Main:z from

Percentage Accurate: 91.6% → 99.2%

Time: 51.4s

Alternatives: 23

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: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{x + 1}\\ t_4 := t\_3 - \sqrt{x}\\ t_5 := \left(t\_2 - \sqrt{z}\right) + \left(t\_4 + \left(t\_1 - \sqrt{y}\right)\right)\\ t_6 := \sqrt{t + 1}\\ \mathbf{if}\;t\_5 \leq 0.004:\\ \;\;\;\;\frac{1}{t\_3 + \sqrt{x}} + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \left(\left(\sqrt{t} + \left(\sqrt{z} - t\_2\right)\right) - t\_6\right)\right)\\ \mathbf{elif}\;t\_5 \leq 2.6:\\ \;\;\;\;t\_4 + \left(\frac{1}{t\_1 + \sqrt{y}} + \frac{1}{t\_2 + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\left(t\_2 + \left(t\_4 - \sqrt{z}\right)\right) + \left(\left(t\_6 - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (- t_3 (sqrt x)))
        (t_5 (+ (- t_2 (sqrt z)) (+ t_4 (- t_1 (sqrt y)))))
        (t_6 (sqrt (+ t 1.0))))
   (if (<= t_5 0.004)
     (+
      (/ 1.0 (+ t_3 (sqrt x)))
      (- (* 0.5 (sqrt (/ 1.0 y))) (- (+ (sqrt t) (- (sqrt z) t_2)) t_6)))
     (if (<= t_5 2.6)
       (+ t_4 (+ (/ 1.0 (+ t_1 (sqrt y))) (/ 1.0 (+ t_2 (sqrt z)))))
       (+ t_1 (+ (+ t_2 (- t_4 (sqrt z))) (- (- t_6 (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((y + 1.0));
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((x + 1.0));
	double t_4 = t_3 - sqrt(x);
	double t_5 = (t_2 - sqrt(z)) + (t_4 + (t_1 - sqrt(y)));
	double t_6 = sqrt((t + 1.0));
	double tmp;
	if (t_5 <= 0.004) {
		tmp = (1.0 / (t_3 + sqrt(x))) + ((0.5 * sqrt((1.0 / y))) - ((sqrt(t) + (sqrt(z) - t_2)) - t_6));
	} else if (t_5 <= 2.6) {
		tmp = t_4 + ((1.0 / (t_1 + sqrt(y))) + (1.0 / (t_2 + sqrt(z))));
	} else {
		tmp = t_1 + ((t_2 + (t_4 - sqrt(z))) + ((t_6 - 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) :: t_6
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((z + 1.0d0))
    t_3 = sqrt((x + 1.0d0))
    t_4 = t_3 - sqrt(x)
    t_5 = (t_2 - sqrt(z)) + (t_4 + (t_1 - sqrt(y)))
    t_6 = sqrt((t + 1.0d0))
    if (t_5 <= 0.004d0) then
        tmp = (1.0d0 / (t_3 + sqrt(x))) + ((0.5d0 * sqrt((1.0d0 / y))) - ((sqrt(t) + (sqrt(z) - t_2)) - t_6))
    else if (t_5 <= 2.6d0) then
        tmp = t_4 + ((1.0d0 / (t_1 + sqrt(y))) + (1.0d0 / (t_2 + sqrt(z))))
    else
        tmp = t_1 + ((t_2 + (t_4 - sqrt(z))) + ((t_6 - 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((y + 1.0));
	double t_2 = Math.sqrt((z + 1.0));
	double t_3 = Math.sqrt((x + 1.0));
	double t_4 = t_3 - Math.sqrt(x);
	double t_5 = (t_2 - Math.sqrt(z)) + (t_4 + (t_1 - Math.sqrt(y)));
	double t_6 = Math.sqrt((t + 1.0));
	double tmp;
	if (t_5 <= 0.004) {
		tmp = (1.0 / (t_3 + Math.sqrt(x))) + ((0.5 * Math.sqrt((1.0 / y))) - ((Math.sqrt(t) + (Math.sqrt(z) - t_2)) - t_6));
	} else if (t_5 <= 2.6) {
		tmp = t_4 + ((1.0 / (t_1 + Math.sqrt(y))) + (1.0 / (t_2 + Math.sqrt(z))));
	} else {
		tmp = t_1 + ((t_2 + (t_4 - Math.sqrt(z))) + ((t_6 - 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((y + 1.0))
	t_2 = math.sqrt((z + 1.0))
	t_3 = math.sqrt((x + 1.0))
	t_4 = t_3 - math.sqrt(x)
	t_5 = (t_2 - math.sqrt(z)) + (t_4 + (t_1 - math.sqrt(y)))
	t_6 = math.sqrt((t + 1.0))
	tmp = 0
	if t_5 <= 0.004:
		tmp = (1.0 / (t_3 + math.sqrt(x))) + ((0.5 * math.sqrt((1.0 / y))) - ((math.sqrt(t) + (math.sqrt(z) - t_2)) - t_6))
	elif t_5 <= 2.6:
		tmp = t_4 + ((1.0 / (t_1 + math.sqrt(y))) + (1.0 / (t_2 + math.sqrt(z))))
	else:
		tmp = t_1 + ((t_2 + (t_4 - math.sqrt(z))) + ((t_6 - 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(y + 1.0))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(t_3 - sqrt(x))
	t_5 = Float64(Float64(t_2 - sqrt(z)) + Float64(t_4 + Float64(t_1 - sqrt(y))))
	t_6 = sqrt(Float64(t + 1.0))
	tmp = 0.0
	if (t_5 <= 0.004)
		tmp = Float64(Float64(1.0 / Float64(t_3 + sqrt(x))) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) - Float64(Float64(sqrt(t) + Float64(sqrt(z) - t_2)) - t_6)));
	elseif (t_5 <= 2.6)
		tmp = Float64(t_4 + Float64(Float64(1.0 / Float64(t_1 + sqrt(y))) + Float64(1.0 / Float64(t_2 + sqrt(z)))));
	else
		tmp = Float64(t_1 + Float64(Float64(t_2 + Float64(t_4 - sqrt(z))) + Float64(Float64(t_6 - 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((y + 1.0));
	t_2 = sqrt((z + 1.0));
	t_3 = sqrt((x + 1.0));
	t_4 = t_3 - sqrt(x);
	t_5 = (t_2 - sqrt(z)) + (t_4 + (t_1 - sqrt(y)));
	t_6 = sqrt((t + 1.0));
	tmp = 0.0;
	if (t_5 <= 0.004)
		tmp = (1.0 / (t_3 + sqrt(x))) + ((0.5 * sqrt((1.0 / y))) - ((sqrt(t) + (sqrt(z) - t_2)) - t_6));
	elseif (t_5 <= 2.6)
		tmp = t_4 + ((1.0 / (t_1 + sqrt(y))) + (1.0 / (t_2 + sqrt(z))));
	else
		tmp = t_1 + ((t_2 + (t_4 - sqrt(z))) + ((t_6 - 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[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, 0.004], N[(N[(1.0 / N[(t$95$3 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.6], N[(t$95$4 + N[(N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(t$95$2 + N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$6 - N[Sqrt[t], $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 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))) < 0.0040000000000000001

   1. Initial program 56.3%

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

      1. associate-+l+ 56.3%

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

      2. associate-+l+ 56.3%

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

      3. +-commutative 56.3%

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

      4. +-commutative 56.3%

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

      5. associate-+l- 52.1%

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

      6. +-commutative 52.1%

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

      7. +-commutative 52.1%

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

   3. Simplified 52.1%

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

      1. flip-- 52.1%

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

      2. add-sqr-sqrt 25.3%

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

      3. +-commutative 25.3%

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

      4. add-sqr-sqrt 52.1%

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

      5. +-commutative 52.1%

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

   6. Applied egg-rr 52.1%

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

      1. associate--l+ 57.6%

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

      2. +-inverses 57.6%

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

      3. metadata-eval 57.6%

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

   8. Simplified 57.6%

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

   9. Taylor expanded in y around inf 66.6%

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

   if 0.0040000000000000001 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))) < 2.60000000000000009

   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. associate-+l+ 96.3%

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

      2. associate-+l+ 96.3%

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

      3. +-commutative 96.3%

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

      4. +-commutative 96.3%

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

      5. associate-+l- 79.1%

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

      6. +-commutative 79.1%

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

      7. +-commutative 79.1%

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

   3. Simplified 79.1%

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

      1. add-log-exp 79.1%

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

      2. associate--r- 96.3%

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

      3. +-commutative 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. flip-- 96.5%

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

      2. add-sqr-sqrt 78.6%

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

      3. add-sqr-sqrt 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+ 97.2%

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

      2. +-inverses 97.2%

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

      3. metadata-eval 97.2%

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

      4. +-commutative 97.2%

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

   10. Simplified 97.2%

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

       1. flip-- 97.6%

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

       2. add-sqr-sqrt 73.8%

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

       3. add-sqr-sqrt 97.8%

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

   12. Applied egg-rr 97.8%

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

       1. associate--l+ 98.2%

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

       2. +-inverses 98.2%

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

       3. metadata-eval 98.2%

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

       4. +-commutative 98.2%

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

   14. Simplified 98.2%

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

   15. Taylor expanded in t around inf 51.3%

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

   if 2.60000000000000009 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z)))

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. associate-+r- 100.0%

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

      4. associate-+l- 100.0%

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

      5. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 0.004:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \left(\left(\sqrt{t} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \sqrt{t + 1}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 2.6:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y + 1} + \left(\left(\sqrt{z + 1} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{1}{\sqrt{y + 1} + \sqrt{y}}\\ t_2 := \sqrt{x + 1}\\ t_3 := t\_2 - \sqrt{x}\\ t_4 := \sqrt{t + 1}\\ t_5 := \sqrt{z + 1}\\ \mathbf{if}\;t\_3 \leq 0.004:\\ \;\;\;\;\frac{1}{t\_2 + \sqrt{x}} + \left(t\_1 + \left(t\_4 + \left(\left(t\_5 - \sqrt{z}\right) - \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_1 + \log \left(e^{\left(t\_4 - \sqrt{t}\right) + \frac{1}{t\_5 + \sqrt{z}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (- t_2 (sqrt x)))
        (t_4 (sqrt (+ t 1.0)))
        (t_5 (sqrt (+ z 1.0))))
   (if (<= t_3 0.004)
     (+ (/ 1.0 (+ t_2 (sqrt x))) (+ t_1 (+ t_4 (- (- t_5 (sqrt z)) (sqrt t)))))
     (+
      t_3
      (+ t_1 (log (exp (+ (- t_4 (sqrt t)) (/ 1.0 (+ t_5 (sqrt z)))))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (sqrt((y + 1.0)) + sqrt(y));
	double t_2 = sqrt((x + 1.0));
	double t_3 = t_2 - sqrt(x);
	double t_4 = sqrt((t + 1.0));
	double t_5 = sqrt((z + 1.0));
	double tmp;
	if (t_3 <= 0.004) {
		tmp = (1.0 / (t_2 + sqrt(x))) + (t_1 + (t_4 + ((t_5 - sqrt(z)) - sqrt(t))));
	} else {
		tmp = t_3 + (t_1 + log(exp(((t_4 - sqrt(t)) + (1.0 / (t_5 + sqrt(z)))))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = 1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y))
    t_2 = sqrt((x + 1.0d0))
    t_3 = t_2 - sqrt(x)
    t_4 = sqrt((t + 1.0d0))
    t_5 = sqrt((z + 1.0d0))
    if (t_3 <= 0.004d0) then
        tmp = (1.0d0 / (t_2 + sqrt(x))) + (t_1 + (t_4 + ((t_5 - sqrt(z)) - sqrt(t))))
    else
        tmp = t_3 + (t_1 + log(exp(((t_4 - sqrt(t)) + (1.0d0 / (t_5 + sqrt(z)))))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y));
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = t_2 - Math.sqrt(x);
	double t_4 = Math.sqrt((t + 1.0));
	double t_5 = Math.sqrt((z + 1.0));
	double tmp;
	if (t_3 <= 0.004) {
		tmp = (1.0 / (t_2 + Math.sqrt(x))) + (t_1 + (t_4 + ((t_5 - Math.sqrt(z)) - Math.sqrt(t))));
	} else {
		tmp = t_3 + (t_1 + Math.log(Math.exp(((t_4 - Math.sqrt(t)) + (1.0 / (t_5 + Math.sqrt(z)))))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y))
	t_2 = math.sqrt((x + 1.0))
	t_3 = t_2 - math.sqrt(x)
	t_4 = math.sqrt((t + 1.0))
	t_5 = math.sqrt((z + 1.0))
	tmp = 0
	if t_3 <= 0.004:
		tmp = (1.0 / (t_2 + math.sqrt(x))) + (t_1 + (t_4 + ((t_5 - math.sqrt(z)) - math.sqrt(t))))
	else:
		tmp = t_3 + (t_1 + math.log(math.exp(((t_4 - math.sqrt(t)) + (1.0 / (t_5 + math.sqrt(z)))))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(t_2 - sqrt(x))
	t_4 = sqrt(Float64(t + 1.0))
	t_5 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (t_3 <= 0.004)
		tmp = Float64(Float64(1.0 / Float64(t_2 + sqrt(x))) + Float64(t_1 + Float64(t_4 + Float64(Float64(t_5 - sqrt(z)) - sqrt(t)))));
	else
		tmp = Float64(t_3 + Float64(t_1 + log(exp(Float64(Float64(t_4 - sqrt(t)) + Float64(1.0 / Float64(t_5 + sqrt(z))))))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / (sqrt((y + 1.0)) + sqrt(y));
	t_2 = sqrt((x + 1.0));
	t_3 = t_2 - sqrt(x);
	t_4 = sqrt((t + 1.0));
	t_5 = sqrt((z + 1.0));
	tmp = 0.0;
	if (t_3 <= 0.004)
		tmp = (1.0 / (t_2 + sqrt(x))) + (t_1 + (t_4 + ((t_5 - sqrt(z)) - sqrt(t))));
	else
		tmp = t_3 + (t_1 + log(exp(((t_4 - sqrt(t)) + (1.0 / (t_5 + sqrt(z)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.004], N[(N[(1.0 / N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(t$95$4 + N[(N[(t$95$5 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$1 + N[Log[N[Exp[N[(N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$5 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 0.0040000000000000001

