Main:z from

Percentage Accurate: 91.7% → 99.1%

Time: 54.5s

Alternatives: 27

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 3.4 \cdot 10^{+30}:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(t\_2 - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{t\_1 + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + t\_2}\right) + \left(t\_1 - \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))) (t_2 (sqrt (+ 1.0 y))))
   (if (<= z 3.4e+30)
     (+
      (+ (- 1.0 (sqrt x)) (- t_2 (sqrt y)))
      (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (/ 1.0 (+ t_1 (sqrt z)))))
     (+
      (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (/ 1.0 (+ (sqrt y) t_2)))
      (- t_1 (sqrt z))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + y));
	double tmp;
	if (z <= 3.4e+30) {
		tmp = ((1.0 - sqrt(x)) + (t_2 - sqrt(y))) + ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 / (t_1 + sqrt(z))));
	} else {
		tmp = ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + t_2))) + (t_1 - 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) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + y))
    if (z <= 3.4d+30) then
        tmp = ((1.0d0 - sqrt(x)) + (t_2 - sqrt(y))) + ((sqrt((1.0d0 + t)) - sqrt(t)) + (1.0d0 / (t_1 + sqrt(z))))
    else
        tmp = ((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (1.0d0 / (sqrt(y) + t_2))) + (t_1 - 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 = Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 3.4e+30) {
		tmp = ((1.0 - Math.sqrt(x)) + (t_2 - Math.sqrt(y))) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (1.0 / (t_1 + Math.sqrt(z))));
	} else {
		tmp = ((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (1.0 / (Math.sqrt(y) + t_2))) + (t_1 - Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 3.4e+30:
		tmp = ((1.0 - math.sqrt(x)) + (t_2 - math.sqrt(y))) + ((math.sqrt((1.0 + t)) - math.sqrt(t)) + (1.0 / (t_1 + math.sqrt(z))))
	else:
		tmp = ((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (1.0 / (math.sqrt(y) + t_2))) + (t_1 - math.sqrt(z))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 3.4e+30)
		tmp = Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(t_2 - sqrt(y))) + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(1.0 / Float64(t_1 + sqrt(z)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(1.0 / Float64(sqrt(y) + t_2))) + Float64(t_1 - sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 3.4e+30)
		tmp = ((1.0 - sqrt(x)) + (t_2 - sqrt(y))) + ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 / (t_1 + sqrt(z))));
	else
		tmp = ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + t_2))) + (t_1 - sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 3.4e+30], N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$1 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 3.4000000000000002e30

   1. Initial program 95.2%

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

      1. associate-+l+ 95.2%

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

      2. sub-neg 95.2%

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

      3. sub-neg 95.2%

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

      4. +-commutative 95.2%

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

      5. +-commutative 95.2%

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

      6. +-commutative 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in x around 0 49.5%

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

      1. flip-- 49.8%

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

      2. div-inv 49.8%

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

      3. add-sqr-sqrt 49.2%

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

      4. add-sqr-sqrt 50.0%

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

      5. associate--l+ 50.4%

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

   7. Applied egg-rr 50.4%

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

      1. +-inverses 50.4%

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

      2. metadata-eval 50.4%

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

      3. *-lft-identity 50.4%

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

      4. +-commutative 50.4%

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

   9. Simplified 50.4%

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

   if 3.4000000000000002e30 < z

   1. Initial program 91.8%

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

      1. associate-+l+ 91.8%

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

      2. sub-neg 91.8%

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

      3. sub-neg 91.8%

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

      4. +-commutative 91.8%

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

      5. +-commutative 91.8%

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

      6. +-commutative 91.8%

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

   3. Simplified 91.8%

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

      1. flip-- 92.0%

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

      2. div-inv 92.0%

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

      3. add-sqr-sqrt 69.5%

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

      4. +-commutative 69.5%

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

      5. add-sqr-sqrt 92.5%

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

      6. +-commutative 92.5%

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

   6. Applied egg-rr 92.5%

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

      1. associate--l+ 95.9%

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

      2. +-inverses 95.9%

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

      3. metadata-eval 95.9%

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

      4. *-lft-identity 95.9%

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

      5. +-commutative 95.9%

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

   8. Simplified 95.9%

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

      1. flip-- 96.5%

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

      2. div-inv 96.5%

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

      3. add-sqr-sqrt 73.7%

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

      4. add-sqr-sqrt 96.5%

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

      5. associate--l+ 99.1%

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

   10. Applied egg-rr 99.1%

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

       1. +-inverses 99.1%

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

       2. metadata-eval 99.1%

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

       3. *-lft-identity 99.1%

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

       4. +-commutative 99.1%

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

   12. Simplified 99.1%

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

   13. Taylor expanded in t around inf 51.3%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{+30}:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

   1. associate-+l+ 93.7%

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

   2. sub-neg 93.7%

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

   3. sub-neg 93.7%

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

   4. +-commutative 93.7%

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

   5. +-commutative 93.7%

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

   6. +-commutative 93.7%

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

3. Simplified 93.7%

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

   1. flip-- 93.8%

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

   2. div-inv 93.8%

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

   3. add-sqr-sqrt 73.2%

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

   4. +-commutative 73.2%

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

   5. add-sqr-sqrt 94.1%

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

   6. +-commutative 94.1%

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

6. Applied egg-rr 94.1%

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

   1. associate--l+ 95.8%

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

   2. +-inverses 95.8%

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

   3. metadata-eval 95.8%

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

   4. *-lft-identity 95.8%

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

   5. +-commutative 95.8%

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

8. Simplified 95.8%

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

   1. flip-- 96.1%

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

   2. div-inv 96.1%

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

   3. add-sqr-sqrt 73.7%

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

   4. add-sqr-sqrt 96.3%

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

   5. associate--l+ 97.5%

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

10. Applied egg-rr 97.5%

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

    1. +-inverses 97.5%

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

    2. metadata-eval 97.5%

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

    3. *-lft-identity 97.5%

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

    4. +-commutative 97.5%

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

12. Simplified 97.5%

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

13. Final simplification 97.5%

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

Alternative 3: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;t \leq 6 \cdot 10^{+30}:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(t\_2 - \sqrt{y}\right)\right) + \left(t\_1 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + t\_2}\right) + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 z)) (sqrt z))) (t_2 (sqrt (+ 1.0 y))))
   (if (<= t 6e+30)
     (+
      (+ (- 1.0 (sqrt x)) (- t_2 (sqrt y)))
      (+ t_1 (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t)))))
     (+
      (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (/ 1.0 (+ (sqrt y) 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((1.0 + z)) - sqrt(z);
	double t_2 = sqrt((1.0 + y));
	double tmp;
	if (t <= 6e+30) {
		tmp = ((1.0 - sqrt(x)) + (t_2 - sqrt(y))) + (t_1 + (1.0 / (sqrt((1.0 + t)) + sqrt(t))));
	} else {
		tmp = ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + 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((1.0d0 + z)) - sqrt(z)
    t_2 = sqrt((1.0d0 + y))
    if (t <= 6d+30) then
        tmp = ((1.0d0 - sqrt(x)) + (t_2 - sqrt(y))) + (t_1 + (1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t))))
    else
        tmp = ((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (1.0d0 / (sqrt(y) + 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((1.0 + z)) - Math.sqrt(z);
	double t_2 = Math.sqrt((1.0 + y));
	double tmp;
	if (t <= 6e+30) {
		tmp = ((1.0 - Math.sqrt(x)) + (t_2 - Math.sqrt(y))) + (t_1 + (1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))));
	} else {
		tmp = ((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (1.0 / (Math.sqrt(y) + 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((1.0 + z)) - math.sqrt(z)
	t_2 = math.sqrt((1.0 + y))
	tmp = 0
	if t <= 6e+30:
		tmp = ((1.0 - math.sqrt(x)) + (t_2 - math.sqrt(y))) + (t_1 + (1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))))
	else:
		tmp = ((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (1.0 / (math.sqrt(y) + t_2))) + t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_2 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (t <= 6e+30)
		tmp = Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(t_2 - sqrt(y))) + Float64(t_1 + Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(1.0 / Float64(sqrt(y) + 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((1.0 + z)) - sqrt(z);
	t_2 = sqrt((1.0 + y));
	tmp = 0.0;
	if (t <= 6e+30)
		tmp = ((1.0 - sqrt(x)) + (t_2 - sqrt(y))) + (t_1 + (1.0 / (sqrt((1.0 + t)) + sqrt(t))));
	else
		tmp = ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + 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[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 6e+30], N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < 5.99999999999999956e30

   1. Initial program 96.9%

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

      1. associate-+l+ 96.9%

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

      2. sub-neg 96.9%

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

      3. sub-neg 96.9%

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

      4. +-commutative 96.9%

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

      5. +-commutative 96.9%

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

      6. +-commutative 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in x around 0 53.9%

