-1.0 / (sqrt(z) + t_2))));
	} else {
		tmp = (1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (t_1 - 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 t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + z));
	double tmp;
	if (z <= 1e-43) {
		tmp = 2.0 + (t_2 + (Math.hypot(1.0, Math.sqrt(t)) - (Math.sqrt(z) + Math.sqrt(t))));
	} else if (z <= 6.8e+19) {
		tmp = t_1 + (1.0 - (Math.sqrt(x) + (-1.0 / (Math.sqrt(z) + t_2))));
	} else {
		tmp = (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) + (t_1 - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + z))
	tmp = 0
	if z <= 1e-43:
		tmp = 2.0 + (t_2 + (math.hypot(1.0, math.sqrt(t)) - (math.sqrt(z) + math.sqrt(t))))
	elif z <= 6.8e+19:
		tmp = t_1 + (1.0 - (math.sqrt(x) + (-1.0 / (math.sqrt(z) + t_2))))
	else:
		tmp = (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) + (t_1 - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (z <= 1e-43)
		tmp = Float64(2.0 + Float64(t_2 + Float64(hypot(1.0, sqrt(t)) - Float64(sqrt(z) + sqrt(t)))));
	elseif (z <= 6.8e+19)
		tmp = Float64(t_1 + Float64(1.0 - Float64(sqrt(x) + Float64(-1.0 / Float64(sqrt(z) + t_2)))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))) + Float64(t_1 - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((1.0 + z));
	tmp = 0.0;
	if (z <= 1e-43)
		tmp = 2.0 + (t_2 + (hypot(1.0, sqrt(t)) - (sqrt(z) + sqrt(t))));
	elseif (z <= 6.8e+19)
		tmp = t_1 + (1.0 - (sqrt(x) + (-1.0 / (sqrt(z) + t_2))));
	else
		tmp = (1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (t_1 - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1e-43], N[(2.0 + N[(t$95$2 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[t], $MachinePrecision] ^ 2], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.8e+19], N[(t$95$1 + N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] + N[(-1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.00000000000000008e-43

   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- 77.0%

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

      4. associate-+l- 46.5%

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

      5. +-commutative 46.5%

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

      6. +-commutative 46.5%

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

      7. associate--l+ 46.5%

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

   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 y around 0 19.2%

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

      1. associate--l+ 39.3%

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

      2. +-commutative 39.3%

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

      3. associate--l+ 42.3%

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

      4. +-commutative 42.3%

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

      5. +-commutative 42.3%

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

      6. +-commutative 42.3%

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

      7. +-commutative 42.3%

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

   6. Simplified 42.3%

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

      1. expm1-log1p-u 18.5%

         \[\leadsto 1 + \left(\left(\sqrt{z + 1} + \sqrt{1 + x}\right) + \left(\sqrt{t + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{t} + \left(\sqrt{z} + \sqrt{x}\right)\right)\right)}\right)\right) \]

   8. Applied egg-rr 18.5%

      \[\leadsto 1 + \left(\left(\sqrt{z + 1} + \sqrt{1 + x}\right) + \left(\sqrt{t + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{t} + \left(\sqrt{z} + \sqrt{x}\right)\right)\right)}\right)\right) \]

   9. Taylor expanded in x around 0 22.6%

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

       1. associate--l+ 46.8%

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

       2. +-commutative 46.8%

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

       3. associate--l+ 37.3%

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

       4. rem-square-sqrt 37.3%

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

       5. hypot-1-def 37.3%

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

   11. Simplified 37.3%

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

   if 1.00000000000000008e-43 < z < 6.8e19

   1. Initial program 85.7%

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

      1. associate-+l+ 85.7%

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

      2. +-commutative 85.7%

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

      3. associate-+r- 69.6%

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

      4. associate-+l- 44.5%

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

      5. +-commutative 44.5%

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

      6. +-commutative 44.5%

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

      7. associate--l+ 44.5%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in t around inf 26.1%