   1. Initial program 88.2%

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

      1. associate-+l+ 88.2%

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

      2. associate-+l+ 88.2%

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

      3. +-commutative 88.2%

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

      4. +-commutative 88.2%

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

      5. associate-+l- 68.8%

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

      6. +-commutative 68.8%

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

      7. +-commutative 68.8%

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

   3. Simplified 68.8%

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

      1. flip-- 68.9%

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

      2. add-sqr-sqrt 37.6%

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

      3. +-commutative 37.6%

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

      4. add-sqr-sqrt 69.3%

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

      5. +-commutative 69.3%

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

   6. Applied egg-rr 69.3%

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

      1. associate--l+ 71.7%

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

      2. +-inverses 71.7%

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

      3. metadata-eval 71.7%

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

   8. Simplified 71.7%

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

      1. flip-- 89.7%

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

      2. add-sqr-sqrt 69.8%

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

      3. add-sqr-sqrt 89.8%

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

   10. Applied egg-rr 71.7%

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

       1. associate--l+ 91.8%

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

       2. +-inverses 91.8%

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

       3. metadata-eval 91.8%

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

       4. +-commutative 91.8%

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

   12. Simplified 73.6%

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

   if 0.0040000000000000001 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 97.7%

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

      1. associate-+l+ 97.7%

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

      2. associate-+l+ 97.7%

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

      3. +-commutative 97.7%

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

      4. +-commutative 97.7%

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

      5. associate-+l- 81.2%

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

      6. +-commutative 81.2%

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

      7. +-commutative 81.2%

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

   3. Simplified 81.2%

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

      1. add-log-exp 81.2%

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

      2. associate--r- 97.7%

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

      3. +-commutative 97.7%

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

   6. Applied egg-rr 97.7%

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

      1. flip-- 97.8%

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

      2. add-sqr-sqrt 78.1%

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

      3. add-sqr-sqrt 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+ 98.4%

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

      2. +-inverses 98.4%

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

      3. metadata-eval 98.4%

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

      4. +-commutative 98.4%

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

   10. Simplified 98.4%

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

       1. flip-- 99.0%

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

       2. add-sqr-sqrt 74.8%

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

       3. add-sqr-sqrt 99.2%

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

   12. Applied egg-rr 99.2%

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

       1. associate--l+ 99.9%

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

       2. +-inverses 99.9%

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

       3. metadata-eval 99.9%

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

       4. +-commutative 99.9%

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

   14. Simplified 99.9%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.004:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \left(\sqrt{t + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) - \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \log \left(e^{\left(\sqrt{t + 1} - \sqrt{t}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{t + 1}\\ t_3 := \sqrt{z + 1}\\ t_4 := t\_3 - \sqrt{z}\\ t_5 := \sqrt{y + 1}\\ \mathbf{if}\;t\_4 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{t\_1 + \sqrt{x}} + \left(\frac{1}{t\_5 + \sqrt{y}} + \left(t\_2 + \left(t\_4 - \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_5 + \left(\left(t\_3 + \left(\left(t\_1 - \sqrt{x}\right) - \sqrt{z}\right)\right) + \left(\left(t\_2 - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0)))
        (t_2 (sqrt (+ t 1.0)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (- t_3 (sqrt z)))
        (t_5 (sqrt (+ y 1.0))))
   (if (<= t_4 2e-8)
     (+
      (/ 1.0 (+ t_1 (sqrt x)))
      (+ (/ 1.0 (+ t_5 (sqrt y))) (+ t_2 (- t_4 (sqrt t)))))
     (+
      t_5
      (+
       (+ t_3 (- (- t_1 (sqrt x)) (sqrt z)))
       (- (- t_2 (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((x + 1.0));
	double t_2 = sqrt((t + 1.0));
	double t_3 = sqrt((z + 1.0));
	double t_4 = t_3 - sqrt(z);
	double t_5 = sqrt((y + 1.0));
	double tmp;
	if (t_4 <= 2e-8) {
		tmp = (1.0 / (t_1 + sqrt(x))) + ((1.0 / (t_5 + sqrt(y))) + (t_2 + (t_4 - sqrt(t))));
	} else {
		tmp = t_5 + ((t_3 + ((t_1 - sqrt(x)) - sqrt(z))) + ((t_2 - 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((x + 1.0d0))
    t_2 = sqrt((t + 1.0d0))
    t_3 = sqrt((z + 1.0d0))
    t_4 = t_3 - sqrt(z)
    t_5 = sqrt((y + 1.0d0))
    if (t_4 <= 2d-8) then
        tmp = (1.0d0 / (t_1 + sqrt(x))) + ((1.0d0 / (t_5 + sqrt(y))) + (t_2 + (t_4 - sqrt(t))))
    else
        tmp = t_5 + ((t_3 + ((t_1 - sqrt(x)) - sqrt(z))) + ((t_2 - 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((x + 1.0));
	double t_2 = Math.sqrt((t + 1.0));
	double t_3 = Math.sqrt((z + 1.0));
	double t_4 = t_3 - Math.sqrt(z);
	double t_5 = Math.sqrt((y + 1.0));
	double tmp;
	if (t_4 <= 2e-8) {
		tmp = (1.0 / (t_1 + Math.sqrt(x))) + ((1.0 / (t_5 + Math.sqrt(y))) + (t_2 + (t_4 - Math.sqrt(t))));
	} else {
		tmp = t_5 + ((t_3 + ((t_1 - Math.sqrt(x)) - Math.sqrt(z))) + ((t_2 - 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((x + 1.0))
	t_2 = math.sqrt((t + 1.0))
	t_3 = math.sqrt((z + 1.0))
	t_4 = t_3 - math.sqrt(z)
	t_5 = math.sqrt((y + 1.0))
	tmp = 0
	if t_4 <= 2e-8:
		tmp = (1.0 / (t_1 + math.sqrt(x))) + ((1.0 / (t_5 + math.sqrt(y))) + (t_2 + (t_4 - math.sqrt(t))))
	else:
		tmp = t_5 + ((t_3 + ((t_1 - math.sqrt(x)) - math.sqrt(z))) + ((t_2 - 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(x + 1.0))
	t_2 = sqrt(Float64(t + 1.0))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(t_3 - sqrt(z))
	t_5 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (t_4 <= 2e-8)
		tmp = Float64(Float64(1.0 / Float64(t_1 + sqrt(x))) + Float64(Float64(1.0 / Float64(t_5 + sqrt(y))) + Float64(t_2 + Float64(t_4 - sqrt(t)))));
	else
		tmp = Float64(t_5 + Float64(Float64(t_3 + Float64(Float64(t_1 - sqrt(x)) - sqrt(z))) + Float64(Float64(t_2 - 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((x + 1.0));
	t_2 = sqrt((t + 1.0));
	t_3 = sqrt((z + 1.0));
	t_4 = t_3 - sqrt(z);
	t_5 = sqrt((y + 1.0));
	tmp = 0.0;
	if (t_4 <= 2e-8)
		tmp = (1.0 / (t_1 + sqrt(x))) + ((1.0 / (t_5 + sqrt(y))) + (t_2 + (t_4 - sqrt(t))));
	else
		tmp = t_5 + ((t_3 + ((t_1 - sqrt(x)) - sqrt(z))) + ((t_2 - 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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 2e-8], N[(N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$5 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$5 + N[(N[(t$95$3 + N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z)) < 2e-8

   1. Initial program 88.9%

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

      1. associate-+l+ 88.9%

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

      2. associate-+l+ 88.9%

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

      3. +-commutative 88.9%

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

      4. +-commutative 88.9%

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

      5. associate-+l- 88.9%

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

      6. +-commutative 88.9%

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

      7. +-commutative 88.9%

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

   3. Simplified 88.9%

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

      1. flip-- 88.9%

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

      2. add-sqr-sqrt 67.0%

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

      3. +-commutative 67.0%

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

      4. add-sqr-sqrt 88.9%

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

      5. +-commutative 88.9%

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

   6. Applied egg-rr 88.9%

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

      1. associate--l+ 90.4%

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

      2. +-inverses 90.4%

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

      3. metadata-eval 90.4%

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

   8. Simplified 90.4%

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

      1. flip-- 90.7%

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

      2. add-sqr-sqrt 67.1%

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

      3. add-sqr-sqrt 91.0%

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

   10. Applied egg-rr 90.8%

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

       1. associate--l+ 93.4%

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

       2. +-inverses 93.4%

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

       3. metadata-eval 93.4%

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

       4. +-commutative 93.4%

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

   12. Simplified 92.9%

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

   if 2e-8 < (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))

   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- 79.0%

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

      4. associate-+l- 68.7%

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

      5. associate-+r- 54.1%

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

   3. Simplified 54.1%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \left(\sqrt{t + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) - \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y + 1} + \left(\left(\sqrt{z + 1} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 3.2 \cdot 10^{+27}:\\ \;\;\;\;\left(1 - \sqrt{x}\right) + \left(\log \left(e^{\left(t\_1 - \sqrt{t}\right) + \frac{1}{t\_2 + \sqrt{z}}}\right) + \left(t\_3 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\frac{1}{t\_3 + \sqrt{y}} + \left(t\_1 + \left(\left(t\_2 - \sqrt{z}\right) - \sqrt{t}\right)\right)\right)\\ \end{array} \end{array} \]

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 (+ t 1.0))) (t_2 (sqrt (+ z 1.0))) (t_3 (sqrt (+ y 1.0))))
   (if (<= z 3.2e+27)
     (+
      (- 1.0 (sqrt x))
      (+
       (log (exp (+ (- t_1 (sqrt t)) (/ 1.0 (+ t_2 (sqrt z))))))
       (- t_3 (sqrt y))))
     (+
      (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
      (+ (/ 1.0 (+ t_3 (sqrt y))) (+ t_1 (- (- t_2 (sqrt z)) (sqrt t))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0));
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((y + 1.0));
	double tmp;
	if (z <= 3.2e+27) {
		tmp = (1.0 - sqrt(x)) + (log(exp(((t_1 - sqrt(t)) + (1.0 / (t_2 + sqrt(z)))))) + (t_3 - sqrt(y)));
	} else {
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((1.0 / (t_3 + sqrt(y))) + (t_1 + ((t_2 - sqrt(z)) - sqrt(t))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((t + 1.0d0))
    t_2 = sqrt((z + 1.0d0))
    t_3 = sqrt((y + 1.0d0))
    if (z <= 3.2d+27) then
        tmp = (1.0d0 - sqrt(x)) + (log(exp(((t_1 - sqrt(t)) + (1.0d0 / (t_2 + sqrt(z)))))) + (t_3 - sqrt(y)))
    else
        tmp = (1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + ((1.0d0 / (t_3 + sqrt(y))) + (t_1 + ((t_2 - sqrt(z)) - 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((t + 1.0));
	double t_2 = Math.sqrt((z + 1.0));
	double t_3 = Math.sqrt((y + 1.0));
	double tmp;
	if (z <= 3.2e+27) {
		tmp = (1.0 - Math.sqrt(x)) + (Math.log(Math.exp(((t_1 - Math.sqrt(t)) + (1.0 / (t_2 + Math.sqrt(z)))))) + (t_3 - Math.sqrt(y)));
	} else {
		tmp = (1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + ((1.0 / (t_3 + Math.sqrt(y))) + (t_1 + ((t_2 - Math.sqrt(z)) - 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((t + 1.0))
	t_2 = math.sqrt((z + 1.0))
	t_3 = math.sqrt((y + 1.0))
	tmp = 0
	if z <= 3.2e+27:
		tmp = (1.0 - math.sqrt(x)) + (math.log(math.exp(((t_1 - math.sqrt(t)) + (1.0 / (t_2 + math.sqrt(z)))))) + (t_3 - math.sqrt(y)))
	else:
		tmp = (1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + ((1.0 / (t_3 + math.sqrt(y))) + (t_1 + ((t_2 - math.sqrt(z)) - math.sqrt(t))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(t + 1.0))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 3.2e+27)
		tmp = Float64(Float64(1.0 - sqrt(x)) + Float64(log(exp(Float64(Float64(t_1 - sqrt(t)) + Float64(1.0 / Float64(t_2 + sqrt(z)))))) + Float64(t_3 - sqrt(y))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(Float64(1.0 / Float64(t_3 + sqrt(y))) + Float64(t_1 + Float64(Float64(t_2 - sqrt(z)) - 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((t + 1.0));
	t_2 = sqrt((z + 1.0));
	t_3 = sqrt((y + 1.0));
	tmp = 0.0;
	if (z <= 3.2e+27)
		tmp = (1.0 - sqrt(x)) + (log(exp(((t_1 - sqrt(t)) + (1.0 / (t_2 + sqrt(z)))))) + (t_3 - sqrt(y)));
	else
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((1.0 / (t_3 + sqrt(y))) + (t_1 + ((t_2 - sqrt(z)) - sqrt(t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 3.2e+27], N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[Exp[N[(N[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$3 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < 3.20000000000000015e27