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

      1. flip-- 27.1%

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

      2. div-inv 27.1%

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

      3. add-sqr-sqrt 26.6%

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

      4. +-commutative 26.6%

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

      5. add-sqr-sqrt 27.1%

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

      6. associate--l+ 27.1%

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

   7. Applied egg-rr 54.0%

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

      1. associate-*r/ 27.1%

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

      2. *-rgt-identity 27.1%

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

      3. associate-+r- 27.1%

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

      4. +-commutative 27.1%

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

      5. associate-+r- 27.2%

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

      6. +-inverses 27.2%

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

      7. metadata-eval 27.2%

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

      8. +-commutative 27.2%

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

   9. Simplified 54.5%

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

   if 5.99999999999999956e30 < t

   1. Initial program 89.8%

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

      1. associate-+l+ 89.8%

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

      2. sub-neg 89.8%

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

      3. sub-neg 89.8%

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

      4. +-commutative 89.8%

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

      5. +-commutative 89.8%

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

      6. +-commutative 89.8%

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

   3. Simplified 89.8%

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

      1. flip-- 89.8%

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

      2. div-inv 89.8%

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

      3. add-sqr-sqrt 73.9%

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

      4. +-commutative 73.9%

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

      5. add-sqr-sqrt 90.2%

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

      6. +-commutative 90.2%

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

   6. Applied egg-rr 90.2%

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

      1. associate--l+ 93.7%

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

      2. +-inverses 93.7%

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

      3. metadata-eval 93.7%

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

      4. *-lft-identity 93.7%

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

      5. +-commutative 93.7%

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

   8. Simplified 93.7%

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

      1. flip-- 93.9%

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

      2. div-inv 93.9%

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

      3. add-sqr-sqrt 69.1%

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

      4. add-sqr-sqrt 94.1%

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

      5. associate--l+ 96.7%

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

   10. Applied egg-rr 96.7%

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

       1. +-inverses 96.7%

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

       2. metadata-eval 96.7%

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

       3. *-lft-identity 96.7%

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

       4. +-commutative 96.7%

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

   12. Simplified 96.7%

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

   13. Taylor expanded in t around inf 96.7%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{+30}:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(1 + \left(z \cdot 0.5 - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + t\_1}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))))
   (if (<= z 4.8e-7)
     (+
      (+ (- 1.0 (sqrt x)) (- t_1 (sqrt y)))
      (+ (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t))) (+ 1.0 (- (* z 0.5) (sqrt z)))))
     (+
      (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (/ 1.0 (+ (sqrt y) t_1)))
      (- (sqrt (+ 1.0 z)) (sqrt z))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double tmp;
	if (z <= 4.8e-7) {
		tmp = ((1.0 - sqrt(x)) + (t_1 - sqrt(y))) + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + (1.0 + ((z * 0.5) - sqrt(z))));
	} else {
		tmp = ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + t_1))) + (sqrt((1.0 + z)) - 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) :: tmp
    t_1 = sqrt((1.0d0 + y))
    if (z <= 4.8d-7) then
        tmp = ((1.0d0 - sqrt(x)) + (t_1 - sqrt(y))) + ((1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t))) + (1.0d0 + ((z * 0.5d0) - sqrt(z))))
    else
        tmp = ((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (1.0d0 / (sqrt(y) + t_1))) + (sqrt((1.0d0 + z)) - 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 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 4.8e-7) {
		tmp = ((1.0 - Math.sqrt(x)) + (t_1 - Math.sqrt(y))) + ((1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))) + (1.0 + ((z * 0.5) - Math.sqrt(z))));
	} else {
		tmp = ((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (1.0 / (Math.sqrt(y) + t_1))) + (Math.sqrt((1.0 + z)) - Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 4.8e-7:
		tmp = ((1.0 - math.sqrt(x)) + (t_1 - math.sqrt(y))) + ((1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))) + (1.0 + ((z * 0.5) - math.sqrt(z))))
	else:
		tmp = ((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (1.0 / (math.sqrt(y) + t_1))) + (math.sqrt((1.0 + z)) - math.sqrt(z))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 4.8e-7)
		tmp = Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))) + Float64(1.0 + Float64(Float64(z * 0.5) - sqrt(z)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(1.0 / Float64(sqrt(y) + t_1))) + Float64(sqrt(Float64(1.0 + z)) - sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 4.8e-7)
		tmp = ((1.0 - sqrt(x)) + (t_1 - sqrt(y))) + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + (1.0 + ((z * 0.5) - sqrt(z))));
	else
		tmp = ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + t_1))) + (sqrt((1.0 + z)) - sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 4.8e-7], N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(z * 0.5), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 4.79999999999999957e-7

   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. sub-neg 97.7%

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

      3. sub-neg 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0 49.7%

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

   6. Taylor expanded in z around 0 49.7%

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

      1. associate--l+ 49.7%

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

   8. Simplified 49.7%

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

      1. flip-- 49.8%

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

      2. div-inv 49.8%

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

      3. add-sqr-sqrt 37.5%

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

      4. +-commutative 37.5%

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

      5. add-sqr-sqrt 49.9%

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

      6. associate--l+ 49.9%

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

   10. Applied egg-rr 49.9%

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

       1. associate-*r/ 49.9%

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

       2. *-rgt-identity 49.9%

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

       3. associate-+r- 49.9%

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

       4. +-commutative 49.9%

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

       5. associate-+r- 50.0%

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

       6. +-inverses 50.0%

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

       7. metadata-eval 50.0%

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

       8. +-commutative 50.0%

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

   12. Simplified 50.0%

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

   if 4.79999999999999957e-7 < z

   1. Initial program 89.8%

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

      1. associate-+l+ 89.8%

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

      2. sub-neg 89.8%

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

      3. sub-neg 89.8%

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

      4. +-commutative 89.8%

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

      5. +-commutative 89.8%

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

      6. +-commutative 89.8%

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

   3. Simplified 89.8%

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

      1. flip-- 90.0%

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

      2. div-inv 90.0%

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

      3. add-sqr-sqrt 69.2%

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

      4. +-commutative 69.2%

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

      5. add-sqr-sqrt 90.4%

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

      6. +-commutative 90.4%

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

   6. Applied egg-rr 90.4%

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

      1. associate--l+ 93.4%

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

      2. +-inverses 93.4%

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

      3. metadata-eval 93.4%

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

      4. *-lft-identity 93.4%

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

      5. +-commutative 93.4%

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

   8. Simplified 93.4%

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

      1. flip-- 93.9%

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

      2. div-inv 93.9%

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

      3. add-sqr-sqrt 70.4%

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

      4. add-sqr-sqrt 93.9%

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

      5. associate--l+ 96.2%

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

   10. Applied egg-rr 96.2%

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

       1. +-inverses 96.2%

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

       2. metadata-eval 96.2%

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

       3. *-lft-identity 96.2%

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

       4. +-commutative 96.2%

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

   12. Simplified 96.2%

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

   13. Taylor expanded in t around inf 50.4%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(1 + \left(z \cdot 0.5 - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(z \cdot 0.5 - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 2.3e-7)
   (+
    (- (- 2.0 (sqrt x)) (sqrt y))
    (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (+ 1.0 (- (* z 0.5) (sqrt z)))))
   (+
    (+
     (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))
     (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y)))))
    (- (sqrt (+ 1.0 z)) (sqrt z)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.3e-7) {
		tmp = ((2.0 - sqrt(x)) - sqrt(y)) + ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 + ((z * 0.5) - sqrt(z))));
	} else {
		tmp = ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) + (sqrt((1.0 + z)) - 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) :: tmp
    if (z <= 2.3d-7) then
        tmp = ((2.0d0 - sqrt(x)) - sqrt(y)) + ((sqrt((1.0d0 + t)) - sqrt(t)) + (1.0d0 + ((z * 0.5d0) - sqrt(z))))
    else
        tmp = ((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (1.0d0 / (sqrt(y) + sqrt((1.0d0 + y))))) + (sqrt((1.0d0 + z)) - 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 tmp;
	if (z <= 2.3e-7) {
		tmp = ((2.0 - Math.sqrt(x)) - Math.sqrt(y)) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (1.0 + ((z * 0.5) - Math.sqrt(z))));
	} else {
		tmp = ((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (1.0 / (Math.sqrt(y) + Math.sqrt((1.0 + y))))) + (Math.sqrt((1.0 + z)) - Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 2.3e-7:
		tmp = ((2.0 - math.sqrt(x)) - math.sqrt(y)) + ((math.sqrt((1.0 + t)) - math.sqrt(t)) + (1.0 + ((z * 0.5) - math.sqrt(z))))
	else:
		tmp = ((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (1.0 / (math.sqrt(y) + math.sqrt((1.0 + y))))) + (math.sqrt((1.0 + z)) - math.sqrt(z))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 2.3e-7)
		tmp = Float64(Float64(Float64(2.0 - sqrt(x)) - sqrt(y)) + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(1.0 + Float64(Float64(z * 0.5) - sqrt(z)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))) + Float64(sqrt(Float64(1.0 + z)) - 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)
	tmp = 0.0;
	if (z <= 2.3e-7)
		tmp = ((2.0 - sqrt(x)) - sqrt(y)) + ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 + ((z * 0.5) - sqrt(z))));
	else
		tmp = ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) + (sqrt((1.0 + z)) - 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_] := If[LessEqual[z, 2.3e-7], N[(N[(N[(2.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(z * 0.5), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.29999999999999995e-7

