Main:z from

Percentage Accurate: 91.8% → 99.5%

Time: 44.0s

Alternatives: 24

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.8% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{z + 1}\\ t_4 := \sqrt{x + 1} - \sqrt{x}\\ t_5 := \left(t\_3 - \sqrt{z}\right) + \left(t\_4 + \left(t\_2 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_5 \leq 0.0001:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \left(\left(t\_1 - \left(\sqrt{t} + \left(\sqrt{z} - t\_3\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \mathbf{elif}\;t\_5 \leq 2.001:\\ \;\;\;\;t\_4 + \left(\frac{1}{\sqrt{z} + t\_3} + \frac{1}{t\_2 + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(\left(t\_3 + \left(t\_4 - \sqrt{z}\right)\right) + \left(\frac{1}{t\_1 + \sqrt{t}} - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (- (sqrt (+ x 1.0)) (sqrt x)))
        (t_5 (+ (- t_3 (sqrt z)) (+ t_4 (- t_2 (sqrt y))))))
   (if (<= t_5 0.0001)
     (+
      (* 0.5 (sqrt (/ 1.0 x)))
      (+ (- t_1 (+ (sqrt t) (- (sqrt z) t_3))) (* 0.5 (sqrt (/ 1.0 y)))))
     (if (<= t_5 2.001)
       (+ t_4 (+ (/ 1.0 (+ (sqrt z) t_3)) (/ 1.0 (+ t_2 (sqrt y)))))
       (+
        t_2
        (+ (+ t_3 (- t_4 (sqrt z))) (- (/ 1.0 (+ t_1 (sqrt t))) (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 t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((z + 1.0));
	double t_4 = sqrt((x + 1.0)) - sqrt(x);
	double t_5 = (t_3 - sqrt(z)) + (t_4 + (t_2 - sqrt(y)));
	double tmp;
	if (t_5 <= 0.0001) {
		tmp = (0.5 * sqrt((1.0 / x))) + ((t_1 - (sqrt(t) + (sqrt(z) - t_3))) + (0.5 * sqrt((1.0 / y))));
	} else if (t_5 <= 2.001) {
		tmp = t_4 + ((1.0 / (sqrt(z) + t_3)) + (1.0 / (t_2 + sqrt(y))));
	} else {
		tmp = t_2 + ((t_3 + (t_4 - sqrt(z))) + ((1.0 / (t_1 + sqrt(t))) - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + t))
    t_2 = sqrt((1.0d0 + y))
    t_3 = sqrt((z + 1.0d0))
    t_4 = sqrt((x + 1.0d0)) - sqrt(x)
    t_5 = (t_3 - sqrt(z)) + (t_4 + (t_2 - sqrt(y)))
    if (t_5 <= 0.0001d0) then
        tmp = (0.5d0 * sqrt((1.0d0 / x))) + ((t_1 - (sqrt(t) + (sqrt(z) - t_3))) + (0.5d0 * sqrt((1.0d0 / y))))
    else if (t_5 <= 2.001d0) then
        tmp = t_4 + ((1.0d0 / (sqrt(z) + t_3)) + (1.0d0 / (t_2 + sqrt(y))))
    else
        tmp = t_2 + ((t_3 + (t_4 - sqrt(z))) + ((1.0d0 / (t_1 + sqrt(t))) - 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 t_2 = Math.sqrt((1.0 + y));
	double t_3 = Math.sqrt((z + 1.0));
	double t_4 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double t_5 = (t_3 - Math.sqrt(z)) + (t_4 + (t_2 - Math.sqrt(y)));
	double tmp;
	if (t_5 <= 0.0001) {
		tmp = (0.5 * Math.sqrt((1.0 / x))) + ((t_1 - (Math.sqrt(t) + (Math.sqrt(z) - t_3))) + (0.5 * Math.sqrt((1.0 / y))));
	} else if (t_5 <= 2.001) {
		tmp = t_4 + ((1.0 / (Math.sqrt(z) + t_3)) + (1.0 / (t_2 + Math.sqrt(y))));
	} else {
		tmp = t_2 + ((t_3 + (t_4 - Math.sqrt(z))) + ((1.0 / (t_1 + Math.sqrt(t))) - 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))
	t_2 = math.sqrt((1.0 + y))
	t_3 = math.sqrt((z + 1.0))
	t_4 = math.sqrt((x + 1.0)) - math.sqrt(x)
	t_5 = (t_3 - math.sqrt(z)) + (t_4 + (t_2 - math.sqrt(y)))
	tmp = 0
	if t_5 <= 0.0001:
		tmp = (0.5 * math.sqrt((1.0 / x))) + ((t_1 - (math.sqrt(t) + (math.sqrt(z) - t_3))) + (0.5 * math.sqrt((1.0 / y))))
	elif t_5 <= 2.001:
		tmp = t_4 + ((1.0 / (math.sqrt(z) + t_3)) + (1.0 / (t_2 + math.sqrt(y))))
	else:
		tmp = t_2 + ((t_3 + (t_4 - math.sqrt(z))) + ((1.0 / (t_1 + math.sqrt(t))) - 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))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_5 = Float64(Float64(t_3 - sqrt(z)) + Float64(t_4 + Float64(t_2 - sqrt(y))))
	tmp = 0.0
	if (t_5 <= 0.0001)
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + Float64(Float64(t_1 - Float64(sqrt(t) + Float64(sqrt(z) - t_3))) + Float64(0.5 * sqrt(Float64(1.0 / y)))));
	elseif (t_5 <= 2.001)
		tmp = Float64(t_4 + Float64(Float64(1.0 / Float64(sqrt(z) + t_3)) + Float64(1.0 / Float64(t_2 + sqrt(y)))));
	else
		tmp = Float64(t_2 + Float64(Float64(t_3 + Float64(t_4 - sqrt(z))) + Float64(Float64(1.0 / Float64(t_1 + sqrt(t))) - 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));
	t_2 = sqrt((1.0 + y));
	t_3 = sqrt((z + 1.0));
	t_4 = sqrt((x + 1.0)) - sqrt(x);
	t_5 = (t_3 - sqrt(z)) + (t_4 + (t_2 - sqrt(y)));
	tmp = 0.0;
	if (t_5 <= 0.0001)
		tmp = (0.5 * sqrt((1.0 / x))) + ((t_1 - (sqrt(t) + (sqrt(z) - t_3))) + (0.5 * sqrt((1.0 / y))));
	elseif (t_5 <= 2.001)
		tmp = t_4 + ((1.0 / (sqrt(z) + t_3)) + (1.0 / (t_2 + sqrt(y))));
	else
		tmp = t_2 + ((t_3 + (t_4 - sqrt(z))) + ((1.0 / (t_1 + sqrt(t))) - 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]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.0001], N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.001], N[(t$95$4 + N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(t$95$3 + N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$1 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00000000000000005e-4

   1. Initial program 48.8%

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

      1. associate-+l+ 48.8%

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

      2. associate-+l+ 48.8%

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

      3. +-commutative 48.8%

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

      4. +-commutative 48.8%

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

      5. associate-+l- 46.6%

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

      6. +-commutative 46.6%

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

      7. +-commutative 46.6%

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

   3. Simplified 46.6%

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

   5. Taylor expanded in x around inf 60.2%

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

   6. Taylor expanded in y around inf 73.2%

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

   if 1.00000000000000005e-4 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00099999999999989

   1. Initial program 96.6%

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

      1. associate-+l+ 96.6%

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

      2. associate-+l+ 96.6%

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

      3. +-commutative 96.6%

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

      4. +-commutative 96.6%

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

      5. associate-+l- 76.7%

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

      6. +-commutative 76.7%

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

      7. +-commutative 76.7%

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

   3. Simplified 76.7%

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

      1. add-log-exp 76.7%

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

      2. associate--r- 96.6%

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

      3. +-commutative 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. flip-- 96.6%

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

      2. add-sqr-sqrt 76.1%

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

      3. add-sqr-sqrt 96.8%

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

   8. Applied egg-rr 96.8%

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

      1. associate--l+ 96.9%

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

      2. +-inverses 96.9%

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

      3. metadata-eval 96.9%

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

      4. +-commutative 96.9%

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

   10. Simplified 96.9%

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

   11. Taylor expanded in t around inf 57.0%

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

       1. flip-- 76.7%

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

       2. add-sqr-sqrt 58.4%

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

       3. add-sqr-sqrt 76.9%

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

   13. Applied egg-rr 57.2%

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

       1. associate--l+ 57.4%

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

       2. +-inverses 57.4%

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

       3. metadata-eval 57.4%

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

       4. +-commutative 57.4%

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

   15. Simplified 57.4%

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

   if 2.00099999999999989 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+r+ 99.6%

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

      3. associate-+r- 99.6%

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

      4. associate-+l- 99.5%

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

      5. associate-+r- 99.5%

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

   3. Simplified 99.5%

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

      1. flip-- 99.5%

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

      2. add-sqr-sqrt 79.5%

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

      3. add-sqr-sqrt 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate--r+ 99.9%

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

      2. +-inverses 99.9%

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

      3. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.0001:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \left(\left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.001:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + y} + \left(\left(\sqrt{z + 1} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{z + 1}\\ t_4 := \sqrt{x + 1} - \sqrt{x}\\ t_5 := \left(t\_3 - \sqrt{z}\right) + \left(t\_4 + \left(t\_2 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_5 \leq 0.0001:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \left(\left(t\_1 - \left(\sqrt{t} + \left(\sqrt{z} - t\_3\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \mathbf{elif}\;t\_5 \leq 2.001:\\ \;\;\;\;t\_4 + \left(\frac{1}{\sqrt{z} + t\_3} + \frac{1}{t\_2 + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(\left(\left(1 + t\_3\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\frac{1}{t\_1 + \sqrt{t}} - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (- (sqrt (+ x 1.0)) (sqrt x)))
        (t_5 (+ (- t_3 (sqrt z)) (+ t_4 (- t_2 (sqrt y))))))
   (if (<= t_5 0.0001)
     (+
      (* 0.5 (sqrt (/ 1.0 x)))
      (+ (- t_1 (+ (sqrt t) (- (sqrt z) t_3))) (* 0.5 (sqrt (/ 1.0 y)))))
     (if (<= t_5 2.001)
       (+ t_4 (+ (/ 1.0 (+ (sqrt z) t_3)) (/ 1.0 (+ t_2 (sqrt y)))))
       (+
        t_2
        (+
         (- (+ 1.0 t_3) (+ (sqrt x) (sqrt z)))
         (- (/ 1.0 (+ t_1 (sqrt t))) (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 t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((z + 1.0));
	double t_4 = sqrt((x + 1.0)) - sqrt(x);
	double t_5 = (t_3 - sqrt(z)) + (t_4 + (t_2 - sqrt(y)));
	double tmp;
	if (t_5 <= 0.0001) {
		tmp = (0.5 * sqrt((1.0 / x))) + ((t_1 - (sqrt(t) + (sqrt(z) - t_3))) + (0.5 * sqrt((1.0 / y))));
	} else if (t_5 <= 2.001) {
		tmp = t_4 + ((1.0 / (sqrt(z) + t_3)) + (1.0 / (t_2 + sqrt(y))));
	} else {
		tmp = t_2 + (((1.0 + t_3) - (sqrt(x) + sqrt(z))) + ((1.0 / (t_1 + sqrt(t))) - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + t))
    t_2 = sqrt((1.0d0 + y))
    t_3 = sqrt((z + 1.0d0))
    t_4 = sqrt((x + 1.0d0)) - sqrt(x)
    t_5 = (t_3 - sqrt(z)) + (t_4 + (t_2 - sqrt(y)))
    if (t_5 <= 0.0001d0) then
        tmp = (0.5d0 * sqrt((1.0d0 / x))) + ((t_1 - (sqrt(t) + (sqrt(z) - t_3))) + (0.5d0 * sqrt((1.0d0 / y))))
    else if (t_5 <= 2.001d0) then
        tmp = t_4 + ((1.0d0 / (sqrt(z) + t_3)) + (1.0d0 / (t_2 + sqrt(y))))
    else
        tmp = t_2 + (((1.0d0 + t_3) - (sqrt(x) + sqrt(z))) + ((1.0d0 / (t_1 + sqrt(t))) - 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 t_2 = Math.sqrt((1.0 + y));
	double t_3 = Math.sqrt((z + 1.0));
	double t_4 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double t_5 = (t_3 - Math.sqrt(z)) + (t_4 + (t_2 - Math.sqrt(y)));
	double tmp;
	if (t_5 <= 0.0001) {
		tmp = (0.5 * Math.sqrt((1.0 / x))) + ((t_1 - (Math.sqrt(t) + (Math.sqrt(z) - t_3))) + (0.5 * Math.sqrt((1.0 / y))));
	} else if (t_5 <= 2.001) {
		tmp = t_4 + ((1.0 / (Math.sqrt(z) + t_3)) + (1.0 / (t_2 + Math.sqrt(y))));
	} else {
		tmp = t_2 + (((1.0 + t_3) - (Math.sqrt(x) + Math.sqrt(z))) + ((1.0 / (t_1 + Math.sqrt(t))) - 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))
	t_2 = math.sqrt((1.0 + y))
	t_3 = math.sqrt((z + 1.0))
	t_4 = math.sqrt((x + 1.0)) - math.sqrt(x)
	t_5 = (t_3 - math.sqrt(z)) + (t_4 + (t_2 - math.sqrt(y)))
	tmp = 0
	if t_5 <= 0.0001:
		tmp = (0.5 * math.sqrt((1.0 / x))) + ((t_1 - (math.sqrt(t) + (math.sqrt(z) - t_3))) + (0.5 * math.sqrt((1.0 / y))))
	elif t_5 <= 2.001:
		tmp = t_4 + ((1.0 / (math.sqrt(z) + t_3)) + (1.0 / (t_2 + math.sqrt(y))))
	else:
		tmp = t_2 + (((1.0 + t_3) - (math.sqrt(x) + math.sqrt(z))) + ((1.0 / (t_1 + math.sqrt(t))) - 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))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_5 = Float64(Float64(t_3 - sqrt(z)) + Float64(t_4 + Float64(t_2 - sqrt(y))))
	tmp = 0.0
	if (t_5 <= 0.0001)
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + Float64(Float64(t_1 - Float64(sqrt(t) + Float64(sqrt(z) - t_3))) + Float64(0.5 * sqrt(Float64(1.0 / y)))));
	elseif (t_5 <= 2.001)
		tmp = Float64(t_4 + Float64(Float64(1.0 / Float64(sqrt(z) + t_3)) + Float64(1.0 / Float64(t_2 + sqrt(y)))));
	else
		tmp = Float64(t_2 + Float64(Float64(Float64(1.0 + t_3) - Float64(sqrt(x) + sqrt(z))) + Float64(Float64(1.0 / Float64(t_1 + sqrt(t))) - 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));
	t_2 = sqrt((1.0 + y));
	t_3 = sqrt((z + 1.0));
	t_4 = sqrt((x + 1.0)) - sqrt(x);
	t_5 = (t_3 - sqrt(z)) + (t_4 + (t_2 - sqrt(y)));
	tmp = 0.0;
	if (t_5 <= 0.0001)
		tmp = (0.5 * sqrt((1.0 / x))) + ((t_1 - (sqrt(t) + (sqrt(z) - t_3))) + (0.5 * sqrt((1.0 / y))));
	elseif (t_5 <= 2.001)
		tmp = t_4 + ((1.0 / (sqrt(z) + t_3)) + (1.0 / (t_2 + sqrt(y))));
	else
		tmp = t_2 + (((1.0 + t_3) - (sqrt(x) + sqrt(z))) + ((1.0 / (t_1 + sqrt(t))) - 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]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.0001], N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.001], N[(t$95$4 + N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(N[(1.0 + t$95$3), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$1 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00000000000000005e-4

