Main:z from

Percentage Accurate: 91.2% → 99.2%

Time: 32.7s

Alternatives: 21

Speedup: 1.1×

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: 91.2% 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: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_1 - \sqrt{x} \leq 0.9:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + t\_1}\right) + \left(t\_2 - \sqrt{z}\right)\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(\left(\left(\left(1 + x \cdot 0.5\right) - \sqrt{x}\right) + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{z} + t\_2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= (- t_1 (sqrt x)) 0.9)
     (+
      (+
       (+ (* 0.5 (sqrt (/ 1.0 y))) (/ 1.0 (+ (sqrt x) t_1)))
       (- t_2 (sqrt z)))
      t_3)
     (+
      t_3
      (+
       (+ (- (+ 1.0 (* x 0.5)) (sqrt x)) (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))
       (/ 1.0 (+ (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((x + 1.0));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if ((t_1 - sqrt(x)) <= 0.9) {
		tmp = (((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + t_1))) + (t_2 - sqrt(z))) + t_3;
	} else {
		tmp = t_3 + ((((1.0 + (x * 0.5)) - sqrt(x)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)))) + (1.0 / (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) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((1.0d0 + z))
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    if ((t_1 - sqrt(x)) <= 0.9d0) then
        tmp = (((0.5d0 * sqrt((1.0d0 / y))) + (1.0d0 / (sqrt(x) + t_1))) + (t_2 - sqrt(z))) + t_3
    else
        tmp = t_3 + ((((1.0d0 + (x * 0.5d0)) - sqrt(x)) + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y)))) + (1.0d0 / (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((x + 1.0));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if ((t_1 - Math.sqrt(x)) <= 0.9) {
		tmp = (((0.5 * Math.sqrt((1.0 / y))) + (1.0 / (Math.sqrt(x) + t_1))) + (t_2 - Math.sqrt(z))) + t_3;
	} else {
		tmp = t_3 + ((((1.0 + (x * 0.5)) - Math.sqrt(x)) + (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y)))) + (1.0 / (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((x + 1.0))
	t_2 = math.sqrt((1.0 + z))
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if (t_1 - math.sqrt(x)) <= 0.9:
		tmp = (((0.5 * math.sqrt((1.0 / y))) + (1.0 / (math.sqrt(x) + t_1))) + (t_2 - math.sqrt(z))) + t_3
	else:
		tmp = t_3 + ((((1.0 + (x * 0.5)) - math.sqrt(x)) + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y)))) + (1.0 / (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(x + 1.0))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (Float64(t_1 - sqrt(x)) <= 0.9)
		tmp = Float64(Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(1.0 / Float64(sqrt(x) + t_1))) + Float64(t_2 - sqrt(z))) + t_3);
	else
		tmp = Float64(t_3 + Float64(Float64(Float64(Float64(1.0 + Float64(x * 0.5)) - sqrt(x)) + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))) + Float64(1.0 / 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((x + 1.0));
	t_2 = sqrt((1.0 + z));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if ((t_1 - sqrt(x)) <= 0.9)
		tmp = (((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + t_1))) + (t_2 - sqrt(z))) + t_3;
	else
		tmp = t_3 + ((((1.0 + (x * 0.5)) - sqrt(x)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)))) + (1.0 / (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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.9], N[(N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(t$95$3 + N[(N[(N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.900000000000000022

   1. Initial program 85.1%

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

      1. flip-- 85.3%

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

      2. add-sqr-sqrt 63.4%

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

      3. +-commutative 63.4%

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

      4. add-sqr-sqrt 86.2%

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

      5. +-commutative 86.2%

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

   4. Applied egg-rr 86.2%

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

      1. associate--l+ 91.0%

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

      2. +-inverses 91.0%

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

      3. metadata-eval 91.0%

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

   6. Simplified 91.0%

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

      1. flip-- 91.1%

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

      2. add-sqr-sqrt 43.9%

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

      3. add-sqr-sqrt 91.1%

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

   8. Applied egg-rr 91.1%

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

   9. Taylor expanded in y around inf 53.5%

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

   if 0.900000000000000022 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 97.8%

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

      1. flip-- 97.8%

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

      2. add-sqr-sqrt 74.2%

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

      3. +-commutative 74.2%

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

      4. add-sqr-sqrt 98.1%

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

      5. +-commutative 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. associate--l+ 98.1%

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

      2. +-inverses 98.1%

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

      3. metadata-eval 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in x around 0 97.7%

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

      1. flip-- 97.7%

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

      2. add-sqr-sqrt 78.0%

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

      3. +-commutative 78.0%

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

      4. add-sqr-sqrt 98.3%

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

      5. +-commutative 98.3%

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

   9. Applied egg-rr 98.3%

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

       1. associate--l+ 98.7%

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

       2. +-inverses 98.7%

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

       3. metadata-eval 98.7%

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

   11. Simplified 98.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.9:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\left(1 + x \cdot 0.5\right) - \sqrt{x}\right) + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + z} - \sqrt{z}\\ \mathbf{if}\;t\_2 - \sqrt{x} \leq 0.9:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + t\_2}\right) + t\_3\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(t\_3 + \left(\left(\left(1 + x \cdot 0.5\right) - \sqrt{x}\right) + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (- (sqrt (+ 1.0 z)) (sqrt z))))
   (if (<= (- t_2 (sqrt x)) 0.9)
     (+ (+ (+ (* 0.5 (sqrt (/ 1.0 y))) (/ 1.0 (+ (sqrt x) t_2))) t_3) t_1)
     (+
      t_1
      (+
       t_3
       (+
        (- (+ 1.0 (* x 0.5)) (sqrt x))
        (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((1.0 + z)) - sqrt(z);
	double tmp;
	if ((t_2 - sqrt(x)) <= 0.9) {
		tmp = (((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + t_2))) + t_3) + t_1;
	} else {
		tmp = t_1 + (t_3 + (((1.0 + (x * 0.5)) - sqrt(x)) + (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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + t)) - sqrt(t)
    t_2 = sqrt((x + 1.0d0))
    t_3 = sqrt((1.0d0 + z)) - sqrt(z)
    if ((t_2 - sqrt(x)) <= 0.9d0) then
        tmp = (((0.5d0 * sqrt((1.0d0 / y))) + (1.0d0 / (sqrt(x) + t_2))) + t_3) + t_1
    else
        tmp = t_1 + (t_3 + (((1.0d0 + (x * 0.5d0)) - sqrt(x)) + (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 t_1 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double tmp;
	if ((t_2 - Math.sqrt(x)) <= 0.9) {
		tmp = (((0.5 * Math.sqrt((1.0 / y))) + (1.0 / (Math.sqrt(x) + t_2))) + t_3) + t_1;
	} else {
		tmp = t_1 + (t_3 + (((1.0 + (x * 0.5)) - Math.sqrt(x)) + (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):
	t_1 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_2 = math.sqrt((x + 1.0))
	t_3 = math.sqrt((1.0 + z)) - math.sqrt(z)
	tmp = 0
	if (t_2 - math.sqrt(x)) <= 0.9:
		tmp = (((0.5 * math.sqrt((1.0 / y))) + (1.0 / (math.sqrt(x) + t_2))) + t_3) + t_1
	else:
		tmp = t_1 + (t_3 + (((1.0 + (x * 0.5)) - math.sqrt(x)) + (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)
	t_1 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	tmp = 0.0
	if (Float64(t_2 - sqrt(x)) <= 0.9)
		tmp = Float64(Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(1.0 / Float64(sqrt(x) + t_2))) + t_3) + t_1);
	else
		tmp = Float64(t_1 + Float64(t_3 + Float64(Float64(Float64(1.0 + Float64(x * 0.5)) - sqrt(x)) + 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)
	t_1 = sqrt((1.0 + t)) - sqrt(t);
	t_2 = sqrt((x + 1.0));
	t_3 = sqrt((1.0 + z)) - sqrt(z);
	tmp = 0.0;
	if ((t_2 - sqrt(x)) <= 0.9)
		tmp = (((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + t_2))) + t_3) + t_1;
	else
		tmp = t_1 + (t_3 + (((1.0 + (x * 0.5)) - sqrt(x)) + (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_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.9], N[(N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$1 + N[(t$95$3 + N[(N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.900000000000000022

   1. Initial program 85.1%

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

      1. flip-- 85.3%

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

      2. add-sqr-sqrt 63.4%

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

      3. +-commutative 63.4%

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

      4. add-sqr-sqrt 86.2%

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

      5. +-commutative 86.2%

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

   4. Applied egg-rr 86.2%

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

      1. associate--l+ 91.0%

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

      2. +-inverses 91.0%

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

      3. metadata-eval 91.0%

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

   6. Simplified 91.0%

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

      1. flip-- 91.1%

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

      2. add-sqr-sqrt 43.9%

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

      3. add-sqr-sqrt 91.1%

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

   8. Applied egg-rr 91.1%

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

   9. Taylor expanded in y around inf 53.5%

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

   if 0.900000000000000022 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 97.8%

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

      1. flip-- 97.8%

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

      2. add-sqr-sqrt 74.2%

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

      3. +-commutative 74.2%

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

      4. add-sqr-sqrt 98.1%

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

      5. +-commutative 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. associate--l+ 98.1%

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

      2. +-inverses 98.1%

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

      3. metadata-eval 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in x around 0 97.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.9:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\left(1 + x \cdot 0.5\right) - \sqrt{x}\right) + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y} - \sqrt{y}\\ t_2 := \sqrt{1 + z} - \sqrt{z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right) + t\_2\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_2 + \left(t\_1 + \left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 y)) (sqrt y)))
        (t_2 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_1 1e-7)
     (+
      (+
       (+ (* 0.5 (sqrt (/ 1.0 y))) (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))))
       t_2)
      t_3)
     (+ t_3 (+ t_2 (+ t_1 (+ 1.0 (- (* x 0.5) (sqrt x)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y)) - sqrt(y);
	double t_2 = sqrt((1.0 + z)) - sqrt(z);
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_1 <= 1e-7) {
		tmp = (((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + sqrt((x + 1.0))))) + t_2) + t_3;
	} else {
		tmp = t_3 + (t_2 + (t_1 + (1.0 + ((x * 0.5) - sqrt(x)))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y)) - sqrt(y)
    t_2 = sqrt((1.0d0 + z)) - sqrt(z)
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    if (t_1 <= 1d-7) then
        tmp = (((0.5d0 * sqrt((1.0d0 / y))) + (1.0d0 / (sqrt(x) + sqrt((x + 1.0d0))))) + t_2) + t_3
    else
        tmp = t_3 + (t_2 + (t_1 + (1.0d0 + ((x * 0.5d0) - sqrt(x)))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y)) - Math.sqrt(y);
	double t_2 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (t_1 <= 1e-7) {
		tmp = (((0.5 * Math.sqrt((1.0 / y))) + (1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0))))) + t_2) + t_3;
	} else {
		tmp = t_3 + (t_2 + (t_1 + (1.0 + ((x * 0.5) - 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 + y)) - math.sqrt(y)
	t_2 = math.sqrt((1.0 + z)) - math.sqrt(z)
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if t_1 <= 1e-7:
		tmp = (((0.5 * math.sqrt((1.0 / y))) + (1.0 / (math.sqrt(x) + math.sqrt((x + 1.0))))) + t_2) + t_3
	else:
		tmp = t_3 + (t_2 + (t_1 + (1.0 + ((x * 0.5) - math.sqrt(x)))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + y)) - sqrt(y))
	t_2 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_1 <= 1e-7)
		tmp = Float64(Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0))))) + t_2) + t_3);
	else
		tmp = Float64(t_3 + Float64(t_2 + Float64(t_1 + Float64(1.0 + Float64(Float64(x * 0.5) - 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 + y)) - sqrt(y);
	t_2 = sqrt((1.0 + z)) - sqrt(z);
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (t_1 <= 1e-7)
		tmp = (((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + sqrt((x + 1.0))))) + t_2) + t_3;
	else
		tmp = t_3 + (t_2 + (t_1 + (1.0 + ((x * 0.5) - sqrt(x)))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 9.9999999999999995e-8

   1. Initial program 87.8%

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

      1. flip-- 88.0%

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

      2. add-sqr-sqrt 44.6%

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

      3. +-commutative 44.6%

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

      4. add-sqr-sqrt 88.3%

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

      5. +-commutative 88.3%

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

   4. Applied egg-rr 88.3%

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

      1. associate--l+ 92.5%

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

      2. +-inverses 92.5%

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

      3. metadata-eval 92.5%

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

   6. Simplified 92.5%

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

      1. flip-- 92.6%

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

      2. add-sqr-sqrt 72.1%

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

      3. add-sqr-sqrt 92.6%

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

   8. Applied egg-rr 92.6%

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

   9. Taylor expanded in y around inf 94.1%

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

   if 9.9999999999999995e-8 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 96.7%