   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. sub-neg 97.7%

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

      3. sub-neg 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0 49.7%

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

   6. Taylor expanded in z around 0 49.7%

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

      1. associate--l+ 49.7%

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

   8. Simplified 49.7%

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

   9. Taylor expanded in y around 0 22.8%

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

       1. associate--r+ 22.8%

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

   11. Simplified 22.8%

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

   if 2.29999999999999995e-7 < z

   1. Initial program 89.8%

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

      1. associate-+l+ 89.8%

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

      2. sub-neg 89.8%

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

      3. sub-neg 89.8%

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

      4. +-commutative 89.8%

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

      5. +-commutative 89.8%

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

      6. +-commutative 89.8%

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

   3. Simplified 89.8%

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

      1. flip-- 90.0%

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

      2. div-inv 90.0%

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

      3. add-sqr-sqrt 69.2%

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

      4. +-commutative 69.2%

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

      5. add-sqr-sqrt 90.4%

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

      6. +-commutative 90.4%

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

   6. Applied egg-rr 90.4%

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

      1. associate--l+ 93.4%

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

      2. +-inverses 93.4%

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

      3. metadata-eval 93.4%

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

      4. *-lft-identity 93.4%

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

      5. +-commutative 93.4%

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

   8. Simplified 93.4%

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

      1. flip-- 93.9%

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

      2. div-inv 93.9%

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

      3. add-sqr-sqrt 70.4%

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

      4. add-sqr-sqrt 93.9%

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

      5. associate--l+ 96.2%

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

   10. Applied egg-rr 96.2%

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

       1. +-inverses 96.2%

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

       2. metadata-eval 96.2%

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

       3. *-lft-identity 96.2%

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

       4. +-commutative 96.2%

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

   12. Simplified 96.2%

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

   13. Taylor expanded in t around inf 50.4%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(z \cdot 0.5 - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ \mathbf{if}\;y \leq 1.4 \cdot 10^{-23}:\\ \;\;\;\;\left(t\_1 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right)\\ \mathbf{elif}\;y \leq 7500000000:\\ \;\;\;\;t\_1 + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{1 + \sqrt{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} \cdot 2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 z)) (sqrt z))))
   (if (<= y 1.4e-23)
     (+
      (+ t_1 (- (sqrt (+ 1.0 t)) (sqrt t)))
      (+ (- 1.0 (sqrt x)) (- 1.0 (sqrt y))))
     (if (<= y 7500000000.0)
       (+
        t_1
        (+ (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y)))) (/ 1.0 (+ 1.0 (sqrt x)))))
       (+
        t_1
        (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (/ 1.0 (* (sqrt y) 2.0))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z)) - sqrt(z);
	double tmp;
	if (y <= 1.4e-23) {
		tmp = (t_1 + (sqrt((1.0 + t)) - sqrt(t))) + ((1.0 - sqrt(x)) + (1.0 - sqrt(y)));
	} else if (y <= 7500000000.0) {
		tmp = t_1 + ((1.0 / (sqrt(y) + sqrt((1.0 + y)))) + (1.0 / (1.0 + sqrt(x))));
	} else {
		tmp = t_1 + ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) * 2.0)));
	}
	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((1.0d0 + z)) - sqrt(z)
    if (y <= 1.4d-23) then
        tmp = (t_1 + (sqrt((1.0d0 + t)) - sqrt(t))) + ((1.0d0 - sqrt(x)) + (1.0d0 - sqrt(y)))
    else if (y <= 7500000000.0d0) then
        tmp = t_1 + ((1.0d0 / (sqrt(y) + sqrt((1.0d0 + y)))) + (1.0d0 / (1.0d0 + sqrt(x))))
    else
        tmp = t_1 + ((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (1.0d0 / (sqrt(y) * 2.0d0)))
    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((1.0 + z)) - Math.sqrt(z);
	double tmp;
	if (y <= 1.4e-23) {
		tmp = (t_1 + (Math.sqrt((1.0 + t)) - Math.sqrt(t))) + ((1.0 - Math.sqrt(x)) + (1.0 - Math.sqrt(y)));
	} else if (y <= 7500000000.0) {
		tmp = t_1 + ((1.0 / (Math.sqrt(y) + Math.sqrt((1.0 + y)))) + (1.0 / (1.0 + Math.sqrt(x))));
	} else {
		tmp = t_1 + ((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (1.0 / (Math.sqrt(y) * 2.0)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z)) - math.sqrt(z)
	tmp = 0
	if y <= 1.4e-23:
		tmp = (t_1 + (math.sqrt((1.0 + t)) - math.sqrt(t))) + ((1.0 - math.sqrt(x)) + (1.0 - math.sqrt(y)))
	elif y <= 7500000000.0:
		tmp = t_1 + ((1.0 / (math.sqrt(y) + math.sqrt((1.0 + y)))) + (1.0 / (1.0 + math.sqrt(x))))
	else:
		tmp = t_1 + ((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (1.0 / (math.sqrt(y) * 2.0)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	tmp = 0.0
	if (y <= 1.4e-23)
		tmp = Float64(Float64(t_1 + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) + Float64(Float64(1.0 - sqrt(x)) + Float64(1.0 - sqrt(y))));
	elseif (y <= 7500000000.0)
		tmp = Float64(t_1 + Float64(Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y)))) + Float64(1.0 / Float64(1.0 + sqrt(x)))));
	else
		tmp = Float64(t_1 + Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(1.0 / Float64(sqrt(y) * 2.0))));
	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((1.0 + z)) - sqrt(z);
	tmp = 0.0;
	if (y <= 1.4e-23)
		tmp = (t_1 + (sqrt((1.0 + t)) - sqrt(t))) + ((1.0 - sqrt(x)) + (1.0 - sqrt(y)));
	elseif (y <= 7500000000.0)
		tmp = t_1 + ((1.0 / (sqrt(y) + sqrt((1.0 + y)))) + (1.0 / (1.0 + sqrt(x))));
	else
		tmp = t_1 + ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) * 2.0)));
	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[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.4e-23], N[(N[(t$95$1 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7500000000.0], N[(t$95$1 + N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.3999999999999999e-23

   1. Initial program 98.1%

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

      1. associate-+l+ 98.1%

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

      2. sub-neg 98.1%

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

      3. sub-neg 98.1%

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

      4. +-commutative 98.1%

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

      5. +-commutative 98.1%

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

      6. +-commutative 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in x around 0 48.7%

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

   6. Taylor expanded in y around 0 48.7%

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

   if 1.3999999999999999e-23 < y < 7.5e9

   1. Initial program 96.6%

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

      1. associate-+l+ 96.6%

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

      2. sub-neg 96.6%

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

      3. sub-neg 96.6%

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

      4. +-commutative 96.6%

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

      5. +-commutative 96.6%

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

      6. +-commutative 96.6%

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

   3. Simplified 96.6%

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

      1. flip-- 96.6%

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

      2. div-inv 96.6%

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

      3. add-sqr-sqrt 82.2%

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

      4. +-commutative 82.2%

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

      5. add-sqr-sqrt 96.6%

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

      6. +-commutative 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. associate--l+ 96.6%

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

      2. +-inverses 96.6%

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

      3. metadata-eval 96.6%

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

      4. *-lft-identity 96.6%

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

      5. +-commutative 96.6%

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

   8. Simplified 96.6%

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

      1. flip-- 98.0%

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

      2. div-inv 98.0%

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

      3. add-sqr-sqrt 99.4%

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

      4. add-sqr-sqrt 99.6%

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

      5. associate--l+ 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. +-inverses 99.6%