   1. Initial program 48.8%

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

      1. associate-+l+ 48.8%

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

      2. associate-+l+ 48.8%

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

      3. +-commutative 48.8%

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

      4. +-commutative 48.8%

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

      5. associate-+l- 46.6%

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

      6. +-commutative 46.6%

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

      7. +-commutative 46.6%

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

   3. Simplified 46.6%

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

   5. Taylor expanded in x around inf 60.2%

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

   6. Taylor expanded in y around inf 73.2%

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

   if 1.00000000000000005e-4 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00099999999999989

   1. Initial program 96.6%

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

      1. associate-+l+ 96.6%

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

      2. associate-+l+ 96.6%

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

      3. +-commutative 96.6%

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

      4. +-commutative 96.6%

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

      5. associate-+l- 76.7%

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

      6. +-commutative 76.7%

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

      7. +-commutative 76.7%

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

   3. Simplified 76.7%

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

      1. add-log-exp 76.7%

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

      2. associate--r- 96.6%

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

      3. +-commutative 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. flip-- 96.6%

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

      2. add-sqr-sqrt 76.1%

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

      3. add-sqr-sqrt 96.8%

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

   8. Applied egg-rr 96.8%

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

      1. associate--l+ 96.9%

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

      2. +-inverses 96.9%

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

      3. metadata-eval 96.9%

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

      4. +-commutative 96.9%

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

   10. Simplified 96.9%

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

   11. Taylor expanded in t around inf 57.0%

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

       1. flip-- 76.7%

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

       2. add-sqr-sqrt 58.4%

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

       3. add-sqr-sqrt 76.9%

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

   13. Applied egg-rr 57.2%

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

       1. associate--l+ 57.4%

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

       2. +-inverses 57.4%

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

       3. metadata-eval 57.4%

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

       4. +-commutative 57.4%

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

   15. Simplified 57.4%

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

   if 2.00099999999999989 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+r+ 99.6%

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

      3. associate-+r- 99.6%

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

      4. associate-+l- 99.5%

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

      5. associate-+r- 99.5%

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

   3. Simplified 99.5%

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

      1. flip-- 99.5%

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

      2. add-sqr-sqrt 79.5%

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

      3. add-sqr-sqrt 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate--r+ 99.9%

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

      2. +-inverses 99.9%

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

      3. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 97.2%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.0001:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \left(\left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.001:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + y} + \left(\left(\left(1 + \sqrt{z + 1}\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{1 + t}\\ t_4 := \sqrt{x + 1}\\ \mathbf{if}\;z \leq 5.8 \cdot 10^{+29}:\\ \;\;\;\;\left(t\_4 - \sqrt{x}\right) + \left(\left(t\_2 - \sqrt{y}\right) + \log \left(e^{\left(t\_3 - \sqrt{t}\right) + \frac{1}{\sqrt{z} + t\_1}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t\_4} + \left(\frac{\left(1 + y\right) - y}{t\_2 + \sqrt{y}} + \left(t\_3 - \left(\sqrt{t} + \left(\sqrt{z} - t\_1\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 (+ z 1.0)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (sqrt (+ 1.0 t)))
        (t_4 (sqrt (+ x 1.0))))
   (if (<= z 5.8e+29)
     (+
      (- t_4 (sqrt x))
      (+
       (- t_2 (sqrt y))
       (log (exp (+ (- t_3 (sqrt t)) (/ 1.0 (+ (sqrt z) t_1)))))))
     (+
      (/ 1.0 (+ (sqrt x) t_4))
      (+
       (/ (- (+ 1.0 y) y) (+ t_2 (sqrt y)))
       (- t_3 (+ (sqrt t) (- (sqrt z) t_1))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((1.0 + t));
	double t_4 = sqrt((x + 1.0));
	double tmp;
	if (z <= 5.8e+29) {
		tmp = (t_4 - sqrt(x)) + ((t_2 - sqrt(y)) + log(exp(((t_3 - sqrt(t)) + (1.0 / (sqrt(z) + t_1))))));
	} else {
		tmp = (1.0 / (sqrt(x) + t_4)) + ((((1.0 + y) - y) / (t_2 + sqrt(y))) + (t_3 - (sqrt(t) + (sqrt(z) - t_1))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = sqrt((1.0d0 + y))
    t_3 = sqrt((1.0d0 + t))
    t_4 = sqrt((x + 1.0d0))
    if (z <= 5.8d+29) then
        tmp = (t_4 - sqrt(x)) + ((t_2 - sqrt(y)) + log(exp(((t_3 - sqrt(t)) + (1.0d0 / (sqrt(z) + t_1))))))
    else
        tmp = (1.0d0 / (sqrt(x) + t_4)) + ((((1.0d0 + y) - y) / (t_2 + sqrt(y))) + (t_3 - (sqrt(t) + (sqrt(z) - t_1))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double t_2 = Math.sqrt((1.0 + y));
	double t_3 = Math.sqrt((1.0 + t));
	double t_4 = Math.sqrt((x + 1.0));
	double tmp;
	if (z <= 5.8e+29) {
		tmp = (t_4 - Math.sqrt(x)) + ((t_2 - Math.sqrt(y)) + Math.log(Math.exp(((t_3 - Math.sqrt(t)) + (1.0 / (Math.sqrt(z) + t_1))))));
	} else {
		tmp = (1.0 / (Math.sqrt(x) + t_4)) + ((((1.0 + y) - y) / (t_2 + Math.sqrt(y))) + (t_3 - (Math.sqrt(t) + (Math.sqrt(z) - t_1))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = math.sqrt((1.0 + y))
	t_3 = math.sqrt((1.0 + t))
	t_4 = math.sqrt((x + 1.0))
	tmp = 0
	if z <= 5.8e+29:
		tmp = (t_4 - math.sqrt(x)) + ((t_2 - math.sqrt(y)) + math.log(math.exp(((t_3 - math.sqrt(t)) + (1.0 / (math.sqrt(z) + t_1))))))
	else:
		tmp = (1.0 / (math.sqrt(x) + t_4)) + ((((1.0 + y) - y) / (t_2 + math.sqrt(y))) + (t_3 - (math.sqrt(t) + (math.sqrt(z) - t_1))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = sqrt(Float64(1.0 + t))
	t_4 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (z <= 5.8e+29)
		tmp = Float64(Float64(t_4 - sqrt(x)) + Float64(Float64(t_2 - sqrt(y)) + log(exp(Float64(Float64(t_3 - sqrt(t)) + Float64(1.0 / Float64(sqrt(z) + t_1)))))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_4)) + Float64(Float64(Float64(Float64(1.0 + y) - y) / Float64(t_2 + sqrt(y))) + Float64(t_3 - Float64(sqrt(t) + Float64(sqrt(z) - t_1)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	t_2 = sqrt((1.0 + y));
	t_3 = sqrt((1.0 + t));
	t_4 = sqrt((x + 1.0));
	tmp = 0.0;
	if (z <= 5.8e+29)
		tmp = (t_4 - sqrt(x)) + ((t_2 - sqrt(y)) + log(exp(((t_3 - sqrt(t)) + (1.0 / (sqrt(z) + t_1))))));
	else
		tmp = (1.0 / (sqrt(x) + t_4)) + ((((1.0 + y) - y) / (t_2 + sqrt(y))) + (t_3 - (sqrt(t) + (sqrt(z) - t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 5.8e+29], N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Log[N[Exp[N[(N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(1.0 + y), $MachinePrecision] - y), $MachinePrecision] / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if z < 5.7999999999999999e29

   1. Initial program 95.5%

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

      1. associate-+l+ 95.5%

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

      2. associate-+l+ 95.5%

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

      3. +-commutative 95.5%

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

      4. +-commutative 95.5%

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

      5. associate-+l- 60.4%

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

      6. +-commutative 60.4%

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

      7. +-commutative 60.4%

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

   3. Simplified 60.4%

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

      1. add-log-exp 60.4%

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

      2. associate--r- 95.5%

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

      3. +-commutative 95.5%

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

   6. Applied egg-rr 95.5%

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

      1. flip-- 95.6%

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

      2. add-sqr-sqrt 95.1%

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

      3. add-sqr-sqrt 96.0%

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

   8. Applied egg-rr 96.0%

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

      1. associate--l+ 96.3%

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

      2. +-inverses 96.3%

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

      3. metadata-eval 96.3%

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

      4. +-commutative 96.3%

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

   10. Simplified 96.3%

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

   if 5.7999999999999999e29 < z

   1. Initial program 87.6%

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

      1. associate-+l+ 87.6%

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

      2. associate-+l+ 87.6%

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

      3. +-commutative 87.6%

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

      4. +-commutative 87.6%

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

      5. associate-+l- 87.6%

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

      6. +-commutative 87.6%

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

      7. +-commutative 87.6%

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

   3. Simplified 87.6%

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

      1. flip-- 87.7%

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

      2. add-sqr-sqrt 70.7%

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

      3. add-sqr-sqrt 88.1%

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

   6. Applied egg-rr 88.1%

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

      1. flip-- 88.3%

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

      2. add-sqr-sqrt 73.6%

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

      3. +-commutative 73.6%

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

      4. add-sqr-sqrt 88.5%

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

      5. +-commutative 88.5%

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

   8. Applied egg-rr 88.5%

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

      1. associate--l+ 91.9%

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

      2. +-inverses 91.9%

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

      3. metadata-eval 91.9%

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

   10. Simplified 91.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+29}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \log \left(e^{\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{x + 1} - \sqrt{x}\\ t_3 := \frac{1}{\sqrt{z} + \sqrt{z + 1}}\\ \mathbf{if}\;t\_2 \leq 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \left(\left(t\_1 - \sqrt{y}\right) + t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(t\_3 + \frac{1}{t\_1 + \sqrt{y}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (- (sqrt (+ x 1.0)) (sqrt x)))
        (t_3 (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0))))))
   (if (<= t_2 1e-5)
     (+ (* 0.5 (sqrt (/ 1.0 x))) (+ (- t_1 (sqrt y)) t_3))
     (+ t_2 (+ t_3 (/ 1.0 (+ t_1 (sqrt y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((x + 1.0)) - sqrt(x);
	double t_3 = 1.0 / (sqrt(z) + sqrt((z + 1.0)));
	double tmp;
	if (t_2 <= 1e-5) {
		tmp = (0.5 * sqrt((1.0 / x))) + ((t_1 - sqrt(y)) + t_3);
	} else {
		tmp = t_2 + (t_3 + (1.0 / (t_1 + sqrt(y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((x + 1.0d0)) - sqrt(x)
    t_3 = 1.0d0 / (sqrt(z) + sqrt((z + 1.0d0)))
    if (t_2 <= 1d-5) then
        tmp = (0.5d0 * sqrt((1.0d0 / x))) + ((t_1 - sqrt(y)) + t_3)
    else
        tmp = t_2 + (t_3 + (1.0d0 / (t_1 + sqrt(y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double t_3 = 1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0)));
	double tmp;
	if (t_2 <= 1e-5) {
		tmp = (0.5 * Math.sqrt((1.0 / x))) + ((t_1 - Math.sqrt(y)) + t_3);
	} else {
		tmp = t_2 + (t_3 + (1.0 / (t_1 + Math.sqrt(y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((x + 1.0)) - math.sqrt(x)
	t_3 = 1.0 / (math.sqrt(z) + math.sqrt((z + 1.0)))
	tmp = 0
	if t_2 <= 1e-5:
		tmp = (0.5 * math.sqrt((1.0 / x))) + ((t_1 - math.sqrt(y)) + t_3)
	else:
		tmp = t_2 + (t_3 + (1.0 / (t_1 + math.sqrt(y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_3 = Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0))))
	tmp = 0.0
	if (t_2 <= 1e-5)
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + Float64(Float64(t_1 - sqrt(y)) + t_3));
	else
		tmp = Float64(t_2 + Float64(t_3 + Float64(1.0 / Float64(t_1 + sqrt(y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((x + 1.0)) - sqrt(x);
	t_3 = 1.0 / (sqrt(z) + sqrt((z + 1.0)));
	tmp = 0.0;
	if (t_2 <= 1e-5)
		tmp = (0.5 * sqrt((1.0 / x))) + ((t_1 - sqrt(y)) + t_3);
	else
		tmp = t_2 + (t_3 + (1.0 / (t_1 + sqrt(y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-5], N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(t$95$3 + N[(1.0 / N[(t$95$1 + N[Sqrt[y], $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)) < 1.00000000000000008e-5