   1. Initial program 94.1%

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

      1. associate-+l+ 94.1%

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

      2. associate-+l+ 94.1%

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

      3. +-commutative 94.1%

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

      4. +-commutative 94.1%

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

      5. associate-+l- 63.2%

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

      6. +-commutative 63.2%

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

      7. +-commutative 63.2%

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

   3. Simplified 63.2%

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

      1. add-log-exp 63.2%

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

      2. associate--r- 94.1%

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

      3. +-commutative 94.1%

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

   6. Applied egg-rr 94.1%

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

      1. flip-- 94.5%

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

      2. add-sqr-sqrt 94.2%

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

      3. add-sqr-sqrt 95.1%

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

   8. Applied egg-rr 95.1%

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

      1. associate--l+ 96.0%

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

      2. +-inverses 96.0%

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

      3. metadata-eval 96.0%

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

      4. +-commutative 96.0%

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

   10. Simplified 96.0%

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

   11. Taylor expanded in x around 0 46.2%

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

   if 3.20000000000000015e27 < z

   1. Initial program 91.3%

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

      1. associate-+l+ 91.3%

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

      2. associate-+l+ 91.3%

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

      3. +-commutative 91.3%

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

      4. +-commutative 91.3%

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

      5. associate-+l- 91.3%

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

      6. +-commutative 91.3%

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

      7. +-commutative 91.3%

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

   3. Simplified 91.3%

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

      1. flip-- 91.2%

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

      2. add-sqr-sqrt 68.7%

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

      3. +-commutative 68.7%

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

      4. add-sqr-sqrt 91.2%

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

      5. +-commutative 91.2%

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

   6. Applied egg-rr 91.2%

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

      1. associate--l+ 92.7%

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

      2. +-inverses 92.7%

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

      3. metadata-eval 92.7%

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

   8. Simplified 92.7%

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

      1. flip-- 91.5%

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

      2. add-sqr-sqrt 68.1%

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

      3. add-sqr-sqrt 91.8%

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

   10. Applied egg-rr 93.2%

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

       1. associate--l+ 94.2%

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

       2. +-inverses 94.2%

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

       3. metadata-eval 94.2%

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

       4. +-commutative 94.2%

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

   12. Simplified 95.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{+27}:\\ \;\;\;\;\left(1 - \sqrt{x}\right) + \left(\log \left(e^{\left(\sqrt{t + 1} - \sqrt{t}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \left(\sqrt{t + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) - \sqrt{t}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{t + 1}\\ \mathbf{if}\;t\_2 - \sqrt{t} \leq 0:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{t\_1 + \sqrt{z}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\left(t\_2 + t\_1\right) - \left(\sqrt{t} + \left(\left(\sqrt{z} + \sqrt{x}\right) + \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0))) (t_2 (sqrt (+ t 1.0))))
   (if (<= (- t_2 (sqrt t)) 0.0)
     (+
      (- (sqrt (+ x 1.0)) (sqrt x))
      (+ (/ 1.0 (+ t_1 (sqrt z))) (- (sqrt (+ y 1.0)) (sqrt y))))
     (+ 2.0 (- (+ t_2 t_1) (+ (sqrt t) (+ (+ (sqrt z) (sqrt x)) (sqrt y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt((t + 1.0));
	double tmp;
	if ((t_2 - sqrt(t)) <= 0.0) {
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((1.0 / (t_1 + sqrt(z))) + (sqrt((y + 1.0)) - sqrt(y)));
	} else {
		tmp = 2.0 + ((t_2 + t_1) - (sqrt(t) + ((sqrt(z) + sqrt(x)) + sqrt(y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = sqrt((t + 1.0d0))
    if ((t_2 - sqrt(t)) <= 0.0d0) then
        tmp = (sqrt((x + 1.0d0)) - sqrt(x)) + ((1.0d0 / (t_1 + sqrt(z))) + (sqrt((y + 1.0d0)) - sqrt(y)))
    else
        tmp = 2.0d0 + ((t_2 + t_1) - (sqrt(t) + ((sqrt(z) + sqrt(x)) + sqrt(y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double t_2 = Math.sqrt((t + 1.0));
	double tmp;
	if ((t_2 - Math.sqrt(t)) <= 0.0) {
		tmp = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + ((1.0 / (t_1 + Math.sqrt(z))) + (Math.sqrt((y + 1.0)) - Math.sqrt(y)));
	} else {
		tmp = 2.0 + ((t_2 + t_1) - (Math.sqrt(t) + ((Math.sqrt(z) + Math.sqrt(x)) + Math.sqrt(y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = math.sqrt((t + 1.0))
	tmp = 0
	if (t_2 - math.sqrt(t)) <= 0.0:
		tmp = (math.sqrt((x + 1.0)) - math.sqrt(x)) + ((1.0 / (t_1 + math.sqrt(z))) + (math.sqrt((y + 1.0)) - math.sqrt(y)))
	else:
		tmp = 2.0 + ((t_2 + t_1) - (math.sqrt(t) + ((math.sqrt(z) + math.sqrt(x)) + math.sqrt(y))))
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$1 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(N[(t$95$2 + t$95$1), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 t 1)) (sqrt.f64 t)) < 0.0

   1. Initial program 86.8%

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

      1. associate-+l+ 86.8%

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

      2. associate-+l+ 86.8%

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

      3. +-commutative 86.8%

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

      4. +-commutative 86.8%

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

      5. associate-+l- 42.9%

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

      6. +-commutative 42.9%

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

      7. +-commutative 42.9%

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

   3. Simplified 42.9%

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

      1. add-log-exp 42.9%

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

      2. associate--r- 86.8%

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

      3. +-commutative 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. flip-- 87.1%

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

      2. add-sqr-sqrt 72.1%

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

      3. add-sqr-sqrt 87.5%

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

   8. Applied egg-rr 87.5%

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

      1. associate--l+ 88.4%

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

      2. +-inverses 88.4%

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

      3. metadata-eval 88.4%

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

      4. +-commutative 88.4%

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

   10. Simplified 88.4%

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

   11. Taylor expanded in t around inf 91.0%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 t 1)) (sqrt.f64 t))

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

      2. associate-+r+ 97.1%

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

      3. associate-+r- 76.3%

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

      4. associate-+l- 56.9%

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

      5. associate-+r- 52.9%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in x around 0 23.8%

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

   6. Taylor expanded in y around 0 14.2%

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

      1. associate--l+ 27.0%

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

      2. +-commutative 27.0%

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

      3. +-commutative 27.0%

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

      4. associate-+r+ 27.0%

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

      5. +-commutative 27.0%

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

      6. associate-+r+ 27.0%

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

   8. Simplified 27.0%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{t + 1} - \sqrt{t} \leq 0:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\left(\sqrt{t + 1} + \sqrt{z + 1}\right) - \left(\sqrt{t} + \left(\left(\sqrt{z} + \sqrt{x}\right) + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1}\\ t_2 := \sqrt{z + 1}\\ \mathbf{if}\;y \leq 5600000000:\\ \;\;\;\;\sqrt{y + 1} + \left(\left(\left(t\_2 - \left(\sqrt{z} + \sqrt{x}\right)\right) + 1\right) + \left(\left(t\_1 - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \left(\left(\sqrt{t} + \left(\sqrt{z} - t\_2\right)\right) - t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ t 1.0))) (t_2 (sqrt (+ z 1.0))))
   (if (<= y 5600000000.0)
     (+
      (sqrt (+ y 1.0))
      (+ (+ (- t_2 (+ (sqrt z) (sqrt x))) 1.0) (- (- t_1 (sqrt t)) (sqrt y))))
     (+
      (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
      (- (* 0.5 (sqrt (/ 1.0 y))) (- (+ (sqrt t) (- (sqrt z) t_2)) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0));
	double t_2 = sqrt((z + 1.0));
	double tmp;
	if (y <= 5600000000.0) {
		tmp = sqrt((y + 1.0)) + (((t_2 - (sqrt(z) + sqrt(x))) + 1.0) + ((t_1 - sqrt(t)) - sqrt(y)));
	} else {
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((0.5 * sqrt((1.0 / y))) - ((sqrt(t) + (sqrt(z) - t_2)) - 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((t + 1.0d0))
    t_2 = sqrt((z + 1.0d0))
    if (y <= 5600000000.0d0) then
        tmp = sqrt((y + 1.0d0)) + (((t_2 - (sqrt(z) + sqrt(x))) + 1.0d0) + ((t_1 - sqrt(t)) - sqrt(y)))
    else
        tmp = (1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + ((0.5d0 * sqrt((1.0d0 / y))) - ((sqrt(t) + (sqrt(z) - t_2)) - 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((t + 1.0));
	double t_2 = Math.sqrt((z + 1.0));
	double tmp;
	if (y <= 5600000000.0) {
		tmp = Math.sqrt((y + 1.0)) + (((t_2 - (Math.sqrt(z) + Math.sqrt(x))) + 1.0) + ((t_1 - Math.sqrt(t)) - Math.sqrt(y)));
	} else {
		tmp = (1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + ((0.5 * Math.sqrt((1.0 / y))) - ((Math.sqrt(t) + (Math.sqrt(z) - t_2)) - t_1));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((t + 1.0))
	t_2 = math.sqrt((z + 1.0))
	tmp = 0
	if y <= 5600000000.0:
		tmp = math.sqrt((y + 1.0)) + (((t_2 - (math.sqrt(z) + math.sqrt(x))) + 1.0) + ((t_1 - math.sqrt(t)) - math.sqrt(y)))
	else:
		tmp = (1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + ((0.5 * math.sqrt((1.0 / y))) - ((math.sqrt(t) + (math.sqrt(z) - t_2)) - t_1))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(t + 1.0))
	t_2 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (y <= 5600000000.0)
		tmp = Float64(sqrt(Float64(y + 1.0)) + Float64(Float64(Float64(t_2 - Float64(sqrt(z) + sqrt(x))) + 1.0) + Float64(Float64(t_1 - sqrt(t)) - sqrt(y))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) - Float64(Float64(sqrt(t) + Float64(sqrt(z) - t_2)) - 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((t + 1.0));
	t_2 = sqrt((z + 1.0));
	tmp = 0.0;
	if (y <= 5600000000.0)
		tmp = sqrt((y + 1.0)) + (((t_2 - (sqrt(z) + sqrt(x))) + 1.0) + ((t_1 - sqrt(t)) - sqrt(y)));
	else
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((0.5 * sqrt((1.0 / y))) - ((sqrt(t) + (sqrt(z) - t_2)) - 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[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 5600000000.0], N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(t$95$2 - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.6e9

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

      2. associate-+r+ 97.1%

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

      3. associate-+r- 97.1%

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

      4. associate-+l- 97.1%

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

      5. associate-+r- 97.1%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in x around 0 41.6%

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

      1. associate--l+ 47.8%

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

   7. Simplified 47.8%

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

   if 5.6e9 < y

   1. Initial program 88.4%

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

      1. associate-+l+ 88.4%

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

      2. associate-+l+ 88.4%

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

      3. +-commutative 88.4%

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

      4. +-commutative 88.4%

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

      5. associate-+l- 70.8%

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

      6. +-commutative 70.8%

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

      7. +-commutative 70.8%

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

   3. Simplified 70.8%

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

      1. flip-- 70.8%

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

      2. add-sqr-sqrt 60.7%

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

      3. +-commutative 60.7%

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

      4. add-sqr-sqrt 71.2%

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

      5. +-commutative 71.2%

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

   6. Applied egg-rr 71.2%

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

      1. associate--l+ 72.7%

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

      2. +-inverses 72.7%

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

      3. metadata-eval 72.7%

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

   8. Simplified 72.7%

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

   9. Taylor expanded in y around inf 75.2%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5600000000:\\ \;\;\;\;\sqrt{y + 1} + \left(\left(\left(\sqrt{z + 1} - \left(\sqrt{z} + \sqrt{x}\right)\right) + 1\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \left(\left(\sqrt{t} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \sqrt{t + 1}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 0.0028:\\ \;\;\;\;t\_4 + \left(\left(\left(t\_1 - \left(\sqrt{z} + \sqrt{x}\right)\right) + 1\right) + \left(t\_3 - \sqrt{y}\right)\right)\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+108}:\\ \;\;\;\;\left(t\_2 - \sqrt{x}\right) + \left(\frac{1}{t\_4 + \sqrt{y}} + \frac{1}{t\_1 + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_2 + \sqrt{x}} + \left(t\_3 + \left(t\_4 - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (+ z 1.0)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4 (sqrt (+ y 1.0))))
   (if (<= z 0.0028)
     (+ t_4 (+ (+ (- t_1 (+ (sqrt z) (sqrt x))) 1.0) (- t_3 (sqrt y))))
     (if (<= z 8.4e+108)
       (+
        (- t_2 (sqrt x))
        (+ (/ 1.0 (+ t_4 (sqrt y))) (/ 1.0 (+ t_1 (sqrt z)))))
       (+ (/ 1.0 (+ t_2 (sqrt x))) (+ t_3 (- t_4 (sqrt 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 t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = sqrt((y + 1.0));
	double tmp;
	if (z <= 0.0028) {
		tmp = t_4 + (((t_1 - (sqrt(z) + sqrt(x))) + 1.0) + (t_3 - sqrt(y)));
	} else if (z <= 8.4e+108) {
		tmp = (t_2 - sqrt(x)) + ((1.0 / (t_4 + sqrt(y))) + (1.0 / (t_1 + sqrt(z))));
	} else {
		tmp = (1.0 / (t_2 + sqrt(x))) + (t_3 + (t_4 - 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) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = sqrt((x + 1.0d0))
    t_3 = sqrt((t + 1.0d0)) - sqrt(t)
    t_4 = sqrt((y + 1.0d0))
    if (z <= 0.0028d0) then
        tmp = t_4 + (((t_1 - (sqrt(z) + sqrt(x))) + 1.0d0) + (t_3 - sqrt(y)))
    else if (z <= 8.4d+108) then
        tmp = (t_2 - sqrt(x)) + ((1.0d0 / (t_4 + sqrt(y))) + (1.0d0 / (t_1 + sqrt(z))))
    else
        tmp = (1.0d0 / (t_2 + sqrt(x))) + (t_3 + (t_4 - 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((z + 1.0));
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_4 = Math.sqrt((y + 1.0));
	double tmp;
	if (z <= 0.0028) {
		tmp = t_4 + (((t_1 - (Math.sqrt(z) + Math.sqrt(x))) + 1.0) + (t_3 - Math.sqrt(y)));
	} else if (z <= 8.4e+108) {
		tmp = (t_2 - Math.sqrt(x)) + ((1.0 / (t_4 + Math.sqrt(y))) + (1.0 / (t_1 + Math.sqrt(z))));
	} else {
		tmp = (1.0 / (t_2 + Math.sqrt(x))) + (t_3 + (t_4 - 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((z + 1.0))
	t_2 = math.sqrt((x + 1.0))
	t_3 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_4 = math.sqrt((y + 1.0))
	tmp = 0
	if z <= 0.0028:
		tmp = t_4 + (((t_1 - (math.sqrt(z) + math.sqrt(x))) + 1.0) + (t_3 - math.sqrt(y)))
	elif z <= 8.4e+108:
		tmp = (t_2 - math.sqrt(x)) + ((1.0 / (t_4 + math.sqrt(y))) + (1.0 / (t_1 + math.sqrt(z))))
	else:
		tmp = (1.0 / (t_2 + math.sqrt(x))) + (t_3 + (t_4 - 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(z + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 0.0028)
		tmp = Float64(t_4 + Float64(Float64(Float64(t_1 - Float64(sqrt(z) + sqrt(x))) + 1.0) + Float64(t_3 - sqrt(y))));
	elseif (z <= 8.4e+108)
		tmp = Float64(Float64(t_2 - sqrt(x)) + Float64(Float64(1.0 / Float64(t_4 + sqrt(y))) + Float64(1.0 / Float64(t_1 + sqrt(z)))));
	else
		tmp = Float64(Float64(1.0 / Float64(t_2 + sqrt(x))) + Float64(t_3 + Float64(t_4 - 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((z + 1.0));
	t_2 = sqrt((x + 1.0));
	t_3 = sqrt((t + 1.0)) - sqrt(t);
	t_4 = sqrt((y + 1.0));
	tmp = 0.0;
	if (z <= 0.0028)
		tmp = t_4 + (((t_1 - (sqrt(z) + sqrt(x))) + 1.0) + (t_3 - sqrt(y)));
	elseif (z <= 8.4e+108)
		tmp = (t_2 - sqrt(x)) + ((1.0 / (t_4 + sqrt(y))) + (1.0 / (t_1 + sqrt(z))));
	else
		tmp = (1.0 / (t_2 + sqrt(x))) + (t_3 + (t_4 - 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[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 0.0028], N[(t$95$4 + N[(N[(N[(t$95$1 - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.4e+108], N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$4 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$1 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 0.00279999999999999997