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

       2. metadata-eval 99.6%

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

       3. *-lft-identity 99.6%

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

       4. +-commutative 99.6%

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

   12. Simplified 99.6%

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

   13. Taylor expanded in t around inf 55.9%

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

   14. Taylor expanded in x around 0 55.9%

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

   if 7.5e9 < y

   1. Initial program 89.4%

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

      1. associate-+l+ 89.4%

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

      2. sub-neg 89.4%

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

      3. sub-neg 89.4%

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

      4. +-commutative 89.4%

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

      5. +-commutative 89.4%

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

      6. +-commutative 89.4%

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

   3. Simplified 89.4%

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

      1. flip-- 89.6%

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

      2. div-inv 89.6%

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

      3. add-sqr-sqrt 72.2%

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

      4. +-commutative 72.2%

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

      5. add-sqr-sqrt 90.1%

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

      6. +-commutative 90.1%

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

   6. Applied egg-rr 90.1%

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

      1. associate--l+ 93.1%

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

      2. +-inverses 93.1%

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

      3. metadata-eval 93.1%

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

      4. *-lft-identity 93.1%

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

      5. +-commutative 93.1%

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

   8. Simplified 93.1%

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

      1. flip-- 93.5%

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

      2. div-inv 93.5%

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

      3. add-sqr-sqrt 48.6%

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

      4. add-sqr-sqrt 93.7%

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

      5. associate--l+ 96.3%

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

   10. Applied egg-rr 96.3%

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

       1. +-inverses 96.3%

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

       2. metadata-eval 96.3%

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

       3. *-lft-identity 96.3%

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

       4. +-commutative 96.3%

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

   12. Simplified 96.3%

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

   13. Taylor expanded in t around inf 55.1%

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

   14. Taylor expanded in y around inf 55.1%

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

       1. *-commutative 55.1%

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

   16. Simplified 55.1%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-23}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right)\\ \mathbf{elif}\;y \leq 7500000000:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{1 + \sqrt{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 18500000000:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \left(\left(\sqrt{x} + \sqrt{t}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{1 + \sqrt{x}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 18500000000.0)
   (+
    3.0
    (-
     (+ (sqrt (+ 1.0 t)) (* z 0.5))
     (+ (+ (sqrt x) (sqrt t)) (+ (sqrt y) (sqrt z)))))
   (+
    (- (sqrt (+ 1.0 z)) (sqrt z))
    (+ (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y)))) (/ 1.0 (+ 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 (t <= 18500000000.0) {
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - ((sqrt(x) + sqrt(t)) + (sqrt(y) + sqrt(z))));
	} else {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 / (sqrt(y) + sqrt((1.0 + y)))) + (1.0 / (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 (t <= 18500000000.0d0) then
        tmp = 3.0d0 + ((sqrt((1.0d0 + t)) + (z * 0.5d0)) - ((sqrt(x) + sqrt(t)) + (sqrt(y) + sqrt(z))))
    else
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + ((1.0d0 / (sqrt(y) + sqrt((1.0d0 + y)))) + (1.0d0 / (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 (t <= 18500000000.0) {
		tmp = 3.0 + ((Math.sqrt((1.0 + t)) + (z * 0.5)) - ((Math.sqrt(x) + Math.sqrt(t)) + (Math.sqrt(y) + Math.sqrt(z))));
	} else {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + ((1.0 / (Math.sqrt(y) + Math.sqrt((1.0 + y)))) + (1.0 / (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 t <= 18500000000.0:
		tmp = 3.0 + ((math.sqrt((1.0 + t)) + (z * 0.5)) - ((math.sqrt(x) + math.sqrt(t)) + (math.sqrt(y) + math.sqrt(z))))
	else:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + ((1.0 / (math.sqrt(y) + math.sqrt((1.0 + y)))) + (1.0 / (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 (t <= 18500000000.0)
		tmp = Float64(3.0 + Float64(Float64(sqrt(Float64(1.0 + t)) + Float64(z * 0.5)) - Float64(Float64(sqrt(x) + sqrt(t)) + Float64(sqrt(y) + sqrt(z)))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y)))) + Float64(1.0 / Float64(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 (t <= 18500000000.0)
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - ((sqrt(x) + sqrt(t)) + (sqrt(y) + sqrt(z))));
	else
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 / (sqrt(y) + sqrt((1.0 + y)))) + (1.0 / (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[t, 18500000000.0], N[(3.0 + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.85e10

   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. sub-neg 97.7%

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

      3. sub-neg 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0 54.1%

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

   6. Taylor expanded in z around 0 27.2%

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

      1. associate--l+ 27.2%

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

   8. Simplified 27.2%

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

   9. Taylor expanded in y around 0 12.9%

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

       1. associate--l+ 12.9%

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

       2. +-commutative 12.9%

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

       3. associate-+r+ 12.9%

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

   11. Simplified 12.9%

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

   if 1.85e10 < t

   1. Initial program 89.4%

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

      1. associate-+l+ 89.4%

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

      2. sub-neg 89.4%

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

      3. sub-neg 89.4%

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

      4. +-commutative 89.4%

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

      5. +-commutative 89.4%

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

      6. +-commutative 89.4%

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

   3. Simplified 89.4%

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

      1. flip-- 89.3%

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

      2. div-inv 89.3%

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

      3. add-sqr-sqrt 73.1%

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

      4. +-commutative 73.1%

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

      5. add-sqr-sqrt 89.7%

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

      6. +-commutative 89.7%

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

   6. Applied egg-rr 89.7%

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

      1. associate--l+ 93.0%

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

      2. +-inverses 93.0%

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

      3. metadata-eval 93.0%

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

      4. *-lft-identity 93.0%

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

      5. +-commutative 93.0%

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

   8. Simplified 93.0%

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

      1. flip-- 93.3%

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

      2. div-inv 93.3%

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

      3. add-sqr-sqrt 69.8%

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

      4. add-sqr-sqrt 93.4%

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

      5. associate--l+ 95.9%

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

   10. Applied egg-rr 95.9%

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

       1. +-inverses 95.9%

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

       2. metadata-eval 95.9%

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

       3. *-lft-identity 95.9%

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

       4. +-commutative 95.9%

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

   12. Simplified 95.9%

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

   13. Taylor expanded in t around inf 95.7%

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

   14. Taylor expanded in x around 0 89.3%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 18500000000:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \left(\left(\sqrt{x} + \sqrt{t}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{1 + \sqrt{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(z \cdot 0.5 - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{1 + \sqrt{x}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 2.3e-7)
   (+
    (- (- 2.0 (sqrt x)) (sqrt y))
    (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (+ 1.0 (- (* z 0.5) (sqrt z)))))
   (+
    (- (sqrt (+ 1.0 z)) (sqrt z))
    (+ (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y)))) (/ 1.0 (+ 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 (z <= 2.3e-7) {
		tmp = ((2.0 - sqrt(x)) - sqrt(y)) + ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 + ((z * 0.5) - sqrt(z))));
	} else {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 / (sqrt(y) + sqrt((1.0 + y)))) + (1.0 / (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 (z <= 2.3d-7) then
        tmp = ((2.0d0 - sqrt(x)) - sqrt(y)) + ((sqrt((1.0d0 + t)) - sqrt(t)) + (1.0d0 + ((z * 0.5d0) - sqrt(z))))
    else
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + ((1.0d0 / (sqrt(y) + sqrt((1.0d0 + y)))) + (1.0d0 / (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 (z <= 2.3e-7) {
		tmp = ((2.0 - Math.sqrt(x)) - Math.sqrt(y)) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (1.0 + ((z * 0.5) - Math.sqrt(z))));
	} else {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + ((1.0 / (Math.sqrt(y) + Math.sqrt((1.0 + y)))) + (1.0 / (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 z <= 2.3e-7:
		tmp = ((2.0 - math.sqrt(x)) - math.sqrt(y)) + ((math.sqrt((1.0 + t)) - math.sqrt(t)) + (1.0 + ((z * 0.5) - math.sqrt(z))))
	else:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + ((1.0 / (math.sqrt(y) + math.sqrt((1.0 + y)))) + (1.0 / (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 (z <= 2.3e-7)
		tmp = Float64(Float64(Float64(2.0 - sqrt(x)) - sqrt(y)) + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(1.0 + Float64(Float64(z * 0.5) - sqrt(z)))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y)))) + Float64(1.0 / Float64(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 (z <= 2.3e-7)
		tmp = ((2.0 - sqrt(x)) - sqrt(y)) + ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 + ((z * 0.5) - sqrt(z))));
	else
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 / (sqrt(y) + sqrt((1.0 + y)))) + (1.0 / (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[z, 2.3e-7], N[(N[(N[(2.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(z * 0.5), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.29999999999999995e-7