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

      1. associate--l+ 26.1%

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

      2. +-commutative 26.1%

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

   6. Simplified 26.1%

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

   7. Taylor expanded in y around 0 23.8%

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

      1. flip-- 24.7%

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

      2. add-sqr-sqrt 24.7%

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

      3. add-sqr-sqrt 24.8%

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

      4. +-commutative 24.8%

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

      5. +-commutative 24.8%

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

   9. Applied egg-rr 24.8%

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

       1. +-commutative 24.8%

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

       2. associate--r+ 24.8%

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

       3. +-inverses 24.8%

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

       4. metadata-eval 24.8%

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

       5. +-commutative 24.8%

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

   11. Simplified 24.8%

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

   if 6.8e19 < z

   1. Initial program 88.6%

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

      1. associate-+l+ 88.6%

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

      2. associate-+l+ 88.6%

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

      3. +-commutative 88.6%

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

      4. +-commutative 88.6%

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

      5. +-commutative 88.6%

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

   3. Simplified 88.6%

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

      1. flip-- 92.2%

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

      2. add-sqr-sqrt 78.4%

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

      3. add-sqr-sqrt 92.2%

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

   5. Applied egg-rr 88.8%

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

      1. associate--l+ 95.2%

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

      2. +-inverses 95.2%

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

      3. metadata-eval 95.2%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in t around inf 43.9%

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

   9. Taylor expanded in z around inf 29.0%

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

       1. +-commutative 29.0%

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

       2. +-commutative 29.0%

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

       3. associate--l+ 43.9%

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

       4. +-commutative 43.9%

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

       5. +-commutative 43.9%

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

   11. Simplified 43.9%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{-43}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} + \left(\mathsf{hypot}\left(1, \sqrt{t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{1 + x} + \left(1 - \left(\sqrt{x} + \frac{-1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \end{array} \]

Alternative 12: 90.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 4.4 \cdot 10^{-32}:\\ \;\;\;\;t_1 + \left(1 - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 10^{+33}:\\ \;\;\;\;\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(t_1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

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 4.4e-32)
     (+ t_1 (- 1.0 (+ (sqrt x) (- (sqrt z) (sqrt (+ 1.0 z))))))
     (if (<= y 1e+33)
       (+ (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))) (- t_1 (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 <= 4.4e-32) {
		tmp = t_1 + (1.0 - (sqrt(x) + (sqrt(z) - sqrt((1.0 + z)))));
	} else if (y <= 1e+33) {
		tmp = (1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (t_1 - 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 <= 4.4d-32) then
        tmp = t_1 + (1.0d0 - (sqrt(x) + (sqrt(z) - sqrt((1.0d0 + z)))))
    else if (y <= 1d+33) then
        tmp = (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))) + (t_1 - 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 <= 4.4e-32) {
		tmp = t_1 + (1.0 - (Math.sqrt(x) + (Math.sqrt(z) - Math.sqrt((1.0 + z)))));
	} else if (y <= 1e+33) {
		tmp = (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) + (t_1 - 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 <= 4.4e-32:
		tmp = t_1 + (1.0 - (math.sqrt(x) + (math.sqrt(z) - math.sqrt((1.0 + z)))))
	elif y <= 1e+33:
		tmp = (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) + (t_1 - 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 <= 4.4e-32)
		tmp = Float64(t_1 + Float64(1.0 - Float64(sqrt(x) + Float64(sqrt(z) - sqrt(Float64(1.0 + z))))));
	elseif (y <= 1e+33)
		tmp = 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_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 <= 4.4e-32)
		tmp = t_1 + (1.0 - (sqrt(x) + (sqrt(z) - sqrt((1.0 + z)))));
	elseif (y <= 1e+33)
		tmp = (1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (t_1 - 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, 4.4e-32], N[(t$95$1 + N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+33], 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], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.4e-32