   1. Initial program 97.2%

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

      1. +-commutative 97.2%

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

      2. associate-+r+ 97.2%

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

      3. associate-+r- 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- 69.5%

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

      5. associate-+r- 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in x around 0 27.8%

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

      1. associate--l+ 27.8%

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

   7. Simplified 27.8%

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

   if 0.00279999999999999997 < z < 8.40000000000000039e108

   1. Initial program 84.7%

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

      1. associate-+l+ 84.7%

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

      2. associate-+l+ 84.7%

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

      3. +-commutative 84.7%

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

      4. +-commutative 84.7%

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

      5. associate-+l- 78.2%

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

      6. +-commutative 78.2%

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

      7. +-commutative 78.2%

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

   3. Simplified 78.2%

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

      1. add-log-exp 78.2%

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

      2. associate--r- 84.7%

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

      3. +-commutative 84.7%

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

   6. Applied egg-rr 84.7%

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

      1. flip-- 86.1%

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

      2. add-sqr-sqrt 61.1%

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

      3. add-sqr-sqrt 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+ 90.9%

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

      2. +-inverses 90.9%

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

      3. metadata-eval 90.9%

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

      4. +-commutative 90.9%

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

   10. Simplified 90.9%

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

       1. flip-- 90.9%

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

       2. add-sqr-sqrt 58.9%

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

       3. add-sqr-sqrt 91.0%

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

   12. Applied egg-rr 91.0%

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

       1. associate--l+ 92.2%

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

       2. +-inverses 92.2%

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

       3. metadata-eval 92.2%

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

       4. +-commutative 92.2%

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

   14. Simplified 92.2%

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

   15. Taylor expanded in t around inf 49.1%

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

   if 8.40000000000000039e108 < z

   1. Initial program 90.7%

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

      1. associate-+l+ 90.7%

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

      2. associate-+l+ 90.7%

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

      3. +-commutative 90.7%

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

      4. +-commutative 90.7%

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

      5. associate-+l- 90.7%

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

      6. +-commutative 90.7%

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

      7. +-commutative 90.7%

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

   3. Simplified 90.7%

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

      1. flip-- 90.7%

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

      2. add-sqr-sqrt 67.4%

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

      3. +-commutative 67.4%

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

      4. add-sqr-sqrt 90.7%

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

      5. +-commutative 90.7%

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

   6. Applied egg-rr 90.7%

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

      1. associate--l+ 92.6%

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

      2. +-inverses 92.6%

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

      3. metadata-eval 92.6%

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

   8. Simplified 92.6%

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

   9. Taylor expanded in z around inf 92.6%

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

       1. +-commutative 90.7%

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

   11. Simplified 92.6%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.0028:\\ \;\;\;\;\sqrt{y + 1} + \left(\left(\left(\sqrt{z + 1} - \left(\sqrt{z} + \sqrt{x}\right)\right) + 1\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+108}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;t\_3 + \left(\left(2 - \left(\sqrt{z} + \sqrt{x}\right)\right) + \left(t\_2 - \sqrt{y}\right)\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+108}:\\ \;\;\;\;\left(t\_1 - \sqrt{x}\right) + \left(\frac{1}{t\_3 + \sqrt{y}} + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1 + \sqrt{x}} + \left(t\_2 + \left(t\_3 - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_3 (sqrt (+ y 1.0))))
   (if (<= z 2.8e-20)
     (+ t_3 (+ (- 2.0 (+ (sqrt z) (sqrt x))) (- t_2 (sqrt y))))
     (if (<= z 3.2e+108)
       (+
        (- t_1 (sqrt x))
        (+ (/ 1.0 (+ t_3 (sqrt y))) (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z)))))
       (+ (/ 1.0 (+ t_1 (sqrt x))) (+ t_2 (- t_3 (sqrt y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double t_3 = sqrt((y + 1.0));
	double tmp;
	if (z <= 2.8e-20) {
		tmp = t_3 + ((2.0 - (sqrt(z) + sqrt(x))) + (t_2 - sqrt(y)));
	} else if (z <= 3.2e+108) {
		tmp = (t_1 - sqrt(x)) + ((1.0 / (t_3 + sqrt(y))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z))));
	} else {
		tmp = (1.0 / (t_1 + sqrt(x))) + (t_2 + (t_3 - 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((x + 1.0d0))
    t_2 = sqrt((t + 1.0d0)) - sqrt(t)
    t_3 = sqrt((y + 1.0d0))
    if (z <= 2.8d-20) then
        tmp = t_3 + ((2.0d0 - (sqrt(z) + sqrt(x))) + (t_2 - sqrt(y)))
    else if (z <= 3.2d+108) then
        tmp = (t_1 - sqrt(x)) + ((1.0d0 / (t_3 + sqrt(y))) + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z))))
    else
        tmp = (1.0d0 / (t_1 + sqrt(x))) + (t_2 + (t_3 - 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((x + 1.0));
	double t_2 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_3 = Math.sqrt((y + 1.0));
	double tmp;
	if (z <= 2.8e-20) {
		tmp = t_3 + ((2.0 - (Math.sqrt(z) + Math.sqrt(x))) + (t_2 - Math.sqrt(y)));
	} else if (z <= 3.2e+108) {
		tmp = (t_1 - Math.sqrt(x)) + ((1.0 / (t_3 + Math.sqrt(y))) + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z))));
	} else {
		tmp = (1.0 / (t_1 + Math.sqrt(x))) + (t_2 + (t_3 - 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((x + 1.0))
	t_2 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_3 = math.sqrt((y + 1.0))
	tmp = 0
	if z <= 2.8e-20:
		tmp = t_3 + ((2.0 - (math.sqrt(z) + math.sqrt(x))) + (t_2 - math.sqrt(y)))
	elif z <= 3.2e+108:
		tmp = (t_1 - math.sqrt(x)) + ((1.0 / (t_3 + math.sqrt(y))) + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z))))
	else:
		tmp = (1.0 / (t_1 + math.sqrt(x))) + (t_2 + (t_3 - 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(x + 1.0))
	t_2 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_3 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 2.8e-20)
		tmp = Float64(t_3 + Float64(Float64(2.0 - Float64(sqrt(z) + sqrt(x))) + Float64(t_2 - sqrt(y))));
	elseif (z <= 3.2e+108)
		tmp = Float64(Float64(t_1 - sqrt(x)) + Float64(Float64(1.0 / Float64(t_3 + sqrt(y))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))));
	else
		tmp = Float64(Float64(1.0 / Float64(t_1 + sqrt(x))) + Float64(t_2 + Float64(t_3 - 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((x + 1.0));
	t_2 = sqrt((t + 1.0)) - sqrt(t);
	t_3 = sqrt((y + 1.0));
	tmp = 0.0;
	if (z <= 2.8e-20)
		tmp = t_3 + ((2.0 - (sqrt(z) + sqrt(x))) + (t_2 - sqrt(y)));
	elseif (z <= 3.2e+108)
		tmp = (t_1 - sqrt(x)) + ((1.0 / (t_3 + sqrt(y))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z))));
	else
		tmp = (1.0 / (t_1 + sqrt(x))) + (t_2 + (t_3 - 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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 2.8e-20], N[(t$95$3 + N[(N[(2.0 - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.2e+108], N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$3 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.8000000000000003e-20