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

   3. Taylor expanded in x around 0 57.0%

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

      1. associate--l+ 57.0%

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

   5. Simplified 57.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 10^{-7}:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.001:\\ \;\;\;\;-0.125 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(0.5 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\left(\left(1 + x \cdot 0.5\right) - \sqrt{x}\right) + \frac{1}{\sqrt{y} + \left(1 + 0.5 \cdot y\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 0.001)
   (+
    (* -0.125 (sqrt (/ 1.0 (pow x 3.0))))
    (+
     (* 0.5 (sqrt (/ 1.0 x)))
     (* 0.5 (+ (sqrt (/ 1.0 t)) (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 z)))))))
   (+
    (- (sqrt (+ 1.0 t)) (sqrt t))
    (+
     (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z))))
     (+
      (- (+ 1.0 (* x 0.5)) (sqrt x))
      (/ 1.0 (+ (sqrt y) (+ 1.0 (* 0.5 y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 0.001) {
		tmp = (-0.125 * sqrt((1.0 / pow(x, 3.0)))) + ((0.5 * sqrt((1.0 / x))) + (0.5 * (sqrt((1.0 / t)) + (sqrt((1.0 / y)) + sqrt((1.0 / z))))));
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + (((1.0 + (x * 0.5)) - sqrt(x)) + (1.0 / (sqrt(y) + (1.0 + (0.5 * 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 ((sqrt((x + 1.0d0)) - sqrt(x)) <= 0.001d0) then
        tmp = ((-0.125d0) * sqrt((1.0d0 / (x ** 3.0d0)))) + ((0.5d0 * sqrt((1.0d0 / x))) + (0.5d0 * (sqrt((1.0d0 / t)) + (sqrt((1.0d0 / y)) + sqrt((1.0d0 / z))))))
    else
        tmp = (sqrt((1.0d0 + t)) - sqrt(t)) + ((1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))) + (((1.0d0 + (x * 0.5d0)) - sqrt(x)) + (1.0d0 / (sqrt(y) + (1.0d0 + (0.5d0 * 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 ((Math.sqrt((x + 1.0)) - Math.sqrt(x)) <= 0.001) {
		tmp = (-0.125 * Math.sqrt((1.0 / Math.pow(x, 3.0)))) + ((0.5 * Math.sqrt((1.0 / x))) + (0.5 * (Math.sqrt((1.0 / t)) + (Math.sqrt((1.0 / y)) + Math.sqrt((1.0 / z))))));
	} else {
		tmp = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) + ((1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)))) + (((1.0 + (x * 0.5)) - Math.sqrt(x)) + (1.0 / (Math.sqrt(y) + (1.0 + (0.5 * y))))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (math.sqrt((x + 1.0)) - math.sqrt(x)) <= 0.001:
		tmp = (-0.125 * math.sqrt((1.0 / math.pow(x, 3.0)))) + ((0.5 * math.sqrt((1.0 / x))) + (0.5 * (math.sqrt((1.0 / t)) + (math.sqrt((1.0 / y)) + math.sqrt((1.0 / z))))))
	else:
		tmp = (math.sqrt((1.0 + t)) - math.sqrt(t)) + ((1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))) + (((1.0 + (x * 0.5)) - math.sqrt(x)) + (1.0 / (math.sqrt(y) + (1.0 + (0.5 * y))))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) <= 0.001)
		tmp = Float64(Float64(-0.125 * sqrt(Float64(1.0 / (x ^ 3.0)))) + Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z)))))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z)))) + Float64(Float64(Float64(1.0 + Float64(x * 0.5)) - sqrt(x)) + Float64(1.0 / Float64(sqrt(y) + Float64(1.0 + Float64(0.5 * 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 ((sqrt((x + 1.0)) - sqrt(x)) <= 0.001)
		tmp = (-0.125 * sqrt((1.0 / (x ^ 3.0)))) + ((0.5 * sqrt((1.0 / x))) + (0.5 * (sqrt((1.0 / t)) + (sqrt((1.0 / y)) + sqrt((1.0 / z))))));
	else
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + (((1.0 + (x * 0.5)) - sqrt(x)) + (1.0 / (sqrt(y) + (1.0 + (0.5 * 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[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.001], N[(N[(-0.125 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[(1.0 + N[(0.5 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 1e-3

   1. Initial program 85.3%

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

   3. Taylor expanded in t around inf 3.8%

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

      1. associate--l+ 5.1%

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

   5. Simplified 5.1%

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

   6. Taylor expanded in z around inf 5.4%

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

      1. associate--l+ 5.4%

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

      2. distribute-lft-out 5.4%

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

      3. +-commutative 5.4%

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

   8. Simplified 5.4%

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

   9. Taylor expanded in y around inf 4.6%

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

       1. distribute-lft-out 4.6%

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

       2. +-commutative 4.6%

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

   11. Simplified 4.6%

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

   12. Taylor expanded in x around inf 19.0%

       \[\leadsto \color{blue}{-0.125 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(0.5 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\right)} \]

   if 1e-3 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 97.5%

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

      1. flip-- 97.5%

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

      2. add-sqr-sqrt 74.1%

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

      3. +-commutative 74.1%

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

      4. add-sqr-sqrt 98.1%

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

      5. +-commutative 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. associate--l+ 98.1%

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

      2. +-inverses 98.1%

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

      3. metadata-eval 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in x around 0 97.3%

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

      1. flip-- 97.3%

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

      2. add-sqr-sqrt 77.5%

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

      3. +-commutative 77.5%

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

      4. add-sqr-sqrt 97.9%

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

      5. +-commutative 97.9%

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

   9. Applied egg-rr 97.9%

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

       1. associate--l+ 98.3%

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

       2. +-inverses 98.3%

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

       3. metadata-eval 98.3%

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

   11. Simplified 98.3%

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

   12. Taylor expanded in y around 0 96.8%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.001:\\ \;\;\;\;-0.125 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(0.5 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\left(\left(1 + x \cdot 0.5\right) - \sqrt{x}\right) + \frac{1}{\sqrt{y} + \left(1 + 0.5 \cdot y\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t\_1 - \sqrt{z} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(2 + \left(\left(t\_1 + 0.5 \cdot \left(x + y\right)\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))))
   (if (<= (- t_1 (sqrt z)) 2e-6)
     (+
      (- (sqrt (+ x 1.0)) (sqrt x))
      (+
       (- (sqrt (+ 1.0 y)) (sqrt y))
       (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 z))))))
     (+
      (- (sqrt (+ 1.0 t)) (sqrt t))
      (+
       2.0
       (- (+ t_1 (* 0.5 (+ x y))) (+ (sqrt z) (+ (sqrt x) (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double tmp;
	if ((t_1 - sqrt(z)) <= 2e-6) {
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((sqrt((1.0 + y)) - sqrt(y)) + (0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / z)))));
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + (2.0 + ((t_1 + (0.5 * (x + y))) - (sqrt(z) + (sqrt(x) + sqrt(y)))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if ((t_1 - sqrt(z)) <= 2d-6) then
        tmp = (sqrt((x + 1.0d0)) - sqrt(x)) + ((sqrt((1.0d0 + y)) - sqrt(y)) + (0.5d0 * (sqrt((1.0d0 / t)) + sqrt((1.0d0 / z)))))
    else
        tmp = (sqrt((1.0d0 + t)) - sqrt(t)) + (2.0d0 + ((t_1 + (0.5d0 * (x + y))) - (sqrt(z) + (sqrt(x) + sqrt(y)))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double tmp;
	if ((t_1 - Math.sqrt(z)) <= 2e-6) {
		tmp = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) + (0.5 * (Math.sqrt((1.0 / t)) + Math.sqrt((1.0 / z)))));
	} else {
		tmp = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (2.0 + ((t_1 + (0.5 * (x + y))) - (Math.sqrt(z) + (Math.sqrt(x) + Math.sqrt(y)))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	tmp = 0
	if (t_1 - math.sqrt(z)) <= 2e-6:
		tmp = (math.sqrt((x + 1.0)) - math.sqrt(x)) + ((math.sqrt((1.0 + y)) - math.sqrt(y)) + (0.5 * (math.sqrt((1.0 / t)) + math.sqrt((1.0 / z)))))
	else:
		tmp = (math.sqrt((1.0 + t)) - math.sqrt(t)) + (2.0 + ((t_1 + (0.5 * (x + y))) - (math.sqrt(z) + (math.sqrt(x) + math.sqrt(y)))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (Float64(t_1 - sqrt(z)) <= 2e-6)
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / z))))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(2.0 + Float64(Float64(t_1 + Float64(0.5 * Float64(x + y))) - Float64(sqrt(z) + Float64(sqrt(x) + sqrt(y))))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	tmp = 0.0;
	if ((t_1 - sqrt(z)) <= 2e-6)
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((sqrt((1.0 + y)) - sqrt(y)) + (0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / z)))));
	else
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + (2.0 + ((t_1 + (0.5 * (x + y))) - (sqrt(z) + (sqrt(x) + sqrt(y)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 2e-6], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(N[(t$95$1 + N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 1.99999999999999991e-6

   1. Initial program 84.9%

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

      1. associate-+l+ 84.9%

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

      2. associate-+l+ 84.9%

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

      3. +-commutative 84.9%

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

      4. +-commutative 84.9%

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

      5. associate-+l- 84.9%

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

      6. +-commutative 84.9%

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

      7. +-commutative 84.9%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in t around inf 41.8%

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

   6. Taylor expanded in z around inf 48.2%

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

      1. distribute-lft-out 48.2%

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

   8. Simplified 48.2%

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

   if 1.99999999999999991e-6 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 98.3%

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

   3. Taylor expanded in x around 0 32.5%

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

      1. associate--l+ 38.9%

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

   5. Simplified 38.9%

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

   6. Taylor expanded in y around 0 28.9%

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

      1. associate--l+ 28.9%

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

      2. distribute-lft-out 28.9%

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

      3. associate-+r+ 28.9%

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

      4. +-commutative 28.9%

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

   8. Simplified 28.9%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(2 + \left(\left(\sqrt{1 + z} + 0.5 \cdot \left(x + y\right)\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;t\_2 + \left(\frac{1}{\sqrt{z} + t\_1} + \left(\left(\left(1 + x \cdot 0.5\right) - \sqrt{x}\right) + \frac{1}{\sqrt{y} + \left(1 + 0.5 \cdot y\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right) + \left(t\_1 - \sqrt{z}\right)\right) + t\_2\\ \end{array} \end{array} \]

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))) (t_2 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= y 2.4)
     (+
      t_2
      (+
       (/ 1.0 (+ (sqrt z) t_1))
       (+
        (- (+ 1.0 (* x 0.5)) (sqrt x))
        (/ 1.0 (+ (sqrt y) (+ 1.0 (* 0.5 y)))))))
     (+
      (+
       (+ (* 0.5 (sqrt (/ 1.0 y))) (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))))
       (- t_1 (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 + z));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (y <= 2.4) {
		tmp = t_2 + ((1.0 / (sqrt(z) + t_1)) + (((1.0 + (x * 0.5)) - sqrt(x)) + (1.0 / (sqrt(y) + (1.0 + (0.5 * y))))));
	} else {
		tmp = (((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + sqrt((x + 1.0))))) + (t_1 - 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) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + t)) - sqrt(t)
    if (y <= 2.4d0) then
        tmp = t_2 + ((1.0d0 / (sqrt(z) + t_1)) + (((1.0d0 + (x * 0.5d0)) - sqrt(x)) + (1.0d0 / (sqrt(y) + (1.0d0 + (0.5d0 * y))))))
    else
        tmp = (((0.5d0 * sqrt((1.0d0 / y))) + (1.0d0 / (sqrt(x) + sqrt((x + 1.0d0))))) + (t_1 - 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 + z));
	double t_2 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (y <= 2.4) {
		tmp = t_2 + ((1.0 / (Math.sqrt(z) + t_1)) + (((1.0 + (x * 0.5)) - Math.sqrt(x)) + (1.0 / (Math.sqrt(y) + (1.0 + (0.5 * y))))));
	} else {
		tmp = (((0.5 * Math.sqrt((1.0 / y))) + (1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0))))) + (t_1 - 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 + z))
	t_2 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if y <= 2.4:
		tmp = t_2 + ((1.0 / (math.sqrt(z) + t_1)) + (((1.0 + (x * 0.5)) - math.sqrt(x)) + (1.0 / (math.sqrt(y) + (1.0 + (0.5 * y))))))
	else:
		tmp = (((0.5 * math.sqrt((1.0 / y))) + (1.0 / (math.sqrt(x) + math.sqrt((x + 1.0))))) + (t_1 - 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 + z))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (y <= 2.4)
		tmp = Float64(t_2 + Float64(Float64(1.0 / Float64(sqrt(z) + t_1)) + Float64(Float64(Float64(1.0 + Float64(x * 0.5)) - sqrt(x)) + Float64(1.0 / Float64(sqrt(y) + Float64(1.0 + Float64(0.5 * y)))))));
	else
		tmp = Float64(Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0))))) + Float64(t_1 - 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 + z));
	t_2 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (y <= 2.4)
		tmp = t_2 + ((1.0 / (sqrt(z) + t_1)) + (((1.0 + (x * 0.5)) - sqrt(x)) + (1.0 / (sqrt(y) + (1.0 + (0.5 * y))))));
	else
		tmp = (((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + sqrt((x + 1.0))))) + (t_1 - 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 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.4], N[(t$95$2 + N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[(1.0 + N[(0.5 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 97.6%

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

      1. flip-- 97.6%

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

      2. add-sqr-sqrt 97.6%

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

      3. +-commutative 97.6%

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

      4. add-sqr-sqrt 97.7%

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

      5. +-commutative 97.7%

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

   4. Applied egg-rr 97.7%

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

      1. associate--l+ 97.6%

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

      2. +-inverses 97.6%

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

      3. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in x around 0 57.9%

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

      1. flip-- 57.9%

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

      2. add-sqr-sqrt 42.9%

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

      3. +-commutative 42.9%

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

      4. add-sqr-sqrt 58.6%

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

      5. +-commutative 58.6%

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

   9. Applied egg-rr 58.6%

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

       1. associate--l+ 58.6%

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

       2. +-inverses 58.6%

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

       3. metadata-eval 58.6%

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

   11. Simplified 58.6%

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

   12. Taylor expanded in y around 0 57.9%

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

   if 2.39999999999999991 < y

   1. Initial program 87.3%

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

      1. flip-- 87.5%

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

      2. add-sqr-sqrt 45.8%

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

      3. +-commutative 45.8%

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

      4. add-sqr-sqrt 88.5%

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

      5. +-commutative 88.5%

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

   4. Applied egg-rr 88.5%

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

      1. associate--l+ 92.5%

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

      2. +-inverses 92.5%

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

      3. metadata-eval 92.5%

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

   6. Simplified 92.5%

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

      1. flip-- 92.6%

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

      2. add-sqr-sqrt 71.5%

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

      3. add-sqr-sqrt 92.6%

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

   8. Applied egg-rr 92.6%

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

   9. Taylor expanded in y around inf 93.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\left(\left(1 + x \cdot 0.5\right) - \sqrt{x}\right) + \frac{1}{\sqrt{y} + \left(1 + 0.5 \cdot y\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.58:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 + 0.5 \cdot \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 0.58)
   (+
    (- (sqrt (+ 1.0 t)) (sqrt t))
    (- (+ 3.0 (* 0.5 (+ z (+ x y)))) (+ (sqrt x) (+ (sqrt z) (sqrt y)))))
   (+
    (- (sqrt (+ x 1.0)) (sqrt x))
    (+
     (- (sqrt (+ 1.0 y)) (sqrt y))
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 z))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.58) {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((3.0 + (0.5 * (z + (x + y)))) - (sqrt(x) + (sqrt(z) + sqrt(y))));
	} else {
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((sqrt((1.0 + y)) - sqrt(y)) + (0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / z)))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 0.58d0) then
        tmp = (sqrt((1.0d0 + t)) - sqrt(t)) + ((3.0d0 + (0.5d0 * (z + (x + y)))) - (sqrt(x) + (sqrt(z) + sqrt(y))))
    else
        tmp = (sqrt((x + 1.0d0)) - sqrt(x)) + ((sqrt((1.0d0 + y)) - sqrt(y)) + (0.5d0 * (sqrt((1.0d0 / t)) + sqrt((1.0d0 / z)))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.58) {
		tmp = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) + ((3.0 + (0.5 * (z + (x + y)))) - (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y))));
	} else {
		tmp = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) + (0.5 * (Math.sqrt((1.0 / t)) + Math.sqrt((1.0 / z)))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 0.58:
		tmp = (math.sqrt((1.0 + t)) - math.sqrt(t)) + ((3.0 + (0.5 * (z + (x + y)))) - (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y))))
	else:
		tmp = (math.sqrt((x + 1.0)) - math.sqrt(x)) + ((math.sqrt((1.0 + y)) - math.sqrt(y)) + (0.5 * (math.sqrt((1.0 / t)) + math.sqrt((1.0 / z)))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 0.58)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(3.0 + Float64(0.5 * Float64(z + Float64(x + y)))) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	else
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / z))))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 0.58)
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((3.0 + (0.5 * (z + (x + y)))) - (sqrt(x) + (sqrt(z) + sqrt(y))));
	else
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((sqrt((1.0 + y)) - sqrt(y)) + (0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / z)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 0.58], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 + N[(0.5 * N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.57999999999999996