   1. Initial program 86.3%

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

      1. associate-+l+ 86.3%

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

      2. associate-+l+ 86.3%

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

      3. +-commutative 86.3%

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

      4. +-commutative 86.3%

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

      5. associate-+l- 66.7%

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

      6. +-commutative 66.7%

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

      7. +-commutative 66.7%

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

   3. Simplified 66.7%

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

      1. add-log-exp 66.7%

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

      2. associate--r- 86.3%

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

      3. +-commutative 86.3%

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

   6. Applied egg-rr 86.3%

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

      1. flip-- 86.4%

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

      2. add-sqr-sqrt 75.7%

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

      3. add-sqr-sqrt 86.6%

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

   8. Applied egg-rr 86.6%

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

      1. associate--l+ 86.7%

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

      2. +-inverses 86.7%

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

      3. metadata-eval 86.7%

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

      4. +-commutative 86.7%

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

   10. Simplified 86.7%

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

   11. Taylor expanded in t around inf 45.2%

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

   12. Taylor expanded in x around inf 47.9%

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

   if 1.00000000000000008e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   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. Step-by-step derivation

      1. associate-+l+ 97.6%

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

      2. associate-+l+ 97.6%

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

      3. +-commutative 97.6%

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

      4. +-commutative 97.6%

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

      5. associate-+l- 76.6%

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

      6. +-commutative 76.6%

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

      7. +-commutative 76.6%

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

   3. Simplified 76.6%

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

      1. add-log-exp 76.6%

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

      2. associate--r- 97.6%

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

      3. +-commutative 97.6%

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

   6. Applied egg-rr 97.6%

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

      1. flip-- 97.6%

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

      2. add-sqr-sqrt 73.5%

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

      3. add-sqr-sqrt 97.9%

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

   8. Applied egg-rr 97.9%

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

      1. associate--l+ 98.1%

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

      2. +-inverses 98.1%

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

      3. metadata-eval 98.1%

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

      4. +-commutative 98.1%

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

   10. Simplified 98.1%

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

   11. Taylor expanded in t around inf 61.1%

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

       1. flip-- 76.6%

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

       2. add-sqr-sqrt 56.7%

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

       3. add-sqr-sqrt 76.7%

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

   13. Applied egg-rr 61.1%

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

       1. associate--l+ 61.1%

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

       2. +-inverses 61.1%

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

       3. metadata-eval 61.1%

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

       4. +-commutative 61.1%

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

   15. Simplified 61.1%

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

4. Final simplification 54.8%

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

Alternative 5: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{x + 1}\\ \mathbf{if}\;t \leq 2950000000000:\\ \;\;\;\;\left(t\_2 - \sqrt{x}\right) + \left(\left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{z} - t\_1\right)\right)\right) + \left(1 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t\_2} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + t\_1}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0))) (t_2 (sqrt (+ x 1.0))))
   (if (<= t 2950000000000.0)
     (+
      (- t_2 (sqrt x))
      (+ (- (sqrt (+ 1.0 t)) (+ (sqrt t) (- (sqrt z) t_1))) (- 1.0 (sqrt y))))
     (+
      (/ 1.0 (+ (sqrt x) t_2))
      (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (/ 1.0 (+ (sqrt z) t_1)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt((x + 1.0));
	double tmp;
	if (t <= 2950000000000.0) {
		tmp = (t_2 - sqrt(x)) + ((sqrt((1.0 + t)) - (sqrt(t) + (sqrt(z) - t_1))) + (1.0 - sqrt(y)));
	} else {
		tmp = (1.0 / (sqrt(x) + t_2)) + ((sqrt((1.0 + y)) - sqrt(y)) + (1.0 / (sqrt(z) + t_1)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = sqrt((x + 1.0d0))
    if (t <= 2950000000000.0d0) then
        tmp = (t_2 - sqrt(x)) + ((sqrt((1.0d0 + t)) - (sqrt(t) + (sqrt(z) - t_1))) + (1.0d0 - sqrt(y)))
    else
        tmp = (1.0d0 / (sqrt(x) + t_2)) + ((sqrt((1.0d0 + y)) - sqrt(y)) + (1.0d0 / (sqrt(z) + t_1)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double t_2 = Math.sqrt((x + 1.0));
	double tmp;
	if (t <= 2950000000000.0) {
		tmp = (t_2 - Math.sqrt(x)) + ((Math.sqrt((1.0 + t)) - (Math.sqrt(t) + (Math.sqrt(z) - t_1))) + (1.0 - Math.sqrt(y)));
	} else {
		tmp = (1.0 / (Math.sqrt(x) + t_2)) + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) + (1.0 / (Math.sqrt(z) + t_1)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = math.sqrt((x + 1.0))
	tmp = 0
	if t <= 2950000000000.0:
		tmp = (t_2 - math.sqrt(x)) + ((math.sqrt((1.0 + t)) - (math.sqrt(t) + (math.sqrt(z) - t_1))) + (1.0 - math.sqrt(y)))
	else:
		tmp = (1.0 / (math.sqrt(x) + t_2)) + ((math.sqrt((1.0 + y)) - math.sqrt(y)) + (1.0 / (math.sqrt(z) + t_1)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (t <= 2950000000000.0)
		tmp = Float64(Float64(t_2 - sqrt(x)) + Float64(Float64(sqrt(Float64(1.0 + t)) - Float64(sqrt(t) + Float64(sqrt(z) - t_1))) + Float64(1.0 - sqrt(y))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_2)) + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(1.0 / Float64(sqrt(z) + t_1))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	t_2 = sqrt((x + 1.0));
	tmp = 0.0;
	if (t <= 2950000000000.0)
		tmp = (t_2 - sqrt(x)) + ((sqrt((1.0 + t)) - (sqrt(t) + (sqrt(z) - t_1))) + (1.0 - sqrt(y)));
	else
		tmp = (1.0 / (sqrt(x) + t_2)) + ((sqrt((1.0 + y)) - sqrt(y)) + (1.0 / (sqrt(z) + t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 2950000000000.0], N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.95e12

   1. Initial program 96.6%

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

      1. associate-+l+ 96.6%

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

      2. associate-+l+ 96.6%

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

      3. +-commutative 96.6%

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

      4. +-commutative 96.6%

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

      5. associate-+l- 96.6%

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

      6. +-commutative 96.6%

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

      7. +-commutative 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in y around 0 50.5%

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

   if 2.95e12 < t

   1. Initial program 87.4%

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

      1. associate-+l+ 87.4%

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

      2. associate-+l+ 87.4%

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

      3. +-commutative 87.4%

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

      4. +-commutative 87.4%

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

      5. associate-+l- 45.5%

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

      6. +-commutative 45.5%

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

      7. +-commutative 45.5%

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

   3. Simplified 45.5%

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

      1. add-log-exp 45.5%

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

      2. associate--r- 87.4%

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

      3. +-commutative 87.4%

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

   6. Applied egg-rr 87.4%

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

      1. flip-- 87.4%

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

      2. add-sqr-sqrt 69.6%

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

      3. add-sqr-sqrt 87.9%

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

   8. Applied egg-rr 87.9%

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

      1. associate--l+ 88.1%

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

      2. +-inverses 88.1%

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

      3. metadata-eval 88.1%

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

      4. +-commutative 88.1%

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

   10. Simplified 88.1%

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

   11. Taylor expanded in t around inf 90.1%

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

       1. flip-- 46.0%

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

       2. add-sqr-sqrt 40.7%

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

       3. +-commutative 40.7%

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

       4. add-sqr-sqrt 46.1%

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

       5. +-commutative 46.1%

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

   13. Applied egg-rr 91.1%

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

       1. associate--l+ 49.2%

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

       2. +-inverses 49.2%

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

       3. metadata-eval 49.2%

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

   15. Simplified 93.4%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2950000000000:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) + \left(1 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{z + 1}\\ \mathbf{if}\;t \leq 3.3 \cdot 10^{+24}:\\ \;\;\;\;t\_1 + \left(\left(t\_2 + \left(\left(1 - \sqrt{x}\right) - \sqrt{z}\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\left(t\_1 - \sqrt{y}\right) + \frac{1}{\sqrt{z} + t\_2}\right)\\ \end{array} \end{array} \]

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 (+ 1.0 y))) (t_2 (sqrt (+ z 1.0))))
   (if (<= t 3.3e+24)
     (+
      t_1
      (+
       (+ t_2 (- (- 1.0 (sqrt x)) (sqrt z)))
       (- (- (sqrt (+ 1.0 t)) (sqrt t)) (sqrt y))))
     (+
      (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0))))
      (+ (- t_1 (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((1.0 + y));
	double t_2 = sqrt((z + 1.0));
	double tmp;
	if (t <= 3.3e+24) {
		tmp = t_1 + ((t_2 + ((1.0 - sqrt(x)) - sqrt(z))) + ((sqrt((1.0 + t)) - sqrt(t)) - sqrt(y)));
	} else {
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) + ((t_1 - 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) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((z + 1.0d0))
    if (t <= 3.3d+24) then
        tmp = t_1 + ((t_2 + ((1.0d0 - sqrt(x)) - sqrt(z))) + ((sqrt((1.0d0 + t)) - sqrt(t)) - sqrt(y)))
    else
        tmp = (1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))) + ((t_1 - 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((1.0 + y));
	double t_2 = Math.sqrt((z + 1.0));
	double tmp;
	if (t <= 3.3e+24) {
		tmp = t_1 + ((t_2 + ((1.0 - Math.sqrt(x)) - Math.sqrt(z))) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) - Math.sqrt(y)));
	} else {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)))) + ((t_1 - 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((1.0 + y))
	t_2 = math.sqrt((z + 1.0))
	tmp = 0
	if t <= 3.3e+24:
		tmp = t_1 + ((t_2 + ((1.0 - math.sqrt(x)) - math.sqrt(z))) + ((math.sqrt((1.0 + t)) - math.sqrt(t)) - math.sqrt(y)))
	else:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))) + ((t_1 - 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(1.0 + y))
	t_2 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (t <= 3.3e+24)
		tmp = Float64(t_1 + Float64(Float64(t_2 + Float64(Float64(1.0 - sqrt(x)) - sqrt(z))) + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) - sqrt(y))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0)))) + Float64(Float64(t_1 - 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((1.0 + y));
	t_2 = sqrt((z + 1.0));
	tmp = 0.0;
	if (t <= 3.3e+24)
		tmp = t_1 + ((t_2 + ((1.0 - sqrt(x)) - sqrt(z))) + ((sqrt((1.0 + t)) - sqrt(t)) - sqrt(y)));
	else
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) + ((t_1 - 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[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 3.3e+24], N[(t$95$1 + N[(N[(t$95$2 + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $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 t < 3.2999999999999999e24

   1. Initial program 96.0%

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

      1. +-commutative 96.0%

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

      2. associate-+r+ 96.0%

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

      3. associate-+r- 79.6%

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

      4. associate-+l- 63.6%

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

      5. associate-+r- 59.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in x around 0 23.0%

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

   if 3.2999999999999999e24 < t

   1. Initial program 87.9%

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

      1. associate-+l+ 87.9%

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

      2. associate-+l+ 87.9%

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

      3. +-commutative 87.9%

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

      4. +-commutative 87.9%

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

      5. associate-+l- 44.9%

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

      6. +-commutative 44.9%

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

      7. +-commutative 44.9%

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

   3. Simplified 44.9%

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

      1. add-log-exp 44.9%

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

      2. associate--r- 87.8%

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

      3. +-commutative 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. flip-- 87.8%