   1. Initial program 97.2%

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

      1. +-commutative 97.2%

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

      2. associate-+r+ 97.2%

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

      3. associate-+r- 80.1%

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

      4. associate-+l- 69.5%

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

      5. associate-+r- 55.7%

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

   3. Simplified 55.7%

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

   5. Taylor expanded in x around 0 29.0%

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

   6. Taylor expanded in z around 0 29.0%

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

      1. +-commutative 29.0%

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

   8. Simplified 29.0%

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

   if 2.8000000000000003e-20 < z < 3.1999999999999999e108

   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- 73.6%

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

      6. +-commutative 73.6%

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

      7. +-commutative 73.6%

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

   3. Simplified 73.6%

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

      1. add-log-exp 73.6%

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

      2. associate--r- 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-- 87.3%

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

      2. add-sqr-sqrt 65.2%

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

      3. add-sqr-sqrt 88.9%

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

   8. Applied egg-rr 88.9%

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

      1. associate--l+ 91.6%

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

      2. +-inverses 91.6%

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

      3. metadata-eval 91.6%

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

      4. +-commutative 91.6%

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

   10. Simplified 91.6%

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

       1. flip-- 91.6%

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

       2. add-sqr-sqrt 58.0%

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

       3. add-sqr-sqrt 91.7%

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

   12. Applied egg-rr 91.7%

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

       1. associate--l+ 92.8%

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

       2. +-inverses 92.8%

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

       3. metadata-eval 92.8%

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

       4. +-commutative 92.8%

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

   14. Simplified 92.8%

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

   15. Taylor expanded in t around inf 51.8%

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

   if 3.1999999999999999e108 < z

   1. Initial program 90.7%

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

      1. associate-+l+ 90.7%

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

      2. associate-+l+ 90.7%

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

      3. +-commutative 90.7%

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

      4. +-commutative 90.7%

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

      5. associate-+l- 90.7%

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

      6. +-commutative 90.7%

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

      7. +-commutative 90.7%

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

   3. Simplified 90.7%

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

      1. flip-- 90.7%

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

      2. add-sqr-sqrt 67.4%

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

      3. +-commutative 67.4%

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

      4. add-sqr-sqrt 90.7%

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

      5. +-commutative 90.7%

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

   6. Applied egg-rr 90.7%

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

      1. associate--l+ 92.6%

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

      2. +-inverses 92.6%

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

      3. metadata-eval 92.6%

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

   8. Simplified 92.6%

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

   9. Taylor expanded in z around inf 92.6%

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

       1. +-commutative 90.7%

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

   11. Simplified 92.6%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{y + 1} + \left(\left(2 - \left(\sqrt{z} + \sqrt{x}\right)\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+108}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{y + 1}\\ t_4 := t\_3 - \sqrt{y}\\ \mathbf{if}\;z \leq 3.4 \cdot 10^{-20}:\\ \;\;\;\;t\_3 + \left(\left(2 - \left(\sqrt{z} + \sqrt{x}\right)\right) + \left(t\_2 - \sqrt{y}\right)\right)\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+28}:\\ \;\;\;\;\left(t\_1 - \sqrt{x}\right) + \left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + t\_4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1 + \sqrt{x}} + \left(t\_2 + t\_4\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_3 (sqrt (+ y 1.0)))
        (t_4 (- t_3 (sqrt y))))
   (if (<= z 3.4e-20)
     (+ t_3 (+ (- 2.0 (+ (sqrt z) (sqrt x))) (- t_2 (sqrt y))))
     (if (<= z 5.6e+28)
       (+ (- t_1 (sqrt x)) (+ (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))) t_4))
       (+ (/ 1.0 (+ t_1 (sqrt x))) (+ t_2 t_4))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double t_3 = sqrt((y + 1.0));
	double t_4 = t_3 - sqrt(y);
	double tmp;
	if (z <= 3.4e-20) {
		tmp = t_3 + ((2.0 - (sqrt(z) + sqrt(x))) + (t_2 - sqrt(y)));
	} else if (z <= 5.6e+28) {
		tmp = (t_1 - sqrt(x)) + ((1.0 / (sqrt((z + 1.0)) + sqrt(z))) + t_4);
	} else {
		tmp = (1.0 / (t_1 + sqrt(x))) + (t_2 + t_4);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((t + 1.0d0)) - sqrt(t)
    t_3 = sqrt((y + 1.0d0))
    t_4 = t_3 - sqrt(y)
    if (z <= 3.4d-20) then
        tmp = t_3 + ((2.0d0 - (sqrt(z) + sqrt(x))) + (t_2 - sqrt(y)))
    else if (z <= 5.6d+28) then
        tmp = (t_1 - sqrt(x)) + ((1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z))) + t_4)
    else
        tmp = (1.0d0 / (t_1 + sqrt(x))) + (t_2 + t_4)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double t_2 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_3 = Math.sqrt((y + 1.0));
	double t_4 = t_3 - Math.sqrt(y);
	double tmp;
	if (z <= 3.4e-20) {
		tmp = t_3 + ((2.0 - (Math.sqrt(z) + Math.sqrt(x))) + (t_2 - Math.sqrt(y)));
	} else if (z <= 5.6e+28) {
		tmp = (t_1 - Math.sqrt(x)) + ((1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z))) + t_4);
	} else {
		tmp = (1.0 / (t_1 + Math.sqrt(x))) + (t_2 + t_4);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	t_2 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_3 = math.sqrt((y + 1.0))
	t_4 = t_3 - math.sqrt(y)
	tmp = 0
	if z <= 3.4e-20:
		tmp = t_3 + ((2.0 - (math.sqrt(z) + math.sqrt(x))) + (t_2 - math.sqrt(y)))
	elif z <= 5.6e+28:
		tmp = (t_1 - math.sqrt(x)) + ((1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z))) + t_4)
	else:
		tmp = (1.0 / (t_1 + math.sqrt(x))) + (t_2 + t_4)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_3 = sqrt(Float64(y + 1.0))
	t_4 = Float64(t_3 - sqrt(y))
	tmp = 0.0
	if (z <= 3.4e-20)
		tmp = Float64(t_3 + Float64(Float64(2.0 - Float64(sqrt(z) + sqrt(x))) + Float64(t_2 - sqrt(y))));
	elseif (z <= 5.6e+28)
		tmp = Float64(Float64(t_1 - sqrt(x)) + Float64(Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z))) + t_4));
	else
		tmp = Float64(Float64(1.0 / Float64(t_1 + sqrt(x))) + Float64(t_2 + t_4));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	t_2 = sqrt((t + 1.0)) - sqrt(t);
	t_3 = sqrt((y + 1.0));
	t_4 = t_3 - sqrt(y);
	tmp = 0.0;
	if (z <= 3.4e-20)
		tmp = t_3 + ((2.0 - (sqrt(z) + sqrt(x))) + (t_2 - sqrt(y)));
	elseif (z <= 5.6e+28)
		tmp = (t_1 - sqrt(x)) + ((1.0 / (sqrt((z + 1.0)) + sqrt(z))) + t_4);
	else
		tmp = (1.0 / (t_1 + sqrt(x))) + (t_2 + t_4);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 3.4e-20], N[(t$95$3 + N[(N[(2.0 - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.6e+28], N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.3999999999999997e-20

   1. Initial program 97.2%

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

      1. +-commutative 97.2%

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

      2. associate-+r+ 97.2%

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

      3. associate-+r- 80.1%

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

      4. associate-+l- 69.5%

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

      5. associate-+r- 55.7%

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

   3. Simplified 55.7%

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

   5. Taylor expanded in x around 0 29.0%

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

   6. Taylor expanded in z around 0 29.0%

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

      1. +-commutative 29.0%

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

   8. Simplified 29.0%

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

   if 3.3999999999999997e-20 < z < 5.6000000000000003e28

   1. Initial program 78.6%

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

      1. associate-+l+ 78.6%

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

      2. associate-+l+ 78.6%

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

      3. +-commutative 78.6%

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

      4. +-commutative 78.6%

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

      5. associate-+l- 52.2%

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

      6. +-commutative 52.2%

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

      7. +-commutative 52.2%

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

   3. Simplified 52.2%

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

      1. add-log-exp 52.2%

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

      2. associate--r- 78.6%

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

      3. +-commutative 78.6%

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

   6. Applied egg-rr 78.6%

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

      1. flip-- 81.2%

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

      2. add-sqr-sqrt 79.1%

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

      3. add-sqr-sqrt 84.6%

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

   8. Applied egg-rr 84.6%

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

      1. associate--l+ 90.1%

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

      2. +-inverses 90.1%

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

      3. metadata-eval 90.1%

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

      4. +-commutative 90.1%

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

   10. Simplified 90.1%

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

   11. Taylor expanded in t around inf 56.8%

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

   if 5.6000000000000003e28 < z

   1. Initial program 91.3%

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

      1. associate-+l+ 91.3%

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

      2. associate-+l+ 91.3%

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

      3. +-commutative 91.3%

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

      4. +-commutative 91.3%

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

      5. associate-+l- 91.3%

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

      6. +-commutative 91.3%

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

      7. +-commutative 91.3%

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

   3. Simplified 91.3%

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

      1. flip-- 91.2%

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

      2. add-sqr-sqrt 68.7%

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

      3. +-commutative 68.7%

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

      4. add-sqr-sqrt 91.2%

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

      5. +-commutative 91.2%

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

   6. Applied egg-rr 91.2%

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

      1. associate--l+ 92.7%

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

      2. +-inverses 92.7%

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

      3. metadata-eval 92.7%

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

   8. Simplified 92.7%

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

   9. Taylor expanded in z around inf 92.7%

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

       1. +-commutative 91.3%

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

   11. Simplified 92.7%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{y + 1} + \left(\left(2 - \left(\sqrt{z} + \sqrt{x}\right)\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+28}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ t_2 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 4 \cdot 10^{-16}:\\ \;\;\;\;t\_2 + \left(\left(2 - \left(\sqrt{z} + \sqrt{x}\right)\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{elif}\;z \leq 10^{+120}:\\ \;\;\;\;\left(t\_2 + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_1 + \left(t\_2 - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (+ t 1.0)) (sqrt t))) (t_2 (sqrt (+ y 1.0))))
   (if (<= z 4e-16)
     (+ t_2 (+ (- 2.0 (+ (sqrt z) (sqrt x))) (- t_1 (sqrt y))))
     (if (<= z 1e+120)
       (+ (+ t_2 (- (- (sqrt (+ z 1.0)) (sqrt z)) (+ (sqrt x) (sqrt y)))) 1.0)
       (+ (- (sqrt (+ x 1.0)) (sqrt x)) (+ t_1 (- t_2 (sqrt y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0)) - sqrt(t);
	double t_2 = sqrt((y + 1.0));
	double tmp;
	if (z <= 4e-16) {
		tmp = t_2 + ((2.0 - (sqrt(z) + sqrt(x))) + (t_1 - sqrt(y)));
	} else if (z <= 1e+120) {
		tmp = (t_2 + ((sqrt((z + 1.0)) - sqrt(z)) - (sqrt(x) + sqrt(y)))) + 1.0;
	} else {
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + (t_1 + (t_2 - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((t + 1.0d0)) - sqrt(t)
    t_2 = sqrt((y + 1.0d0))
    if (z <= 4d-16) then
        tmp = t_2 + ((2.0d0 - (sqrt(z) + sqrt(x))) + (t_1 - sqrt(y)))
    else if (z <= 1d+120) then
        tmp = (t_2 + ((sqrt((z + 1.0d0)) - sqrt(z)) - (sqrt(x) + sqrt(y)))) + 1.0d0
    else
        tmp = (sqrt((x + 1.0d0)) - sqrt(x)) + (t_1 + (t_2 - sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_2 = Math.sqrt((y + 1.0));
	double tmp;
	if (z <= 4e-16) {
		tmp = t_2 + ((2.0 - (Math.sqrt(z) + Math.sqrt(x))) + (t_1 - Math.sqrt(y)));
	} else if (z <= 1e+120) {
		tmp = (t_2 + ((Math.sqrt((z + 1.0)) - Math.sqrt(z)) - (Math.sqrt(x) + Math.sqrt(y)))) + 1.0;
	} else {
		tmp = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (t_1 + (t_2 - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_2 = math.sqrt((y + 1.0))
	tmp = 0
	if z <= 4e-16:
		tmp = t_2 + ((2.0 - (math.sqrt(z) + math.sqrt(x))) + (t_1 - math.sqrt(y)))
	elif z <= 1e+120:
		tmp = (t_2 + ((math.sqrt((z + 1.0)) - math.sqrt(z)) - (math.sqrt(x) + math.sqrt(y)))) + 1.0
	else:
		tmp = (math.sqrt((x + 1.0)) - math.sqrt(x)) + (t_1 + (t_2 - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_2 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 4e-16)
		tmp = Float64(t_2 + Float64(Float64(2.0 - Float64(sqrt(z) + sqrt(x))) + Float64(t_1 - sqrt(y))));
	elseif (z <= 1e+120)
		tmp = Float64(Float64(t_2 + Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) - Float64(sqrt(x) + sqrt(y)))) + 1.0);
	else
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_1 + Float64(t_2 - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((t + 1.0)) - sqrt(t);
	t_2 = sqrt((y + 1.0));
	tmp = 0.0;
	if (z <= 4e-16)
		tmp = t_2 + ((2.0 - (sqrt(z) + sqrt(x))) + (t_1 - sqrt(y)));
	elseif (z <= 1e+120)
		tmp = (t_2 + ((sqrt((z + 1.0)) - sqrt(z)) - (sqrt(x) + sqrt(y)))) + 1.0;
	else
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + (t_1 + (t_2 - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 4e-16], N[(t$95$2 + N[(N[(2.0 - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e+120], N[(N[(t$95$2 + N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.9999999999999999e-16

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

      2. associate-+r+ 97.1%

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

      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- 69.2%

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

      5. associate-+r- 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in x around 0 28.6%

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

   6. Taylor expanded in z around 0 28.6%

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

      1. +-commutative 28.6%

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

   8. Simplified 28.6%

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

   if 3.9999999999999999e-16 < z < 9.9999999999999998e119

   1. Initial program 86.5%

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

      1. +-commutative 86.5%

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

      2. associate-+r+ 86.5%

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

      3. associate-+r- 66.8%

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

      4. associate-+l- 53.6%

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

      5. associate-+r- 46.9%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in t around inf 7.7%

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

      1. associate--l+ 23.0%

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

      2. associate--l+ 25.1%

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

      3. associate-+r+ 25.1%

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

   7. Simplified 25.1%

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

   8. Taylor expanded in x around 0 7.0%

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

      1. associate--l+ 27.2%

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

      2. associate-+r+ 27.2%

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

      3. +-commutative 27.2%

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

      4. associate--l+ 29.4%

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

      5. associate--r+ 31.0%

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

   10. Simplified 31.0%

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

   if 9.9999999999999998e119 < z

   1. Initial program 90.5%

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

      1. associate-+l+ 90.5%

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

      2. associate-+l+ 90.5%

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

      3. +-commutative 90.5%

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

      4. +-commutative 90.5%

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

      5. associate-+l- 90.5%

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

      6. +-commutative 90.5%

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

      7. +-commutative 90.5%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in z around inf 90.5%

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

      1. +-commutative 90.5%

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

   7. Simplified 90.5%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{y + 1} + \left(\left(2 - \left(\sqrt{z} + \sqrt{x}\right)\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \mathbf{elif}\;z \leq 10^{+120}:\\ \;\;\;\;\left(\sqrt{y + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ \mathbf{if}\;t \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;2 + \left(\left(\sqrt{t + 1} + t\_1\right) - \left(\sqrt{t} + \left(\left(\sqrt{z} + \sqrt{x}\right) + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{y + 1} + \left(\left(t\_1 - \sqrt{z}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + 1\\ \end{array} \end{array} \]

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 (+ z 1.0))))
   (if (<= t 3.7e+15)
     (+
      2.0
      (-
       (+ (sqrt (+ t 1.0)) t_1)
       (+ (sqrt t) (+ (+ (sqrt z) (sqrt x)) (sqrt y)))))
     (+ (+ (sqrt (+ y 1.0)) (- (- t_1 (sqrt z)) (+ (sqrt x) (sqrt y)))) 1.0))))

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 (t <= 3.7e+15) {
		tmp = 2.0 + ((sqrt((t + 1.0)) + t_1) - (sqrt(t) + ((sqrt(z) + sqrt(x)) + sqrt(y))));
	} else {
		tmp = (sqrt((y + 1.0)) + ((t_1 - sqrt(z)) - (sqrt(x) + sqrt(y)))) + 1.0;
	}
	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 (t <= 3.7d+15) then
        tmp = 2.0d0 + ((sqrt((t + 1.0d0)) + t_1) - (sqrt(t) + ((sqrt(z) + sqrt(x)) + sqrt(y))))
    else
        tmp = (sqrt((y + 1.0d0)) + ((t_1 - sqrt(z)) - (sqrt(x) + sqrt(y)))) + 1.0d0
    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 (t <= 3.7e+15) {
		tmp = 2.0 + ((Math.sqrt((t + 1.0)) + t_1) - (Math.sqrt(t) + ((Math.sqrt(z) + Math.sqrt(x)) + Math.sqrt(y))));
	} else {
		tmp = (Math.sqrt((y + 1.0)) + ((t_1 - Math.sqrt(z)) - (Math.sqrt(x) + Math.sqrt(y)))) + 1.0;
	}
	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 t <= 3.7e+15:
		tmp = 2.0 + ((math.sqrt((t + 1.0)) + t_1) - (math.sqrt(t) + ((math.sqrt(z) + math.sqrt(x)) + math.sqrt(y))))
	else:
		tmp = (math.sqrt((y + 1.0)) + ((t_1 - math.sqrt(z)) - (math.sqrt(x) + math.sqrt(y)))) + 1.0
	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 (t <= 3.7e+15)
		tmp = Float64(2.0 + Float64(Float64(sqrt(Float64(t + 1.0)) + t_1) - Float64(sqrt(t) + Float64(Float64(sqrt(z) + sqrt(x)) + sqrt(y)))));
	else
		tmp = Float64(Float64(sqrt(Float64(y + 1.0)) + Float64(Float64(t_1 - sqrt(z)) - Float64(sqrt(x) + sqrt(y)))) + 1.0);
	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 (t <= 3.7e+15)
		tmp = 2.0 + ((sqrt((t + 1.0)) + t_1) - (sqrt(t) + ((sqrt(z) + sqrt(x)) + sqrt(y))));
	else
		tmp = (sqrt((y + 1.0)) + ((t_1 - sqrt(z)) - (sqrt(x) + sqrt(y)))) + 1.0;
	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[t, 3.7e+15], N[(2.0 + N[(N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 3.7e15