   1. Initial program 98.3%

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

   3. Taylor expanded in x around 0 32.2%

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

      1. associate--l+ 38.7%

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

   5. Simplified 38.7%

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

   6. Taylor expanded in y around 0 28.6%

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

      1. associate--l+ 28.6%

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

      2. metadata-eval 28.6%

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

      3. rem-square-sqrt 28.6%

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

      4. hypot-undefine 28.6%

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

      5. distribute-lft-out 28.6%

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

      6. associate-+r+ 28.6%

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

      7. +-commutative 28.6%

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

   8. Simplified 28.6%

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

   9. Taylor expanded in z around 0 28.6%

      \[\leadsto \color{blue}{\left(\left(3 + \left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out 28.6%

          \[\leadsto \left(\left(3 + \color{blue}{0.5 \cdot \left(z + \left(x + y\right)\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 28.6%

       \[\leadsto \color{blue}{\left(\left(3 + 0.5 \cdot \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.57999999999999996 < z

   1. Initial program 85.1%

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

      1. associate-+l+ 85.1%

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

      2. associate-+l+ 85.1%

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

      3. +-commutative 85.1%

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

      4. +-commutative 85.1%

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

      5. associate-+l- 85.1%

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

      6. +-commutative 85.1%

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

      7. +-commutative 85.1%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in t around inf 41.2%

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

   6. Taylor expanded in z around inf 47.5%

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

      1. distribute-lft-out 47.5%

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

   8. Simplified 47.5%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.58:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 + 0.5 \cdot \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.39:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 + 0.5 \cdot \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{1 + y} + \left(0.5 \cdot \left({t}^{-0.5} + {z}^{-0.5}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 0.39)
   (+
    (- (sqrt (+ 1.0 t)) (sqrt t))
    (- (+ 3.0 (* 0.5 (+ z (+ x y)))) (+ (sqrt x) (+ (sqrt z) (sqrt y)))))
   (+
    (sqrt (+ x 1.0))
    (+
     (sqrt (+ 1.0 y))
     (- (* 0.5 (+ (pow t -0.5) (pow z -0.5))) (+ (sqrt x) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.39) {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((3.0 + (0.5 * (z + (x + y)))) - (sqrt(x) + (sqrt(z) + sqrt(y))));
	} else {
		tmp = sqrt((x + 1.0)) + (sqrt((1.0 + y)) + ((0.5 * (pow(t, -0.5) + pow(z, -0.5))) - (sqrt(x) + sqrt(y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 0.39d0) then
        tmp = (sqrt((1.0d0 + t)) - sqrt(t)) + ((3.0d0 + (0.5d0 * (z + (x + y)))) - (sqrt(x) + (sqrt(z) + sqrt(y))))
    else
        tmp = sqrt((x + 1.0d0)) + (sqrt((1.0d0 + y)) + ((0.5d0 * ((t ** (-0.5d0)) + (z ** (-0.5d0)))) - (sqrt(x) + sqrt(y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.39) {
		tmp = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) + ((3.0 + (0.5 * (z + (x + y)))) - (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y))));
	} else {
		tmp = Math.sqrt((x + 1.0)) + (Math.sqrt((1.0 + y)) + ((0.5 * (Math.pow(t, -0.5) + Math.pow(z, -0.5))) - (Math.sqrt(x) + Math.sqrt(y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 0.39:
		tmp = (math.sqrt((1.0 + t)) - math.sqrt(t)) + ((3.0 + (0.5 * (z + (x + y)))) - (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y))))
	else:
		tmp = math.sqrt((x + 1.0)) + (math.sqrt((1.0 + y)) + ((0.5 * (math.pow(t, -0.5) + math.pow(z, -0.5))) - (math.sqrt(x) + math.sqrt(y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 0.39)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(3.0 + Float64(0.5 * Float64(z + Float64(x + y)))) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(sqrt(Float64(1.0 + y)) + Float64(Float64(0.5 * Float64((t ^ -0.5) + (z ^ -0.5))) - Float64(sqrt(x) + sqrt(y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 0.39)
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((3.0 + (0.5 * (z + (x + y)))) - (sqrt(x) + (sqrt(z) + sqrt(y))));
	else
		tmp = sqrt((x + 1.0)) + (sqrt((1.0 + y)) + ((0.5 * ((t ^ -0.5) + (z ^ -0.5))) - (sqrt(x) + sqrt(y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 0.39], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 + N[(0.5 * N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[(N[Power[t, -0.5], $MachinePrecision] + N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.39000000000000001

   1. Initial program 98.3%

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

   3. Taylor expanded in x around 0 32.2%

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

      1. associate--l+ 38.7%

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

   5. Simplified 38.7%

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

   6. Taylor expanded in y around 0 28.6%

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

      1. associate--l+ 28.6%

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

      2. metadata-eval 28.6%

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

      3. rem-square-sqrt 28.6%

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

      4. hypot-undefine 28.6%

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

      5. distribute-lft-out 28.6%

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

      6. associate-+r+ 28.6%

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

      7. +-commutative 28.6%

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

   8. Simplified 28.6%

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

   9. Taylor expanded in z around 0 28.6%

      \[\leadsto \color{blue}{\left(\left(3 + \left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out 28.6%

          \[\leadsto \left(\left(3 + \color{blue}{0.5 \cdot \left(z + \left(x + y\right)\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 28.6%

       \[\leadsto \color{blue}{\left(\left(3 + 0.5 \cdot \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.39000000000000001 < z

   1. Initial program 85.1%

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

   3. Taylor expanded in t around inf 4.2%

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

      1. associate--l+ 15.9%

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

   5. Simplified 15.9%

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

   6. Taylor expanded in z around inf 21.1%

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

      1. associate--l+ 28.7%

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

      2. distribute-lft-in 28.7%

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

      3. +-commutative 28.7%

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

      4. associate-+r- 28.7%

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

      5. associate-+r+ 21.1%

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

   8. Applied egg-rr 21.1%

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

      1. associate-+l+ 28.7%

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

      2. +-commutative 28.7%

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

   10. Simplified 28.7%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.39:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 + 0.5 \cdot \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{1 + y} + \left(0.5 \cdot \left({t}^{-0.5} + {z}^{-0.5}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-244}:\\ \;\;\;\;\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot \left(x + y\right) + 0.5 \cdot z\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.3:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) + \left(\left(1 + \left(0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right) + y \cdot \left(0.5 + y \cdot \left(y \cdot \left(0.0625 + y \cdot -0.0390625\right) - 0.125\right)\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + 0.5 \cdot \left({y}^{-0.5} + \left({t}^{-0.5} + {z}^{-0.5}\right)\right)\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.28e-244)
   (-
    (+ 4.0 (+ (* 0.5 t) (+ (* 0.5 (+ x y)) (* 0.5 z))))
    (+ (sqrt t) (+ (sqrt x) (+ (sqrt z) (sqrt y)))))
   (if (<= y 1.3)
     (+
      (+ 1.0 (* x 0.5))
      (-
       (+
        1.0
        (+
         (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 z))))
         (* y (+ 0.5 (* y (- (* y (+ 0.0625 (* y -0.0390625))) 0.125))))))
       (+ (sqrt x) (sqrt y))))
     (-
      (+
       (sqrt (+ x 1.0))
       (* 0.5 (+ (pow y -0.5) (+ (pow t -0.5) (pow z -0.5)))))
      (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.28e-244) {
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))));
	} else if (y <= 1.3) {
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / z)))) + (y * (0.5 + (y * ((y * (0.0625 + (y * -0.0390625))) - 0.125)))))) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = (sqrt((x + 1.0)) + (0.5 * (pow(y, -0.5) + (pow(t, -0.5) + pow(z, -0.5))))) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.28d-244) then
        tmp = (4.0d0 + ((0.5d0 * t) + ((0.5d0 * (x + y)) + (0.5d0 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))))
    else if (y <= 1.3d0) then
        tmp = (1.0d0 + (x * 0.5d0)) + ((1.0d0 + ((0.5d0 * (sqrt((1.0d0 / t)) + sqrt((1.0d0 / z)))) + (y * (0.5d0 + (y * ((y * (0.0625d0 + (y * (-0.0390625d0)))) - 0.125d0)))))) - (sqrt(x) + sqrt(y)))
    else
        tmp = (sqrt((x + 1.0d0)) + (0.5d0 * ((y ** (-0.5d0)) + ((t ** (-0.5d0)) + (z ** (-0.5d0)))))) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.28e-244) {
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (Math.sqrt(t) + (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y))));
	} else if (y <= 1.3) {
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (Math.sqrt((1.0 / t)) + Math.sqrt((1.0 / z)))) + (y * (0.5 + (y * ((y * (0.0625 + (y * -0.0390625))) - 0.125)))))) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = (Math.sqrt((x + 1.0)) + (0.5 * (Math.pow(y, -0.5) + (Math.pow(t, -0.5) + Math.pow(z, -0.5))))) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 1.28e-244:
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (math.sqrt(t) + (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y))))
	elif y <= 1.3:
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (math.sqrt((1.0 / t)) + math.sqrt((1.0 / z)))) + (y * (0.5 + (y * ((y * (0.0625 + (y * -0.0390625))) - 0.125)))))) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = (math.sqrt((x + 1.0)) + (0.5 * (math.pow(y, -0.5) + (math.pow(t, -0.5) + math.pow(z, -0.5))))) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.28e-244)
		tmp = Float64(Float64(4.0 + Float64(Float64(0.5 * t) + Float64(Float64(0.5 * Float64(x + y)) + Float64(0.5 * z)))) - Float64(sqrt(t) + Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	elseif (y <= 1.3)
		tmp = Float64(Float64(1.0 + Float64(x * 0.5)) + Float64(Float64(1.0 + Float64(Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / z)))) + Float64(y * Float64(0.5 + Float64(y * Float64(Float64(y * Float64(0.0625 + Float64(y * -0.0390625))) - 0.125)))))) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) + Float64(0.5 * Float64((y ^ -0.5) + Float64((t ^ -0.5) + (z ^ -0.5))))) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.28e-244)
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))));
	elseif (y <= 1.3)
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / z)))) + (y * (0.5 + (y * ((y * (0.0625 + (y * -0.0390625))) - 0.125)))))) - (sqrt(x) + sqrt(y)));
	else
		tmp = (sqrt((x + 1.0)) + (0.5 * ((y ^ -0.5) + ((t ^ -0.5) + (z ^ -0.5))))) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 1.28e-244], N[(N[(4.0 + N[(N[(0.5 * t), $MachinePrecision] + N[(N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(0.5 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3], N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * N[(0.5 + N[(y * N[(N[(y * N[(0.0625 + N[(y * -0.0390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(N[Power[y, -0.5], $MachinePrecision] + N[(N[Power[t, -0.5], $MachinePrecision] + N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.27999999999999994e-244

   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. Add Preprocessing

   3. Taylor expanded in x around 0 32.5%

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

      1. associate--l+ 37.1%

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

   5. Simplified 37.1%

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

   6. Taylor expanded in y around 0 32.5%

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

      1. associate--l+ 53.5%

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

      2. metadata-eval 53.5%

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

      3. rem-square-sqrt 53.5%

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

      4. hypot-undefine 53.5%

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

      5. distribute-lft-out 53.5%

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

      6. associate-+r+ 53.5%

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

      7. +-commutative 53.5%

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

   8. Simplified 53.5%

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

   9. Taylor expanded in t around 0 30.3%

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

   10. Taylor expanded in z around 0 9.7%

       \[\leadsto \color{blue}{\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   if 1.27999999999999994e-244 < y < 1.30000000000000004

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf 19.7%

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

      1. associate--l+ 23.5%

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

   5. Simplified 23.5%

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

   6. Taylor expanded in z around inf 21.6%

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

      1. associate--l+ 21.6%

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

      2. distribute-lft-out 21.6%

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

      3. +-commutative 21.6%

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

   8. Simplified 21.6%

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

   9. Taylor expanded in x around 0 21.8%

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

   10. Taylor expanded in y around 0 21.8%

       \[\leadsto \left(1 + 0.5 \cdot x\right) + \color{blue}{\left(\left(1 + \left(0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right) + y \cdot \left(0.5 + y \cdot \left(y \cdot \left(0.0625 + -0.0390625 \cdot y\right) - 0.125\right)\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 1.30000000000000004 < y