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

      2. add-sqr-sqrt 70.1%

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

      3. add-sqr-sqrt 88.3%

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

   8. Applied egg-rr 88.3%

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

      1. associate--l+ 88.6%

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

      2. +-inverses 88.6%

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

      3. metadata-eval 88.6%

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

      4. +-commutative 88.6%

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

   10. Simplified 88.6%

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

   11. Taylor expanded in t around inf 90.6%

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

       1. flip-- 45.4%

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

       2. add-sqr-sqrt 41.1%

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

       3. +-commutative 41.1%

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

       4. add-sqr-sqrt 45.6%

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

       5. +-commutative 45.6%

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

   13. Applied egg-rr 91.7%

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

       1. associate--l+ 48.7%

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

       2. +-inverses 48.7%

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

       3. metadata-eval 48.7%

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

   15. Simplified 94.0%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{1 + y} + \left(\left(\sqrt{z + 1} + \left(\left(1 - \sqrt{x}\right) - \sqrt{z}\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{x + 1}\\ \mathbf{if}\;t \leq 160000000000:\\ \;\;\;\;1 + \left(\left(t\_2 + \left(\sqrt{1 + t} + t\_1\right)\right) - \left(\sqrt{t} + \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t\_2} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + t\_1}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0))) (t_2 (sqrt (+ x 1.0))))
   (if (<= t 160000000000.0)
     (+
      1.0
      (-
       (+ t_2 (+ (sqrt (+ 1.0 t)) t_1))
       (+ (sqrt t) (+ (sqrt z) (+ (sqrt x) (sqrt y))))))
     (+
      (/ 1.0 (+ (sqrt x) t_2))
      (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (/ 1.0 (+ (sqrt z) t_1)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt((x + 1.0));
	double tmp;
	if (t <= 160000000000.0) {
		tmp = 1.0 + ((t_2 + (sqrt((1.0 + t)) + t_1)) - (sqrt(t) + (sqrt(z) + (sqrt(x) + sqrt(y)))));
	} else {
		tmp = (1.0 / (sqrt(x) + t_2)) + ((sqrt((1.0 + y)) - sqrt(y)) + (1.0 / (sqrt(z) + t_1)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = sqrt((x + 1.0d0))
    if (t <= 160000000000.0d0) then
        tmp = 1.0d0 + ((t_2 + (sqrt((1.0d0 + t)) + t_1)) - (sqrt(t) + (sqrt(z) + (sqrt(x) + sqrt(y)))))
    else
        tmp = (1.0d0 / (sqrt(x) + t_2)) + ((sqrt((1.0d0 + y)) - sqrt(y)) + (1.0d0 / (sqrt(z) + t_1)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double t_2 = Math.sqrt((x + 1.0));
	double tmp;
	if (t <= 160000000000.0) {
		tmp = 1.0 + ((t_2 + (Math.sqrt((1.0 + t)) + t_1)) - (Math.sqrt(t) + (Math.sqrt(z) + (Math.sqrt(x) + Math.sqrt(y)))));
	} else {
		tmp = (1.0 / (Math.sqrt(x) + t_2)) + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) + (1.0 / (Math.sqrt(z) + t_1)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = math.sqrt((x + 1.0))
	tmp = 0
	if t <= 160000000000.0:
		tmp = 1.0 + ((t_2 + (math.sqrt((1.0 + t)) + t_1)) - (math.sqrt(t) + (math.sqrt(z) + (math.sqrt(x) + math.sqrt(y)))))
	else:
		tmp = (1.0 / (math.sqrt(x) + t_2)) + ((math.sqrt((1.0 + y)) - math.sqrt(y)) + (1.0 / (math.sqrt(z) + t_1)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (t <= 160000000000.0)
		tmp = Float64(1.0 + Float64(Float64(t_2 + Float64(sqrt(Float64(1.0 + t)) + t_1)) - Float64(sqrt(t) + Float64(sqrt(z) + Float64(sqrt(x) + sqrt(y))))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_2)) + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(1.0 / Float64(sqrt(z) + t_1))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	t_2 = sqrt((x + 1.0));
	tmp = 0.0;
	if (t <= 160000000000.0)
		tmp = 1.0 + ((t_2 + (sqrt((1.0 + t)) + t_1)) - (sqrt(t) + (sqrt(z) + (sqrt(x) + sqrt(y)))));
	else
		tmp = (1.0 / (sqrt(x) + t_2)) + ((sqrt((1.0 + y)) - sqrt(y)) + (1.0 / (sqrt(z) + t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 160000000000.0], N[(1.0 + N[(N[(t$95$2 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + t$95$1), $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[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.6e11

   1. Initial program 96.6%

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

      1. +-commutative 96.6%

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

      2. associate-+r+ 96.6%

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

      3. associate-+r- 80.4%

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

      4. associate-+l- 64.0%

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

      5. associate-+r- 60.0%

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

   3. Simplified 50.9%

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

   5. Taylor expanded in y around 0 16.3%

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

      1. associate--l+ 26.5%

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

      2. +-commutative 26.5%

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

      3. associate-+r+ 26.5%

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

      4. +-commutative 26.5%

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

      5. associate-+r+ 26.5%

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

   7. Simplified 26.5%

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

   if 1.6e11 < t

   1. Initial program 87.4%

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

      1. associate-+l+ 87.4%

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

      2. associate-+l+ 87.4%

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

      3. +-commutative 87.4%

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

      4. +-commutative 87.4%

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

      5. associate-+l- 45.5%

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

      6. +-commutative 45.5%

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

      7. +-commutative 45.5%

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

   3. Simplified 45.5%

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

      1. add-log-exp 45.5%

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

      2. associate--r- 87.4%

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

      3. +-commutative 87.4%

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

   6. Applied egg-rr 87.4%

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

      1. flip-- 87.4%

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

      2. add-sqr-sqrt 69.6%

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

      3. add-sqr-sqrt 87.9%

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

   8. Applied egg-rr 87.9%

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

      1. associate--l+ 88.1%

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

      2. +-inverses 88.1%

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

      3. metadata-eval 88.1%

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

      4. +-commutative 88.1%

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

   10. Simplified 88.1%

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

   11. Taylor expanded in t around inf 90.1%

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

       1. flip-- 46.0%

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

       2. add-sqr-sqrt 40.7%

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

       3. +-commutative 40.7%

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

       4. add-sqr-sqrt 46.1%

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

       5. +-commutative 46.1%

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

   13. Applied egg-rr 91.1%

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

       1. associate--l+ 49.2%

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

       2. +-inverses 49.2%

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

       3. metadata-eval 49.2%

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

   15. Simplified 93.4%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 160000000000:\\ \;\;\;\;1 + \left(\left(\sqrt{x + 1} + \left(\sqrt{1 + t} + \sqrt{z + 1}\right)\right) - \left(\sqrt{t} + \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ \mathbf{if}\;x \leq 41000000:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{z} + t\_1} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \left(\left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{z} - t\_1\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0))))
   (if (<= x 41000000.0)
     (+
      (- (sqrt (+ x 1.0)) (sqrt x))
      (+ (/ 1.0 (+ (sqrt z) t_1)) (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))))
     (+
      (* 0.5 (sqrt (/ 1.0 x)))
      (+
       (- (sqrt (+ 1.0 t)) (+ (sqrt t) (- (sqrt z) t_1)))
       (* 0.5 (sqrt (/ 1.0 y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double tmp;
	if (x <= 41000000.0) {
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((1.0 / (sqrt(z) + t_1)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y))));
	} else {
		tmp = (0.5 * sqrt((1.0 / x))) + ((sqrt((1.0 + t)) - (sqrt(t) + (sqrt(z) - t_1))) + (0.5 * sqrt((1.0 / 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((z + 1.0d0))
    if (x <= 41000000.0d0) then
        tmp = (sqrt((x + 1.0d0)) - sqrt(x)) + ((1.0d0 / (sqrt(z) + t_1)) + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))))
    else
        tmp = (0.5d0 * sqrt((1.0d0 / x))) + ((sqrt((1.0d0 + t)) - (sqrt(t) + (sqrt(z) - t_1))) + (0.5d0 * sqrt((1.0d0 / 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((z + 1.0));
	double tmp;
	if (x <= 41000000.0) {
		tmp = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + ((1.0 / (Math.sqrt(z) + t_1)) + (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))));
	} else {
		tmp = (0.5 * Math.sqrt((1.0 / x))) + ((Math.sqrt((1.0 + t)) - (Math.sqrt(t) + (Math.sqrt(z) - t_1))) + (0.5 * Math.sqrt((1.0 / y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	tmp = 0
	if x <= 41000000.0:
		tmp = (math.sqrt((x + 1.0)) - math.sqrt(x)) + ((1.0 / (math.sqrt(z) + t_1)) + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))))
	else:
		tmp = (0.5 * math.sqrt((1.0 / x))) + ((math.sqrt((1.0 + t)) - (math.sqrt(t) + (math.sqrt(z) - t_1))) + (0.5 * math.sqrt((1.0 / y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (x <= 41000000.0)
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(Float64(1.0 / Float64(sqrt(z) + t_1)) + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + Float64(Float64(sqrt(Float64(1.0 + t)) - Float64(sqrt(t) + Float64(sqrt(z) - t_1))) + Float64(0.5 * sqrt(Float64(1.0 / 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((z + 1.0));
	tmp = 0.0;
	if (x <= 41000000.0)
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((1.0 / (sqrt(z) + t_1)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y))));
	else
		tmp = (0.5 * sqrt((1.0 / x))) + ((sqrt((1.0 + t)) - (sqrt(t) + (sqrt(z) - t_1))) + (0.5 * sqrt((1.0 / 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[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 41000000.0], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.1e7

   1. Initial program 98.4%

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

      1. associate-+l+ 98.4%

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

      2. associate-+l+ 98.4%

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

      3. +-commutative 98.4%

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

      4. +-commutative 98.4%

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

      5. associate-+l- 77.3%

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

      6. +-commutative 77.3%

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

      7. +-commutative 77.3%

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

   3. Simplified 77.3%

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

      1. add-log-exp 77.3%

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

      2. associate--r- 98.4%

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

      3. +-commutative 98.4%

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

   6. Applied egg-rr 98.4%

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

      1. flip-- 98.4%

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

      2. add-sqr-sqrt 74.2%

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

      3. add-sqr-sqrt 98.7%

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

   8. Applied egg-rr 98.7%

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

      1. associate--l+ 98.9%

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

      2. +-inverses 98.9%

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

      3. metadata-eval 98.9%

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

      4. +-commutative 98.9%

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

   10. Simplified 98.9%

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

   11. Taylor expanded in t around inf 60.8%

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

       1. flip-- 77.2%

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

       2. add-sqr-sqrt 56.9%

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

       3. add-sqr-sqrt 77.4%

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

   13. Applied egg-rr 60.8%

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

       1. associate--l+ 60.8%

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

       2. +-inverses 60.8%

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

       3. metadata-eval 60.8%

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

       4. +-commutative 60.8%

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

   15. Simplified 60.8%

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

   if 4.1e7 < x

   1. Initial program 85.8%

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

      1. associate-+l+ 85.8%

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

      2. associate-+l+ 85.8%

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

      3. +-commutative 85.8%

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

      4. +-commutative 85.8%

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

      5. associate-+l- 66.4%

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

      6. +-commutative 66.4%

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

      7. +-commutative 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in x around inf 70.6%

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

   6. Taylor expanded in y around inf 35.5%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 41000000:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \left(\left(\sqrt{1 + t} - \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y} - \sqrt{y}\\ t_2 := \frac{1}{\sqrt{z} + \sqrt{z + 1}}\\ \mathbf{if}\;x \leq 2000000000:\\ \;\;\;\;t\_2 + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \left(t\_1 + t\_2\right)\\ \end{array} \end{array} \]

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 (+ 1.0 y)) (sqrt y)))
        (t_2 (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0))))))
   (if (<= x 2000000000.0)
     (+ t_2 (+ (- (sqrt (+ x 1.0)) (sqrt x)) t_1))
     (+ (* 0.5 (sqrt (/ 1.0 x))) (+ t_1 t_2)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y)) - sqrt(y);
	double t_2 = 1.0 / (sqrt(z) + sqrt((z + 1.0)));
	double tmp;
	if (x <= 2000000000.0) {
		tmp = t_2 + ((sqrt((x + 1.0)) - sqrt(x)) + t_1);
	} else {
		tmp = (0.5 * sqrt((1.0 / x))) + (t_1 + 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 + y)) - sqrt(y)
    t_2 = 1.0d0 / (sqrt(z) + sqrt((z + 1.0d0)))
    if (x <= 2000000000.0d0) then
        tmp = t_2 + ((sqrt((x + 1.0d0)) - sqrt(x)) + t_1)
    else
        tmp = (0.5d0 * sqrt((1.0d0 / x))) + (t_1 + t_2)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y)) - Math.sqrt(y);
	double t_2 = 1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0)));
	double tmp;
	if (x <= 2000000000.0) {
		tmp = t_2 + ((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + t_1);
	} else {
		tmp = (0.5 * Math.sqrt((1.0 / x))) + (t_1 + t_2);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y)) - math.sqrt(y)
	t_2 = 1.0 / (math.sqrt(z) + math.sqrt((z + 1.0)))
	tmp = 0
	if x <= 2000000000.0:
		tmp = t_2 + ((math.sqrt((x + 1.0)) - math.sqrt(x)) + t_1)
	else:
		tmp = (0.5 * math.sqrt((1.0 / x))) + (t_1 + t_2)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + y)) - sqrt(y))
	t_2 = Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0))))
	tmp = 0.0
	if (x <= 2000000000.0)
		tmp = Float64(t_2 + Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + t_1));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + Float64(t_1 + t_2));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y)) - sqrt(y);
	t_2 = 1.0 / (sqrt(z) + sqrt((z + 1.0)));
	tmp = 0.0;
	if (x <= 2000000000.0)
		tmp = t_2 + ((sqrt((x + 1.0)) - sqrt(x)) + t_1);
	else
		tmp = (0.5 * sqrt((1.0 / x))) + (t_1 + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2000000000.0], N[(t$95$2 + N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2e9

   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. Step-by-step derivation

      1. associate-+l+ 97.6%

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

      2. associate-+l+ 97.6%

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

      3. +-commutative 97.6%

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

      4. +-commutative 97.6%

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

      5. associate-+l- 76.6%

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

      6. +-commutative 76.6%

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

      7. +-commutative 76.6%

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

   3. Simplified 76.6%

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

      1. add-log-exp 76.6%

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

      2. associate--r- 97.6%

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

      3. +-commutative 97.6%

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

   6. Applied egg-rr 97.6%

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

      1. flip-- 97.6%

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

      2. add-sqr-sqrt 73.5%

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

      3. add-sqr-sqrt 97.9%

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

   8. Applied egg-rr 97.9%

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

      1. associate--l+ 98.1%

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

      2. +-inverses 98.1%

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

      3. metadata-eval 98.1%

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

      4. +-commutative 98.1%

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

   10. Simplified 98.1%

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

   11. Taylor expanded in t around inf 61.1%

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

       1. associate-+l- 61.1%

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

       2. +-commutative 61.1%

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

       3. +-commutative 61.1%

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

   13. Applied egg-rr 61.1%

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

       1. associate-+l- 61.1%

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

       2. +-commutative 61.1%

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

       3. associate-+l+ 61.1%

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

   15. Simplified 61.1%

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

   if 2e9 < x

   1. Initial program 86.3%

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

      1. associate-+l+ 86.3%

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

      2. associate-+l+ 86.3%

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

      3. +-commutative 86.3%

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

      4. +-commutative 86.3%

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

      5. associate-+l- 66.7%

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

      6. +-commutative 66.7%

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

      7. +-commutative 66.7%

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

   3. Simplified 66.7%

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

      1. add-log-exp 66.7%

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

      2. associate--r- 86.3%

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

      3. +-commutative 86.3%

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

   6. Applied egg-rr 86.3%

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

      1. flip-- 86.4%

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

      2. add-sqr-sqrt 75.7%

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

      3. add-sqr-sqrt 86.6%

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

   8. Applied egg-rr 86.6%

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

      1. associate--l+ 86.7%

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

      2. +-inverses 86.7%

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

      3. metadata-eval 86.7%

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

      4. +-commutative 86.7%

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

   10. Simplified 86.7%

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

   11. Taylor expanded in t around inf 45.2%

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

   12. Taylor expanded in x around inf 47.9%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2000000000:\\ \;\;\;\;\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\\ \mathbf{if}\;x \leq 2000000000:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1
         (+
          (- (sqrt (+ 1.0 y)) (sqrt y))
          (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0)))))))
   (if (<= x 2000000000.0)
     (+ (- (sqrt (+ x 1.0)) (sqrt x)) t_1)
     (+ (* 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 + y)) - sqrt(y)) + (1.0 / (sqrt(z) + sqrt((z + 1.0))));
	double tmp;
	if (x <= 2000000000.0) {
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + t_1;
	} 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 + y)) - sqrt(y)) + (1.0d0 / (sqrt(z) + sqrt((z + 1.0d0))))
    if (x <= 2000000000.0d0) then
        tmp = (sqrt((x + 1.0d0)) - sqrt(x)) + t_1
    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 + y)) - Math.sqrt(y)) + (1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0))));
	double tmp;
	if (x <= 2000000000.0) {
		tmp = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + t_1;
	} 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 + y)) - math.sqrt(y)) + (1.0 / (math.sqrt(z) + math.sqrt((z + 1.0))))
	tmp = 0
	if x <= 2000000000.0:
		tmp = (math.sqrt((x + 1.0)) - math.sqrt(x)) + t_1
	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(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0)))))
	tmp = 0.0
	if (x <= 2000000000.0)
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + t_1);
	else
		tmp = Float64(Float64(0.5 * 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 + y)) - sqrt(y)) + (1.0 / (sqrt(z) + sqrt((z + 1.0))));
	tmp = 0.0;
	if (x <= 2000000000.0)
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + t_1;
	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[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2000000000.0], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2e9