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

      2. associate-+r+ 97.1%

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

      3. associate-+r- 76.3%

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

      4. associate-+l- 56.9%

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

      5. associate-+r- 52.9%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in x around 0 23.8%

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

   6. Taylor expanded in y around 0 14.2%

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

      1. associate--l+ 27.0%

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

      2. +-commutative 27.0%

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

      3. +-commutative 27.0%

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

      4. associate-+r+ 27.0%

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

      5. +-commutative 27.0%

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

      6. associate-+r+ 27.0%

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

   8. Simplified 27.0%

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

   if 3.7e15 < t

   1. Initial program 86.8%

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

      1. +-commutative 86.8%

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

      2. associate-+r+ 86.8%

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

      3. associate-+r- 70.0%

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

      4. associate-+l- 70.0%

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

      5. associate-+r- 55.7%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in t around inf 20.9%

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

      1. associate--l+ 34.6%

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

      2. associate--l+ 35.8%

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

      3. associate-+r+ 35.8%

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

   7. Simplified 35.8%

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

   8. Taylor expanded in x around 0 18.4%

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

      1. associate--l+ 43.1%

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

      2. associate-+r+ 43.1%

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

      3. +-commutative 43.1%

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

      4. associate--l+ 40.0%

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

      5. associate--r+ 45.3%

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

   10. Simplified 45.3%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;2 + \left(\left(\sqrt{t + 1} + \sqrt{z + 1}\right) - \left(\sqrt{t} + \left(\left(\sqrt{z} + \sqrt{x}\right) + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{y + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7400000:\\ \;\;\;\;\left(\sqrt{y + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(0.5 \cdot \sqrt{\frac{1}{y}} + -0.125 \cdot \sqrt{\frac{1}{{y}^{3}}}\right)\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 7400000.0)
   (+
    (+
     (sqrt (+ y 1.0))
     (- (- (sqrt (+ z 1.0)) (sqrt z)) (+ (sqrt x) (sqrt y))))
    1.0)
   (-
    (+
     (sqrt (+ x 1.0))
     (+ (* 0.5 (sqrt (/ 1.0 y))) (* -0.125 (sqrt (/ 1.0 (pow y 3.0))))))
    (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7400000.0) {
		tmp = (sqrt((y + 1.0)) + ((sqrt((z + 1.0)) - sqrt(z)) - (sqrt(x) + sqrt(y)))) + 1.0;
	} else {
		tmp = (sqrt((x + 1.0)) + ((0.5 * sqrt((1.0 / y))) + (-0.125 * sqrt((1.0 / pow(y, 3.0)))))) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 7400000.0d0) then
        tmp = (sqrt((y + 1.0d0)) + ((sqrt((z + 1.0d0)) - sqrt(z)) - (sqrt(x) + sqrt(y)))) + 1.0d0
    else
        tmp = (sqrt((x + 1.0d0)) + ((0.5d0 * sqrt((1.0d0 / y))) + ((-0.125d0) * sqrt((1.0d0 / (y ** 3.0d0)))))) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7400000.0) {
		tmp = (Math.sqrt((y + 1.0)) + ((Math.sqrt((z + 1.0)) - Math.sqrt(z)) - (Math.sqrt(x) + Math.sqrt(y)))) + 1.0;
	} else {
		tmp = (Math.sqrt((x + 1.0)) + ((0.5 * Math.sqrt((1.0 / y))) + (-0.125 * Math.sqrt((1.0 / Math.pow(y, 3.0)))))) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 7400000.0:
		tmp = (math.sqrt((y + 1.0)) + ((math.sqrt((z + 1.0)) - math.sqrt(z)) - (math.sqrt(x) + math.sqrt(y)))) + 1.0
	else:
		tmp = (math.sqrt((x + 1.0)) + ((0.5 * math.sqrt((1.0 / y))) + (-0.125 * math.sqrt((1.0 / math.pow(y, 3.0)))))) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 7400000.0)
		tmp = Float64(Float64(sqrt(Float64(y + 1.0)) + Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) - Float64(sqrt(x) + sqrt(y)))) + 1.0);
	else
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(-0.125 * sqrt(Float64(1.0 / (y ^ 3.0)))))) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 7400000.0)
		tmp = (sqrt((y + 1.0)) + ((sqrt((z + 1.0)) - sqrt(z)) - (sqrt(x) + sqrt(y)))) + 1.0;
	else
		tmp = (sqrt((x + 1.0)) + ((0.5 * sqrt((1.0 / y))) + (-0.125 * sqrt((1.0 / (y ^ 3.0)))))) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 7400000.0], N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-0.125 * N[Sqrt[N[(1.0 / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.4e6

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

      2. associate-+r+ 97.1%

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

      3. associate-+r- 97.1%

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

      4. associate-+l- 97.1%

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

      5. associate-+r- 97.1%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in t around inf 19.0%

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

      1. associate--l+ 22.3%

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

      2. associate--l+ 27.8%

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

      3. associate-+r+ 27.8%

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

   7. Simplified 27.8%

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

   8. Taylor expanded in x around 0 16.1%

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

      1. associate--l+ 24.3%

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

      2. associate-+r+ 24.3%

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

      3. +-commutative 24.3%

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

      4. associate--l+ 30.9%

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

      5. associate--r+ 24.6%

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

   10. Simplified 24.6%

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

   if 7.4e6 < y

   1. Initial program 88.4%

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

      1. +-commutative 88.4%

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

      2. associate-+r+ 88.5%

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

      3. associate-+r- 48.9%

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

      4. associate-+l- 25.2%

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

      5. associate-+r- 8.3%

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

   3. Simplified 7.3%

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

   5. Taylor expanded in t around inf 5.0%

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

      1. associate--l+ 20.8%

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

      2. associate--l+ 15.3%

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

      3. associate-+r+ 15.3%

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

   7. Simplified 15.3%

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

   8. Taylor expanded in z around inf 5.1%

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

   9. Taylor expanded in y around inf 19.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(-0.125 \cdot \sqrt{\frac{1}{{y}^{3}}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7400000:\\ \;\;\;\;\left(\sqrt{y + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(0.5 \cdot \sqrt{\frac{1}{y}} + -0.125 \cdot \sqrt{\frac{1}{{y}^{3}}}\right)\right) - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 86.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3700000000:\\ \;\;\;\;\left(\sqrt{y + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 3700000000.0)
   (+
    (+
     (sqrt (+ y 1.0))
     (- (- (sqrt (+ z 1.0)) (sqrt z)) (+ (sqrt x) (sqrt y))))
    1.0)
   (- (+ (sqrt (+ x 1.0)) (* 0.5 (sqrt (/ 1.0 y)))) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3700000000.0) {
		tmp = (sqrt((y + 1.0)) + ((sqrt((z + 1.0)) - sqrt(z)) - (sqrt(x) + sqrt(y)))) + 1.0;
	} else {
		tmp = (sqrt((x + 1.0)) + (0.5 * sqrt((1.0 / y)))) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 3700000000.0d0) then
        tmp = (sqrt((y + 1.0d0)) + ((sqrt((z + 1.0d0)) - sqrt(z)) - (sqrt(x) + sqrt(y)))) + 1.0d0
    else
        tmp = (sqrt((x + 1.0d0)) + (0.5d0 * sqrt((1.0d0 / y)))) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3700000000.0) {
		tmp = (Math.sqrt((y + 1.0)) + ((Math.sqrt((z + 1.0)) - Math.sqrt(z)) - (Math.sqrt(x) + Math.sqrt(y)))) + 1.0;
	} else {
		tmp = (Math.sqrt((x + 1.0)) + (0.5 * Math.sqrt((1.0 / y)))) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 3700000000.0:
		tmp = (math.sqrt((y + 1.0)) + ((math.sqrt((z + 1.0)) - math.sqrt(z)) - (math.sqrt(x) + math.sqrt(y)))) + 1.0
	else:
		tmp = (math.sqrt((x + 1.0)) + (0.5 * math.sqrt((1.0 / y)))) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3700000000.0)
		tmp = Float64(Float64(sqrt(Float64(y + 1.0)) + Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) - Float64(sqrt(x) + sqrt(y)))) + 1.0);
	else
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) + Float64(0.5 * sqrt(Float64(1.0 / y)))) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3700000000.0)
		tmp = (sqrt((y + 1.0)) + ((sqrt((z + 1.0)) - sqrt(z)) - (sqrt(x) + sqrt(y)))) + 1.0;
	else
		tmp = (sqrt((x + 1.0)) + (0.5 * sqrt((1.0 / y)))) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 3700000000.0], N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.7e9

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

      2. associate-+r+ 97.1%

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

      3. associate-+r- 97.1%

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

      4. associate-+l- 97.1%

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

      5. associate-+r- 97.1%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in t around inf 19.0%

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

      1. associate--l+ 22.3%

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

      2. associate--l+ 27.8%

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

      3. associate-+r+ 27.8%

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

   7. Simplified 27.8%

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

   8. Taylor expanded in x around 0 16.1%

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

      1. associate--l+ 24.3%

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

      2. associate-+r+ 24.3%

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

      3. +-commutative 24.3%

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

      4. associate--l+ 30.9%

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

      5. associate--r+ 24.6%

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

   10. Simplified 24.6%

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

   if 3.7e9 < y

   1. Initial program 88.4%

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

      1. +-commutative 88.4%

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

      2. associate-+r+ 88.5%

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

      3. associate-+r- 48.9%

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

      4. associate-+l- 25.2%

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

      5. associate-+r- 8.3%

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

   3. Simplified 7.3%

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

   5. Taylor expanded in t around inf 5.0%

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

      1. associate--l+ 20.8%

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

      2. associate--l+ 15.3%

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

      3. associate-+r+ 15.3%

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

   7. Simplified 15.3%

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

   8. Taylor expanded in z around inf 5.1%

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

   9. Taylor expanded in y around inf 19.7%

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

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3700000000:\\ \;\;\;\;\left(\sqrt{y + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 86.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{y + 1}\\ \mathbf{if}\;y \leq 1.4 \cdot 10^{-25}:\\ \;\;\;\;t\_1 + \left(t\_2 + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 3200000:\\ \;\;\;\;\left(t\_2 - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

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))) (t_2 (sqrt (+ y 1.0))))
   (if (<= y 1.4e-25)
     (+ t_1 (+ t_2 (- (sqrt (+ z 1.0)) (sqrt z))))
     (if (<= y 3200000.0)
       (+ (- t_2 (+ (sqrt x) (sqrt y))) 1.0)
       (+ 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 t_2 = sqrt((y + 1.0));
	double tmp;
	if (y <= 1.4e-25) {
		tmp = t_1 + (t_2 + (sqrt((z + 1.0)) - sqrt(z)));
	} else if (y <= 3200000.0) {
		tmp = (t_2 - (sqrt(x) + sqrt(y))) + 1.0;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((y + 1.0d0))
    if (y <= 1.4d-25) then
        tmp = t_1 + (t_2 + (sqrt((z + 1.0d0)) - sqrt(z)))
    else if (y <= 3200000.0d0) then
        tmp = (t_2 - (sqrt(x) + sqrt(y))) + 1.0d0
    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 t_2 = Math.sqrt((y + 1.0));
	double tmp;
	if (y <= 1.4e-25) {
		tmp = t_1 + (t_2 + (Math.sqrt((z + 1.0)) - Math.sqrt(z)));
	} else if (y <= 3200000.0) {
		tmp = (t_2 - (Math.sqrt(x) + Math.sqrt(y))) + 1.0;
	} 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))
	t_2 = math.sqrt((y + 1.0))
	tmp = 0
	if y <= 1.4e-25:
		tmp = t_1 + (t_2 + (math.sqrt((z + 1.0)) - math.sqrt(z)))
	elif y <= 3200000.0:
		tmp = (t_2 - (math.sqrt(x) + math.sqrt(y))) + 1.0
	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))
	t_2 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (y <= 1.4e-25)
		tmp = Float64(t_1 + Float64(t_2 + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))));
	elseif (y <= 3200000.0)
		tmp = Float64(Float64(t_2 - Float64(sqrt(x) + sqrt(y))) + 1.0);
	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));
	t_2 = sqrt((y + 1.0));
	tmp = 0.0;
	if (y <= 1.4e-25)
		tmp = t_1 + (t_2 + (sqrt((z + 1.0)) - sqrt(z)));
	elseif (y <= 3200000.0)
		tmp = (t_2 - (sqrt(x) + sqrt(y))) + 1.0;
	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]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.4e-25], N[(t$95$1 + N[(t$95$2 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3200000.0], N[(N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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 < 1.39999999999999994e-25

   1. Initial program 97.4%

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

      1. +-commutative 97.4%

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

      2. associate-+r+ 97.4%

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

      3. associate-+r- 97.4%

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

      4. associate-+l- 97.4%

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

      5. associate-+r- 97.4%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in t around inf 19.9%

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

      1. associate--l+ 23.0%

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

      2. associate--l+ 28.9%

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

      3. associate-+r+ 28.9%

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

   7. Simplified 28.9%

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

   8. Taylor expanded in z around inf 28.1%

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

   if 1.39999999999999994e-25 < y < 3.2e6

   1. Initial program 95.1%

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

      1. +-commutative 95.1%

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

      2. associate-+r+ 95.1%

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

      3. associate-+r- 95.0%

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

      4. associate-+l- 94.9%

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

      5. associate-+r- 94.9%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in t around inf 12.4%

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

      1. associate--l+ 17.6%

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

      2. associate--l+ 20.0%

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

      3. associate-+r+ 20.0%

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

   7. Simplified 20.0%

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

   8. Taylor expanded in z around inf 17.2%

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

   9. Taylor expanded in x around 0 15.4%

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

       1. associate--l+ 15.5%

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

   11. Simplified 15.5%

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

   if 3.2e6 < y

   1. Initial program 88.4%

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

      1. +-commutative 88.4%

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

      2. associate-+r+ 88.5%

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

      3. associate-+r- 48.9%

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

      4. associate-+l- 25.2%

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

      5. associate-+r- 8.3%

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

   3. Simplified 7.3%

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

   5. Taylor expanded in t around inf 5.0%

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

      1. associate--l+ 20.8%

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

      2. associate--l+ 15.3%

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

      3. associate-+r+ 15.3%

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

   7. Simplified 15.3%

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

   8. Taylor expanded in z around inf 18.3%

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

   9. Taylor expanded in y around inf 17.7%

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

       1. distribute-lft-out 17.7%

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

   11. Simplified 17.7%

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

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 3200000:\\ \;\;\;\;\left(\sqrt{y + 1} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 480000:\\ \;\;\;\;t\_1 + \left(t\_2 + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\left(t\_2 + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{y}\right)\\ \end{array} \end{array} \]