   1. Initial program 87.4%

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

   3. Taylor expanded in t around inf 4.4%

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

      1. associate--l+ 15.2%

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

   5. Simplified 15.2%

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

   6. Taylor expanded in z around inf 14.1%

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

      1. associate--l+ 14.1%

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

      2. distribute-lft-out 14.1%

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

      3. +-commutative 14.1%

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

   8. Simplified 14.1%

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

   9. Taylor expanded in y around inf 13.5%

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

       1. distribute-lft-out 13.5%

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

       2. +-commutative 13.5%

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

   11. Simplified 13.5%

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

       1. associate-+r- 13.5%

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

       2. pow1/2 13.5%

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

       3. inv-pow 13.5%

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

       4. pow-pow 13.5%

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

       5. metadata-eval 13.5%

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

       6. pow1/2 13.5%

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

       7. inv-pow 13.5%

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

       8. pow-pow 13.5%

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

       9. metadata-eval 13.5%

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

       10. pow1/2 13.5%

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

       11. inv-pow 13.5%

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

       12. pow-pow 13.5%

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

       13. metadata-eval 13.5%

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

   13. Applied egg-rr 13.5%

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

4. Final simplification 16.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-244}:\\ \;\;\;\;\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot \left(x + y\right) + 0.5 \cdot z\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.3:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) + \left(\left(1 + \left(0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right) + y \cdot \left(0.5 + y \cdot \left(y \cdot \left(0.0625 + y \cdot -0.0390625\right) - 0.125\right)\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + 0.5 \cdot \left({y}^{-0.5} + \left({t}^{-0.5} + {z}^{-0.5}\right)\right)\right) - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.8:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 + 0.5 \cdot \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} + 0.5 \cdot \left(x + \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 0.8)
   (+
    (- (sqrt (+ 1.0 t)) (sqrt t))
    (- (+ 3.0 (* 0.5 (+ z (+ x y)))) (+ (sqrt x) (+ (sqrt z) (sqrt y)))))
   (+
    1.0
    (-
     (+ (sqrt (+ 1.0 y)) (* 0.5 (+ x (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 z))))))
     (+ (sqrt x) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.8) {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((3.0 + (0.5 * (z + (x + y)))) - (sqrt(x) + (sqrt(z) + sqrt(y))));
	} else {
		tmp = 1.0 + ((sqrt((1.0 + y)) + (0.5 * (x + (sqrt((1.0 / t)) + sqrt((1.0 / z)))))) - (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 0.8d0) then
        tmp = (sqrt((1.0d0 + t)) - sqrt(t)) + ((3.0d0 + (0.5d0 * (z + (x + y)))) - (sqrt(x) + (sqrt(z) + sqrt(y))))
    else
        tmp = 1.0d0 + ((sqrt((1.0d0 + y)) + (0.5d0 * (x + (sqrt((1.0d0 / t)) + sqrt((1.0d0 / z)))))) - (sqrt(x) + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.8) {
		tmp = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) + ((3.0 + (0.5 * (z + (x + y)))) - (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y))));
	} else {
		tmp = 1.0 + ((Math.sqrt((1.0 + y)) + (0.5 * (x + (Math.sqrt((1.0 / t)) + Math.sqrt((1.0 / z)))))) - (Math.sqrt(x) + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 0.8:
		tmp = (math.sqrt((1.0 + t)) - math.sqrt(t)) + ((3.0 + (0.5 * (z + (x + y)))) - (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y))))
	else:
		tmp = 1.0 + ((math.sqrt((1.0 + y)) + (0.5 * (x + (math.sqrt((1.0 / t)) + math.sqrt((1.0 / z)))))) - (math.sqrt(x) + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 0.8)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(3.0 + Float64(0.5 * Float64(z + Float64(x + y)))) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	else
		tmp = Float64(1.0 + Float64(Float64(sqrt(Float64(1.0 + y)) + Float64(0.5 * Float64(x + Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / z)))))) - Float64(sqrt(x) + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 0.8)
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((3.0 + (0.5 * (z + (x + y)))) - (sqrt(x) + (sqrt(z) + sqrt(y))));
	else
		tmp = 1.0 + ((sqrt((1.0 + y)) + (0.5 * (x + (sqrt((1.0 / t)) + sqrt((1.0 / z)))))) - (sqrt(x) + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 0.8], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 + N[(0.5 * N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(x + N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.80000000000000004

   1. Initial program 98.3%

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

   3. Taylor expanded in x around 0 32.2%

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

      1. associate--l+ 38.7%

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

   5. Simplified 38.7%

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

   6. Taylor expanded in y around 0 28.6%

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

      1. associate--l+ 28.6%

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

      2. metadata-eval 28.6%

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

      3. rem-square-sqrt 28.6%

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

      4. hypot-undefine 28.6%

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

      5. distribute-lft-out 28.6%

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

      6. associate-+r+ 28.6%

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

      7. +-commutative 28.6%

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

   8. Simplified 28.6%

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

   9. Taylor expanded in z around 0 28.6%

      \[\leadsto \color{blue}{\left(\left(3 + \left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out 28.6%

          \[\leadsto \left(\left(3 + \color{blue}{0.5 \cdot \left(z + \left(x + y\right)\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 28.6%

       \[\leadsto \color{blue}{\left(\left(3 + 0.5 \cdot \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.80000000000000004 < z

   1. Initial program 85.1%

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

   3. Taylor expanded in t around inf 4.2%

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

      1. associate--l+ 15.9%

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

   5. Simplified 15.9%

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

   6. Taylor expanded in z around inf 28.7%

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

      1. associate--l+ 28.7%

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

      2. distribute-lft-out 28.7%

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

      3. +-commutative 28.7%

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

   8. Simplified 28.7%

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

   9. Taylor expanded in x around 0 20.6%

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

       1. associate--l+ 32.0%

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

       2. distribute-lft-in 32.0%

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

   11. Simplified 32.0%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.8:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(3 + 0.5 \cdot \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} + 0.5 \cdot \left(x + \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{t}}\\ \mathbf{if}\;z \leq 10500000:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + z} + 0.5 \cdot \left(t\_1 + \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} + 0.5 \cdot \left(x + \left(t\_1 + \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (/ 1.0 t))))
   (if (<= z 10500000.0)
     (+
      2.0
      (-
       (+ (sqrt (+ 1.0 z)) (* 0.5 (+ t_1 (+ x y))))
       (+ (sqrt x) (+ (sqrt z) (sqrt y)))))
     (+
      1.0
      (-
       (+ (sqrt (+ 1.0 y)) (* 0.5 (+ x (+ t_1 (sqrt (/ 1.0 z))))))
       (+ (sqrt x) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 / t));
	double tmp;
	if (z <= 10500000.0) {
		tmp = 2.0 + ((sqrt((1.0 + z)) + (0.5 * (t_1 + (x + y)))) - (sqrt(x) + (sqrt(z) + sqrt(y))));
	} else {
		tmp = 1.0 + ((sqrt((1.0 + y)) + (0.5 * (x + (t_1 + sqrt((1.0 / z)))))) - (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 / t))
    if (z <= 10500000.0d0) then
        tmp = 2.0d0 + ((sqrt((1.0d0 + z)) + (0.5d0 * (t_1 + (x + y)))) - (sqrt(x) + (sqrt(z) + sqrt(y))))
    else
        tmp = 1.0d0 + ((sqrt((1.0d0 + y)) + (0.5d0 * (x + (t_1 + sqrt((1.0d0 / z)))))) - (sqrt(x) + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 / t));
	double tmp;
	if (z <= 10500000.0) {
		tmp = 2.0 + ((Math.sqrt((1.0 + z)) + (0.5 * (t_1 + (x + y)))) - (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y))));
	} else {
		tmp = 1.0 + ((Math.sqrt((1.0 + y)) + (0.5 * (x + (t_1 + Math.sqrt((1.0 / z)))))) - (Math.sqrt(x) + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 / t))
	tmp = 0
	if z <= 10500000.0:
		tmp = 2.0 + ((math.sqrt((1.0 + z)) + (0.5 * (t_1 + (x + y)))) - (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y))))
	else:
		tmp = 1.0 + ((math.sqrt((1.0 + y)) + (0.5 * (x + (t_1 + math.sqrt((1.0 / z)))))) - (math.sqrt(x) + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 / t))
	tmp = 0.0
	if (z <= 10500000.0)
		tmp = Float64(2.0 + Float64(Float64(sqrt(Float64(1.0 + z)) + Float64(0.5 * Float64(t_1 + Float64(x + y)))) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	else
		tmp = Float64(1.0 + Float64(Float64(sqrt(Float64(1.0 + y)) + Float64(0.5 * Float64(x + Float64(t_1 + sqrt(Float64(1.0 / z)))))) - Float64(sqrt(x) + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 / t));
	tmp = 0.0;
	if (z <= 10500000.0)
		tmp = 2.0 + ((sqrt((1.0 + z)) + (0.5 * (t_1 + (x + y)))) - (sqrt(x) + (sqrt(z) + sqrt(y))));
	else
		tmp = 1.0 + ((sqrt((1.0 + y)) + (0.5 * (x + (t_1 + sqrt((1.0 / z)))))) - (sqrt(x) + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 10500000.0], N[(2.0 + N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(t$95$1 + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(x + N[(t$95$1 + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.05e7

   1. Initial program 98.3%

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

   3. Taylor expanded in x around 0 32.5%

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

      1. associate--l+ 38.9%

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

   5. Simplified 38.9%

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

   6. Taylor expanded in y around 0 28.9%

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

      1. associate--l+ 28.9%

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

      2. metadata-eval 28.9%

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

      3. rem-square-sqrt 28.9%

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

      4. hypot-undefine 28.9%

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

      5. distribute-lft-out 28.9%

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

      6. associate-+r+ 28.9%

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

      7. +-commutative 28.9%

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

   8. Simplified 28.9%

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

   9. Taylor expanded in t around inf 17.4%

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

       1. associate--l+ 17.4%

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

       2. distribute-lft-out 17.4%

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

   11. Simplified 17.4%

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

   if 1.05e7 < z

   1. Initial program 84.9%

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

   3. Taylor expanded in t around inf 4.2%

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

      1. associate--l+ 16.0%

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

   5. Simplified 16.0%

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

   6. Taylor expanded in z around inf 29.1%

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

      1. associate--l+ 29.1%

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

      2. distribute-lft-out 29.1%

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

      3. +-commutative 29.1%

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

   8. Simplified 29.1%

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

   9. Taylor expanded in x around 0 20.9%

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

       1. associate--l+ 32.4%

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

       2. distribute-lft-in 32.4%

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

   11. Simplified 32.4%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10500000:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + z} + 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} + 0.5 \cdot \left(x + \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{+24}:\\ \;\;\;\;\left(\left(3 + 0.5 \cdot \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right) + \left(\left(1 + 0.5 \cdot t\right) - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} + 0.5 \cdot \left(x + \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (<= t 4.4e+24)
   (+
    (- (+ 3.0 (* 0.5 (+ z (+ x y)))) (+ (sqrt x) (+ (sqrt z) (sqrt y))))
    (- (+ 1.0 (* 0.5 t)) (sqrt t)))
   (+
    1.0
    (-
     (+ (sqrt (+ 1.0 y)) (* 0.5 (+ x (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 z))))))
     (+ (sqrt x) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 4.4e+24) {
		tmp = ((3.0 + (0.5 * (z + (x + y)))) - (sqrt(x) + (sqrt(z) + sqrt(y)))) + ((1.0 + (0.5 * t)) - sqrt(t));
	} else {
		tmp = 1.0 + ((sqrt((1.0 + y)) + (0.5 * (x + (sqrt((1.0 / t)) + sqrt((1.0 / z)))))) - (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 4.4d+24) then
        tmp = ((3.0d0 + (0.5d0 * (z + (x + y)))) - (sqrt(x) + (sqrt(z) + sqrt(y)))) + ((1.0d0 + (0.5d0 * t)) - sqrt(t))
    else
        tmp = 1.0d0 + ((sqrt((1.0d0 + y)) + (0.5d0 * (x + (sqrt((1.0d0 / t)) + sqrt((1.0d0 / z)))))) - (sqrt(x) + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 4.4e+24) {
		tmp = ((3.0 + (0.5 * (z + (x + y)))) - (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y)))) + ((1.0 + (0.5 * t)) - Math.sqrt(t));
	} else {
		tmp = 1.0 + ((Math.sqrt((1.0 + y)) + (0.5 * (x + (Math.sqrt((1.0 / t)) + Math.sqrt((1.0 / z)))))) - (Math.sqrt(x) + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 4.4e+24:
		tmp = ((3.0 + (0.5 * (z + (x + y)))) - (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y)))) + ((1.0 + (0.5 * t)) - math.sqrt(t))
	else:
		tmp = 1.0 + ((math.sqrt((1.0 + y)) + (0.5 * (x + (math.sqrt((1.0 / t)) + math.sqrt((1.0 / z)))))) - (math.sqrt(x) + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 4.4e+24)
		tmp = Float64(Float64(Float64(3.0 + Float64(0.5 * Float64(z + Float64(x + y)))) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))) + Float64(Float64(1.0 + Float64(0.5 * t)) - sqrt(t)));
	else
		tmp = Float64(1.0 + Float64(Float64(sqrt(Float64(1.0 + y)) + Float64(0.5 * Float64(x + Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / z)))))) - Float64(sqrt(x) + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 4.4e+24)
		tmp = ((3.0 + (0.5 * (z + (x + y)))) - (sqrt(x) + (sqrt(z) + sqrt(y)))) + ((1.0 + (0.5 * t)) - sqrt(t));
	else
		tmp = 1.0 + ((sqrt((1.0 + y)) + (0.5 * (x + (sqrt((1.0 / t)) + sqrt((1.0 / z)))))) - (sqrt(x) + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 4.4e+24], N[(N[(N[(3.0 + N[(0.5 * N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(0.5 * t), $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(x + N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.40000000000000003e24

   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. Add Preprocessing

   3. Taylor expanded in x around 0 27.2%

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

      1. associate--l+ 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in y around 0 18.9%

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

      1. associate--l+ 26.1%

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

      2. metadata-eval 26.1%

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

      3. rem-square-sqrt 26.2%

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

      4. hypot-undefine 26.2%

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

      5. distribute-lft-out 26.2%

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

      6. associate-+r+ 26.2%

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

      7. +-commutative 26.2%

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

   8. Simplified 26.2%

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

   9. Taylor expanded in t around 0 24.4%

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

   10. Taylor expanded in z around 0 15.4%

       \[\leadsto \color{blue}{\left(\left(3 + \left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\left(1 + 0.5 \cdot t\right) - \sqrt{t}\right) \]
   11. Step-by-step derivation