   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- 60.5%

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

      4. associate-+l- 55.1%

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

      5. +-commutative 55.1%

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

      6. +-commutative 55.1%

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

      7. associate--l+ 55.1%

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

   3. Simplified 39.4%

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

   4. Taylor expanded in t around inf 30.5%

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

      1. associate--l+ 30.9%

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

      2. +-commutative 30.9%

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

   6. Simplified 30.9%

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

   7. Taylor expanded in y around 0 30.9%

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

   if 4.4e-32 < y < 9.9999999999999995e32

   1. Initial program 90.2%

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

      1. associate-+l+ 90.2%

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

      2. associate-+l+ 90.2%

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

      3. +-commutative 90.2%

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

      4. +-commutative 90.2%

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

      5. +-commutative 90.2%

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

   3. Simplified 90.2%

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

      1. flip-- 91.4%

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

      2. add-sqr-sqrt 80.2%

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

      3. add-sqr-sqrt 91.4%

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

   5. Applied egg-rr 90.9%

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

      1. associate--l+ 95.1%

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

      2. +-inverses 95.1%

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

      3. metadata-eval 95.1%

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

   7. Simplified 93.8%

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

   8. Taylor expanded in t around inf 41.5%

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

   9. Taylor expanded in z around inf 18.7%

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

       1. +-commutative 18.7%

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

       2. +-commutative 18.7%

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

       3. associate--l+ 26.2%

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

       4. +-commutative 26.2%

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

       5. +-commutative 26.2%

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

   11. Simplified 26.2%

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

   if 9.9999999999999995e32 < y

   1. Initial program 88.4%

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

      1. associate-+l+ 88.4%

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

      2. +-commutative 88.4%

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

      3. associate-+r- 88.4%

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

      4. associate-+l- 45.8%

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

      5. +-commutative 45.8%

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

      6. +-commutative 45.8%

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

      7. associate--l+ 45.8%

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

   3. Simplified 27.7%

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

   4. Taylor expanded in t around inf 28.9%

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

      1. associate--l+ 29.6%

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

      2. +-commutative 29.6%

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

   6. Simplified 29.6%

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

   7. Taylor expanded in z around inf 16.7%

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

      1. +-commutative 16.7%

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

   9. Simplified 16.7%

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

   10. Taylor expanded in y around inf 16.7%

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

       1. flip-- 16.7%

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

       2. add-sqr-sqrt 17.1%

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

       3. add-sqr-sqrt 16.8%

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

   12. Applied egg-rr 16.8%

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

       1. associate--l+ 21.3%

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

       2. +-inverses 21.3%

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

       3. metadata-eval 21.3%

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

   14. Simplified 21.3%

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

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{1 + x} + \left(1 - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 10^{+33}:\\ \;\;\;\;\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 13: 91.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 7.2 \cdot 10^{-32}:\\ \;\;\;\;t_1 + \left(1 - \left(\sqrt{x} + \frac{-1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(t_1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

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 7.2e-32)
     (+ t_1 (- 1.0 (+ (sqrt x) (/ -1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))))))
     (if (<= y 2e+30)
       (+ (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))) (- t_1 (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 <= 7.2e-32) {
		tmp = t_1 + (1.0 - (sqrt(x) + (-1.0 / (sqrt(z) + sqrt((1.0 + z))))));
	} else if (y <= 2e+30) {
		tmp = (1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (t_1 - 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 <= 7.2d-32) then
        tmp = t_1 + (1.0d0 - (sqrt(x) + ((-1.0d0) / (sqrt(z) + sqrt((1.0d0 + z))))))
    else if (y <= 2d+30) then
        tmp = (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))) + (t_1 - 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 <= 7.2e-32) {
		tmp = t_1 + (1.0 - (Math.sqrt(x) + (-1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z))))));
	} else if (y <= 2e+30) {
		tmp = (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) + (t_1 - 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 <= 7.2e-32:
		tmp = t_1 + (1.0 - (math.sqrt(x) + (-1.0 / (math.sqrt(z) + math.sqrt((1.0 + z))))))
	elif y <= 2e+30:
		tmp = (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) + (t_1 - 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 <= 7.2e-32)
		tmp = Float64(t_1 + Float64(1.0 - Float64(sqrt(x) + Float64(-1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z)))))));
	elseif (y <= 2e+30)
		tmp = 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_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 <= 7.2e-32)
		tmp = t_1 + (1.0 - (sqrt(x) + (-1.0 / (sqrt(z) + sqrt((1.0 + z))))));
	elseif (y <= 2e+30)
		tmp = (1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (t_1 - 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, 7.2e-32], N[(t$95$1 + N[(1.0 - 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], If[LessEqual[y, 2e+30], 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], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.19999999999999986e-32