   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. sub-neg 97.7%

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

      3. sub-neg 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0 49.7%

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

   6. Taylor expanded in z around 0 49.7%

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

      1. associate--l+ 49.7%

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

   8. Simplified 49.7%

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

   9. Taylor expanded in y around 0 22.8%

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

       1. associate--r+ 22.8%

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

   11. Simplified 22.8%

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

   if 2.29999999999999995e-7 < z

   1. Initial program 89.8%

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

      1. associate-+l+ 89.8%

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

      2. sub-neg 89.8%

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

      3. sub-neg 89.8%

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

      4. +-commutative 89.8%

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

      5. +-commutative 89.8%

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

      6. +-commutative 89.8%

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

   3. Simplified 89.8%

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

      1. flip-- 90.0%

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

      2. div-inv 90.0%

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

      3. add-sqr-sqrt 69.2%

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

      4. +-commutative 69.2%

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

      5. add-sqr-sqrt 90.4%

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

      6. +-commutative 90.4%

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

   6. Applied egg-rr 90.4%

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

      1. associate--l+ 93.4%

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

      2. +-inverses 93.4%

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

      3. metadata-eval 93.4%

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

      4. *-lft-identity 93.4%

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

      5. +-commutative 93.4%

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

   8. Simplified 93.4%

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

      1. flip-- 93.9%

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

      2. div-inv 93.9%

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

      3. add-sqr-sqrt 70.4%

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

      4. add-sqr-sqrt 93.9%

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

      5. associate--l+ 96.2%

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

   10. Applied egg-rr 96.2%

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

       1. +-inverses 96.2%

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

       2. metadata-eval 96.2%

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

       3. *-lft-identity 96.2%

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

       4. +-commutative 96.2%

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

   12. Simplified 96.2%

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

   13. Taylor expanded in t around inf 50.4%

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

   14. Taylor expanded in x around 0 46.0%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(z \cdot 0.5 - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{1 + \sqrt{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 18500000000:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \left(\left(\sqrt{x} + \sqrt{t}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 18500000000.0)
   (+
    3.0
    (-
     (+ (sqrt (+ 1.0 t)) (* z 0.5))
     (+ (+ (sqrt x) (sqrt t)) (+ (sqrt y) (sqrt z)))))
   (+
    (- (sqrt (+ 1.0 z)) (sqrt z))
    (+ (- 1.0 (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 18500000000.0) {
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - ((sqrt(x) + sqrt(t)) + (sqrt(y) + sqrt(z))));
	} else {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 18500000000.0d0) then
        tmp = 3.0d0 + ((sqrt((1.0d0 + t)) + (z * 0.5d0)) - ((sqrt(x) + sqrt(t)) + (sqrt(y) + sqrt(z))))
    else
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + ((1.0d0 - sqrt(x)) + (sqrt((1.0d0 + y)) - sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 18500000000.0) {
		tmp = 3.0 + ((Math.sqrt((1.0 + t)) + (z * 0.5)) - ((Math.sqrt(x) + Math.sqrt(t)) + (Math.sqrt(y) + Math.sqrt(z))));
	} else {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + ((1.0 - Math.sqrt(x)) + (Math.sqrt((1.0 + y)) - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 18500000000.0:
		tmp = 3.0 + ((math.sqrt((1.0 + t)) + (z * 0.5)) - ((math.sqrt(x) + math.sqrt(t)) + (math.sqrt(y) + math.sqrt(z))))
	else:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + ((1.0 - math.sqrt(x)) + (math.sqrt((1.0 + y)) - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 18500000000.0)
		tmp = Float64(3.0 + Float64(Float64(sqrt(Float64(1.0 + t)) + Float64(z * 0.5)) - Float64(Float64(sqrt(x) + sqrt(t)) + Float64(sqrt(y) + sqrt(z)))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(1.0 - sqrt(x)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 18500000000.0)
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - ((sqrt(x) + sqrt(t)) + (sqrt(y) + sqrt(z))));
	else
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 18500000000.0], N[(3.0 + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.85e10

   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. sub-neg 97.7%

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

      3. sub-neg 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0 54.1%

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

   6. Taylor expanded in z around 0 27.2%

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

      1. associate--l+ 27.2%

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

   8. Simplified 27.2%

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

   9. Taylor expanded in y around 0 12.9%

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

       1. associate--l+ 12.9%

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

       2. +-commutative 12.9%

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

       3. associate-+r+ 12.9%

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

   11. Simplified 12.9%

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

   if 1.85e10 < t

   1. Initial program 89.4%

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

      1. associate-+l+ 89.4%

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

      2. sub-neg 89.4%

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

      3. sub-neg 89.4%

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

      4. +-commutative 89.4%

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

      5. +-commutative 89.4%

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

      6. +-commutative 89.4%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in x around 0 50.6%

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

   6. Taylor expanded in t around inf 50.6%

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

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 18500000000:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \left(\left(\sqrt{x} + \sqrt{t}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 18500000000:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \left(\left(\sqrt{x} + \sqrt{t}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 18500000000.0)
   (+
    3.0
    (-
     (+ (sqrt (+ 1.0 t)) (* z 0.5))
     (+ (+ (sqrt x) (sqrt t)) (+ (sqrt y) (sqrt z)))))
   (+
    (+ (- 1.0 (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y)))
    (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 18500000000.0) {
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - ((sqrt(x) + sqrt(t)) + (sqrt(y) + sqrt(z))));
	} else {
		tmp = ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + (1.0 / (sqrt((1.0 + z)) + 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) :: tmp
    if (t <= 18500000000.0d0) then
        tmp = 3.0d0 + ((sqrt((1.0d0 + t)) + (z * 0.5d0)) - ((sqrt(x) + sqrt(t)) + (sqrt(y) + sqrt(z))))
    else
        tmp = ((1.0d0 - sqrt(x)) + (sqrt((1.0d0 + y)) - sqrt(y))) + (1.0d0 / (sqrt((1.0d0 + z)) + 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 tmp;
	if (t <= 18500000000.0) {
		tmp = 3.0 + ((Math.sqrt((1.0 + t)) + (z * 0.5)) - ((Math.sqrt(x) + Math.sqrt(t)) + (Math.sqrt(y) + Math.sqrt(z))));
	} else {
		tmp = ((1.0 - Math.sqrt(x)) + (Math.sqrt((1.0 + y)) - Math.sqrt(y))) + (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 18500000000.0:
		tmp = 3.0 + ((math.sqrt((1.0 + t)) + (z * 0.5)) - ((math.sqrt(x) + math.sqrt(t)) + (math.sqrt(y) + math.sqrt(z))))
	else:
		tmp = ((1.0 - math.sqrt(x)) + (math.sqrt((1.0 + y)) - math.sqrt(y))) + (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 18500000000.0)
		tmp = Float64(3.0 + Float64(Float64(sqrt(Float64(1.0 + t)) + Float64(z * 0.5)) - Float64(Float64(sqrt(x) + sqrt(t)) + Float64(sqrt(y) + sqrt(z)))));
	else
		tmp = Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + 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)
	tmp = 0.0;
	if (t <= 18500000000.0)
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - ((sqrt(x) + sqrt(t)) + (sqrt(y) + sqrt(z))));
	else
		tmp = ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + (1.0 / (sqrt((1.0 + z)) + 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_] := If[LessEqual[t, 18500000000.0], N[(3.0 + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.85e10

   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. sub-neg 97.7%

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

      3. sub-neg 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0 54.1%

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

   6. Taylor expanded in z around 0 27.2%

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

      1. associate--l+ 27.2%

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

   8. Simplified 27.2%

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

   9. Taylor expanded in y around 0 12.9%

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

       1. associate--l+ 12.9%

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

       2. +-commutative 12.9%

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

       3. associate-+r+ 12.9%

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

   11. Simplified 12.9%

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

   if 1.85e10 < t

   1. Initial program 89.4%

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

      1. associate-+l+ 89.4%

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

      2. sub-neg 89.4%

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

      3. sub-neg 89.4%

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

      4. +-commutative 89.4%

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

      5. +-commutative 89.4%

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

      6. +-commutative 89.4%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in x around 0 50.6%