       1. distribute-lft-out 16.0%

          \[\leadsto \left(\left(3 + \color{blue}{0.5 \cdot \left(z + \left(x + y\right)\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Simplified 15.4%

       \[\leadsto \color{blue}{\left(\left(3 + 0.5 \cdot \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\left(1 + 0.5 \cdot t\right) - \sqrt{t}\right) \]

   if 4.40000000000000003e24 < t

   1. Initial program 86.7%

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

   3. Taylor expanded in t around inf 18.7%

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

      1. associate--l+ 30.5%

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

   5. Simplified 30.5%

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

   6. Taylor expanded in z around inf 27.4%

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

      1. associate--l+ 27.4%

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

      2. distribute-lft-out 27.4%

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

      3. +-commutative 27.4%

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

   8. Simplified 27.4%

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

   9. Taylor expanded in x around 0 19.7%

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

       1. associate--l+ 27.3%

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

       2. distribute-lft-in 27.3%

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

   11. Simplified 27.3%

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

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{+24}:\\ \;\;\;\;\left(\left(3 + 0.5 \cdot \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right) + \left(\left(1 + 0.5 \cdot t\right) - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} + 0.5 \cdot \left(x + \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{t}}\\ t_2 := \sqrt{\frac{1}{z}}\\ \mathbf{if}\;y \leq 1.28 \cdot 10^{-244}:\\ \;\;\;\;\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot \left(x + y\right) + 0.5 \cdot z\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.3:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) + \left(\left(1 + \left(0.5 \cdot \left(t\_1 + t\_2\right) + y \cdot \left(0.5 + y \cdot \left(y \cdot \left(0.0625 + y \cdot -0.0390625\right) - 0.125\right)\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \left(x + \left(t\_1 + \left(\sqrt{\frac{1}{y}} + t\_2\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (/ 1.0 t))) (t_2 (sqrt (/ 1.0 z))))
   (if (<= y 1.28e-244)
     (-
      (+ 4.0 (+ (* 0.5 t) (+ (* 0.5 (+ x y)) (* 0.5 z))))
      (+ (sqrt t) (+ (sqrt x) (+ (sqrt z) (sqrt y)))))
     (if (<= y 1.3)
       (+
        (+ 1.0 (* x 0.5))
        (-
         (+
          1.0
          (+
           (* 0.5 (+ t_1 t_2))
           (* y (+ 0.5 (* y (- (* y (+ 0.0625 (* y -0.0390625))) 0.125))))))
         (+ (sqrt x) (sqrt y))))
       (+ 1.0 (- (* 0.5 (+ x (+ t_1 (+ (sqrt (/ 1.0 y)) t_2)))) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 / t));
	double t_2 = sqrt((1.0 / z));
	double tmp;
	if (y <= 1.28e-244) {
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))));
	} else if (y <= 1.3) {
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (t_1 + t_2)) + (y * (0.5 + (y * ((y * (0.0625 + (y * -0.0390625))) - 0.125)))))) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 + ((0.5 * (x + (t_1 + (sqrt((1.0 / y)) + t_2)))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 / t))
    t_2 = sqrt((1.0d0 / z))
    if (y <= 1.28d-244) then
        tmp = (4.0d0 + ((0.5d0 * t) + ((0.5d0 * (x + y)) + (0.5d0 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))))
    else if (y <= 1.3d0) then
        tmp = (1.0d0 + (x * 0.5d0)) + ((1.0d0 + ((0.5d0 * (t_1 + t_2)) + (y * (0.5d0 + (y * ((y * (0.0625d0 + (y * (-0.0390625d0)))) - 0.125d0)))))) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 + ((0.5d0 * (x + (t_1 + (sqrt((1.0d0 / y)) + t_2)))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 / t));
	double t_2 = Math.sqrt((1.0 / z));
	double tmp;
	if (y <= 1.28e-244) {
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (Math.sqrt(t) + (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y))));
	} else if (y <= 1.3) {
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (t_1 + t_2)) + (y * (0.5 + (y * ((y * (0.0625 + (y * -0.0390625))) - 0.125)))))) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 + ((0.5 * (x + (t_1 + (Math.sqrt((1.0 / y)) + t_2)))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 / t))
	t_2 = math.sqrt((1.0 / z))
	tmp = 0
	if y <= 1.28e-244:
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (math.sqrt(t) + (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y))))
	elif y <= 1.3:
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (t_1 + t_2)) + (y * (0.5 + (y * ((y * (0.0625 + (y * -0.0390625))) - 0.125)))))) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 + ((0.5 * (x + (t_1 + (math.sqrt((1.0 / y)) + t_2)))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 / t))
	t_2 = sqrt(Float64(1.0 / z))
	tmp = 0.0
	if (y <= 1.28e-244)
		tmp = Float64(Float64(4.0 + Float64(Float64(0.5 * t) + Float64(Float64(0.5 * Float64(x + y)) + Float64(0.5 * z)))) - Float64(sqrt(t) + Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	elseif (y <= 1.3)
		tmp = Float64(Float64(1.0 + Float64(x * 0.5)) + Float64(Float64(1.0 + Float64(Float64(0.5 * Float64(t_1 + t_2)) + Float64(y * Float64(0.5 + Float64(y * Float64(Float64(y * Float64(0.0625 + Float64(y * -0.0390625))) - 0.125)))))) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 * Float64(x + Float64(t_1 + Float64(sqrt(Float64(1.0 / y)) + t_2)))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 / t));
	t_2 = sqrt((1.0 / z));
	tmp = 0.0;
	if (y <= 1.28e-244)
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))));
	elseif (y <= 1.3)
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (t_1 + t_2)) + (y * (0.5 + (y * ((y * (0.0625 + (y * -0.0390625))) - 0.125)))))) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 + ((0.5 * (x + (t_1 + (sqrt((1.0 / y)) + t_2)))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.28e-244], N[(N[(4.0 + N[(N[(0.5 * t), $MachinePrecision] + N[(N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(0.5 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3], N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(N[(0.5 * N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(y * N[(0.5 + N[(y * N[(N[(y * N[(0.0625 + N[(y * -0.0390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 * N[(x + N[(t$95$1 + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.27999999999999994e-244

   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. Add Preprocessing

   3. Taylor expanded in x around 0 32.5%

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

      1. associate--l+ 37.1%

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

   5. Simplified 37.1%

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

   6. Taylor expanded in y around 0 32.5%

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

      1. associate--l+ 53.5%

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

      2. metadata-eval 53.5%

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

      3. rem-square-sqrt 53.5%

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

      4. hypot-undefine 53.5%

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

      5. distribute-lft-out 53.5%

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

      6. associate-+r+ 53.5%

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

      7. +-commutative 53.5%

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

   8. Simplified 53.5%

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

   9. Taylor expanded in t around 0 30.3%

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

   10. Taylor expanded in z around 0 9.7%

       \[\leadsto \color{blue}{\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   if 1.27999999999999994e-244 < y < 1.30000000000000004

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf 19.7%

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

      1. associate--l+ 23.5%

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

   5. Simplified 23.5%

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

   6. Taylor expanded in z around inf 21.6%

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

      1. associate--l+ 21.6%

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

      2. distribute-lft-out 21.6%

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

      3. +-commutative 21.6%

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

   8. Simplified 21.6%

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

   9. Taylor expanded in x around 0 21.8%

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

   10. Taylor expanded in y around 0 21.8%

       \[\leadsto \left(1 + 0.5 \cdot x\right) + \color{blue}{\left(\left(1 + \left(0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right) + y \cdot \left(0.5 + y \cdot \left(y \cdot \left(0.0625 + -0.0390625 \cdot y\right) - 0.125\right)\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 1.30000000000000004 < y

   1. Initial program 87.4%

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

   3. Taylor expanded in t around inf 4.4%

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

      1. associate--l+ 15.2%

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

   5. Simplified 15.2%

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

   6. Taylor expanded in z around inf 14.1%

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

      1. associate--l+ 14.1%

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

      2. distribute-lft-out 14.1%

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

      3. +-commutative 14.1%

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

   8. Simplified 14.1%

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

   9. Taylor expanded in y around inf 13.5%

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

       1. distribute-lft-out 13.5%

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

       2. +-commutative 13.5%

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

   11. Simplified 13.5%

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

   12. Taylor expanded in x around 0 13.1%

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

       1. associate--l+ 13.1%

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

       2. distribute-lft-out 13.1%

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

       3. +-commutative 13.1%

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

   14. Simplified 13.1%

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

4. Final simplification 16.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-244}:\\ \;\;\;\;\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot \left(x + y\right) + 0.5 \cdot z\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.3:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) + \left(\left(1 + \left(0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right) + y \cdot \left(0.5 + y \cdot \left(y \cdot \left(0.0625 + y \cdot -0.0390625\right) - 0.125\right)\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \left(x + \left(\sqrt{\frac{1}{t}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{t}}\\ t_2 := \sqrt{\frac{1}{z}}\\ \mathbf{if}\;y \leq 1.28 \cdot 10^{-244}:\\ \;\;\;\;\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot \left(x + y\right) + 0.5 \cdot z\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.3:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) + \left(\left(1 + \left(0.5 \cdot \left(t\_1 + t\_2\right) + y \cdot \left(0.5 + y \cdot \left(y \cdot 0.0625 - 0.125\right)\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \left(x + \left(t\_1 + \left(\sqrt{\frac{1}{y}} + t\_2\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (/ 1.0 t))) (t_2 (sqrt (/ 1.0 z))))
   (if (<= y 1.28e-244)
     (-
      (+ 4.0 (+ (* 0.5 t) (+ (* 0.5 (+ x y)) (* 0.5 z))))
      (+ (sqrt t) (+ (sqrt x) (+ (sqrt z) (sqrt y)))))
     (if (<= y 1.3)
       (+
        (+ 1.0 (* x 0.5))
        (-
         (+
          1.0
          (+ (* 0.5 (+ t_1 t_2)) (* y (+ 0.5 (* y (- (* y 0.0625) 0.125))))))
         (+ (sqrt x) (sqrt y))))
       (+ 1.0 (- (* 0.5 (+ x (+ t_1 (+ (sqrt (/ 1.0 y)) t_2)))) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 / t));
	double t_2 = sqrt((1.0 / z));
	double tmp;
	if (y <= 1.28e-244) {
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))));
	} else if (y <= 1.3) {
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (t_1 + t_2)) + (y * (0.5 + (y * ((y * 0.0625) - 0.125)))))) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 + ((0.5 * (x + (t_1 + (sqrt((1.0 / y)) + t_2)))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 / t))
    t_2 = sqrt((1.0d0 / z))
    if (y <= 1.28d-244) then
        tmp = (4.0d0 + ((0.5d0 * t) + ((0.5d0 * (x + y)) + (0.5d0 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))))
    else if (y <= 1.3d0) then
        tmp = (1.0d0 + (x * 0.5d0)) + ((1.0d0 + ((0.5d0 * (t_1 + t_2)) + (y * (0.5d0 + (y * ((y * 0.0625d0) - 0.125d0)))))) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 + ((0.5d0 * (x + (t_1 + (sqrt((1.0d0 / y)) + t_2)))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 / t));
	double t_2 = Math.sqrt((1.0 / z));
	double tmp;
	if (y <= 1.28e-244) {
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (Math.sqrt(t) + (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y))));
	} else if (y <= 1.3) {
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (t_1 + t_2)) + (y * (0.5 + (y * ((y * 0.0625) - 0.125)))))) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 + ((0.5 * (x + (t_1 + (Math.sqrt((1.0 / y)) + t_2)))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 / t))
	t_2 = math.sqrt((1.0 / z))
	tmp = 0
	if y <= 1.28e-244:
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (math.sqrt(t) + (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y))))
	elif y <= 1.3:
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (t_1 + t_2)) + (y * (0.5 + (y * ((y * 0.0625) - 0.125)))))) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 + ((0.5 * (x + (t_1 + (math.sqrt((1.0 / y)) + t_2)))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 / t))
	t_2 = sqrt(Float64(1.0 / z))
	tmp = 0.0
	if (y <= 1.28e-244)
		tmp = Float64(Float64(4.0 + Float64(Float64(0.5 * t) + Float64(Float64(0.5 * Float64(x + y)) + Float64(0.5 * z)))) - Float64(sqrt(t) + Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	elseif (y <= 1.3)
		tmp = Float64(Float64(1.0 + Float64(x * 0.5)) + Float64(Float64(1.0 + Float64(Float64(0.5 * Float64(t_1 + t_2)) + Float64(y * Float64(0.5 + Float64(y * Float64(Float64(y * 0.0625) - 0.125)))))) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 * Float64(x + Float64(t_1 + Float64(sqrt(Float64(1.0 / y)) + t_2)))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 / t));
	t_2 = sqrt((1.0 / z));
	tmp = 0.0;
	if (y <= 1.28e-244)
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))));
	elseif (y <= 1.3)
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (t_1 + t_2)) + (y * (0.5 + (y * ((y * 0.0625) - 0.125)))))) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 + ((0.5 * (x + (t_1 + (sqrt((1.0 / y)) + t_2)))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.28e-244], N[(N[(4.0 + N[(N[(0.5 * t), $MachinePrecision] + N[(N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(0.5 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3], N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(N[(0.5 * N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(y * N[(0.5 + N[(y * N[(N[(y * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 * N[(x + N[(t$95$1 + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.27999999999999994e-244

   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. Add Preprocessing

   3. Taylor expanded in x around 0 32.5%

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

      1. associate--l+ 37.1%

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

   5. Simplified 37.1%

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

   6. Taylor expanded in y around 0 32.5%

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

      1. associate--l+ 53.5%

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

      2. metadata-eval 53.5%

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

      3. rem-square-sqrt 53.5%

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

      4. hypot-undefine 53.5%

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

      5. distribute-lft-out 53.5%

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

      6. associate-+r+ 53.5%

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

      7. +-commutative 53.5%

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

   8. Simplified 53.5%

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

   9. Taylor expanded in t around 0 30.3%

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

   10. Taylor expanded in z around 0 9.7%

       \[\leadsto \color{blue}{\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   if 1.27999999999999994e-244 < y < 1.30000000000000004