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 (+ x 1.0))) (t_2 (sqrt (+ y 1.0))))
   (if (<= z 480000.0)
     (+ t_1 (+ t_2 (- (sqrt (+ z 1.0)) (sqrt z))))
     (+ t_1 (- (+ t_2 (* 0.5 (sqrt (/ 1.0 z)))) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((y + 1.0));
	double tmp;
	if (z <= 480000.0) {
		tmp = t_1 + (t_2 + (sqrt((z + 1.0)) - sqrt(z)));
	} else {
		tmp = t_1 + ((t_2 + (0.5 * sqrt((1.0 / z)))) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((y + 1.0d0))
    if (z <= 480000.0d0) then
        tmp = t_1 + (t_2 + (sqrt((z + 1.0d0)) - sqrt(z)))
    else
        tmp = t_1 + ((t_2 + (0.5d0 * sqrt((1.0d0 / z)))) - 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((x + 1.0));
	double t_2 = Math.sqrt((y + 1.0));
	double tmp;
	if (z <= 480000.0) {
		tmp = t_1 + (t_2 + (Math.sqrt((z + 1.0)) - Math.sqrt(z)));
	} else {
		tmp = t_1 + ((t_2 + (0.5 * Math.sqrt((1.0 / z)))) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	t_2 = math.sqrt((y + 1.0))
	tmp = 0
	if z <= 480000.0:
		tmp = t_1 + (t_2 + (math.sqrt((z + 1.0)) - math.sqrt(z)))
	else:
		tmp = t_1 + ((t_2 + (0.5 * math.sqrt((1.0 / z)))) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 480000.0)
		tmp = Float64(t_1 + Float64(t_2 + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))));
	else
		tmp = Float64(t_1 + Float64(Float64(t_2 + Float64(0.5 * sqrt(Float64(1.0 / z)))) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	t_2 = sqrt((y + 1.0));
	tmp = 0.0;
	if (z <= 480000.0)
		tmp = t_1 + (t_2 + (sqrt((z + 1.0)) - sqrt(z)));
	else
		tmp = t_1 + ((t_2 + (0.5 * sqrt((1.0 / z)))) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 480000.0], N[(t$95$1 + N[(t$95$2 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(t$95$2 + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 4.8e5

   1. Initial program 96.8%

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

      1. +-commutative 96.8%

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

      2. associate-+r+ 96.8%

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

      3. associate-+r- 80.3%

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

      4. associate-+l- 69.7%

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

      5. associate-+r- 55.0%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in t around inf 19.2%

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

      1. associate--l+ 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+ 22.9%

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

      3. associate-+r+ 22.9%

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

   7. Simplified 22.9%

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

   8. Taylor expanded in z around inf 17.8%

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

   if 4.8e5 < z

   1. Initial program 88.3%

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

      1. +-commutative 88.3%

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

      2. associate-+r+ 88.3%

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

      3. associate-+r- 65.8%

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

      4. associate-+l- 53.4%

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

      5. associate-+r- 52.9%

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

   3. Simplified 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 3.9%

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

      1. associate--l+ 20.2%

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

      2. associate--l+ 20.4%

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

      3. associate-+r+ 20.4%

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

   7. Simplified 20.4%

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

   8. Taylor expanded in z around inf 27.4%

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

   9. Taylor expanded in y around inf 28.0%

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

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 480000:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 86.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{y + 1}\\ \mathbf{if}\;y \leq 3 \cdot 10^{-25}:\\ \;\;\;\;t\_1 + \left(t\_2 + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 45000000:\\ \;\;\;\;\left(t\_2 - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))) (t_2 (sqrt (+ y 1.0))))
   (if (<= y 3e-25)
     (+ t_1 (+ t_2 (- (sqrt (+ z 1.0)) (sqrt z))))
     (if (<= y 45000000.0)
       (+ (- t_2 (+ (sqrt x) (sqrt y))) 1.0)
       (- (+ t_1 (* 0.5 (sqrt (/ 1.0 y)))) (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 t_2 = sqrt((y + 1.0));
	double tmp;
	if (y <= 3e-25) {
		tmp = t_1 + (t_2 + (sqrt((z + 1.0)) - sqrt(z)));
	} else if (y <= 45000000.0) {
		tmp = (t_2 - (sqrt(x) + sqrt(y))) + 1.0;
	} else {
		tmp = (t_1 + (0.5 * sqrt((1.0 / y)))) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((y + 1.0d0))
    if (y <= 3d-25) then
        tmp = t_1 + (t_2 + (sqrt((z + 1.0d0)) - sqrt(z)))
    else if (y <= 45000000.0d0) then
        tmp = (t_2 - (sqrt(x) + sqrt(y))) + 1.0d0
    else
        tmp = (t_1 + (0.5d0 * sqrt((1.0d0 / y)))) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double t_2 = Math.sqrt((y + 1.0));
	double tmp;
	if (y <= 3e-25) {
		tmp = t_1 + (t_2 + (Math.sqrt((z + 1.0)) - Math.sqrt(z)));
	} else if (y <= 45000000.0) {
		tmp = (t_2 - (Math.sqrt(x) + Math.sqrt(y))) + 1.0;
	} else {
		tmp = (t_1 + (0.5 * Math.sqrt((1.0 / y)))) - 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))
	t_2 = math.sqrt((y + 1.0))
	tmp = 0
	if y <= 3e-25:
		tmp = t_1 + (t_2 + (math.sqrt((z + 1.0)) - math.sqrt(z)))
	elif y <= 45000000.0:
		tmp = (t_2 - (math.sqrt(x) + math.sqrt(y))) + 1.0
	else:
		tmp = (t_1 + (0.5 * math.sqrt((1.0 / y)))) - 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))
	t_2 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (y <= 3e-25)
		tmp = Float64(t_1 + Float64(t_2 + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))));
	elseif (y <= 45000000.0)
		tmp = Float64(Float64(t_2 - Float64(sqrt(x) + sqrt(y))) + 1.0);
	else
		tmp = Float64(Float64(t_1 + Float64(0.5 * sqrt(Float64(1.0 / y)))) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	t_2 = sqrt((y + 1.0));
	tmp = 0.0;
	if (y <= 3e-25)
		tmp = t_1 + (t_2 + (sqrt((z + 1.0)) - sqrt(z)));
	elseif (y <= 45000000.0)
		tmp = (t_2 - (sqrt(x) + sqrt(y))) + 1.0;
	else
		tmp = (t_1 + (0.5 * sqrt((1.0 / y)))) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 3e-25], N[(t$95$1 + N[(t$95$2 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 45000000.0], N[(N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(t$95$1 + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.9999999999999998e-25

   1. Initial program 97.4%

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

      1. +-commutative 97.4%

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

      2. associate-+r+ 97.4%

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

      3. associate-+r- 97.4%

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

      4. associate-+l- 97.4%

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

      5. associate-+r- 97.4%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in t around inf 19.9%

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

      1. associate--l+ 23.0%

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

      2. associate--l+ 28.9%

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

      3. associate-+r+ 28.9%

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

   7. Simplified 28.9%

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

   8. Taylor expanded in z around inf 28.1%

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

   if 2.9999999999999998e-25 < y < 4.5e7

   1. Initial program 95.1%

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

      1. +-commutative 95.1%

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

      2. associate-+r+ 95.1%

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

      3. associate-+r- 95.0%

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

      4. associate-+l- 94.9%

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

      5. associate-+r- 94.9%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in t around inf 12.4%

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

      1. associate--l+ 17.6%

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

      2. associate--l+ 20.0%

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

      3. associate-+r+ 20.0%

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

   7. Simplified 20.0%

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

   8. Taylor expanded in z around inf 17.2%

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

   9. Taylor expanded in x around 0 15.4%

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

       1. associate--l+ 15.5%

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

   11. Simplified 15.5%

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

   if 4.5e7 < y

   1. Initial program 88.4%

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

      1. +-commutative 88.4%

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

      2. associate-+r+ 88.5%

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

      3. associate-+r- 48.9%

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

      4. associate-+l- 25.2%

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

      5. associate-+r- 8.3%

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

   3. Simplified 7.3%

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

   5. Taylor expanded in t around inf 5.0%

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

      1. associate--l+ 20.8%

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

      2. associate--l+ 15.3%

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

      3. associate-+r+ 15.3%

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

   7. Simplified 15.3%

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

   8. Taylor expanded in z around inf 5.1%

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

   9. Taylor expanded in y around inf 19.7%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 45000000:\\ \;\;\;\;\left(\sqrt{y + 1} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 82.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{x + 1}\\ \mathbf{if}\;z \leq 1.76:\\ \;\;\;\;\left(\left(t\_2 + 1\right) + \left(t\_1 + z \cdot 0.5\right)\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(t\_1 - \sqrt{y}\right)\\ \end{array} \end{array} \]

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 (+ y 1.0))) (t_2 (sqrt (+ x 1.0))))
   (if (<= z 1.76)
     (- (+ (+ t_2 1.0) (+ t_1 (* z 0.5))) (sqrt z))
     (+ t_2 (- 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((y + 1.0));
	double t_2 = sqrt((x + 1.0));
	double tmp;
	if (z <= 1.76) {
		tmp = ((t_2 + 1.0) + (t_1 + (z * 0.5))) - sqrt(z);
	} else {
		tmp = t_2 + (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) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((x + 1.0d0))
    if (z <= 1.76d0) then
        tmp = ((t_2 + 1.0d0) + (t_1 + (z * 0.5d0))) - sqrt(z)
    else
        tmp = t_2 + (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((y + 1.0));
	double t_2 = Math.sqrt((x + 1.0));
	double tmp;
	if (z <= 1.76) {
		tmp = ((t_2 + 1.0) + (t_1 + (z * 0.5))) - Math.sqrt(z);
	} else {
		tmp = t_2 + (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((y + 1.0))
	t_2 = math.sqrt((x + 1.0))
	tmp = 0
	if z <= 1.76:
		tmp = ((t_2 + 1.0) + (t_1 + (z * 0.5))) - math.sqrt(z)
	else:
		tmp = t_2 + (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(y + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (z <= 1.76)
		tmp = Float64(Float64(Float64(t_2 + 1.0) + Float64(t_1 + Float64(z * 0.5))) - sqrt(z));
	else
		tmp = Float64(t_2 + 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((y + 1.0));
	t_2 = sqrt((x + 1.0));
	tmp = 0.0;
	if (z <= 1.76)
		tmp = ((t_2 + 1.0) + (t_1 + (z * 0.5))) - sqrt(z);
	else
		tmp = t_2 + (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[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1.76], N[(N[(N[(t$95$2 + 1.0), $MachinePrecision] + N[(t$95$1 + N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.76000000000000001

   1. Initial program 97.3%

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

      1. +-commutative 97.3%

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

      2. associate-+r+ 97.3%

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

      3. associate-+r- 80.1%

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

      4. associate-+l- 69.0%

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

      5. associate-+r- 54.2%

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

   3. Simplified 54.2%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   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+ 22.6%

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

      2. associate--l+ 22.6%

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

      3. associate-+r+ 22.6%

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

   7. Simplified 22.6%

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

   8. Taylor expanded in z around 0 18.8%

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

      1. associate-+r+ 18.8%

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

      2. +-commutative 18.8%

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

      3. associate-+r+ 18.8%

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

      4. +-commutative 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in z around inf 17.6%

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

   if 1.76000000000000001 < z

   1. Initial program 88.2%

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

      1. +-commutative 88.2%

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

      2. associate-+r+ 88.2%

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

      3. associate-+r- 66.8%

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

      4. associate-+l- 55.0%

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

      5. associate-+r- 53.9%

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

   3. Simplified 36.6%

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

   5. Taylor expanded in t around inf 5.1%

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

      1. associate--l+ 20.6%

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

      2. associate--l+ 20.8%

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

      3. associate-+r+ 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in y around inf 27.2%

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

      1. mul-1-neg 27.2%

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

   10. Simplified 27.2%

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

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.76:\\ \;\;\;\;\left(\left(\sqrt{x + 1} + 1\right) + \left(\sqrt{y + 1} + z \cdot 0.5\right)\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 66.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 110000000:\\ \;\;\;\;\left(\sqrt{y + 1} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 110000000.0)
   (+ (- (sqrt (+ y 1.0)) (+ (sqrt x) (sqrt y))) 1.0)
   (- (+ (sqrt (+ x 1.0)) (* 0.5 (sqrt (/ 1.0 y)))) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 110000000.0) {
		tmp = (sqrt((y + 1.0)) - (sqrt(x) + sqrt(y))) + 1.0;
	} else {
		tmp = (sqrt((x + 1.0)) + (0.5 * sqrt((1.0 / y)))) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 110000000.0d0) then
        tmp = (sqrt((y + 1.0d0)) - (sqrt(x) + sqrt(y))) + 1.0d0
    else
        tmp = (sqrt((x + 1.0d0)) + (0.5d0 * sqrt((1.0d0 / y)))) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 110000000.0) {
		tmp = (Math.sqrt((y + 1.0)) - (Math.sqrt(x) + Math.sqrt(y))) + 1.0;
	} else {
		tmp = (Math.sqrt((x + 1.0)) + (0.5 * Math.sqrt((1.0 / y)))) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 110000000.0:
		tmp = (math.sqrt((y + 1.0)) - (math.sqrt(x) + math.sqrt(y))) + 1.0
	else:
		tmp = (math.sqrt((x + 1.0)) + (0.5 * math.sqrt((1.0 / y)))) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 110000000.0)
		tmp = Float64(Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(x) + sqrt(y))) + 1.0);
	else
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) + Float64(0.5 * sqrt(Float64(1.0 / y)))) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 110000000.0)
		tmp = (sqrt((y + 1.0)) - (sqrt(x) + sqrt(y))) + 1.0;
	else
		tmp = (sqrt((x + 1.0)) + (0.5 * sqrt((1.0 / y)))) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 110000000.0], N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e8