   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- 60.5%

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

      4. associate-+l- 55.1%

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

      5. +-commutative 55.1%

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

      6. +-commutative 55.1%

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

      7. associate--l+ 55.1%

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

   3. Simplified 39.4%

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

   4. Taylor expanded in t around inf 30.5%

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

      1. associate--l+ 30.9%

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

      2. +-commutative 30.9%

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

   6. Simplified 30.9%

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

   7. Taylor expanded in y around 0 30.9%

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

      1. flip-- 31.1%

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

      2. add-sqr-sqrt 23.1%

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

      3. add-sqr-sqrt 31.1%

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

      4. +-commutative 31.1%

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

      5. +-commutative 31.1%

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

   9. Applied egg-rr 31.1%

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

       1. +-commutative 31.1%

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

       2. associate--r+ 31.1%

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

       3. +-inverses 31.1%

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

       4. metadata-eval 31.1%

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

       5. +-commutative 31.1%

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

   11. Simplified 31.1%

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

   if 7.19999999999999986e-32 < y < 2e30

   1. Initial program 90.7%

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

      1. associate-+l+ 90.7%

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

      2. associate-+l+ 90.7%

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

      3. +-commutative 90.7%

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

      4. +-commutative 90.7%

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

      5. +-commutative 90.7%

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

   3. Simplified 90.7%

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

      1. flip-- 92.0%

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

      2. add-sqr-sqrt 85.4%

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

      3. add-sqr-sqrt 92.0%

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

   5. Applied egg-rr 91.5%

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

      1. associate--l+ 96.1%

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

      2. +-inverses 96.1%

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

      3. metadata-eval 96.1%

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

   7. Simplified 94.6%

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

   8. Taylor expanded in t around inf 41.3%

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

   9. Taylor expanded in z around inf 17.2%

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

       1. +-commutative 17.2%

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

       2. +-commutative 17.2%

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

       3. associate--l+ 25.0%

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

       4. +-commutative 25.0%

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

       5. +-commutative 25.0%

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

   11. Simplified 25.0%

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

   if 2e30 < y

   1. Initial program 88.3%

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

      1. associate-+l+ 88.3%

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

      2. +-commutative 88.3%

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

      3. associate-+r- 88.3%

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

      4. associate-+l- 45.6%

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

      5. +-commutative 45.6%

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

      6. +-commutative 45.6%

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

      7. associate--l+ 45.6%

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

   3. Simplified 27.3%

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

   4. Taylor expanded in t around inf 28.8%

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

      1. associate--l+ 29.8%

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

      2. +-commutative 29.8%

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

   6. Simplified 29.8%

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

   7. Taylor expanded in z around inf 16.8%

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

      1. +-commutative 16.8%

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

   9. Simplified 16.8%

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

   10. Taylor expanded in y around inf 17.0%

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

       1. flip-- 17.0%

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

       2. add-sqr-sqrt 17.4%

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

       3. add-sqr-sqrt 17.1%

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

   12. Applied egg-rr 17.1%

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

       1. associate--l+ 21.6%

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

       2. +-inverses 21.6%

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

       3. metadata-eval 21.6%

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

   14. Simplified 21.6%

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

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{1 + x} + \left(1 - \left(\sqrt{x} + \frac{-1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 14: 89.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 6.4 \cdot 10^{-32}:\\ \;\;\;\;t_1 + \left(1 - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + y} + \left(t_1 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

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 6.4e-32)
     (+ t_1 (- 1.0 (+ (sqrt x) (- (sqrt z) (sqrt (+ 1.0 z))))))
     (if (<= y 5e+15)
       (+ (sqrt (+ 1.0 y)) (- t_1 (+ (sqrt x) (sqrt y))))
       (/ 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 <= 6.4e-32) {
		tmp = t_1 + (1.0 - (sqrt(x) + (sqrt(z) - sqrt((1.0 + z)))));
	} else if (y <= 5e+15) {
		tmp = sqrt((1.0 + y)) + (t_1 - (sqrt(x) + sqrt(y)));
	} 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 <= 6.4d-32) then
        tmp = t_1 + (1.0d0 - (sqrt(x) + (sqrt(z) - sqrt((1.0d0 + z)))))
    else if (y <= 5d+15) then
        tmp = sqrt((1.0d0 + y)) + (t_1 - (sqrt(x) + sqrt(y)))
    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 <= 6.4e-32) {
		tmp = t_1 + (1.0 - (Math.sqrt(x) + (Math.sqrt(z) - Math.sqrt((1.0 + z)))));
	} else if (y <= 5e+15) {
		tmp = Math.sqrt((1.0 + y)) + (t_1 - (Math.sqrt(x) + Math.sqrt(y)));
	} 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 <= 6.4e-32:
		tmp = t_1 + (1.0 - (math.sqrt(x) + (math.sqrt(z) - math.sqrt((1.0 + z)))))
	elif y <= 5e+15:
		tmp = math.sqrt((1.0 + y)) + (t_1 - (math.sqrt(x) + math.sqrt(y)))
	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 <= 6.4e-32)
		tmp = Float64(t_1 + Float64(1.0 - Float64(sqrt(x) + Float64(sqrt(z) - sqrt(Float64(1.0 + z))))));
	elseif (y <= 5e+15)
		tmp = Float64(sqrt(Float64(1.0 + y)) + Float64(t_1 - Float64(sqrt(x) + sqrt(y))));
	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 <= 6.4e-32)
		tmp = t_1 + (1.0 - (sqrt(x) + (sqrt(z) - sqrt((1.0 + z)))));
	elseif (y <= 5e+15)
		tmp = sqrt((1.0 + y)) + (t_1 - (sqrt(x) + sqrt(y)));
	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, 6.4e-32], N[(t$95$1 + N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+15], N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $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 < 6.4000000000000004e-32