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf 19.7%

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

      1. associate--l+ 23.5%

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

   5. Simplified 23.5%

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

   6. Taylor expanded in z around inf 21.6%

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

      1. associate--l+ 21.6%

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

      2. distribute-lft-out 21.6%

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

      3. +-commutative 21.6%

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

   8. Simplified 21.6%

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

   9. Taylor expanded in x around 0 21.8%

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

   10. Taylor expanded in y around 0 21.8%

       \[\leadsto \left(1 + 0.5 \cdot x\right) + \color{blue}{\left(\left(1 + \left(0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right) + y \cdot \left(0.5 + y \cdot \left(0.0625 \cdot y - 0.125\right)\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 1.30000000000000004 < y

   1. Initial program 87.4%

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

   3. Taylor expanded in t around inf 4.4%

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

      1. associate--l+ 15.2%

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

   5. Simplified 15.2%

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

   6. Taylor expanded in z around inf 14.1%

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

      1. associate--l+ 14.1%

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

      2. distribute-lft-out 14.1%

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

      3. +-commutative 14.1%

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

   8. Simplified 14.1%

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

   9. Taylor expanded in y around inf 13.5%

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

       1. distribute-lft-out 13.5%

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

       2. +-commutative 13.5%

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

   11. Simplified 13.5%

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

   12. Taylor expanded in x around 0 13.1%

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

       1. associate--l+ 13.1%

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

       2. distribute-lft-out 13.1%

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

       3. +-commutative 13.1%

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

   14. Simplified 13.1%

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

4. Final simplification 16.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-244}:\\ \;\;\;\;\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot \left(x + y\right) + 0.5 \cdot z\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.3:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) + \left(\left(1 + \left(0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right) + y \cdot \left(0.5 + y \cdot \left(y \cdot 0.0625 - 0.125\right)\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \left(x + \left(\sqrt{\frac{1}{t}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{t}}\\ t_2 := \sqrt{\frac{1}{z}}\\ \mathbf{if}\;y \leq 1.28 \cdot 10^{-244}:\\ \;\;\;\;\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot \left(x + y\right) + 0.5 \cdot z\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.25:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) + \left(\left(1 + \left(0.5 \cdot \left(t\_1 + t\_2\right) + y \cdot \left(0.5 + y \cdot -0.125\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \left(x + \left(t\_1 + \left(\sqrt{\frac{1}{y}} + t\_2\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (/ 1.0 t))) (t_2 (sqrt (/ 1.0 z))))
   (if (<= y 1.28e-244)
     (-
      (+ 4.0 (+ (* 0.5 t) (+ (* 0.5 (+ x y)) (* 0.5 z))))
      (+ (sqrt t) (+ (sqrt x) (+ (sqrt z) (sqrt y)))))
     (if (<= y 1.25)
       (+
        (+ 1.0 (* x 0.5))
        (-
         (+ 1.0 (+ (* 0.5 (+ t_1 t_2)) (* y (+ 0.5 (* y -0.125)))))
         (+ (sqrt x) (sqrt y))))
       (+ 1.0 (- (* 0.5 (+ x (+ t_1 (+ (sqrt (/ 1.0 y)) t_2)))) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 / t));
	double t_2 = sqrt((1.0 / z));
	double tmp;
	if (y <= 1.28e-244) {
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))));
	} else if (y <= 1.25) {
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (t_1 + t_2)) + (y * (0.5 + (y * -0.125))))) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 + ((0.5 * (x + (t_1 + (sqrt((1.0 / y)) + t_2)))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 / t))
    t_2 = sqrt((1.0d0 / z))
    if (y <= 1.28d-244) then
        tmp = (4.0d0 + ((0.5d0 * t) + ((0.5d0 * (x + y)) + (0.5d0 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))))
    else if (y <= 1.25d0) then
        tmp = (1.0d0 + (x * 0.5d0)) + ((1.0d0 + ((0.5d0 * (t_1 + t_2)) + (y * (0.5d0 + (y * (-0.125d0)))))) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 + ((0.5d0 * (x + (t_1 + (sqrt((1.0d0 / y)) + t_2)))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 / t));
	double t_2 = Math.sqrt((1.0 / z));
	double tmp;
	if (y <= 1.28e-244) {
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (Math.sqrt(t) + (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y))));
	} else if (y <= 1.25) {
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (t_1 + t_2)) + (y * (0.5 + (y * -0.125))))) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 + ((0.5 * (x + (t_1 + (Math.sqrt((1.0 / y)) + t_2)))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 / t))
	t_2 = math.sqrt((1.0 / z))
	tmp = 0
	if y <= 1.28e-244:
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (math.sqrt(t) + (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y))))
	elif y <= 1.25:
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (t_1 + t_2)) + (y * (0.5 + (y * -0.125))))) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 + ((0.5 * (x + (t_1 + (math.sqrt((1.0 / y)) + t_2)))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 / t))
	t_2 = sqrt(Float64(1.0 / z))
	tmp = 0.0
	if (y <= 1.28e-244)
		tmp = Float64(Float64(4.0 + Float64(Float64(0.5 * t) + Float64(Float64(0.5 * Float64(x + y)) + Float64(0.5 * z)))) - Float64(sqrt(t) + Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	elseif (y <= 1.25)
		tmp = Float64(Float64(1.0 + Float64(x * 0.5)) + Float64(Float64(1.0 + Float64(Float64(0.5 * Float64(t_1 + t_2)) + Float64(y * Float64(0.5 + Float64(y * -0.125))))) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 * Float64(x + Float64(t_1 + Float64(sqrt(Float64(1.0 / y)) + t_2)))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 / t));
	t_2 = sqrt((1.0 / z));
	tmp = 0.0;
	if (y <= 1.28e-244)
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))));
	elseif (y <= 1.25)
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * (t_1 + t_2)) + (y * (0.5 + (y * -0.125))))) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 + ((0.5 * (x + (t_1 + (sqrt((1.0 / y)) + t_2)))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.28e-244], N[(N[(4.0 + N[(N[(0.5 * t), $MachinePrecision] + N[(N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(0.5 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25], N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(N[(0.5 * N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(y * N[(0.5 + N[(y * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 * N[(x + N[(t$95$1 + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.27999999999999994e-244

   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. Add Preprocessing

   3. Taylor expanded in x around 0 32.5%

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

      1. associate--l+ 37.1%

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

   5. Simplified 37.1%

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

   6. Taylor expanded in y around 0 32.5%

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

      1. associate--l+ 53.5%

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

      2. metadata-eval 53.5%

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

      3. rem-square-sqrt 53.5%

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

      4. hypot-undefine 53.5%

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

      5. distribute-lft-out 53.5%

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

      6. associate-+r+ 53.5%

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

      7. +-commutative 53.5%

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

   8. Simplified 53.5%

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

   9. Taylor expanded in t around 0 30.3%

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

   10. Taylor expanded in z around 0 9.7%

       \[\leadsto \color{blue}{\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   if 1.27999999999999994e-244 < y < 1.25

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf 19.7%

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

      1. associate--l+ 23.5%

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

   5. Simplified 23.5%

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

   6. Taylor expanded in z around inf 21.6%

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

      1. associate--l+ 21.6%

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

      2. distribute-lft-out 21.6%

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

      3. +-commutative 21.6%

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

   8. Simplified 21.6%

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

   9. Taylor expanded in x around 0 21.8%

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

   10. Taylor expanded in y around 0 21.8%

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

   if 1.25 < y

   1. Initial program 87.4%

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

   3. Taylor expanded in t around inf 4.4%

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

      1. associate--l+ 15.2%

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

   5. Simplified 15.2%

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

   6. Taylor expanded in z around inf 14.1%

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

      1. associate--l+ 14.1%

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

      2. distribute-lft-out 14.1%

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

      3. +-commutative 14.1%

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

   8. Simplified 14.1%

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

   9. Taylor expanded in y around inf 13.5%

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

       1. distribute-lft-out 13.5%

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

       2. +-commutative 13.5%

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

   11. Simplified 13.5%

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

   12. Taylor expanded in x around 0 13.1%

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

       1. associate--l+ 13.1%

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

       2. distribute-lft-out 13.1%

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

       3. +-commutative 13.1%

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

   14. Simplified 13.1%

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

4. Final simplification 16.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-244}:\\ \;\;\;\;\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot \left(x + y\right) + 0.5 \cdot z\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.25:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) + \left(\left(1 + \left(0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right) + y \cdot \left(0.5 + y \cdot -0.125\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \left(x + \left(\sqrt{\frac{1}{t}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{t}}\\ t_2 := \sqrt{\frac{1}{z}}\\ \mathbf{if}\;y \leq 1.28 \cdot 10^{-244}:\\ \;\;\;\;\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot \left(x + y\right) + 0.5 \cdot z\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) + \left(\left(1 + \left(0.5 \cdot y + 0.5 \cdot \left(t\_1 + t\_2\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \left(x + \left(t\_1 + \left(\sqrt{\frac{1}{y}} + t\_2\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (/ 1.0 t))) (t_2 (sqrt (/ 1.0 z))))
   (if (<= y 1.28e-244)
     (-
      (+ 4.0 (+ (* 0.5 t) (+ (* 0.5 (+ x y)) (* 0.5 z))))
      (+ (sqrt t) (+ (sqrt x) (+ (sqrt z) (sqrt y)))))
     (if (<= y 1.0)
       (+
        (+ 1.0 (* x 0.5))
        (- (+ 1.0 (+ (* 0.5 y) (* 0.5 (+ t_1 t_2)))) (+ (sqrt x) (sqrt y))))
       (+ 1.0 (- (* 0.5 (+ x (+ t_1 (+ (sqrt (/ 1.0 y)) t_2)))) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 / t));
	double t_2 = sqrt((1.0 / z));
	double tmp;
	if (y <= 1.28e-244) {
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))));
	} else if (y <= 1.0) {
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * y) + (0.5 * (t_1 + t_2)))) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 + ((0.5 * (x + (t_1 + (sqrt((1.0 / y)) + t_2)))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 / t))
    t_2 = sqrt((1.0d0 / z))
    if (y <= 1.28d-244) then
        tmp = (4.0d0 + ((0.5d0 * t) + ((0.5d0 * (x + y)) + (0.5d0 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))))
    else if (y <= 1.0d0) then
        tmp = (1.0d0 + (x * 0.5d0)) + ((1.0d0 + ((0.5d0 * y) + (0.5d0 * (t_1 + t_2)))) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 + ((0.5d0 * (x + (t_1 + (sqrt((1.0d0 / y)) + t_2)))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 / t));
	double t_2 = Math.sqrt((1.0 / z));
	double tmp;
	if (y <= 1.28e-244) {
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (Math.sqrt(t) + (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y))));
	} else if (y <= 1.0) {
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * y) + (0.5 * (t_1 + t_2)))) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 + ((0.5 * (x + (t_1 + (Math.sqrt((1.0 / y)) + t_2)))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 / t))
	t_2 = math.sqrt((1.0 / z))
	tmp = 0
	if y <= 1.28e-244:
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (math.sqrt(t) + (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y))))
	elif y <= 1.0:
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * y) + (0.5 * (t_1 + t_2)))) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 + ((0.5 * (x + (t_1 + (math.sqrt((1.0 / y)) + t_2)))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 / t))
	t_2 = sqrt(Float64(1.0 / z))
	tmp = 0.0
	if (y <= 1.28e-244)
		tmp = Float64(Float64(4.0 + Float64(Float64(0.5 * t) + Float64(Float64(0.5 * Float64(x + y)) + Float64(0.5 * z)))) - Float64(sqrt(t) + Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	elseif (y <= 1.0)
		tmp = Float64(Float64(1.0 + Float64(x * 0.5)) + Float64(Float64(1.0 + Float64(Float64(0.5 * y) + Float64(0.5 * Float64(t_1 + t_2)))) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 * Float64(x + Float64(t_1 + Float64(sqrt(Float64(1.0 / y)) + t_2)))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 / t));
	t_2 = sqrt((1.0 / z));
	tmp = 0.0;
	if (y <= 1.28e-244)
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))));
	elseif (y <= 1.0)
		tmp = (1.0 + (x * 0.5)) + ((1.0 + ((0.5 * y) + (0.5 * (t_1 + t_2)))) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 + ((0.5 * (x + (t_1 + (sqrt((1.0 / y)) + t_2)))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.28e-244], N[(N[(4.0 + N[(N[(0.5 * t), $MachinePrecision] + N[(N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(0.5 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(N[(0.5 * y), $MachinePrecision] + N[(0.5 * N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 * N[(x + N[(t$95$1 + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.27999999999999994e-244

   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. Add Preprocessing

   3. Taylor expanded in x around 0 32.5%

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

      1. associate--l+ 37.1%

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

   5. Simplified 37.1%

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

   6. Taylor expanded in y around 0 32.5%

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

      1. associate--l+ 53.5%

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

      2. metadata-eval 53.5%

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

      3. rem-square-sqrt 53.5%

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

      4. hypot-undefine 53.5%

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

      5. distribute-lft-out 53.5%

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

      6. associate-+r+ 53.5%

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

      7. +-commutative 53.5%

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

   8. Simplified 53.5%

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

   9. Taylor expanded in t around 0 30.3%

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

   10. Taylor expanded in z around 0 9.7%

       \[\leadsto \color{blue}{\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   if 1.27999999999999994e-244 < y < 1