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

      1. flip-- 50.8%

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

      2. div-inv 50.8%

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

      3. add-sqr-sqrt 40.7%

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

      4. add-sqr-sqrt 51.1%

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

      5. associate--l+ 51.2%

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

   7. Applied egg-rr 51.2%

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

      1. +-inverses 51.2%

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

      2. metadata-eval 51.2%

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

      3. *-lft-identity 51.2%

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

      4. +-commutative 51.2%

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

   9. Simplified 51.2%

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

   10. Taylor expanded in t around inf 51.2%

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

       1. +-commutative 51.2%

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

   12. Simplified 51.2%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 18500000000:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \left(\left(\sqrt{x} + \sqrt{t}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 18500000000:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \left(\left(\sqrt{x} + \sqrt{t}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(1 - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 18500000000.0)
   (+
    3.0
    (-
     (+ (sqrt (+ 1.0 t)) (* z 0.5))
     (+ (+ (sqrt x) (sqrt t)) (+ (sqrt y) (sqrt z)))))
   (+
    (- (sqrt (+ 1.0 z)) (sqrt z))
    (+ (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y)))) (- 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 (t <= 18500000000.0) {
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - ((sqrt(x) + sqrt(t)) + (sqrt(y) + sqrt(z))));
	} else {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 / (sqrt(y) + sqrt((1.0 + y)))) + (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 (t <= 18500000000.0d0) then
        tmp = 3.0d0 + ((sqrt((1.0d0 + t)) + (z * 0.5d0)) - ((sqrt(x) + sqrt(t)) + (sqrt(y) + sqrt(z))))
    else
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + ((1.0d0 / (sqrt(y) + sqrt((1.0d0 + y)))) + (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 (t <= 18500000000.0) {
		tmp = 3.0 + ((Math.sqrt((1.0 + t)) + (z * 0.5)) - ((Math.sqrt(x) + Math.sqrt(t)) + (Math.sqrt(y) + Math.sqrt(z))));
	} else {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + ((1.0 / (Math.sqrt(y) + Math.sqrt((1.0 + y)))) + (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 t <= 18500000000.0:
		tmp = 3.0 + ((math.sqrt((1.0 + t)) + (z * 0.5)) - ((math.sqrt(x) + math.sqrt(t)) + (math.sqrt(y) + math.sqrt(z))))
	else:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + ((1.0 / (math.sqrt(y) + math.sqrt((1.0 + y)))) + (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 (t <= 18500000000.0)
		tmp = Float64(3.0 + Float64(Float64(sqrt(Float64(1.0 + t)) + Float64(z * 0.5)) - Float64(Float64(sqrt(x) + sqrt(t)) + Float64(sqrt(y) + sqrt(z)))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y)))) + Float64(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 (t <= 18500000000.0)
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - ((sqrt(x) + sqrt(t)) + (sqrt(y) + sqrt(z))));
	else
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 / (sqrt(y) + sqrt((1.0 + y)))) + (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[t, 18500000000.0], N[(3.0 + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.85e10

   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. sub-neg 97.7%

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

      3. sub-neg 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0 54.1%

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

   6. Taylor expanded in z around 0 27.2%

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

      1. associate--l+ 27.2%

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

   8. Simplified 27.2%

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

   9. Taylor expanded in y around 0 12.9%

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

       1. associate--l+ 12.9%

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

       2. +-commutative 12.9%

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

       3. associate-+r+ 12.9%

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

   11. Simplified 12.9%

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

   if 1.85e10 < t

   1. Initial program 89.4%

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

      1. associate-+l+ 89.4%

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

      2. sub-neg 89.4%

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

      3. sub-neg 89.4%

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

      4. +-commutative 89.4%

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

      5. +-commutative 89.4%

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

      6. +-commutative 89.4%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in x around 0 50.6%

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

   6. Taylor expanded in t around inf 50.6%

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

      1. flip-- 93.3%

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

      2. div-inv 93.3%

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

      3. add-sqr-sqrt 69.8%

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

      4. add-sqr-sqrt 93.4%

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

      5. associate--l+ 95.9%

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

   8. Applied egg-rr 51.0%

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

      1. +-inverses 95.9%

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

      2. metadata-eval 95.9%

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

      3. *-lft-identity 95.9%

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

      4. +-commutative 95.9%

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

   10. Simplified 51.0%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 18500000000:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \left(\left(\sqrt{x} + \sqrt{t}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(1 - \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 2.5e-34)
   (+ 3.0 (- (+ (sqrt (+ 1.0 t)) (* z 0.5)) (sqrt t)))
   (+
    (- (sqrt (+ 1.0 z)) (sqrt z))
    (+ (- 1.0 (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.5e-34) {
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - sqrt(t));
	} else {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 2.5d-34) then
        tmp = 3.0d0 + ((sqrt((1.0d0 + t)) + (z * 0.5d0)) - sqrt(t))
    else
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + ((1.0d0 - sqrt(x)) + (sqrt((1.0d0 + y)) - sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.5e-34) {
		tmp = 3.0 + ((Math.sqrt((1.0 + t)) + (z * 0.5)) - Math.sqrt(t));
	} else {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + ((1.0 - Math.sqrt(x)) + (Math.sqrt((1.0 + y)) - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 2.5e-34:
		tmp = 3.0 + ((math.sqrt((1.0 + t)) + (z * 0.5)) - math.sqrt(t))
	else:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + ((1.0 - math.sqrt(x)) + (math.sqrt((1.0 + y)) - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 2.5e-34)
		tmp = Float64(3.0 + Float64(Float64(sqrt(Float64(1.0 + t)) + Float64(z * 0.5)) - sqrt(t)));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(1.0 - sqrt(x)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 2.5e-34)
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - sqrt(t));
	else
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 2.5e-34], N[(3.0 + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.5000000000000001e-34

   1. Initial program 97.8%

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

      1. associate-+l+ 97.8%

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

      2. sub-neg 97.8%

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

      3. sub-neg 97.8%

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

      4. +-commutative 97.8%

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

      5. +-commutative 97.8%

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

      6. +-commutative 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in x around 0 50.1%

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

   6. Taylor expanded in z around 0 50.1%

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

      1. associate--l+ 50.1%

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

   8. Simplified 50.1%

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

   9. Taylor expanded in y around 0 14.1%

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

       1. associate--l+ 26.5%

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

       2. +-commutative 26.5%

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

       3. associate-+r+ 26.5%

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

   11. Simplified 26.5%

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

   12. Taylor expanded in t around inf 37.0%

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

   if 2.5000000000000001e-34 < z

   1. Initial program 90.3%

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

      1. associate-+l+ 90.3%

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

      2. sub-neg 90.3%

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

      3. sub-neg 90.3%

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

      4. +-commutative 90.3%

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

      5. +-commutative 90.3%

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

      6. +-commutative 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in x around 0 54.3%

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

   6. Taylor expanded in t around inf 29.8%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 660000000:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(2 + \left(y \cdot 0.5 - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.05e-32)
   (+ 3.0 (- (+ (sqrt (+ 1.0 t)) (* z 0.5)) (sqrt t)))
   (if (<= z 660000000.0)
     (+
      (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z)))
      (+ 2.0 (- (* y 0.5) (+ (sqrt x) (sqrt y)))))
     (+
      (+ (- 1.0 (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y)))
      (* 0.5 (sqrt (/ 1.0 z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.05e-32) {
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - sqrt(t));
	} else if (z <= 660000000.0) {
		tmp = (1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (2.0 + ((y * 0.5) - (sqrt(x) + sqrt(y))));
	} else {
		tmp = ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + (0.5 * sqrt((1.0 / z)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.05d-32) then
        tmp = 3.0d0 + ((sqrt((1.0d0 + t)) + (z * 0.5d0)) - sqrt(t))
    else if (z <= 660000000.0d0) then
        tmp = (1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + (2.0d0 + ((y * 0.5d0) - (sqrt(x) + sqrt(y))))
    else
        tmp = ((1.0d0 - sqrt(x)) + (sqrt((1.0d0 + y)) - sqrt(y))) + (0.5d0 * sqrt((1.0d0 / z)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.05e-32) {
		tmp = 3.0 + ((Math.sqrt((1.0 + t)) + (z * 0.5)) - Math.sqrt(t));
	} else if (z <= 660000000.0) {
		tmp = (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + (2.0 + ((y * 0.5) - (Math.sqrt(x) + Math.sqrt(y))));
	} else {
		tmp = ((1.0 - Math.sqrt(x)) + (Math.sqrt((1.0 + y)) - Math.sqrt(y))) + (0.5 * Math.sqrt((1.0 / z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.05e-32:
		tmp = 3.0 + ((math.sqrt((1.0 + t)) + (z * 0.5)) - math.sqrt(t))
	elif z <= 660000000.0:
		tmp = (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + (2.0 + ((y * 0.5) - (math.sqrt(x) + math.sqrt(y))))
	else:
		tmp = ((1.0 - math.sqrt(x)) + (math.sqrt((1.0 + y)) - math.sqrt(y))) + (0.5 * math.sqrt((1.0 / z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.05e-32)
		tmp = Float64(3.0 + Float64(Float64(sqrt(Float64(1.0 + t)) + Float64(z * 0.5)) - sqrt(t)));
	elseif (z <= 660000000.0)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + Float64(2.0 + Float64(Float64(y * 0.5) - Float64(sqrt(x) + sqrt(y)))));
	else
		tmp = Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))) + Float64(0.5 * sqrt(Float64(1.0 / z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.05e-32)
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - sqrt(t));
	elseif (z <= 660000000.0)
		tmp = (1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (2.0 + ((y * 0.5) - (sqrt(x) + sqrt(y))));
	else
		tmp = ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + (0.5 * sqrt((1.0 / z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 1.05e-32], N[(3.0 + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 660000000.0], N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(N[(y * 0.5), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.05e-32