   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- 60.5%

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

      4. associate-+l- 55.1%

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

      5. +-commutative 55.1%

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

      6. +-commutative 55.1%

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

      7. associate--l+ 55.1%

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

   3. Simplified 39.4%

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

   4. Taylor expanded in t around inf 30.5%

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

      1. associate--l+ 30.9%

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

      2. +-commutative 30.9%

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

   6. Simplified 30.9%

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

   7. Taylor expanded in y around 0 30.9%

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

   if 6.4000000000000004e-32 < y < 5e15

   1. Initial program 93.9%

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

      1. associate-+l+ 93.9%

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

      2. +-commutative 93.9%

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

      3. associate-+r- 56.8%

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

      4. associate-+l- 48.4%

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

      5. +-commutative 48.4%

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

      6. +-commutative 48.4%

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

      7. associate--l+ 48.4%

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

   3. Simplified 44.5%

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

   4. Taylor expanded in t around inf 19.6%

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

      1. associate--l+ 19.6%

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

      2. +-commutative 19.6%

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

   6. Simplified 19.6%

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

   7. Taylor expanded in z around inf 14.9%

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

      1. +-commutative 14.9%

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

   9. Simplified 14.9%

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

       1. associate-+r- 14.9%

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

       2. +-commutative 14.9%

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

   11. Applied egg-rr 14.9%

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

       1. +-commutative 14.9%

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

       2. associate--l+ 21.9%

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

       3. +-commutative 21.9%

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

   13. Simplified 21.9%

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

   if 5e15 < y

   1. Initial program 87.9%

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

      1. associate-+l+ 87.9%

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

      2. +-commutative 87.9%

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

      3. associate-+r- 87.9%

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

      4. associate-+l- 45.0%

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

      5. +-commutative 45.0%

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

      6. +-commutative 45.0%

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

      7. associate--l+ 45.0%

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

   3. Simplified 26.8%

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

   4. Taylor expanded in t around inf 28.4%

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

      1. associate--l+ 29.3%

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

      2. +-commutative 29.3%

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

   6. Simplified 29.3%

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

   7. Taylor expanded in z around inf 17.0%

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

      1. +-commutative 17.0%

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

   9. Simplified 17.0%

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

   10. Taylor expanded in y around inf 17.1%

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

       1. flip-- 17.1%

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

       2. add-sqr-sqrt 17.5%

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

       3. add-sqr-sqrt 17.2%

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

   12. Applied egg-rr 17.2%

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

       1. associate--l+ 21.6%

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

       2. +-inverses 21.6%

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

       3. metadata-eval 21.6%

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

   14. Simplified 21.6%

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

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{1 + x} + \left(1 - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + y} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 15: 84.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.6e+14)
   (- (+ (sqrt (+ 1.0 z)) 2.0) (sqrt z))
   (+ (sqrt (+ 1.0 x)) (- (sqrt (+ 1.0 y)) (+ (sqrt x) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.6e+14) {
		tmp = (sqrt((1.0 + z)) + 2.0) - sqrt(z);
	} else {
		tmp = sqrt((1.0 + x)) + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.6d+14) then
        tmp = (sqrt((1.0d0 + z)) + 2.0d0) - sqrt(z)
    else
        tmp = sqrt((1.0d0 + x)) + (sqrt((1.0d0 + y)) - (sqrt(x) + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.6e+14) {
		tmp = (Math.sqrt((1.0 + z)) + 2.0) - Math.sqrt(z);
	} else {
		tmp = Math.sqrt((1.0 + x)) + (Math.sqrt((1.0 + y)) - (Math.sqrt(x) + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.6e+14:
		tmp = (math.sqrt((1.0 + z)) + 2.0) - math.sqrt(z)
	else:
		tmp = math.sqrt((1.0 + x)) + (math.sqrt((1.0 + y)) - (math.sqrt(x) + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.6e+14)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) + 2.0) - sqrt(z));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(sqrt(Float64(1.0 + y)) - Float64(sqrt(x) + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.6e+14)
		tmp = (sqrt((1.0 + z)) + 2.0) - sqrt(z);
	else
		tmp = sqrt((1.0 + x)) + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 1.6e+14], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.6e14