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf 19.7%

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

      1. associate--l+ 23.5%

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

   5. Simplified 23.5%

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

   6. Taylor expanded in z around inf 21.6%

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

      1. associate--l+ 21.6%

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

      2. distribute-lft-out 21.6%

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

      3. +-commutative 21.6%

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

   8. Simplified 21.6%

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

   9. Taylor expanded in x around 0 21.8%

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

   10. Taylor expanded in y around 0 21.8%

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

   if 1 < y

   1. Initial program 87.4%

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

   3. Taylor expanded in t around inf 4.4%

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

      1. associate--l+ 15.2%

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

   5. Simplified 15.2%

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

   6. Taylor expanded in z around inf 14.1%

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

      1. associate--l+ 14.1%

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

      2. distribute-lft-out 14.1%

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

      3. +-commutative 14.1%

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

   8. Simplified 14.1%

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

   9. Taylor expanded in y around inf 13.5%

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

       1. distribute-lft-out 13.5%

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

       2. +-commutative 13.5%

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

   11. Simplified 13.5%

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

   12. Taylor expanded in x around 0 13.1%

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

       1. associate--l+ 13.1%

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

       2. distribute-lft-out 13.1%

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

       3. +-commutative 13.1%

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

   14. Simplified 13.1%

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

4. Final simplification 16.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-244}:\\ \;\;\;\;\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot \left(x + y\right) + 0.5 \cdot z\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) + \left(\left(1 + \left(0.5 \cdot y + 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \left(x + \left(\sqrt{\frac{1}{t}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 62.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{t}}\\ t_2 := \sqrt{\frac{1}{z}}\\ \mathbf{if}\;y \leq 1.28 \cdot 10^{-244}:\\ \;\;\;\;\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot \left(x + y\right) + 0.5 \cdot z\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) + \left(1 + \left(0.5 \cdot \left(y + \left(t\_1 + t\_2\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \left(x + \left(t\_1 + \left(\sqrt{\frac{1}{y}} + t\_2\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (/ 1.0 t))) (t_2 (sqrt (/ 1.0 z))))
   (if (<= y 1.28e-244)
     (-
      (+ 4.0 (+ (* 0.5 t) (+ (* 0.5 (+ x y)) (* 0.5 z))))
      (+ (sqrt t) (+ (sqrt x) (+ (sqrt z) (sqrt y)))))
     (if (<= y 1.0)
       (+
        (+ 1.0 (* x 0.5))
        (+ 1.0 (- (* 0.5 (+ y (+ t_1 t_2))) (+ (sqrt x) (sqrt y)))))
       (+ 1.0 (- (* 0.5 (+ x (+ t_1 (+ (sqrt (/ 1.0 y)) t_2)))) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 / t));
	double t_2 = sqrt((1.0 / z));
	double tmp;
	if (y <= 1.28e-244) {
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))));
	} else if (y <= 1.0) {
		tmp = (1.0 + (x * 0.5)) + (1.0 + ((0.5 * (y + (t_1 + t_2))) - (sqrt(x) + sqrt(y))));
	} else {
		tmp = 1.0 + ((0.5 * (x + (t_1 + (sqrt((1.0 / y)) + t_2)))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 / t))
    t_2 = sqrt((1.0d0 / z))
    if (y <= 1.28d-244) then
        tmp = (4.0d0 + ((0.5d0 * t) + ((0.5d0 * (x + y)) + (0.5d0 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))))
    else if (y <= 1.0d0) then
        tmp = (1.0d0 + (x * 0.5d0)) + (1.0d0 + ((0.5d0 * (y + (t_1 + t_2))) - (sqrt(x) + sqrt(y))))
    else
        tmp = 1.0d0 + ((0.5d0 * (x + (t_1 + (sqrt((1.0d0 / y)) + t_2)))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 / t));
	double t_2 = Math.sqrt((1.0 / z));
	double tmp;
	if (y <= 1.28e-244) {
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (Math.sqrt(t) + (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y))));
	} else if (y <= 1.0) {
		tmp = (1.0 + (x * 0.5)) + (1.0 + ((0.5 * (y + (t_1 + t_2))) - (Math.sqrt(x) + Math.sqrt(y))));
	} else {
		tmp = 1.0 + ((0.5 * (x + (t_1 + (Math.sqrt((1.0 / y)) + t_2)))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 / t))
	t_2 = math.sqrt((1.0 / z))
	tmp = 0
	if y <= 1.28e-244:
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (math.sqrt(t) + (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y))))
	elif y <= 1.0:
		tmp = (1.0 + (x * 0.5)) + (1.0 + ((0.5 * (y + (t_1 + t_2))) - (math.sqrt(x) + math.sqrt(y))))
	else:
		tmp = 1.0 + ((0.5 * (x + (t_1 + (math.sqrt((1.0 / y)) + t_2)))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 / t))
	t_2 = sqrt(Float64(1.0 / z))
	tmp = 0.0
	if (y <= 1.28e-244)
		tmp = Float64(Float64(4.0 + Float64(Float64(0.5 * t) + Float64(Float64(0.5 * Float64(x + y)) + Float64(0.5 * z)))) - Float64(sqrt(t) + Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	elseif (y <= 1.0)
		tmp = Float64(Float64(1.0 + Float64(x * 0.5)) + Float64(1.0 + Float64(Float64(0.5 * Float64(y + Float64(t_1 + t_2))) - Float64(sqrt(x) + sqrt(y)))));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 * Float64(x + Float64(t_1 + Float64(sqrt(Float64(1.0 / y)) + t_2)))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 / t));
	t_2 = sqrt((1.0 / z));
	tmp = 0.0;
	if (y <= 1.28e-244)
		tmp = (4.0 + ((0.5 * t) + ((0.5 * (x + y)) + (0.5 * z)))) - (sqrt(t) + (sqrt(x) + (sqrt(z) + sqrt(y))));
	elseif (y <= 1.0)
		tmp = (1.0 + (x * 0.5)) + (1.0 + ((0.5 * (y + (t_1 + t_2))) - (sqrt(x) + sqrt(y))));
	else
		tmp = 1.0 + ((0.5 * (x + (t_1 + (sqrt((1.0 / y)) + t_2)))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.28e-244], N[(N[(4.0 + N[(N[(0.5 * t), $MachinePrecision] + N[(N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(0.5 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(0.5 * N[(y + N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 * N[(x + N[(t$95$1 + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.27999999999999994e-244

   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. Add Preprocessing

   3. Taylor expanded in x around 0 32.5%

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

      1. associate--l+ 37.1%

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

   5. Simplified 37.1%

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

   6. Taylor expanded in y around 0 32.5%

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

      1. associate--l+ 53.5%

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

      2. metadata-eval 53.5%

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

      3. rem-square-sqrt 53.5%

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

      4. hypot-undefine 53.5%

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

      5. distribute-lft-out 53.5%

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

      6. associate-+r+ 53.5%

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

      7. +-commutative 53.5%

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

   8. Simplified 53.5%

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

   9. Taylor expanded in t around 0 30.3%

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

   10. Taylor expanded in z around 0 9.7%

       \[\leadsto \color{blue}{\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   if 1.27999999999999994e-244 < y < 1

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf 19.7%

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

      1. associate--l+ 23.5%

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

   5. Simplified 23.5%

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

   6. Taylor expanded in z around inf 21.6%

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

      1. associate--l+ 21.6%

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

      2. distribute-lft-out 21.6%

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

      3. +-commutative 21.6%

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

   8. Simplified 21.6%

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

   9. Taylor expanded in x around 0 21.8%

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

   10. Taylor expanded in y around 0 21.8%

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

       1. associate--l+ 21.8%

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

       2. distribute-lft-out 21.8%

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

       3. +-commutative 21.8%

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

   12. Simplified 21.8%

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

   if 1 < y

   1. Initial program 87.4%

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

   3. Taylor expanded in t around inf 4.4%

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

      1. associate--l+ 15.2%

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

   5. Simplified 15.2%

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

   6. Taylor expanded in z around inf 14.1%

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

      1. associate--l+ 14.1%

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

      2. distribute-lft-out 14.1%

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

      3. +-commutative 14.1%

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

   8. Simplified 14.1%

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

   9. Taylor expanded in y around inf 13.5%

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

       1. distribute-lft-out 13.5%

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

       2. +-commutative 13.5%

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

   11. Simplified 13.5%

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

   12. Taylor expanded in x around 0 13.1%

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

       1. associate--l+ 13.1%

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

       2. distribute-lft-out 13.1%

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

       3. +-commutative 13.1%

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

   14. Simplified 13.1%

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

4. Final simplification 16.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-244}:\\ \;\;\;\;\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot \left(x + y\right) + 0.5 \cdot z\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) + \left(1 + \left(0.5 \cdot \left(y + \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \left(x + \left(\sqrt{\frac{1}{t}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{t}}\\ t_2 := \sqrt{x} + \sqrt{y}\\ t_3 := \sqrt{\frac{1}{z}}\\ \mathbf{if}\;y \leq 1.28 \cdot 10^{-244}:\\ \;\;\;\;4 + \left(0.5 \cdot \left(t + \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{z} + t\_2\right)\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) + \left(1 + \left(0.5 \cdot \left(y + \left(t\_1 + t\_3\right)\right) - t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \left(x + \left(t\_1 + \left(\sqrt{\frac{1}{y}} + t\_3\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (/ 1.0 t)))
        (t_2 (+ (sqrt x) (sqrt y)))
        (t_3 (sqrt (/ 1.0 z))))
   (if (<= y 1.28e-244)
     (+ 4.0 (- (* 0.5 (+ t (+ z (+ x y)))) (+ (sqrt t) (+ (sqrt z) t_2))))
     (if (<= y 1.0)
       (+ (+ 1.0 (* x 0.5)) (+ 1.0 (- (* 0.5 (+ y (+ t_1 t_3))) t_2)))
       (+ 1.0 (- (* 0.5 (+ x (+ t_1 (+ (sqrt (/ 1.0 y)) t_3)))) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 / t));
	double t_2 = sqrt(x) + sqrt(y);
	double t_3 = sqrt((1.0 / z));
	double tmp;
	if (y <= 1.28e-244) {
		tmp = 4.0 + ((0.5 * (t + (z + (x + y)))) - (sqrt(t) + (sqrt(z) + t_2)));
	} else if (y <= 1.0) {
		tmp = (1.0 + (x * 0.5)) + (1.0 + ((0.5 * (y + (t_1 + t_3))) - t_2));
	} else {
		tmp = 1.0 + ((0.5 * (x + (t_1 + (sqrt((1.0 / y)) + t_3)))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 / t))
    t_2 = sqrt(x) + sqrt(y)
    t_3 = sqrt((1.0d0 / z))
    if (y <= 1.28d-244) then
        tmp = 4.0d0 + ((0.5d0 * (t + (z + (x + y)))) - (sqrt(t) + (sqrt(z) + t_2)))
    else if (y <= 1.0d0) then
        tmp = (1.0d0 + (x * 0.5d0)) + (1.0d0 + ((0.5d0 * (y + (t_1 + t_3))) - t_2))
    else
        tmp = 1.0d0 + ((0.5d0 * (x + (t_1 + (sqrt((1.0d0 / y)) + t_3)))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 / t));
	double t_2 = Math.sqrt(x) + Math.sqrt(y);
	double t_3 = Math.sqrt((1.0 / z));
	double tmp;
	if (y <= 1.28e-244) {
		tmp = 4.0 + ((0.5 * (t + (z + (x + y)))) - (Math.sqrt(t) + (Math.sqrt(z) + t_2)));
	} else if (y <= 1.0) {
		tmp = (1.0 + (x * 0.5)) + (1.0 + ((0.5 * (y + (t_1 + t_3))) - t_2));
	} else {
		tmp = 1.0 + ((0.5 * (x + (t_1 + (Math.sqrt((1.0 / y)) + t_3)))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 / t))
	t_2 = math.sqrt(x) + math.sqrt(y)
	t_3 = math.sqrt((1.0 / z))
	tmp = 0
	if y <= 1.28e-244:
		tmp = 4.0 + ((0.5 * (t + (z + (x + y)))) - (math.sqrt(t) + (math.sqrt(z) + t_2)))
	elif y <= 1.0:
		tmp = (1.0 + (x * 0.5)) + (1.0 + ((0.5 * (y + (t_1 + t_3))) - t_2))
	else:
		tmp = 1.0 + ((0.5 * (x + (t_1 + (math.sqrt((1.0 / y)) + t_3)))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 / t))
	t_2 = Float64(sqrt(x) + sqrt(y))
	t_3 = sqrt(Float64(1.0 / z))
	tmp = 0.0
	if (y <= 1.28e-244)
		tmp = Float64(4.0 + Float64(Float64(0.5 * Float64(t + Float64(z + Float64(x + y)))) - Float64(sqrt(t) + Float64(sqrt(z) + t_2))));
	elseif (y <= 1.0)
		tmp = Float64(Float64(1.0 + Float64(x * 0.5)) + Float64(1.0 + Float64(Float64(0.5 * Float64(y + Float64(t_1 + t_3))) - t_2)));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 * Float64(x + Float64(t_1 + Float64(sqrt(Float64(1.0 / y)) + t_3)))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 / t));
	t_2 = sqrt(x) + sqrt(y);
	t_3 = sqrt((1.0 / z));
	tmp = 0.0;
	if (y <= 1.28e-244)
		tmp = 4.0 + ((0.5 * (t + (z + (x + y)))) - (sqrt(t) + (sqrt(z) + t_2)));
	elseif (y <= 1.0)
		tmp = (1.0 + (x * 0.5)) + (1.0 + ((0.5 * (y + (t_1 + t_3))) - t_2));
	else
		tmp = 1.0 + ((0.5 * (x + (t_1 + (sqrt((1.0 / y)) + t_3)))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.28e-244], N[(4.0 + N[(N[(0.5 * N[(t + N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(0.5 * N[(y + N[(t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 * N[(x + N[(t$95$1 + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.27999999999999994e-244

   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. Add Preprocessing

   3. Taylor expanded in x around 0 32.5%

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

      1. associate--l+ 37.1%

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

   5. Simplified 37.1%

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

   6. Taylor expanded in y around 0 32.5%

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

      1. associate--l+ 53.5%

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

      2. metadata-eval 53.5%

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

      3. rem-square-sqrt 53.5%

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

      4. hypot-undefine 53.5%

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

      5. distribute-lft-out 53.5%

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

      6. associate-+r+ 53.5%

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

      7. +-commutative 53.5%

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

   8. Simplified 53.5%

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

   9. Taylor expanded in t around 0 30.3%

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

   10. Taylor expanded in z around 0 9.7%

       \[\leadsto \color{blue}{\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   11. Step-by-step derivation

       1. associate--l+ 9.7%

          \[\leadsto \color{blue}{4 + \left(\left(0.5 \cdot t + \left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       2. +-commutative 9.7%

          \[\leadsto 4 + \left(\color{blue}{\left(\left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right) + 0.5 \cdot t\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       3. distribute-lft-out 9.7%

          \[\leadsto 4 + \left(\left(\color{blue}{0.5 \cdot \left(z + \left(x + y\right)\right)} + 0.5 \cdot t\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       4. distribute-lft-out 9.7%

          \[\leadsto 4 + \left(\color{blue}{0.5 \cdot \left(\left(z + \left(x + y\right)\right) + t\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       5. +-commutative 9.7%

          \[\leadsto 4 + \left(0.5 \cdot \left(\left(z + \color{blue}{\left(y + x\right)}\right) + t\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       6. +-commutative 9.7%

          \[\leadsto 4 + \left(0.5 \cdot \left(\left(z + \left(y + x\right)\right) + t\right) - \left(\sqrt{t} + \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)}\right)\right) \]