   1. Initial program 97.8%

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

      1. associate-+l+ 97.8%

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

      2. sub-neg 97.8%

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

      3. sub-neg 97.8%

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

      4. +-commutative 97.8%

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

      5. +-commutative 97.8%

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

      6. +-commutative 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in x around 0 50.1%

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

   6. Taylor expanded in z around 0 50.1%

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

      1. associate--l+ 50.1%

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

   8. Simplified 50.1%

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

   9. Taylor expanded in y around 0 14.1%

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

       1. associate--l+ 26.5%

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

       2. +-commutative 26.5%

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

       3. associate-+r+ 26.5%

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

   11. Simplified 26.5%

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

   12. Taylor expanded in t around inf 37.0%

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

   if 1.05e-32 < z < 6.6e8

   1. Initial program 94.4%

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

      1. associate-+l+ 94.4%

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

      2. sub-neg 94.4%

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

      3. sub-neg 94.4%

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

      4. +-commutative 94.4%

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

      5. +-commutative 94.4%

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

      6. +-commutative 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in x around 0 54.8%

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

   6. Taylor expanded in t around inf 20.2%

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

   7. Taylor expanded in y around 0 16.0%

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

      1. associate--l+ 16.0%

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

      2. *-commutative 16.0%

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

      3. +-commutative 16.0%

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

   9. Simplified 16.0%

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

       1. flip-- 55.1%

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

       2. div-inv 55.0%

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

       3. add-sqr-sqrt 54.3%

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

       4. add-sqr-sqrt 55.3%

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

       5. associate--l+ 55.2%

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

   11. Applied egg-rr 16.0%

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

       1. +-inverses 55.2%

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

       2. metadata-eval 55.2%

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

       3. *-lft-identity 55.2%

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

       4. +-commutative 55.2%

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

   13. Simplified 16.0%

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

   if 6.6e8 < z

   1. Initial program 89.7%

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

      1. associate-+l+ 89.7%

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

      2. sub-neg 89.7%

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

      3. sub-neg 89.7%

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

      4. +-commutative 89.7%

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

      5. +-commutative 89.7%

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

      6. +-commutative 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in x around 0 54.3%

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

   6. Taylor expanded in t around inf 31.2%

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

   7. Taylor expanded in z around inf 31.8%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 660000000:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(2 + \left(y \cdot 0.5 - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 180000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.05e-32)
   (+ 3.0 (- (+ (sqrt (+ 1.0 t)) (* z 0.5)) (sqrt t)))
   (if (<= z 180000.0)
     (- (+ (sqrt (+ 1.0 z)) 2.0) (+ (sqrt x) (+ (sqrt y) (sqrt z))))
     (+
      (+ (- 1.0 (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y)))
      (* 0.5 (sqrt (/ 1.0 z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.05e-32) {
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - sqrt(t));
	} else if (z <= 180000.0) {
		tmp = (sqrt((1.0 + z)) + 2.0) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	} else {
		tmp = ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + (0.5 * sqrt((1.0 / z)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.05d-32) then
        tmp = 3.0d0 + ((sqrt((1.0d0 + t)) + (z * 0.5d0)) - sqrt(t))
    else if (z <= 180000.0d0) then
        tmp = (sqrt((1.0d0 + z)) + 2.0d0) - (sqrt(x) + (sqrt(y) + sqrt(z)))
    else
        tmp = ((1.0d0 - sqrt(x)) + (sqrt((1.0d0 + y)) - sqrt(y))) + (0.5d0 * sqrt((1.0d0 / z)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.05e-32) {
		tmp = 3.0 + ((Math.sqrt((1.0 + t)) + (z * 0.5)) - Math.sqrt(t));
	} else if (z <= 180000.0) {
		tmp = (Math.sqrt((1.0 + z)) + 2.0) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z)));
	} else {
		tmp = ((1.0 - Math.sqrt(x)) + (Math.sqrt((1.0 + y)) - Math.sqrt(y))) + (0.5 * Math.sqrt((1.0 / z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.05e-32:
		tmp = 3.0 + ((math.sqrt((1.0 + t)) + (z * 0.5)) - math.sqrt(t))
	elif z <= 180000.0:
		tmp = (math.sqrt((1.0 + z)) + 2.0) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z)))
	else:
		tmp = ((1.0 - math.sqrt(x)) + (math.sqrt((1.0 + y)) - math.sqrt(y))) + (0.5 * math.sqrt((1.0 / z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.05e-32)
		tmp = Float64(3.0 + Float64(Float64(sqrt(Float64(1.0 + t)) + Float64(z * 0.5)) - sqrt(t)));
	elseif (z <= 180000.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) + 2.0) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))));
	else
		tmp = Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))) + Float64(0.5 * sqrt(Float64(1.0 / z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.05e-32)
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - sqrt(t));
	elseif (z <= 180000.0)
		tmp = (sqrt((1.0 + z)) + 2.0) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	else
		tmp = ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + (0.5 * sqrt((1.0 / z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 1.05e-32], N[(3.0 + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 180000.0], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.05e-32

   1. Initial program 97.8%

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

      1. associate-+l+ 97.8%

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

      2. sub-neg 97.8%

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

      3. sub-neg 97.8%

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

      4. +-commutative 97.8%

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

      5. +-commutative 97.8%

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

      6. +-commutative 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in x around 0 50.1%

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

   6. Taylor expanded in z around 0 50.1%

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

      1. associate--l+ 50.1%

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

   8. Simplified 50.1%

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

   9. Taylor expanded in y around 0 14.1%

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

       1. associate--l+ 26.5%

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

       2. +-commutative 26.5%

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

       3. associate-+r+ 26.5%

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

   11. Simplified 26.5%

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

   12. Taylor expanded in t around inf 37.0%

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

   if 1.05e-32 < z < 1.8e5

   1. Initial program 96.6%

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

      1. associate-+l+ 96.6%

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

      2. sub-neg 96.6%

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

      3. sub-neg 96.6%

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

      4. +-commutative 96.6%

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

      5. +-commutative 96.6%

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

      6. +-commutative 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in x around 0 55.7%

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

   6. Taylor expanded in t around inf 21.4%

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

   7. Taylor expanded in y around 0 14.4%

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

   if 1.8e5 < z

   1. Initial program 89.4%

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

      1. associate-+l+ 89.4%

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

      2. sub-neg 89.4%

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

      3. sub-neg 89.4%

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

      4. +-commutative 89.4%

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

      5. +-commutative 89.4%

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

      6. +-commutative 89.4%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in x around 0 54.2%

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

   6. Taylor expanded in t around inf 30.8%

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

   7. Taylor expanded in z around inf 31.5%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 180000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 89.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ \mathbf{if}\;y \leq 4 \cdot 10^{-6}:\\ \;\;\;\;t\_1 + \left(2 + \left(y \cdot 0.5 - \sqrt{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(\frac{\sqrt{1 + y}}{x} + \left(\frac{1}{x} - \left(\sqrt{\frac{1}{x}} + \frac{\sqrt{y}}{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 z)) (sqrt z))))
   (if (<= y 4e-6)
     (+ t_1 (+ 2.0 (- (* y 0.5) (sqrt y))))
     (if (<= y 1.35e+14)
       (*
        x
        (+
         (/ (sqrt (+ 1.0 y)) x)
         (- (/ 1.0 x) (+ (sqrt (/ 1.0 x)) (/ (sqrt y) x)))))
       (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) t_1)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z)) - sqrt(z);
	double tmp;
	if (y <= 4e-6) {
		tmp = t_1 + (2.0 + ((y * 0.5) - sqrt(y)));
	} else if (y <= 1.35e+14) {
		tmp = x * ((sqrt((1.0 + y)) / x) + ((1.0 / x) - (sqrt((1.0 / x)) + (sqrt(y) / x))));
	} else {
		tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) + t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z)) - sqrt(z)
    if (y <= 4d-6) then
        tmp = t_1 + (2.0d0 + ((y * 0.5d0) - sqrt(y)))
    else if (y <= 1.35d+14) then
        tmp = x * ((sqrt((1.0d0 + y)) / x) + ((1.0d0 / x) - (sqrt((1.0d0 / x)) + (sqrt(y) / x))))
    else
        tmp = (1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double tmp;
	if (y <= 4e-6) {
		tmp = t_1 + (2.0 + ((y * 0.5) - Math.sqrt(y)));
	} else if (y <= 1.35e+14) {
		tmp = x * ((Math.sqrt((1.0 + y)) / x) + ((1.0 / x) - (Math.sqrt((1.0 / x)) + (Math.sqrt(y) / x))));
	} else {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z)) - math.sqrt(z)
	tmp = 0
	if y <= 4e-6:
		tmp = t_1 + (2.0 + ((y * 0.5) - math.sqrt(y)))
	elif y <= 1.35e+14:
		tmp = x * ((math.sqrt((1.0 + y)) / x) + ((1.0 / x) - (math.sqrt((1.0 / x)) + (math.sqrt(y) / x))))
	else:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	tmp = 0.0
	if (y <= 4e-6)
		tmp = Float64(t_1 + Float64(2.0 + Float64(Float64(y * 0.5) - sqrt(y))));
	elseif (y <= 1.35e+14)
		tmp = Float64(x * Float64(Float64(sqrt(Float64(1.0 + y)) / x) + Float64(Float64(1.0 / x) - Float64(sqrt(Float64(1.0 / x)) + Float64(sqrt(y) / x)))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + t_1);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z)) - sqrt(z);
	tmp = 0.0;
	if (y <= 4e-6)
		tmp = t_1 + (2.0 + ((y * 0.5) - sqrt(y)));
	elseif (y <= 1.35e+14)
		tmp = x * ((sqrt((1.0 + y)) / x) + ((1.0 / x) - (sqrt((1.0 / x)) + (sqrt(y) / x))));
	else
		tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4e-6], N[(t$95$1 + N[(2.0 + N[(N[(y * 0.5), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+14], N[(x * N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.99999999999999982e-6