   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. Step-by-step derivation

      1. associate-+l+ 97.6%

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

      2. associate-+l+ 97.6%

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

      3. +-commutative 97.6%

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

      4. +-commutative 97.6%

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

      5. associate-+l- 76.6%

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

      6. +-commutative 76.6%

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

      7. +-commutative 76.6%

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

   3. Simplified 76.6%

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

      1. add-log-exp 76.6%

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

      2. associate--r- 97.6%

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

      3. +-commutative 97.6%

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

   6. Applied egg-rr 97.6%

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

      1. flip-- 97.6%

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

      2. add-sqr-sqrt 73.5%

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

      3. add-sqr-sqrt 97.9%

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

   8. Applied egg-rr 97.9%

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

      1. associate--l+ 98.1%

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

      2. +-inverses 98.1%

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

      3. metadata-eval 98.1%

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

      4. +-commutative 98.1%

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

   10. Simplified 98.1%

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

   11. Taylor expanded in t around inf 61.1%

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

   if 2e9 < x

   1. Initial program 86.3%

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

      1. associate-+l+ 86.3%

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

      2. associate-+l+ 86.3%

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

      3. +-commutative 86.3%

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

      4. +-commutative 86.3%

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

      5. associate-+l- 66.7%

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

      6. +-commutative 66.7%

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

      7. +-commutative 66.7%

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

   3. Simplified 66.7%

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

      1. add-log-exp 66.7%

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

      2. associate--r- 86.3%

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

      3. +-commutative 86.3%

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

   6. Applied egg-rr 86.3%

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

      1. flip-- 86.4%

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

      2. add-sqr-sqrt 75.7%

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

      3. add-sqr-sqrt 86.6%

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

   8. Applied egg-rr 86.6%

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

      1. associate--l+ 86.7%

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

      2. +-inverses 86.7%

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

      3. metadata-eval 86.7%

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

      4. +-commutative 86.7%

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

   10. Simplified 86.7%

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

   11. Taylor expanded in t around inf 45.2%

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

   12. Taylor expanded in x around inf 47.9%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2000000000:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))) + ((sqrt((1.0d0 + y)) - sqrt(y)) + (1.0d0 / (sqrt(z) + sqrt((z + 1.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

   1. associate-+l+ 92.2%

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

   2. associate-+l+ 92.2%

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

   3. +-commutative 92.2%

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

   4. +-commutative 92.2%

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

   5. associate-+l- 71.9%

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

   6. +-commutative 71.9%

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

   7. +-commutative 71.9%

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

3. Simplified 71.9%

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

   1. add-log-exp 71.9%

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

   2. associate--r- 92.2%

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

   3. +-commutative 92.2%

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

6. Applied egg-rr 92.2%

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

   1. flip-- 92.2%

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

   2. add-sqr-sqrt 74.6%

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

   3. add-sqr-sqrt 92.5%

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

8. Applied egg-rr 92.5%

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

   1. associate--l+ 92.6%

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

   2. +-inverses 92.6%

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

   3. metadata-eval 92.6%

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

   4. +-commutative 92.6%

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

10. Simplified 92.6%

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

11. Taylor expanded in t around inf 53.5%

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

    1. flip-- 72.4%

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

    2. add-sqr-sqrt 58.2%

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

    3. +-commutative 58.2%

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

    4. add-sqr-sqrt 72.7%

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

    5. +-commutative 72.7%

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

13. Applied egg-rr 54.0%

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

    1. associate--l+ 74.5%

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

    2. +-inverses 74.5%

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

    3. metadata-eval 74.5%

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

15. Simplified 55.1%

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

16. Final simplification 55.1%

    \[\leadsto \frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right) \]
17. Add Preprocessing

Alternative 12: 91.0% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 98.6%

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

      1. associate-+l+ 98.6%

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

      2. associate-+l+ 98.6%

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

      3. +-commutative 98.6%

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

      4. +-commutative 98.6%

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

      5. associate-+l- 78.0%

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

      6. +-commutative 78.0%

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

      7. +-commutative 78.0%

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

   3. Simplified 78.0%

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

      1. add-log-exp 78.0%

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

      2. associate--r- 98.5%

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

      3. +-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. flip-- 98.5%

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

      2. add-sqr-sqrt 73.6%

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

      3. add-sqr-sqrt 98.9%

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

   8. Applied egg-rr 98.9%

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

      1. associate--l+ 99.0%

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

      2. +-inverses 99.0%

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

      3. metadata-eval 99.0%

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

      4. +-commutative 99.0%

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

   10. Simplified 99.0%

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

   11. Taylor expanded in t around inf 60.9%

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

   12. Taylor expanded in x around 0 60.6%

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

       1. associate--l+ 60.6%

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

   14. Simplified 60.6%

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

   if 1 < x

   1. Initial program 86.1%

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

      1. associate-+l+ 86.1%

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

      2. associate-+l+ 86.1%

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

      3. +-commutative 86.1%

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

      4. +-commutative 86.1%

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

      5. associate-+l- 66.0%

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

      6. +-commutative 66.0%

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

      7. +-commutative 66.0%

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

   3. Simplified 66.0%

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

      1. add-log-exp 66.0%

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

      2. associate--r- 86.1%

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

      3. +-commutative 86.1%

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

   6. Applied egg-rr 86.1%

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

      1. flip-- 86.2%

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

      2. add-sqr-sqrt 75.5%

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

      3. add-sqr-sqrt 86.4%

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

   8. Applied egg-rr 86.4%

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

      1. associate--l+ 86.5%

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

      2. +-inverses 86.5%

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

      3. metadata-eval 86.5%

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

      4. +-commutative 86.5%

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

   10. Simplified 86.5%

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

   11. Taylor expanded in t around inf 46.4%

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

   12. Taylor expanded in x around inf 48.8%

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

4. Final simplification 54.6%

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

Alternative 13: 90.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\\ \mathbf{if}\;x \leq 0.38:\\ \;\;\;\;t\_1 + \left(1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1
         (+
          (- (sqrt (+ 1.0 y)) (sqrt y))
          (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0)))))))
   (if (<= x 0.38) (+ t_1 (- 1.0 (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 + y)) - sqrt(y)) + (1.0 / (sqrt(z) + sqrt((z + 1.0))));
	double tmp;
	if (x <= 0.38) {
		tmp = t_1 + (1.0 - 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 + y)) - sqrt(y)) + (1.0d0 / (sqrt(z) + sqrt((z + 1.0d0))))
    if (x <= 0.38d0) then
        tmp = t_1 + (1.0d0 - 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 + y)) - Math.sqrt(y)) + (1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0))));
	double tmp;
	if (x <= 0.38) {
		tmp = t_1 + (1.0 - 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 + y)) - math.sqrt(y)) + (1.0 / (math.sqrt(z) + math.sqrt((z + 1.0))))
	tmp = 0
	if x <= 0.38:
		tmp = t_1 + (1.0 - 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(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0)))))
	tmp = 0.0
	if (x <= 0.38)
		tmp = Float64(t_1 + Float64(1.0 - sqrt(x)));
	else
		tmp = Float64(Float64(0.5 * 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 + y)) - sqrt(y)) + (1.0 / (sqrt(z) + sqrt((z + 1.0))));
	tmp = 0.0;
	if (x <= 0.38)
		tmp = t_1 + (1.0 - 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[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.38], N[(t$95$1 + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.38

   1. Initial program 98.6%

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

      1. associate-+l+ 98.6%

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

      2. associate-+l+ 98.6%

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

      3. +-commutative 98.6%

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

      4. +-commutative 98.6%

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

      5. associate-+l- 78.0%

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

      6. +-commutative 78.0%

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

      7. +-commutative 78.0%

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

   3. Simplified 78.0%

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

      1. add-log-exp 78.0%

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

      2. associate--r- 98.5%

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

      3. +-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. flip-- 98.5%

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

      2. add-sqr-sqrt 73.6%

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

      3. add-sqr-sqrt 98.9%

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

   8. Applied egg-rr 98.9%

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

      1. associate--l+ 99.0%

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

      2. +-inverses 99.0%

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

      3. metadata-eval 99.0%

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

      4. +-commutative 99.0%

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

   10. Simplified 99.0%

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

   11. Taylor expanded in t around inf 60.9%

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

   12. Taylor expanded in x around 0 59.8%

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

   if 0.38 < x

   1. Initial program 86.1%

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

      1. associate-+l+ 86.1%

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

      2. associate-+l+ 86.1%

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

      3. +-commutative 86.1%

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

      4. +-commutative 86.1%

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

      5. associate-+l- 66.0%

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

      6. +-commutative 66.0%

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

      7. +-commutative 66.0%

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

   3. Simplified 66.0%

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

      1. add-log-exp 66.0%

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

      2. associate--r- 86.1%

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

      3. +-commutative 86.1%

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

   6. Applied egg-rr 86.1%

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

      1. flip-- 86.2%

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

      2. add-sqr-sqrt 75.5%

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

      3. add-sqr-sqrt 86.4%

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

   8. Applied egg-rr 86.4%

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

      1. associate--l+ 86.5%

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

      2. +-inverses 86.5%

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

      3. metadata-eval 86.5%

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

      4. +-commutative 86.5%

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

   10. Simplified 86.5%

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

   11. Taylor expanded in t around inf 46.4%

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

   12. Taylor expanded in x around inf 48.8%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.38:\\ \;\;\;\;\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right) + \left(1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y} - \sqrt{y}\\ \mathbf{if}\;x \leq 0.45:\\ \;\;\;\;\left(t\_1 + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right) + \left(1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \left(t\_1 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 y)) (sqrt y))))
   (if (<= x 0.45)
     (+ (+ t_1 (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0))))) (- 1.0 (sqrt x)))
     (+ (* 0.5 (sqrt (/ 1.0 x))) (+ t_1 (- (sqrt (+ 1.0 t)) (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y)) - sqrt(y);
	double tmp;
	if (x <= 0.45) {
		tmp = (t_1 + (1.0 / (sqrt(z) + sqrt((z + 1.0))))) + (1.0 - sqrt(x));
	} else {
		tmp = (0.5 * sqrt((1.0 / x))) + (t_1 + (sqrt((1.0 + t)) - sqrt(t)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y)) - sqrt(y)
    if (x <= 0.45d0) then
        tmp = (t_1 + (1.0d0 / (sqrt(z) + sqrt((z + 1.0d0))))) + (1.0d0 - sqrt(x))
    else
        tmp = (0.5d0 * sqrt((1.0d0 / x))) + (t_1 + (sqrt((1.0d0 + t)) - sqrt(t)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y)) - Math.sqrt(y);
	double tmp;
	if (x <= 0.45) {
		tmp = (t_1 + (1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0))))) + (1.0 - Math.sqrt(x));
	} else {
		tmp = (0.5 * Math.sqrt((1.0 / x))) + (t_1 + (Math.sqrt((1.0 + t)) - Math.sqrt(t)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y)) - math.sqrt(y)
	tmp = 0
	if x <= 0.45:
		tmp = (t_1 + (1.0 / (math.sqrt(z) + math.sqrt((z + 1.0))))) + (1.0 - math.sqrt(x))
	else:
		tmp = (0.5 * math.sqrt((1.0 / x))) + (t_1 + (math.sqrt((1.0 + t)) - math.sqrt(t)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + y)) - sqrt(y))
	tmp = 0.0
	if (x <= 0.45)
		tmp = Float64(Float64(t_1 + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0))))) + Float64(1.0 - sqrt(x)));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + Float64(t_1 + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y)) - sqrt(y);
	tmp = 0.0;
	if (x <= 0.45)
		tmp = (t_1 + (1.0 / (sqrt(z) + sqrt((z + 1.0))))) + (1.0 - sqrt(x));
	else
		tmp = (0.5 * sqrt((1.0 / x))) + (t_1 + (sqrt((1.0 + t)) - sqrt(t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.45], N[(N[(t$95$1 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.450000000000000011