   1. Initial program 95.3%

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

      1. associate-+l+ 95.3%

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

      2. +-commutative 95.3%

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

      3. associate-+r- 75.9%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 46.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 46.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 46.1%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 46.1%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 29.2%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 29.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 29.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 29.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 29.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in y around 0 24.8%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{1} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

   8. Taylor expanded in x around 0 40.8%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \sqrt{z}} \]
   9. Step-by-step derivation

      1. +-commutative 40.8%

         \[\leadsto \color{blue}{\left(\sqrt{1 + z} + 2\right)} - \sqrt{z} \]

   10. Simplified 40.8%

       \[\leadsto \color{blue}{\left(\sqrt{1 + z} + 2\right) - \sqrt{z}} \]

   if 1.6e14 < z

   1. Initial program 88.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 88.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 88.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 68.7%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 55.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 55.0%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 55.0%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 55.0%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 40.1%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 27.3%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 28.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 28.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 28.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 27.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 27.9%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 27.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 16: 61.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;y \leq 800:\\ \;\;\;\;1 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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 800.0) (+ 1.0 t_1) t_1)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (y <= 800.0) {
		tmp = 1.0 + t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x)) - sqrt(x)
    if (y <= 800.0d0) then
        tmp = 1.0d0 + t_1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (y <= 800.0) {
		tmp = 1.0 + t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if y <= 800.0:
		tmp = 1.0 + t_1
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (y <= 800.0)
		tmp = Float64(1.0 + t_1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (y <= 800.0)
		tmp = 1.0 + t_1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 800.0], N[(1.0 + t$95$1), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if y < 800

   1. Initial program 96.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 96.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 96.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 60.1%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 54.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 54.5%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 54.5%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 54.5%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   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 29.1%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 29.5%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 29.5%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 29.5%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 19.5%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 19.5%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 19.5%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around 0 19.5%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + x}\right) - \sqrt{x}} \]
   11. Step-by-step derivation

       1. associate--l+ 33.4%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

   12. Simplified 33.4%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

   if 800 < y

   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- 86.9%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 45.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 45.0%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 45.0%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 45.0%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 27.6%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 28.4%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 29.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 29.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 29.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 17.3%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 17.3%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 17.3%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 17.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 800:\\ \;\;\;\;1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 17: 84.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{1 + z} + \left(2 - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

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 1.6e+14)
   (+ (sqrt (+ 1.0 z)) (- 2.0 (sqrt z)))
   (+ 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 <= 1.6e+14) {
		tmp = sqrt((1.0 + z)) + (2.0 - sqrt(z));
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.6d+14) then
        tmp = sqrt((1.0d0 + z)) + (2.0d0 - sqrt(z))
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.6e+14) {
		tmp = Math.sqrt((1.0 + z)) + (2.0 - Math.sqrt(z));
	} 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 <= 1.6e+14:
		tmp = math.sqrt((1.0 + z)) + (2.0 - math.sqrt(z))
	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 <= 1.6e+14)
		tmp = Float64(sqrt(Float64(1.0 + z)) + Float64(2.0 - sqrt(z)));
	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 <= 1.6e+14)
		tmp = sqrt((1.0 + z)) + (2.0 - sqrt(z));
	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, 1.6e+14], N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(2.0 - N[Sqrt[z], $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 2 regimes

2. if z < 1.6e14

   1. Initial program 95.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 95.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 95.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 75.9%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 46.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 46.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 46.1%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 46.1%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 29.2%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 29.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 29.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 29.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 29.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in y around 0 24.8%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{1} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

   8. Taylor expanded in x around 0 40.8%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \sqrt{z}} \]
   9. Step-by-step derivation