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

      2. associate-+r+ 97.1%

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

      3. associate-+r- 97.1%

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

      4. associate-+l- 97.1%

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

      5. associate-+r- 97.1%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in t around inf 19.0%

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

      1. associate--l+ 22.3%

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

      2. associate--l+ 27.8%

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

      3. associate-+r+ 27.8%

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

   7. Simplified 27.8%

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

   8. Taylor expanded in z around inf 18.0%

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

   9. Taylor expanded in x around 0 15.8%

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

       1. associate--l+ 15.8%

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

   11. Simplified 15.8%

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

   if 1.1e8 < y

   1. Initial program 88.4%

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

      1. +-commutative 88.4%

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

      2. associate-+r+ 88.5%

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

      3. associate-+r- 48.9%

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

      4. associate-+l- 25.2%

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

      5. associate-+r- 8.3%

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

   3. Simplified 7.3%

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

   5. Taylor expanded in t around inf 5.0%

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

      1. associate--l+ 20.8%

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

      2. associate--l+ 15.3%

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

      3. associate-+r+ 15.3%

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

   7. Simplified 15.3%

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

   8. Taylor expanded in z around inf 5.1%

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

   9. Taylor expanded in y around inf 19.7%

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

4. Final simplification 17.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 110000000:\\ \;\;\;\;\left(\sqrt{y + 1} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 65.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.06 \cdot 10^{-10}:\\ \;\;\;\;\left(\sqrt{y + 1} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.06e-10)
   (+ (- (sqrt (+ y 1.0)) (+ (sqrt x) (sqrt y))) 1.0)
   (- (sqrt (+ x 1.0)) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.06e-10) {
		tmp = (sqrt((y + 1.0)) - (sqrt(x) + sqrt(y))) + 1.0;
	} else {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 1.06d-10) then
        tmp = (sqrt((y + 1.0d0)) - (sqrt(x) + sqrt(y))) + 1.0d0
    else
        tmp = sqrt((x + 1.0d0)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.06e-10) {
		tmp = (Math.sqrt((y + 1.0)) - (Math.sqrt(x) + Math.sqrt(y))) + 1.0;
	} else {
		tmp = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 1.06e-10:
		tmp = (math.sqrt((y + 1.0)) - (math.sqrt(x) + math.sqrt(y))) + 1.0
	else:
		tmp = math.sqrt((x + 1.0)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.06e-10)
		tmp = Float64(Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(x) + sqrt(y))) + 1.0);
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.06e-10)
		tmp = (sqrt((y + 1.0)) - (sqrt(x) + sqrt(y))) + 1.0;
	else
		tmp = sqrt((x + 1.0)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 1.06e-10], N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.06e-10

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

      2. associate-+r+ 97.5%

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

      3. associate-+r- 60.4%

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

      4. associate-+l- 57.8%

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

      5. associate-+r- 54.5%

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

   3. Simplified 36.7%

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

   5. Taylor expanded in t around inf 19.6%

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

      1. associate--l+ 37.5%

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

      2. associate--l+ 37.9%

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

      3. associate-+r+ 37.9%

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

   7. Simplified 37.9%

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

   8. Taylor expanded in z around inf 19.8%

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

   9. Taylor expanded in x around 0 19.8%

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

       1. associate--l+ 33.8%

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

   11. Simplified 33.8%

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

   if 1.06e-10 < x

   1. Initial program 88.8%

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

      1. +-commutative 88.8%

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

      2. associate-+r+ 88.8%

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

      3. associate-+r- 85.5%

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

      4. associate-+l- 66.2%

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

      5. associate-+r- 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 53.7%

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

   5. Taylor expanded in t around inf 5.7%

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

      1. associate--l+ 7.6%

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

      2. associate--l+ 7.5%

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

      3. associate-+r+ 7.5%

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

   7. Simplified 7.5%

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

   8. Taylor expanded in z around inf 4.6%

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

   9. Taylor expanded in y around inf 4.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.06 \cdot 10^{-10}:\\ \;\;\;\;\left(\sqrt{y + 1} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 65.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 6.6 \cdot 10^{+25}:\\ \;\;\;\;t\_1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))))
   (if (<= y 6.6e+25) (+ t_1 (- (sqrt (+ y 1.0)) (sqrt y))) (- t_1 (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double tmp;
	if (y <= 6.6e+25) {
		tmp = t_1 + (sqrt((y + 1.0)) - sqrt(y));
	} else {
		tmp = 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((x + 1.0d0))
    if (y <= 6.6d+25) then
        tmp = t_1 + (sqrt((y + 1.0d0)) - sqrt(y))
    else
        tmp = 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((x + 1.0));
	double tmp;
	if (y <= 6.6e+25) {
		tmp = t_1 + (Math.sqrt((y + 1.0)) - Math.sqrt(y));
	} else {
		tmp = 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((x + 1.0))
	tmp = 0
	if y <= 6.6e+25:
		tmp = t_1 + (math.sqrt((y + 1.0)) - math.sqrt(y))
	else:
		tmp = 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(x + 1.0))
	tmp = 0.0
	if (y <= 6.6e+25)
		tmp = Float64(t_1 + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)));
	else
		tmp = Float64(t_1 - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	tmp = 0.0;
	if (y <= 6.6e+25)
		tmp = t_1 + (sqrt((y + 1.0)) - sqrt(y));
	else
		tmp = 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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 6.6e+25], N[(t$95$1 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 6.6000000000000002e25

   1. Initial program 95.6%

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

      1. +-commutative 95.6%

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

      2. associate-+r+ 95.6%

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

      3. associate-+r- 95.6%

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

      4. associate-+l- 95.6%

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

      5. associate-+r- 95.6%

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

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 19.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 23.6%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate--l+ 28.2%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. associate-+r+ 28.2%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

   7. Simplified 28.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in y around inf 19.7%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{-1 \cdot \sqrt{y}}\right) \]
   9. Step-by-step derivation

      1. mul-1-neg 19.7%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{\left(-\sqrt{y}\right)}\right) \]

   10. Simplified 19.7%

       \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{\left(-\sqrt{y}\right)}\right) \]

   if 6.6000000000000002e25 < y

   1. Initial program 89.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 89.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ 89.6%

         \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      3. associate-+r- 47.3%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 22.0%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. associate-+r- 4.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. Simplified 4.0%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   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+ 19.3%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate--l+ 14.0%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. associate-+r+ 14.0%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

   7. Simplified 14.0%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in z around inf 3.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   9. Taylor expanded in y around inf 18.4%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{+25}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 35.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{x + 1} - \sqrt{x} \end{array} \]

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 1.0)) (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt((x + 1.0)) - 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 + 1.0d0)) - 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 + 1.0)) - Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt((x + 1.0)) - math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt((x + 1.0)) - sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.9%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. +-commutative 92.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate-+r+ 92.9%

      \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. associate-+r- 73.7%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   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- 54.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

3. Simplified 45.7%

   \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 12.2%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
6. Step-by-step derivation

   1. associate--l+ 21.6%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. associate--l+ 21.7%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   3. associate-+r+ 21.7%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

7. Simplified 21.7%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

8. Taylor expanded in z around inf 11.7%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

9. Taylor expanded in y around inf 14.3%

   \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

10. Final simplification 14.3%

    \[\leadsto \sqrt{x + 1} - \sqrt{x} \]
11. Add Preprocessing

Alternative 22: 7.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 0.5 \cdot {z}^{-0.5} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (* 0.5 (pow z -0.5)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.5 * pow(z, -0.5);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.5d0 * (z ** (-0.5d0))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 0.5 * Math.pow(z, -0.5);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 0.5 * math.pow(z, -0.5)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.5 * (z ^ -0.5))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 0.5 * (z ^ -0.5);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(0.5 * N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.9%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. +-commutative 92.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate-+r+ 92.9%

      \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. associate-+r- 73.7%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   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- 54.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

3. Simplified 45.7%

   \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 28.3%

   \[\leadsto \sqrt{1 + y} + \left(\color{blue}{\left(\left(\sqrt{1 + x} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)} - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right) \]

6. Taylor expanded in z around 0 8.4%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 8.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{z}}\right)} \]

   2. pow1/2 8.4%

      \[\leadsto 0.5 \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{z}\right)}^{0.5}}\right) \]

   3. inv-pow 8.4%

      \[\leadsto 0.5 \cdot \left(1 \cdot {\color{blue}{\left({z}^{-1}\right)}}^{0.5}\right) \]

   4. pow-pow 8.4%

      \[\leadsto 0.5 \cdot \left(1 \cdot \color{blue}{{z}^{\left(-1 \cdot 0.5\right)}}\right) \]

   5. metadata-eval 8.4%

      \[\leadsto 0.5 \cdot \left(1 \cdot {z}^{\color{blue}{-0.5}}\right) \]

8. Applied egg-rr 8.4%

   \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot {z}^{-0.5}\right)} \]
9. Step-by-step derivation

   1. *-lft-identity 8.4%

      \[\leadsto 0.5 \cdot \color{blue}{{z}^{-0.5}} \]

10. Simplified 8.4%

    \[\leadsto 0.5 \cdot \color{blue}{{z}^{-0.5}} \]

11. Final simplification 8.4%

    \[\leadsto 0.5 \cdot {z}^{-0.5} \]
12. Add Preprocessing

Alternative 23: 3.1% accurate, 274.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ x \cdot 0 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (* x 0.0))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return x * 0.0;
}

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 = x * 0.0d0
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return x * 0.0;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return x * 0.0

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(x * 0.0)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = x * 0.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(x * 0.0), $MachinePrecision]

Derivation

1. Initial program 92.9%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. +-commutative 92.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate-+r+ 92.9%

      \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. associate-+r- 73.7%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   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- 54.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

3. Simplified 45.7%

   \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 12.2%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
6. Step-by-step derivation

   1. associate--l+ 21.6%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. associate--l+ 21.7%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   3. associate-+r+ 21.7%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

7. Simplified 21.7%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)\right)} \]

8. Taylor expanded in y around inf 19.7%

   \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{y \cdot \left(\frac{1}{y} \cdot \sqrt{1 + z} - \left(\sqrt{\frac{1}{y}} + \left(\sqrt{x} \cdot \frac{1}{y} + \frac{1}{y} \cdot \sqrt{z}\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. associate-*l/ 18.6%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + y \cdot \left(\color{blue}{\frac{1 \cdot \sqrt{1 + z}}{y}} - \left(\sqrt{\frac{1}{y}} + \left(\sqrt{x} \cdot \frac{1}{y} + \frac{1}{y} \cdot \sqrt{z}\right)\right)\right)\right) \]

   2. *-lft-identity 18.6%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + y \cdot \left(\frac{\color{blue}{\sqrt{1 + z}}}{y} - \left(\sqrt{\frac{1}{y}} + \left(\sqrt{x} \cdot \frac{1}{y} + \frac{1}{y} \cdot \sqrt{z}\right)\right)\right)\right) \]

   3. associate-+r+ 18.6%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + y \cdot \left(\frac{\sqrt{1 + z}}{y} - \color{blue}{\left(\left(\sqrt{\frac{1}{y}} + \sqrt{x} \cdot \frac{1}{y}\right) + \frac{1}{y} \cdot \sqrt{z}\right)}\right)\right) \]

   4. associate-*r/ 18.6%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + y \cdot \left(\frac{\sqrt{1 + z}}{y} - \left(\left(\sqrt{\frac{1}{y}} + \color{blue}{\frac{\sqrt{x} \cdot 1}{y}}\right) + \frac{1}{y} \cdot \sqrt{z}\right)\right)\right) \]

   5. *-rgt-identity 18.6%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + y \cdot \left(\frac{\sqrt{1 + z}}{y} - \left(\left(\sqrt{\frac{1}{y}} + \frac{\color{blue}{\sqrt{x}}}{y}\right) + \frac{1}{y} \cdot \sqrt{z}\right)\right)\right) \]

   6. associate-*l/ 19.7%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + y \cdot \left(\frac{\sqrt{1 + z}}{y} - \left(\left(\sqrt{\frac{1}{y}} + \frac{\sqrt{x}}{y}\right) + \color{blue}{\frac{1 \cdot \sqrt{z}}{y}}\right)\right)\right) \]

   7. *-lft-identity 19.7%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + y \cdot \left(\frac{\sqrt{1 + z}}{y} - \left(\left(\sqrt{\frac{1}{y}} + \frac{\sqrt{x}}{y}\right) + \frac{\color{blue}{\sqrt{z}}}{y}\right)\right)\right) \]

10. Simplified 19.7%

    \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} + \color{blue}{y \cdot \left(\frac{\sqrt{1 + z}}{y} - \left(\left(\sqrt{\frac{1}{y}} + \frac{\sqrt{x}}{y}\right) + \frac{\sqrt{z}}{y}\right)\right)}\right) \]

11. Taylor expanded in x around inf 3.1%

    \[\leadsto \color{blue}{x \cdot \left(\sqrt{\frac{1}{x}} + -1 \cdot \sqrt{\frac{1}{x}}\right)} \]
12. Step-by-step derivation

    1. distribute-rgt1-in 3.1%

       \[\leadsto x \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \sqrt{\frac{1}{x}}\right)} \]

    2. metadata-eval 3.1%

       \[\leadsto x \cdot \left(\color{blue}{0} \cdot \sqrt{\frac{1}{x}}\right) \]

    3. mul0-lft 3.1%

       \[\leadsto x \cdot \color{blue}{0} \]

13. Simplified 3.1%

    \[\leadsto \color{blue}{x \cdot 0} \]

14. Final simplification 3.1%

    \[\leadsto x \cdot 0 \]
15. 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 2024054 
(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))))