   1. Initial program 98.6%

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

      1. associate-+l+ 98.6%

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

      2. associate-+l+ 98.6%

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

      3. +-commutative 98.6%

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

      4. +-commutative 98.6%

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

      5. associate-+l- 78.0%

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

      6. +-commutative 78.0%

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

      7. +-commutative 78.0%

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

   3. Simplified 78.0%

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

      1. add-log-exp 78.0%

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

      2. associate--r- 98.5%

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

      3. +-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. flip-- 98.5%

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

      2. add-sqr-sqrt 73.6%

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

      3. add-sqr-sqrt 98.9%

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

   8. Applied egg-rr 98.9%

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

      1. associate--l+ 99.0%

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

      2. +-inverses 99.0%

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

      3. metadata-eval 99.0%

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

      4. +-commutative 99.0%

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

   10. Simplified 99.0%

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

   11. Taylor expanded in t around inf 60.9%

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

   12. Taylor expanded in x around 0 59.8%

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

   if 0.450000000000000011 < x

   1. Initial program 86.1%

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

      1. associate-+l+ 86.1%

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

      2. associate-+l+ 86.1%

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

      3. +-commutative 86.1%

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

      4. +-commutative 86.1%

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

      5. associate-+l- 66.0%

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

      6. +-commutative 66.0%

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

      7. +-commutative 66.0%

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

   3. Simplified 66.0%

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

   5. Taylor expanded in x around inf 69.3%

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

   6. Taylor expanded in z around inf 44.6%

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

      1. +-commutative 44.6%

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

   8. Simplified 44.6%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.45:\\ \;\;\;\;\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right) + \left(1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 5.8 \cdot 10^{+17}:\\ \;\;\;\;1 + \left(t\_1 + \left(\sqrt{z + 1} - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \frac{1}{t\_1 + \sqrt{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))))
   (if (<= z 5.8e+17)
     (+ 1.0 (+ t_1 (- (sqrt (+ z 1.0)) (+ (sqrt z) (+ (sqrt x) (sqrt y))))))
     (+ (sqrt (+ x 1.0)) (/ 1.0 (+ t_1 (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double tmp;
	if (z <= 5.8e+17) {
		tmp = 1.0 + (t_1 + (sqrt((z + 1.0)) - (sqrt(z) + (sqrt(x) + sqrt(y)))));
	} else {
		tmp = sqrt((x + 1.0)) + (1.0 / (t_1 + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    if (z <= 5.8d+17) then
        tmp = 1.0d0 + (t_1 + (sqrt((z + 1.0d0)) - (sqrt(z) + (sqrt(x) + sqrt(y)))))
    else
        tmp = sqrt((x + 1.0d0)) + (1.0d0 / (t_1 + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 5.8e+17) {
		tmp = 1.0 + (t_1 + (Math.sqrt((z + 1.0)) - (Math.sqrt(z) + (Math.sqrt(x) + Math.sqrt(y)))));
	} else {
		tmp = Math.sqrt((x + 1.0)) + (1.0 / (t_1 + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 5.8e+17:
		tmp = 1.0 + (t_1 + (math.sqrt((z + 1.0)) - (math.sqrt(z) + (math.sqrt(x) + math.sqrt(y)))))
	else:
		tmp = math.sqrt((x + 1.0)) + (1.0 / (t_1 + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 5.8e+17)
		tmp = Float64(1.0 + Float64(t_1 + Float64(sqrt(Float64(z + 1.0)) - Float64(sqrt(z) + Float64(sqrt(x) + sqrt(y))))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(1.0 / Float64(t_1 + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 5.8e+17)
		tmp = 1.0 + (t_1 + (sqrt((z + 1.0)) - (sqrt(z) + (sqrt(x) + sqrt(y)))));
	else
		tmp = sqrt((x + 1.0)) + (1.0 / (t_1 + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 5.8e+17], N[(1.0 + N[(t$95$1 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 5.8e17

   1. Initial program 96.3%

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

      1. +-commutative 96.3%

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

      2. associate-+r+ 96.3%

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

      3. associate-+r- 77.1%

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

      4. associate-+l- 68.8%

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

      5. associate-+r- 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around inf 18.7%

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

      1. associate--l+ 22.8%

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

      2. associate--l+ 22.8%

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

      3. associate-+r+ 22.8%

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

   7. Simplified 22.8%

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

   8. Taylor expanded in x around 0 16.4%

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

      1. associate--l+ 23.1%

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

      2. associate-+r+ 23.1%

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

      3. +-commutative 23.1%

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

      4. associate--l+ 23.1%

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

   10. Simplified 23.1%

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

   if 5.8e17 < z

   1. Initial program 86.9%

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

      1. +-commutative 86.9%

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

      2. associate-+r+ 86.9%

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

      3. associate-+r- 62.4%

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

      4. associate-+l- 53.7%

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

      5. associate-+r- 53.7%

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

   3. Simplified 35.0%

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

   5. Taylor expanded in t around inf 3.2%

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

      1. associate--l+ 26.1%

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

      2. associate--l+ 24.9%

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

      3. associate-+r+ 24.9%

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

   7. Simplified 24.9%

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

   8. Taylor expanded in y around inf 33.9%

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

      1. neg-mul-1 33.9%

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

   10. Simplified 33.9%

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

       1. flip-+ 33.9%

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

       2. add-sqr-sqrt 24.6%

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

       3. pow2 24.6%

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

   12. Applied egg-rr 24.6%

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

       1. associate--l+ 24.6%

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

       2. unpow2 24.6%

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

       3. sqr-neg 24.6%

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

       4. rem-square-sqrt 33.9%

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

       5. +-inverses 33.9%

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

       6. metadata-eval 33.9%

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

       7. sub-neg 33.9%

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

       8. remove-double-neg 33.9%

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

   14. Simplified 33.9%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+17}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(\sqrt{z + 1} - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 86.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{z}}\\ \mathbf{if}\;y \leq 3.1 \cdot 10^{-21}:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 95000000:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} + 0.5 \cdot \left(x + t\_1\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(0.5 \cdot \left(\sqrt{\frac{1}{y}} + t\_1\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

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 z))))
   (if (<= y 3.1e-21)
     (+ 2.0 (- (sqrt (+ z 1.0)) (+ (sqrt y) (+ (sqrt x) (sqrt z)))))
     (if (<= y 95000000.0)
       (+ 1.0 (- (+ (sqrt (+ 1.0 y)) (* 0.5 (+ x t_1))) (+ (sqrt x) (sqrt y))))
       (+ (sqrt (+ x 1.0)) (- (* 0.5 (+ (sqrt (/ 1.0 y)) t_1)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 / z));
	double tmp;
	if (y <= 3.1e-21) {
		tmp = 2.0 + (sqrt((z + 1.0)) - (sqrt(y) + (sqrt(x) + sqrt(z))));
	} else if (y <= 95000000.0) {
		tmp = 1.0 + ((sqrt((1.0 + y)) + (0.5 * (x + t_1))) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = sqrt((x + 1.0)) + ((0.5 * (sqrt((1.0 / y)) + t_1)) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 / z))
    if (y <= 3.1d-21) then
        tmp = 2.0d0 + (sqrt((z + 1.0d0)) - (sqrt(y) + (sqrt(x) + sqrt(z))))
    else if (y <= 95000000.0d0) then
        tmp = 1.0d0 + ((sqrt((1.0d0 + y)) + (0.5d0 * (x + t_1))) - (sqrt(x) + sqrt(y)))
    else
        tmp = sqrt((x + 1.0d0)) + ((0.5d0 * (sqrt((1.0d0 / y)) + t_1)) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 / z));
	double tmp;
	if (y <= 3.1e-21) {
		tmp = 2.0 + (Math.sqrt((z + 1.0)) - (Math.sqrt(y) + (Math.sqrt(x) + Math.sqrt(z))));
	} else if (y <= 95000000.0) {
		tmp = 1.0 + ((Math.sqrt((1.0 + y)) + (0.5 * (x + t_1))) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = Math.sqrt((x + 1.0)) + ((0.5 * (Math.sqrt((1.0 / y)) + t_1)) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 / z))
	tmp = 0
	if y <= 3.1e-21:
		tmp = 2.0 + (math.sqrt((z + 1.0)) - (math.sqrt(y) + (math.sqrt(x) + math.sqrt(z))))
	elif y <= 95000000.0:
		tmp = 1.0 + ((math.sqrt((1.0 + y)) + (0.5 * (x + t_1))) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = math.sqrt((x + 1.0)) + ((0.5 * (math.sqrt((1.0 / y)) + t_1)) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 / z))
	tmp = 0.0
	if (y <= 3.1e-21)
		tmp = Float64(2.0 + Float64(sqrt(Float64(z + 1.0)) - Float64(sqrt(y) + Float64(sqrt(x) + sqrt(z)))));
	elseif (y <= 95000000.0)
		tmp = Float64(1.0 + Float64(Float64(sqrt(Float64(1.0 + y)) + Float64(0.5 * Float64(x + t_1))) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(Float64(0.5 * Float64(sqrt(Float64(1.0 / y)) + t_1)) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 / z));
	tmp = 0.0;
	if (y <= 3.1e-21)
		tmp = 2.0 + (sqrt((z + 1.0)) - (sqrt(y) + (sqrt(x) + sqrt(z))));
	elseif (y <= 95000000.0)
		tmp = 1.0 + ((sqrt((1.0 + y)) + (0.5 * (x + t_1))) - (sqrt(x) + sqrt(y)));
	else
		tmp = sqrt((x + 1.0)) + ((0.5 * (sqrt((1.0 / y)) + t_1)) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 3.1e-21], N[(2.0 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 95000000.0], N[(1.0 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(x + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.0999999999999998e-21

   1. Initial program 98.6%

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

      1. +-commutative 98.6%

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

      2. associate-+r+ 98.6%

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

      3. associate-+r- 98.6%

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

      4. associate-+l- 98.6%

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

      5. associate-+r- 98.6%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in t around inf 20.8%

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

      1. associate--l+ 24.8%

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

      2. associate--l+ 32.0%

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

      3. associate-+r+ 32.0%

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

   7. Simplified 32.0%

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

   8. Taylor expanded in x around 0 18.2%

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

   9. Taylor expanded in y around 0 18.2%

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

       1. associate--l+ 33.7%

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

       2. associate-+r+ 33.7%

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

       3. +-commutative 33.7%

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

       4. associate-+l+ 33.7%

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

       5. +-commutative 33.7%

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

   11. Simplified 33.7%

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

   if 3.0999999999999998e-21 < y < 9.5e7

   1. Initial program 94.2%

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

      1. +-commutative 94.2%

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

      2. associate-+r+ 94.2%

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

      3. associate-+r- 94.2%

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

      4. associate-+l- 94.2%

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

      5. associate-+r- 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in t around inf 14.1%

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

      1. associate--l+ 14.6%

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

      2. associate--l+ 14.6%

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

      3. associate-+r+ 14.6%

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

   7. Simplified 14.6%

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

   8. Taylor expanded in z around inf 4.5%

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

   9. Taylor expanded in x around 0 5.1%

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

       1. associate--l+ 5.1%

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

       2. distribute-lft-out 5.1%

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

   11. Simplified 5.1%

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

   if 9.5e7 < y

   1. Initial program 86.4%

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

      1. +-commutative 86.4%

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

      2. associate-+r+ 86.4%

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

      3. associate-+r- 44.1%

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

      4. associate-+l- 27.5%

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

      5. associate-+r- 8.4%

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

   3. Simplified 8.1%

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

   5. Taylor expanded in t around inf 4.0%

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

      1. associate--l+ 24.8%

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

      2. associate--l+ 17.7%

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

      3. associate-+r+ 17.7%

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

   7. Simplified 17.7%

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

   8. Taylor expanded in z around inf 4.2%

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

   9. Taylor expanded in y around inf 22.0%

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

       1. associate--l+ 22.0%

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

       2. distribute-lft-out 22.0%

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

   11. Simplified 22.0%

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

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-21}:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 95000000:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} + 0.5 \cdot \left(x + \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 86.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{z}}\\ \mathbf{if}\;y \leq 3.25 \cdot 10^{-21}:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 36000000:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + y} + 0.5 \cdot t\_1\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(0.5 \cdot \left(\sqrt{\frac{1}{y}} + t\_1\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

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 z))))
   (if (<= y 3.25e-21)
     (+ 2.0 (- (sqrt (+ z 1.0)) (+ (sqrt y) (+ (sqrt x) (sqrt z)))))
     (if (<= y 36000000.0)
       (- (+ 1.0 (+ (sqrt (+ 1.0 y)) (* 0.5 t_1))) (+ (sqrt x) (sqrt y)))
       (+ (sqrt (+ x 1.0)) (- (* 0.5 (+ (sqrt (/ 1.0 y)) t_1)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 / z));
	double tmp;
	if (y <= 3.25e-21) {
		tmp = 2.0 + (sqrt((z + 1.0)) - (sqrt(y) + (sqrt(x) + sqrt(z))));
	} else if (y <= 36000000.0) {
		tmp = (1.0 + (sqrt((1.0 + y)) + (0.5 * t_1))) - (sqrt(x) + sqrt(y));
	} else {
		tmp = sqrt((x + 1.0)) + ((0.5 * (sqrt((1.0 / y)) + t_1)) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 / z))
    if (y <= 3.25d-21) then
        tmp = 2.0d0 + (sqrt((z + 1.0d0)) - (sqrt(y) + (sqrt(x) + sqrt(z))))
    else if (y <= 36000000.0d0) then
        tmp = (1.0d0 + (sqrt((1.0d0 + y)) + (0.5d0 * t_1))) - (sqrt(x) + sqrt(y))
    else
        tmp = sqrt((x + 1.0d0)) + ((0.5d0 * (sqrt((1.0d0 / y)) + t_1)) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 / z));
	double tmp;
	if (y <= 3.25e-21) {
		tmp = 2.0 + (Math.sqrt((z + 1.0)) - (Math.sqrt(y) + (Math.sqrt(x) + Math.sqrt(z))));
	} else if (y <= 36000000.0) {
		tmp = (1.0 + (Math.sqrt((1.0 + y)) + (0.5 * t_1))) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = Math.sqrt((x + 1.0)) + ((0.5 * (Math.sqrt((1.0 / y)) + t_1)) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 / z))
	tmp = 0
	if y <= 3.25e-21:
		tmp = 2.0 + (math.sqrt((z + 1.0)) - (math.sqrt(y) + (math.sqrt(x) + math.sqrt(z))))
	elif y <= 36000000.0:
		tmp = (1.0 + (math.sqrt((1.0 + y)) + (0.5 * t_1))) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = math.sqrt((x + 1.0)) + ((0.5 * (math.sqrt((1.0 / y)) + t_1)) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 / z))
	tmp = 0.0
	if (y <= 3.25e-21)
		tmp = Float64(2.0 + Float64(sqrt(Float64(z + 1.0)) - Float64(sqrt(y) + Float64(sqrt(x) + sqrt(z)))));
	elseif (y <= 36000000.0)
		tmp = Float64(Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) + Float64(0.5 * t_1))) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(Float64(0.5 * Float64(sqrt(Float64(1.0 / y)) + t_1)) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 / z));
	tmp = 0.0;
	if (y <= 3.25e-21)
		tmp = 2.0 + (sqrt((z + 1.0)) - (sqrt(y) + (sqrt(x) + sqrt(z))));
	elseif (y <= 36000000.0)
		tmp = (1.0 + (sqrt((1.0 + y)) + (0.5 * t_1))) - (sqrt(x) + sqrt(y));
	else
		tmp = sqrt((x + 1.0)) + ((0.5 * (sqrt((1.0 / y)) + t_1)) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 3.25e-21], N[(2.0 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 36000000.0], N[(N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.24999999999999993e-21