      1. +-commutative 40.8%

         \[\leadsto \color{blue}{\left(\sqrt{1 + z} + 2\right)} - \sqrt{z} \]

      2. associate--l+ 40.8%

         \[\leadsto \color{blue}{\sqrt{1 + z} + \left(2 - \sqrt{z}\right)} \]

   10. Simplified 40.8%

       \[\leadsto \color{blue}{\sqrt{1 + z} + \left(2 - \sqrt{z}\right)} \]

   if 1.6e14 < z

   1. Initial program 88.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 88.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 88.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 68.7%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 55.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 55.0%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 55.0%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 55.0%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 40.1%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 27.3%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 28.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 28.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 28.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 27.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 27.9%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 27.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in x around 0 31.9%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   11. Step-by-step derivation

       1. associate--l+ 52.6%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   12. Simplified 52.6%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{1 + z} + \left(2 - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 18: 84.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

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+14)
   (- (+ (sqrt (+ 1.0 z)) 2.0) (sqrt z))
   (+ 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+14) {
		tmp = (sqrt((1.0 + z)) + 2.0) - sqrt(z);
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.7d+14) then
        tmp = (sqrt((1.0d0 + z)) + 2.0d0) - sqrt(z)
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.7e+14) {
		tmp = (Math.sqrt((1.0 + z)) + 2.0) - Math.sqrt(z);
	} 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+14:
		tmp = (math.sqrt((1.0 + z)) + 2.0) - math.sqrt(z)
	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+14)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) + 2.0) - sqrt(z));
	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+14)
		tmp = (sqrt((1.0 + z)) + 2.0) - sqrt(z);
	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+14], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[Sqrt[z], $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 2 regimes

2. if z < 1.7e14

   1. Initial program 95.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 95.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 95.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 75.9%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 46.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 46.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 46.1%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 46.1%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 29.2%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 29.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 29.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 29.9%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 29.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in y around 0 24.8%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{1} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

   8. Taylor expanded in x around 0 40.8%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \sqrt{z}} \]
   9. Step-by-step derivation

      1. +-commutative 40.8%

         \[\leadsto \color{blue}{\left(\sqrt{1 + z} + 2\right)} - \sqrt{z} \]

   10. Simplified 40.8%

       \[\leadsto \color{blue}{\left(\sqrt{1 + z} + 2\right) - \sqrt{z}} \]

   if 1.7e14 < z

   1. Initial program 88.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 88.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 88.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 68.7%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 55.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 55.0%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 55.0%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 55.0%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 40.1%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 27.3%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 28.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 28.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 28.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 27.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 27.9%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 27.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in x around 0 31.9%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   11. Step-by-step derivation

       1. associate--l+ 52.6%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   12. Simplified 52.6%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 19: 64.2% 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 92.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 92.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. +-commutative 92.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. associate-+r- 72.7%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 50.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. +-commutative 50.0%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

   6. +-commutative 50.0%

      \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

   7. associate--l+ 50.0%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

3. Simplified 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 28.8%

   \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
5. Step-by-step derivation

   1. associate--l+ 29.4%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

   2. +-commutative 29.4%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

6. Simplified 29.4%

   \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

7. Taylor expanded in z around inf 18.5%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative 18.5%

      \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

9. Simplified 18.5%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

10. Taylor expanded in x around 0 26.9%

    \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
11. Step-by-step derivation

    1. associate--l+ 44.1%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

12. Simplified 44.1%

    \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

13. Final simplification 44.1%

    \[\leadsto 1 + \left(\sqrt{1 + y} - \sqrt{y}\right) \]

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 92.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 92.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. +-commutative 92.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. associate-+r- 72.7%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 50.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. +-commutative 50.0%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

   6. +-commutative 50.0%

      \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

   7. associate--l+ 50.0%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

3. Simplified 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 28.8%

   \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
5. Step-by-step derivation

   1. associate--l+ 29.4%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

   2. +-commutative 29.4%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

6. Simplified 29.4%

   \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

7. Taylor expanded in z around inf 18.5%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative 18.5%

      \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

9. Simplified 18.5%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

10. Taylor expanded in y around inf 13.7%

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

11. Final simplification 13.7%

    \[\leadsto \sqrt{1 + x} - \sqrt{x} \]

Developer target: 99.4% 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 2023271 
(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))))