       7. +-commutative 9.7%

          \[\leadsto 4 + \left(0.5 \cdot \left(\left(z + \left(y + x\right)\right) + t\right) - \left(\sqrt{t} + \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right)\right)\right) \]

       8. associate-+r+ 9.7%

          \[\leadsto 4 + \left(0.5 \cdot \left(\left(z + \left(y + x\right)\right) + t\right) - \left(\sqrt{t} + \color{blue}{\left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)}\right)\right) \]

   12. Simplified 9.7%

       \[\leadsto \color{blue}{4 + \left(0.5 \cdot \left(\left(z + \left(y + x\right)\right) + t\right) - \left(\sqrt{t} + \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\right)} \]

   if 1.27999999999999994e-244 < y < 1

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf 19.7%

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

      1. associate--l+ 23.5%

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

   5. Simplified 23.5%

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

   6. Taylor expanded in z around inf 21.6%

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

      1. associate--l+ 21.6%

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

      2. distribute-lft-out 21.6%

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

      3. +-commutative 21.6%

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

   8. Simplified 21.6%

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

   9. Taylor expanded in x around 0 21.8%

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

   10. Taylor expanded in y around 0 21.8%

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

       1. associate--l+ 21.8%

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

       2. distribute-lft-out 21.8%

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

       3. +-commutative 21.8%

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

   12. Simplified 21.8%

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

   if 1 < y

   1. Initial program 87.4%

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

   3. Taylor expanded in t around inf 4.4%

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

      1. associate--l+ 15.2%

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

   5. Simplified 15.2%

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

   6. Taylor expanded in z around inf 14.1%

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

      1. associate--l+ 14.1%

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

      2. distribute-lft-out 14.1%

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

      3. +-commutative 14.1%

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

   8. Simplified 14.1%

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

   9. Taylor expanded in y around inf 13.5%

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

       1. distribute-lft-out 13.5%

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

       2. +-commutative 13.5%

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

   11. Simplified 13.5%

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

   12. Taylor expanded in x around 0 13.1%

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

       1. associate--l+ 13.1%

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

       2. distribute-lft-out 13.1%

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

       3. +-commutative 13.1%

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

   14. Simplified 13.1%

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

4. Final simplification 16.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-244}:\\ \;\;\;\;4 + \left(0.5 \cdot \left(t + \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) + \left(1 + \left(0.5 \cdot \left(y + \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \left(x + \left(\sqrt{\frac{1}{t}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{+33}:\\ \;\;\;\;4 + \left(0.5 \cdot \left(t + \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \left(x + \left(\sqrt{\frac{1}{t}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 8.5e+33)
   (+
    4.0
    (-
     (* 0.5 (+ t (+ z (+ x y))))
     (+ (sqrt t) (+ (sqrt z) (+ (sqrt x) (sqrt y))))))
   (+
    1.0
    (-
     (* 0.5 (+ x (+ (sqrt (/ 1.0 t)) (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 z))))))
     (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 8.5e+33) {
		tmp = 4.0 + ((0.5 * (t + (z + (x + y)))) - (sqrt(t) + (sqrt(z) + (sqrt(x) + sqrt(y)))));
	} else {
		tmp = 1.0 + ((0.5 * (x + (sqrt((1.0 / t)) + (sqrt((1.0 / y)) + sqrt((1.0 / z)))))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 8.5d+33) then
        tmp = 4.0d0 + ((0.5d0 * (t + (z + (x + y)))) - (sqrt(t) + (sqrt(z) + (sqrt(x) + sqrt(y)))))
    else
        tmp = 1.0d0 + ((0.5d0 * (x + (sqrt((1.0d0 / t)) + (sqrt((1.0d0 / y)) + sqrt((1.0d0 / z)))))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 8.5e+33) {
		tmp = 4.0 + ((0.5 * (t + (z + (x + y)))) - (Math.sqrt(t) + (Math.sqrt(z) + (Math.sqrt(x) + Math.sqrt(y)))));
	} else {
		tmp = 1.0 + ((0.5 * (x + (Math.sqrt((1.0 / t)) + (Math.sqrt((1.0 / y)) + Math.sqrt((1.0 / z)))))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 8.5e+33:
		tmp = 4.0 + ((0.5 * (t + (z + (x + y)))) - (math.sqrt(t) + (math.sqrt(z) + (math.sqrt(x) + math.sqrt(y)))))
	else:
		tmp = 1.0 + ((0.5 * (x + (math.sqrt((1.0 / t)) + (math.sqrt((1.0 / y)) + math.sqrt((1.0 / z)))))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 8.5e+33)
		tmp = Float64(4.0 + Float64(Float64(0.5 * Float64(t + Float64(z + Float64(x + y)))) - Float64(sqrt(t) + Float64(sqrt(z) + Float64(sqrt(x) + sqrt(y))))));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 * Float64(x + Float64(sqrt(Float64(1.0 / t)) + Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z)))))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 8.5e+33)
		tmp = 4.0 + ((0.5 * (t + (z + (x + y)))) - (sqrt(t) + (sqrt(z) + (sqrt(x) + sqrt(y)))));
	else
		tmp = 1.0 + ((0.5 * (x + (sqrt((1.0 / t)) + (sqrt((1.0 / y)) + sqrt((1.0 / z)))))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 8.5e+33], N[(4.0 + N[(N[(0.5 * N[(t + N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 * N[(x + N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.4999999999999998e33

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0 26.9%

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

      1. associate--l+ 38.5%

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

   5. Simplified 38.5%

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

   6. Taylor expanded in y around 0 18.9%

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

      1. associate--l+ 26.8%

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

      2. metadata-eval 26.8%

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

      3. rem-square-sqrt 26.8%

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

      4. hypot-undefine 26.8%

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

      5. distribute-lft-out 26.8%

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

      6. associate-+r+ 26.8%

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

      7. +-commutative 26.8%

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

   8. Simplified 26.8%

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

   9. Taylor expanded in t around 0 23.7%

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

   10. Taylor expanded in z around 0 15.0%

       \[\leadsto \color{blue}{\left(4 + \left(0.5 \cdot t + \left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   11. Step-by-step derivation

       1. associate--l+ 15.0%

          \[\leadsto \color{blue}{4 + \left(\left(0.5 \cdot t + \left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       2. +-commutative 15.0%

          \[\leadsto 4 + \left(\color{blue}{\left(\left(0.5 \cdot z + 0.5 \cdot \left(x + y\right)\right) + 0.5 \cdot t\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       3. distribute-lft-out 15.0%

          \[\leadsto 4 + \left(\left(\color{blue}{0.5 \cdot \left(z + \left(x + y\right)\right)} + 0.5 \cdot t\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       4. distribute-lft-out 15.0%

          \[\leadsto 4 + \left(\color{blue}{0.5 \cdot \left(\left(z + \left(x + y\right)\right) + t\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       5. +-commutative 15.0%

          \[\leadsto 4 + \left(0.5 \cdot \left(\left(z + \color{blue}{\left(y + x\right)}\right) + t\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       6. +-commutative 15.0%

          \[\leadsto 4 + \left(0.5 \cdot \left(\left(z + \left(y + x\right)\right) + t\right) - \left(\sqrt{t} + \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)}\right)\right) \]

       7. +-commutative 15.0%

          \[\leadsto 4 + \left(0.5 \cdot \left(\left(z + \left(y + x\right)\right) + t\right) - \left(\sqrt{t} + \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right)\right)\right) \]

       8. associate-+r+ 15.0%

          \[\leadsto 4 + \left(0.5 \cdot \left(\left(z + \left(y + x\right)\right) + t\right) - \left(\sqrt{t} + \color{blue}{\left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)}\right)\right) \]

   12. Simplified 15.0%

       \[\leadsto \color{blue}{4 + \left(0.5 \cdot \left(\left(z + \left(y + x\right)\right) + t\right) - \left(\sqrt{t} + \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\right)} \]

   if 8.4999999999999998e33 < t

   1. Initial program 87.0%

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

   3. Taylor expanded in t around inf 18.7%

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

      1. associate--l+ 30.8%

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

   5. Simplified 30.8%

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

   6. Taylor expanded in z around inf 26.9%

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

      1. associate--l+ 26.9%

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

      2. distribute-lft-out 26.9%

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

      3. +-commutative 26.9%

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

   8. Simplified 26.9%

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

   9. Taylor expanded in y around inf 14.7%

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

       1. distribute-lft-out 14.7%

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

       2. +-commutative 14.7%

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

   11. Simplified 14.7%

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

   12. Taylor expanded in x around 0 14.2%

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

       1. associate--l+ 14.2%

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

       2. distribute-lft-out 14.2%

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

       3. +-commutative 14.2%

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

   14. Simplified 14.2%

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

4. Final simplification 14.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{+33}:\\ \;\;\;\;4 + \left(0.5 \cdot \left(t + \left(z + \left(x + y\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \left(x + \left(\sqrt{\frac{1}{t}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 35.7% accurate, 1.9× speedup

Math

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

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 t)) (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 z))))))
   (if (<= x 1.0)
     (+ 1.0 (- (* 0.5 (+ x t_1)) (sqrt x)))
     (* 0.5 (+ (sqrt (/ 1.0 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 / t)) + (sqrt((1.0 / y)) + sqrt((1.0 / z)));
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + ((0.5 * (x + t_1)) - sqrt(x));
	} else {
		tmp = 0.5 * (sqrt((1.0 / 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 / t)) + (sqrt((1.0d0 / y)) + sqrt((1.0d0 / z)))
    if (x <= 1.0d0) then
        tmp = 1.0d0 + ((0.5d0 * (x + t_1)) - sqrt(x))
    else
        tmp = 0.5d0 * (sqrt((1.0d0 / 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 / t)) + (Math.sqrt((1.0 / y)) + Math.sqrt((1.0 / z)));
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + ((0.5 * (x + t_1)) - Math.sqrt(x));
	} else {
		tmp = 0.5 * (Math.sqrt((1.0 / 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 / t)) + (math.sqrt((1.0 / y)) + math.sqrt((1.0 / z)))
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 + ((0.5 * (x + t_1)) - math.sqrt(x))
	else:
		tmp = 0.5 * (math.sqrt((1.0 / x)) + 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 / t)) + Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z))))
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1.0 + Float64(Float64(0.5 * Float64(x + t_1)) - sqrt(x)));
	else
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / 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 / t)) + (sqrt((1.0 / y)) + sqrt((1.0 / z)));
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0 + ((0.5 * (x + t_1)) - sqrt(x));
	else
		tmp = 0.5 * (sqrt((1.0 / 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[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.0], N[(1.0 + N[(N[(0.5 * N[(x + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 97.5%

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

   3. Taylor expanded in t around inf 17.7%

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

      1. associate--l+ 30.4%

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

   5. Simplified 30.4%

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

   6. Taylor expanded in z around inf 27.0%

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

      1. associate--l+ 27.0%

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

      2. distribute-lft-out 27.0%

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

      3. +-commutative 27.0%

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

   8. Simplified 27.0%

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

   9. Taylor expanded in y around inf 14.3%

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

       1. distribute-lft-out 14.3%

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

       2. +-commutative 14.3%

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

   11. Simplified 14.3%

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

   12. Taylor expanded in x around 0 13.9%

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

       1. associate--l+ 13.9%

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

       2. distribute-lft-out 13.9%

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

       3. +-commutative 13.9%

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

   14. Simplified 13.9%

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

   if 1 < x

   1. Initial program 85.3%

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

   3. Taylor expanded in t around inf 3.8%

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

      1. associate--l+ 5.1%

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

   5. Simplified 5.1%

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

   6. Taylor expanded in z around inf 5.4%

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

      1. associate--l+ 5.4%

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

      2. distribute-lft-out 5.4%

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

      3. +-commutative 5.4%

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

   8. Simplified 5.4%

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

   9. Taylor expanded in y around inf 4.6%

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

       1. distribute-lft-out 4.6%

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

       2. +-commutative 4.6%

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

   11. Simplified 4.6%

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

   12. Taylor expanded in x around inf 19.0%

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

       1. distribute-lft-out 19.0%

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

       2. +-commutative 19.0%

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

   14. Simplified 19.0%

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

4. Final simplification 16.2%

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

Alternative 21: 11.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{t}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\right) \end{array} \]

FPCore

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

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.5 * (sqrt((1.0 / x)) + (sqrt((1.0 / t)) + (sqrt((1.0 / y)) + sqrt((1.0 / z)))));
}

Fortran

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

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 0.5 * (Math.sqrt((1.0 / x)) + (Math.sqrt((1.0 / t)) + (Math.sqrt((1.0 / y)) + Math.sqrt((1.0 / z)))));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 0.5 * (math.sqrt((1.0 / x)) + (math.sqrt((1.0 / t)) + (math.sqrt((1.0 / y)) + math.sqrt((1.0 / z)))))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + Float64(sqrt(Float64(1.0 / t)) + Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z))))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 0.5 * (sqrt((1.0 / x)) + (sqrt((1.0 / t)) + (sqrt((1.0 / y)) + sqrt((1.0 / z)))));
end

Wolfram

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

Derivation

1. Initial program 92.0%

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

3. Taylor expanded in t around inf 11.4%

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

   1. associate--l+ 19.0%

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

5. Simplified 19.0%

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

6. Taylor expanded in z around inf 17.2%

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

   1. associate--l+ 17.2%

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

   2. distribute-lft-out 17.2%

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

   3. +-commutative 17.2%

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

8. Simplified 17.2%

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

9. Taylor expanded in y around inf 9.9%

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

    1. distribute-lft-out 9.9%

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

    2. +-commutative 9.9%

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

11. Simplified 9.9%

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

12. Taylor expanded in x around inf 11.8%

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

    1. distribute-lft-out 11.8%

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

    2. +-commutative 11.8%

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

14. Simplified 11.8%

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

15. Final simplification 11.8%

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

Developer Target 1: 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 2024179 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (/ 1 (+ (sqrt (+ x 1)) (sqrt x))) (/ 1 (+ (sqrt (+ y 1)) (sqrt y)))) (/ 1 (+ (sqrt (+ z 1)) (sqrt z)))) (- (sqrt (+ t 1)) (sqrt t))))

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