   1. Initial program 98.6%

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

      1. +-commutative 98.6%

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

      2. associate-+r+ 98.6%

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

      3. associate-+r- 98.6%

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

      4. associate-+l- 98.6%

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

      5. associate-+r- 98.6%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in t around inf 20.8%

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

      1. associate--l+ 24.8%

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

      2. associate--l+ 32.0%

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

      3. associate-+r+ 32.0%

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

   7. Simplified 32.0%

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

   8. Taylor expanded in x around 0 18.2%

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

   9. Taylor expanded in y around 0 18.2%

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

       1. associate--l+ 33.7%

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

       2. associate-+r+ 33.7%

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

       3. +-commutative 33.7%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right) \]

       4. associate-+l+ 33.7%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)}\right) \]

       5. +-commutative 33.7%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \color{blue}{\left(\sqrt{z} + \sqrt{x}\right)}\right)\right) \]

   11. Simplified 33.7%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \left(\sqrt{z} + \sqrt{x}\right)\right)\right)} \]

   if 3.24999999999999993e-21 < y < 3.6e7

   1. Initial program 94.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 94.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ 94.2%

         \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      3. associate-+r- 94.2%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 94.2%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. associate-+r- 94.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. Simplified 94.1%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 14.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 14.6%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate--l+ 14.6%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. associate-+r+ 14.6%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

   7. Simplified 14.6%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in z around inf 4.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   9. Taylor expanded in x around 0 3.2%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   if 3.6e7 < y

   1. Initial program 86.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 86.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ 86.4%

         \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      3. associate-+r- 44.1%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 27.5%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. associate-+r- 8.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. Simplified 8.1%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 4.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 24.8%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate--l+ 17.7%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. associate-+r+ 17.7%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

   7. Simplified 17.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in z around inf 4.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   9. Taylor expanded in y around inf 22.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(0.5 \cdot \sqrt{\frac{1}{y}} + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. associate--l+ 22.0%

          \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(0.5 \cdot \sqrt{\frac{1}{y}} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)} \]

       2. distribute-lft-out 22.0%

          \[\leadsto \sqrt{1 + x} + \left(\color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} - \sqrt{x}\right) \]

   11. Simplified 22.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} + \left(0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.25 \cdot 10^{-21}:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 36000000:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + y} + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 86.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 3.25 \cdot 10^{-21}:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 105000000:\\ \;\;\;\;\left(\sqrt{1 + y} + t\_1\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

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 (+ x 1.0))))
   (if (<= y 3.25e-21)
     (+ 2.0 (- (sqrt (+ z 1.0)) (+ (sqrt y) (+ (sqrt x) (sqrt z)))))
     (if (<= y 105000000.0)
       (- (+ (sqrt (+ 1.0 y)) t_1) (+ (sqrt x) (sqrt y)))
       (+ t_1 (- (* 0.5 (+ (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 t_1 = sqrt((x + 1.0));
	double tmp;
	if (y <= 3.25e-21) {
		tmp = 2.0 + (sqrt((z + 1.0)) - (sqrt(y) + (sqrt(x) + sqrt(z))));
	} else if (y <= 105000000.0) {
		tmp = (sqrt((1.0 + y)) + t_1) - (sqrt(x) + sqrt(y));
	} else {
		tmp = t_1 + ((0.5 * (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) :: t_1
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    if (y <= 3.25d-21) then
        tmp = 2.0d0 + (sqrt((z + 1.0d0)) - (sqrt(y) + (sqrt(x) + sqrt(z))))
    else if (y <= 105000000.0d0) then
        tmp = (sqrt((1.0d0 + y)) + t_1) - (sqrt(x) + sqrt(y))
    else
        tmp = t_1 + ((0.5d0 * (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 t_1 = Math.sqrt((x + 1.0));
	double tmp;
	if (y <= 3.25e-21) {
		tmp = 2.0 + (Math.sqrt((z + 1.0)) - (Math.sqrt(y) + (Math.sqrt(x) + Math.sqrt(z))));
	} else if (y <= 105000000.0) {
		tmp = (Math.sqrt((1.0 + y)) + t_1) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = t_1 + ((0.5 * (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):
	t_1 = math.sqrt((x + 1.0))
	tmp = 0
	if y <= 3.25e-21:
		tmp = 2.0 + (math.sqrt((z + 1.0)) - (math.sqrt(y) + (math.sqrt(x) + math.sqrt(z))))
	elif y <= 105000000.0:
		tmp = (math.sqrt((1.0 + y)) + t_1) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = t_1 + ((0.5 * (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)
	t_1 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (y <= 3.25e-21)
		tmp = Float64(2.0 + Float64(sqrt(Float64(z + 1.0)) - Float64(sqrt(y) + Float64(sqrt(x) + sqrt(z)))));
	elseif (y <= 105000000.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 + y)) + t_1) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(t_1 + Float64(Float64(0.5 * 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)
	t_1 = sqrt((x + 1.0));
	tmp = 0.0;
	if (y <= 3.25e-21)
		tmp = 2.0 + (sqrt((z + 1.0)) - (sqrt(y) + (sqrt(x) + sqrt(z))));
	elseif (y <= 105000000.0)
		tmp = (sqrt((1.0 + y)) + t_1) - (sqrt(x) + sqrt(y));
	else
		tmp = t_1 + ((0.5 * (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_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 3.25e-21], N[(2.0 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 105000000.0], N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.24999999999999993e-21

   1. Initial program 98.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 98.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ 98.6%

         \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      3. associate-+r- 98.6%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 98.6%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. associate-+r- 98.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. Simplified 80.0%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 20.8%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 24.8%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate--l+ 32.0%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. associate-+r+ 32.0%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

   7. Simplified 32.0%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in x around 0 18.2%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in y around 0 18.2%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 33.7%

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

       2. associate-+r+ 33.7%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

       3. +-commutative 33.7%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right) \]

       4. associate-+l+ 33.7%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)}\right) \]

       5. +-commutative 33.7%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \color{blue}{\left(\sqrt{z} + \sqrt{x}\right)}\right)\right) \]

   11. Simplified 33.7%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \left(\sqrt{z} + \sqrt{x}\right)\right)\right)} \]

   if 3.24999999999999993e-21 < y < 1.05e8

   1. Initial program 94.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 94.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ 94.2%

         \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      3. associate-+r- 94.2%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 94.2%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. associate-+r- 94.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. Simplified 94.1%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 14.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 14.6%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate--l+ 14.6%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. associate-+r+ 14.6%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

   7. Simplified 14.6%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in z around inf 7.9%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   if 1.05e8 < y

   1. Initial program 86.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 86.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ 86.4%

         \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      3. associate-+r- 44.1%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 27.5%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. associate-+r- 8.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. Simplified 8.1%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 4.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 24.8%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate--l+ 17.7%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. associate-+r+ 17.7%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

   7. Simplified 17.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in z around inf 4.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   9. Taylor expanded in y around inf 22.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(0.5 \cdot \sqrt{\frac{1}{y}} + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. associate--l+ 22.0%

          \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(0.5 \cdot \sqrt{\frac{1}{y}} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)} \]

       2. distribute-lft-out 22.0%

          \[\leadsto \sqrt{1 + x} + \left(\color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} - \sqrt{x}\right) \]

   11. Simplified 22.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} + \left(0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.25 \cdot 10^{-21}:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 105000000:\\ \;\;\;\;\left(\sqrt{1 + y} + \sqrt{x + 1}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 85.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 58000000000000:\\ \;\;\;\;2 + \left(\left(\sqrt{z + 1} - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right) + y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 58000000000000.0)
   (+
    2.0
    (+ (- (sqrt (+ z 1.0)) (+ (sqrt y) (+ (sqrt x) (sqrt z)))) (* y 0.5)))
   (+ (sqrt (+ x 1.0)) (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 58000000000000.0) {
		tmp = 2.0 + ((sqrt((z + 1.0)) - (sqrt(y) + (sqrt(x) + sqrt(z)))) + (y * 0.5));
	} else {
		tmp = sqrt((x + 1.0)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 58000000000000.0d0) then
        tmp = 2.0d0 + ((sqrt((z + 1.0d0)) - (sqrt(y) + (sqrt(x) + sqrt(z)))) + (y * 0.5d0))
    else
        tmp = sqrt((x + 1.0d0)) + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 58000000000000.0) {
		tmp = 2.0 + ((Math.sqrt((z + 1.0)) - (Math.sqrt(y) + (Math.sqrt(x) + Math.sqrt(z)))) + (y * 0.5));
	} else {
		tmp = Math.sqrt((x + 1.0)) + (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 58000000000000.0:
		tmp = 2.0 + ((math.sqrt((z + 1.0)) - (math.sqrt(y) + (math.sqrt(x) + math.sqrt(z)))) + (y * 0.5))
	else:
		tmp = math.sqrt((x + 1.0)) + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 58000000000000.0)
		tmp = Float64(2.0 + Float64(Float64(sqrt(Float64(z + 1.0)) - Float64(sqrt(y) + Float64(sqrt(x) + sqrt(z)))) + Float64(y * 0.5)));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 58000000000000.0)
		tmp = 2.0 + ((sqrt((z + 1.0)) - (sqrt(y) + (sqrt(x) + sqrt(z)))) + (y * 0.5));
	else
		tmp = sqrt((x + 1.0)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 58000000000000.0], N[(2.0 + N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.8e13

   1. Initial program 96.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 96.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ 96.5%

         \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      3. associate-+r- 77.6%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 69.3%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. associate-+r- 51.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. Simplified 51.9%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 18.8%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 22.7%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate--l+ 22.7%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. associate-+r+ 22.7%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

   7. Simplified 22.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in x around 0 16.4%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in y around 0 16.1%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} + 0.5 \cdot y\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 16.0%

          \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + z} + 0.5 \cdot y\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

       2. +-commutative 16.0%

          \[\leadsto 2 + \left(\color{blue}{\left(0.5 \cdot y + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

       3. +-commutative 16.0%

          \[\leadsto 2 + \left(\left(0.5 \cdot y + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right) \]

       4. associate--l+ 16.0%

          \[\leadsto 2 + \color{blue}{\left(0.5 \cdot y + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)} \]

       5. +-commutative 16.0%

          \[\leadsto 2 + \left(0.5 \cdot y + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right)\right) \]

       6. associate-+r+ 16.0%

          \[\leadsto 2 + \left(0.5 \cdot y + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

       7. +-commutative 16.0%

          \[\leadsto 2 + \left(0.5 \cdot y + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right)\right) \]

       8. associate-+l+ 16.0%

          \[\leadsto 2 + \left(0.5 \cdot y + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)}\right)\right) \]

       9. +-commutative 16.0%

          \[\leadsto 2 + \left(0.5 \cdot y + \left(\sqrt{1 + z} - \left(\sqrt{y} + \color{blue}{\left(\sqrt{z} + \sqrt{x}\right)}\right)\right)\right) \]

   11. Simplified 16.0%

       \[\leadsto \color{blue}{2 + \left(0.5 \cdot y + \left(\sqrt{1 + z} - \left(\sqrt{y} + \left(\sqrt{z} + \sqrt{x}\right)\right)\right)\right)} \]

   if 5.8e13 < z

   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. Step-by-step derivation

      1. +-commutative 86.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ 86.7%

         \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      3. associate-+r- 61.9%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 53.3%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. associate-+r- 53.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. Simplified 34.7%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 3.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 26.1%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate--l+ 25.0%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. associate-+r+ 25.0%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

   7. Simplified 25.0%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in y around inf 33.9%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{-1 \cdot \sqrt{y}}\right) \]
   9. Step-by-step derivation

      1. neg-mul-1 33.9%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{\left(-\sqrt{y}\right)}\right) \]

   10. Simplified 33.9%

       \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{\left(-\sqrt{y}\right)}\right) \]
   11. Step-by-step derivation

       1. flip-+ 33.9%

          \[\leadsto \sqrt{1 + x} + \color{blue}{\frac{\sqrt{1 + y} \cdot \sqrt{1 + y} - \left(-\sqrt{y}\right) \cdot \left(-\sqrt{y}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}} \]

       2. add-sqr-sqrt 24.4%

          \[\leadsto \sqrt{1 + x} + \frac{\color{blue}{\left(1 + y\right)} - \left(-\sqrt{y}\right) \cdot \left(-\sqrt{y}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

       3. pow2 24.4%

          \[\leadsto \sqrt{1 + x} + \frac{\left(1 + y\right) - \color{blue}{{\left(-\sqrt{y}\right)}^{2}}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

   12. Applied egg-rr 24.4%

       \[\leadsto \sqrt{1 + x} + \color{blue}{\frac{\left(1 + y\right) - {\left(-\sqrt{y}\right)}^{2}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}} \]
   13. Step-by-step derivation

       1. associate--l+ 24.4%

          \[\leadsto \sqrt{1 + x} + \frac{\color{blue}{1 + \left(y - {\left(-\sqrt{y}\right)}^{2}\right)}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

       2. unpow2 24.4%

          \[\leadsto \sqrt{1 + x} + \frac{1 + \left(y - \color{blue}{\left(-\sqrt{y}\right) \cdot \left(-\sqrt{y}\right)}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

       3. sqr-neg 24.4%

          \[\leadsto \sqrt{1 + x} + \frac{1 + \left(y - \color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

       4. rem-square-sqrt 33.9%

          \[\leadsto \sqrt{1 + x} + \frac{1 + \left(y - \color{blue}{y}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

       5. +-inverses 33.9%

          \[\leadsto \sqrt{1 + x} + \frac{1 + \color{blue}{0}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

       6. metadata-eval 33.9%

          \[\leadsto \sqrt{1 + x} + \frac{\color{blue}{1}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

       7. sub-neg 33.9%

          \[\leadsto \sqrt{1 + x} + \frac{1}{\color{blue}{\sqrt{1 + y} + \left(-\left(-\sqrt{y}\right)\right)}} \]

       8. remove-double-neg 33.9%

          \[\leadsto \sqrt{1 + x} + \frac{1}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}} \]

   14. Simplified 33.9%

       \[\leadsto \sqrt{1 + x} + \color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 58000000000000:\\ \;\;\;\;2 + \left(\left(\sqrt{z + 1} - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right) + y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 85.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+17}:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 5.8e+17)
   (+ 2.0 (- (sqrt (+ z 1.0)) (+ (sqrt y) (+ (sqrt x) (sqrt z)))))
   (+ (sqrt (+ x 1.0)) (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5.8e+17) {
		tmp = 2.0 + (sqrt((z + 1.0)) - (sqrt(y) + (sqrt(x) + sqrt(z))));
	} else {
		tmp = sqrt((x + 1.0)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 5.8d+17) then
        tmp = 2.0d0 + (sqrt((z + 1.0d0)) - (sqrt(y) + (sqrt(x) + sqrt(z))))
    else
        tmp = sqrt((x + 1.0d0)) + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5.8e+17) {
		tmp = 2.0 + (Math.sqrt((z + 1.0)) - (Math.sqrt(y) + (Math.sqrt(x) + Math.sqrt(z))));
	} else {
		tmp = Math.sqrt((x + 1.0)) + (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 5.8e+17:
		tmp = 2.0 + (math.sqrt((z + 1.0)) - (math.sqrt(y) + (math.sqrt(x) + math.sqrt(z))))
	else:
		tmp = math.sqrt((x + 1.0)) + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 5.8e+17)
		tmp = Float64(2.0 + Float64(sqrt(Float64(z + 1.0)) - Float64(sqrt(y) + Float64(sqrt(x) + sqrt(z)))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 5.8e+17)
		tmp = 2.0 + (sqrt((z + 1.0)) - (sqrt(y) + (sqrt(x) + sqrt(z))));
	else
		tmp = sqrt((x + 1.0)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 5.8e+17], N[(2.0 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.8e17

   1. Initial program 96.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 96.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ 96.3%

         \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      3. associate-+r- 77.1%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 68.8%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. associate-+r- 51.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. Simplified 51.6%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 18.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 22.8%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate--l+ 22.8%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. associate-+r+ 22.8%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

   7. Simplified 22.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in x around 0 16.4%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in y around 0 14.2%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 14.2%

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

       2. associate-+r+ 14.2%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

       3. +-commutative 14.2%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right) \]

       4. associate-+l+ 14.2%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)}\right) \]

       5. +-commutative 14.2%

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \color{blue}{\left(\sqrt{z} + \sqrt{x}\right)}\right)\right) \]

   11. Simplified 14.2%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \left(\sqrt{z} + \sqrt{x}\right)\right)\right)} \]

   if 5.8e17 < z

   1. Initial program 86.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 86.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ 86.9%

         \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      3. associate-+r- 62.4%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 53.7%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. associate-+r- 53.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. Simplified 35.0%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 3.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 26.1%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate--l+ 24.9%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. associate-+r+ 24.9%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

   7. Simplified 24.9%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in y around inf 33.9%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{-1 \cdot \sqrt{y}}\right) \]
   9. Step-by-step derivation

      1. neg-mul-1 33.9%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{\left(-\sqrt{y}\right)}\right) \]

   10. Simplified 33.9%

       \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{\left(-\sqrt{y}\right)}\right) \]
   11. Step-by-step derivation

       1. flip-+ 33.9%

          \[\leadsto \sqrt{1 + x} + \color{blue}{\frac{\sqrt{1 + y} \cdot \sqrt{1 + y} - \left(-\sqrt{y}\right) \cdot \left(-\sqrt{y}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}} \]

       2. add-sqr-sqrt 24.6%

          \[\leadsto \sqrt{1 + x} + \frac{\color{blue}{\left(1 + y\right)} - \left(-\sqrt{y}\right) \cdot \left(-\sqrt{y}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

       3. pow2 24.6%

          \[\leadsto \sqrt{1 + x} + \frac{\left(1 + y\right) - \color{blue}{{\left(-\sqrt{y}\right)}^{2}}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

   12. Applied egg-rr 24.6%

       \[\leadsto \sqrt{1 + x} + \color{blue}{\frac{\left(1 + y\right) - {\left(-\sqrt{y}\right)}^{2}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}} \]
   13. Step-by-step derivation

       1. associate--l+ 24.6%

          \[\leadsto \sqrt{1 + x} + \frac{\color{blue}{1 + \left(y - {\left(-\sqrt{y}\right)}^{2}\right)}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

       2. unpow2 24.6%

          \[\leadsto \sqrt{1 + x} + \frac{1 + \left(y - \color{blue}{\left(-\sqrt{y}\right) \cdot \left(-\sqrt{y}\right)}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

       3. sqr-neg 24.6%

          \[\leadsto \sqrt{1 + x} + \frac{1 + \left(y - \color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

       4. rem-square-sqrt 33.9%

          \[\leadsto \sqrt{1 + x} + \frac{1 + \left(y - \color{blue}{y}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

       5. +-inverses 33.9%

          \[\leadsto \sqrt{1 + x} + \frac{1 + \color{blue}{0}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

       6. metadata-eval 33.9%

          \[\leadsto \sqrt{1 + x} + \frac{\color{blue}{1}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

       7. sub-neg 33.9%

          \[\leadsto \sqrt{1 + x} + \frac{1}{\color{blue}{\sqrt{1 + y} + \left(-\left(-\sqrt{y}\right)\right)}} \]

       8. remove-double-neg 33.9%

          \[\leadsto \sqrt{1 + x} + \frac{1}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}} \]

   14. Simplified 33.9%

       \[\leadsto \sqrt{1 + x} + \color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+17}:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 65.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{x + 1} + \frac{1}{\sqrt{1 + y} + \sqrt{y}} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (sqrt (+ x 1.0)) (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt((x + 1.0)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt((x + 1.0d0)) + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y)))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.sqrt((x + 1.0)) + (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y)));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt((x + 1.0)) + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y)))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(sqrt(Float64(x + 1.0)) + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt((x + 1.0)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. +-commutative 92.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate-+r+ 92.2%

      \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. associate-+r- 70.7%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 62.2%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. associate-+r- 52.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

3. Simplified 44.3%

   \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 11.9%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
6. Step-by-step derivation

   1. associate--l+ 24.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. associate--l+ 23.7%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   3. associate-+r+ 23.7%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

7. Simplified 23.7%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

8. Taylor expanded in y around inf 22.1%

   \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{-1 \cdot \sqrt{y}}\right) \]
9. Step-by-step derivation

   1. neg-mul-1 22.1%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{\left(-\sqrt{y}\right)}\right) \]

10. Simplified 22.1%

    \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{\left(-\sqrt{y}\right)}\right) \]
11. Step-by-step derivation

    1. flip-+ 22.1%

       \[\leadsto \sqrt{1 + x} + \color{blue}{\frac{\sqrt{1 + y} \cdot \sqrt{1 + y} - \left(-\sqrt{y}\right) \cdot \left(-\sqrt{y}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}} \]

    2. add-sqr-sqrt 16.8%

       \[\leadsto \sqrt{1 + x} + \frac{\color{blue}{\left(1 + y\right)} - \left(-\sqrt{y}\right) \cdot \left(-\sqrt{y}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

    3. pow2 16.8%

       \[\leadsto \sqrt{1 + x} + \frac{\left(1 + y\right) - \color{blue}{{\left(-\sqrt{y}\right)}^{2}}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

12. Applied egg-rr 16.8%

    \[\leadsto \sqrt{1 + x} + \color{blue}{\frac{\left(1 + y\right) - {\left(-\sqrt{y}\right)}^{2}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)}} \]
13. Step-by-step derivation

    1. associate--l+ 16.8%

       \[\leadsto \sqrt{1 + x} + \frac{\color{blue}{1 + \left(y - {\left(-\sqrt{y}\right)}^{2}\right)}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

    2. unpow2 16.8%

       \[\leadsto \sqrt{1 + x} + \frac{1 + \left(y - \color{blue}{\left(-\sqrt{y}\right) \cdot \left(-\sqrt{y}\right)}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

    3. sqr-neg 16.8%

       \[\leadsto \sqrt{1 + x} + \frac{1 + \left(y - \color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

    4. rem-square-sqrt 22.1%

       \[\leadsto \sqrt{1 + x} + \frac{1 + \left(y - \color{blue}{y}\right)}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

    5. +-inverses 22.1%

       \[\leadsto \sqrt{1 + x} + \frac{1 + \color{blue}{0}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

    6. metadata-eval 22.1%

       \[\leadsto \sqrt{1 + x} + \frac{\color{blue}{1}}{\sqrt{1 + y} - \left(-\sqrt{y}\right)} \]

    7. sub-neg 22.1%

       \[\leadsto \sqrt{1 + x} + \frac{1}{\color{blue}{\sqrt{1 + y} + \left(-\left(-\sqrt{y}\right)\right)}} \]

    8. remove-double-neg 22.1%

       \[\leadsto \sqrt{1 + x} + \frac{1}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}} \]

14. Simplified 22.1%

    \[\leadsto \sqrt{1 + x} + \color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}} \]

15. Final simplification 22.1%

    \[\leadsto \sqrt{x + 1} + \frac{1}{\sqrt{1 + y} + \sqrt{y}} \]
16. Add Preprocessing

Alternative 22: 64.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + \left(\sqrt{1 + y} - \sqrt{y}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + (sqrt((1.0 + y)) - sqrt(y));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. +-commutative 92.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate-+r+ 92.2%

      \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. associate-+r- 70.7%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 62.2%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. associate-+r- 52.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

3. Simplified 44.3%

   \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 11.9%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
6. Step-by-step derivation

   1. associate--l+ 24.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. associate--l+ 23.7%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   3. associate-+r+ 23.7%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

7. Simplified 23.7%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

8. Taylor expanded in y around inf 22.1%

   \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{-1 \cdot \sqrt{y}}\right) \]
9. Step-by-step derivation

   1. neg-mul-1 22.1%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{\left(-\sqrt{y}\right)}\right) \]

10. Simplified 22.1%

    \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{\left(-\sqrt{y}\right)}\right) \]

11. Taylor expanded in x around 0 23.3%

    \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
12. Step-by-step derivation

    1. associate--l+ 43.5%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

13. Simplified 43.5%

    \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]
14. Add Preprocessing

Alternative 23: 33.7% accurate, 8.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{x + 1} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (sqrt (+ x 1.0)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt((x + 1.0));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt((x + 1.0d0))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.sqrt((x + 1.0));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt((x + 1.0))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return sqrt(Float64(x + 1.0))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt((x + 1.0));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 92.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. +-commutative 92.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate-+r+ 92.2%

      \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. associate-+r- 70.7%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 62.2%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. associate-+r- 52.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

3. Simplified 44.3%

   \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 11.9%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
6. Step-by-step derivation

   1. associate--l+ 24.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. associate--l+ 23.7%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   3. associate-+r+ 23.7%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

7. Simplified 23.7%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

8. Taylor expanded in y around inf 22.1%

   \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{-1 \cdot \sqrt{y}}\right) \]
9. Step-by-step derivation

   1. neg-mul-1 22.1%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{\left(-\sqrt{y}\right)}\right) \]

10. Simplified 22.1%

    \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{\left(-\sqrt{y}\right)}\right) \]

11. Taylor expanded in y around inf 18.7%

    \[\leadsto \color{blue}{\sqrt{1 + x}} \]

12. Final simplification 18.7%

    \[\leadsto \sqrt{x + 1} \]
13. Add Preprocessing

Alternative 24: 6.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (sqrt x))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt(x);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt(x)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return sqrt(x)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[Sqrt[x], $MachinePrecision]

Derivation

1. Initial program 92.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. +-commutative 92.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate-+r+ 92.2%

      \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. associate-+r- 70.7%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 62.2%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. associate-+r- 52.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

3. Simplified 44.3%

   \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 11.9%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
6. Step-by-step derivation

   1. associate--l+ 24.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. associate--l+ 23.7%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   3. associate-+r+ 23.7%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

7. Simplified 23.7%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

8. Taylor expanded in y around inf 22.1%

   \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{-1 \cdot \sqrt{y}}\right) \]
9. Step-by-step derivation

   1. neg-mul-1 22.1%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{\left(-\sqrt{y}\right)}\right) \]

10. Simplified 22.1%

    \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{\left(-\sqrt{y}\right)}\right) \]

11. Taylor expanded in x around inf 7.0%

    \[\leadsto \color{blue}{\sqrt{x}} \]
12. Add Preprocessing

Developer target: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
    (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y)))) + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + (1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y)))) + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024086 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (+ (+ (+ (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))) (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y)))) (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z)))) (- (sqrt (+ t 1.0)) (sqrt t)))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))