   1. Initial program 98.2%

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

      1. associate-+l+ 98.2%

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

      2. sub-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. +-commutative 98.2%

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

      5. +-commutative 98.2%

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

      6. +-commutative 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in x around 0 50.0%

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

   6. Taylor expanded in t around inf 29.0%

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

   7. Taylor expanded in y around 0 29.0%

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

      1. associate--l+ 29.0%

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

      2. *-commutative 29.0%

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

      3. +-commutative 29.0%

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

   9. Simplified 29.0%

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

   10. Taylor expanded in y around inf 56.4%

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

   if 3.99999999999999982e-6 < y < 1.35e14

   1. Initial program 89.1%

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

      1. associate-+l+ 89.1%

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

      2. sub-neg 89.1%

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

      3. sub-neg 89.1%

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

      4. +-commutative 89.1%

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

      5. +-commutative 89.1%

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

      6. +-commutative 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in x around 0 46.1%

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

   6. Taylor expanded in t around inf 28.9%

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

   7. Taylor expanded in x around inf 3.3%

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

   8. Taylor expanded in z around inf 21.4%

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

      1. associate--l+ 21.0%

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

      2. associate-*l/ 21.0%

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

      3. +-commutative 21.0%

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

      4. *-lft-identity 21.0%

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

      5. +-commutative 21.0%

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

      6. associate-*l/ 21.0%

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

      7. *-lft-identity 21.0%

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

   10. Simplified 21.0%

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

   if 1.35e14 < y

   1. Initial program 89.8%

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

      1. associate-+l+ 89.8%

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

      2. sub-neg 89.8%

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

      3. sub-neg 89.8%

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

      4. +-commutative 89.8%

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

      5. +-commutative 89.8%

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

      6. +-commutative 89.8%

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

   3. Simplified 89.8%

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

      1. flip-- 89.9%

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

      2. div-inv 89.9%

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

      3. add-sqr-sqrt 72.2%

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

      4. +-commutative 72.2%

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

      5. add-sqr-sqrt 90.5%

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

      6. +-commutative 90.5%

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

   6. Applied egg-rr 90.5%

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

      1. associate--l+ 93.6%

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

      2. +-inverses 93.6%

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

      3. metadata-eval 93.6%

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

      4. *-lft-identity 93.6%

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

      5. +-commutative 93.6%

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

   8. Simplified 93.6%

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

      1. flip-- 93.6%

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

      2. div-inv 93.6%

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

      3. add-sqr-sqrt 47.6%

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

      4. add-sqr-sqrt 93.6%

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

      5. associate--l+ 96.2%

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

   10. Applied egg-rr 96.2%

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

       1. +-inverses 96.2%

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

       2. metadata-eval 96.2%

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

       3. *-lft-identity 96.2%

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

       4. +-commutative 96.2%

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

   12. Simplified 96.2%

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

   13. Taylor expanded in t around inf 55.3%

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

   14. Taylor expanded in y around inf 52.8%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(2 + \left(y \cdot 0.5 - \sqrt{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(\frac{\sqrt{1 + y}}{x} + \left(\frac{1}{x} - \left(\sqrt{\frac{1}{x}} + \frac{\sqrt{y}}{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 68.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-33}:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 4.5:\\ \;\;\;\;3 + z \cdot \left(0.5 - \sqrt{\frac{1}{z}}\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+122} \lor \neg \left(z \leq 2.3 \cdot 10^{+144}\right) \land z \leq 5.4 \cdot 10^{+180}:\\ \;\;\;\;2 + \left(y \cdot 0.5 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 2e-33)
   (+ 3.0 (- (+ (sqrt (+ 1.0 t)) (* z 0.5)) (sqrt t)))
   (if (<= z 4.5)
     (+ 3.0 (* z (- 0.5 (sqrt (/ 1.0 z)))))
     (if (or (<= z 1.2e+122) (and (not (<= z 2.3e+144)) (<= z 5.4e+180)))
       (+ 2.0 (- (* y 0.5) (+ (sqrt x) (sqrt y))))
       (+ 1.0 (- (sqrt (+ 1.0 z)) (+ (sqrt x) (sqrt z))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2e-33) {
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - sqrt(t));
	} else if (z <= 4.5) {
		tmp = 3.0 + (z * (0.5 - sqrt((1.0 / z))));
	} else if ((z <= 1.2e+122) || (!(z <= 2.3e+144) && (z <= 5.4e+180))) {
		tmp = 2.0 + ((y * 0.5) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 + (sqrt((1.0 + z)) - (sqrt(x) + 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) :: tmp
    if (z <= 2d-33) then
        tmp = 3.0d0 + ((sqrt((1.0d0 + t)) + (z * 0.5d0)) - sqrt(t))
    else if (z <= 4.5d0) then
        tmp = 3.0d0 + (z * (0.5d0 - sqrt((1.0d0 / z))))
    else if ((z <= 1.2d+122) .or. (.not. (z <= 2.3d+144)) .and. (z <= 5.4d+180)) then
        tmp = 2.0d0 + ((y * 0.5d0) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + z)) - (sqrt(x) + 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 tmp;
	if (z <= 2e-33) {
		tmp = 3.0 + ((Math.sqrt((1.0 + t)) + (z * 0.5)) - Math.sqrt(t));
	} else if (z <= 4.5) {
		tmp = 3.0 + (z * (0.5 - Math.sqrt((1.0 / z))));
	} else if ((z <= 1.2e+122) || (!(z <= 2.3e+144) && (z <= 5.4e+180))) {
		tmp = 2.0 + ((y * 0.5) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(x) + Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 2e-33:
		tmp = 3.0 + ((math.sqrt((1.0 + t)) + (z * 0.5)) - math.sqrt(t))
	elif z <= 4.5:
		tmp = 3.0 + (z * (0.5 - math.sqrt((1.0 / z))))
	elif (z <= 1.2e+122) or (not (z <= 2.3e+144) and (z <= 5.4e+180)):
		tmp = 2.0 + ((y * 0.5) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 + (math.sqrt((1.0 + z)) - (math.sqrt(x) + math.sqrt(z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 2e-33)
		tmp = Float64(3.0 + Float64(Float64(sqrt(Float64(1.0 + t)) + Float64(z * 0.5)) - sqrt(t)));
	elseif (z <= 4.5)
		tmp = Float64(3.0 + Float64(z * Float64(0.5 - sqrt(Float64(1.0 / z)))));
	elseif ((z <= 1.2e+122) || (!(z <= 2.3e+144) && (z <= 5.4e+180)))
		tmp = Float64(2.0 + Float64(Float64(y * 0.5) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(x) + 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)
	tmp = 0.0;
	if (z <= 2e-33)
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - sqrt(t));
	elseif (z <= 4.5)
		tmp = 3.0 + (z * (0.5 - sqrt((1.0 / z))));
	elseif ((z <= 1.2e+122) || (~((z <= 2.3e+144)) && (z <= 5.4e+180)))
		tmp = 2.0 + ((y * 0.5) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 + (sqrt((1.0 + z)) - (sqrt(x) + 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_] := If[LessEqual[z, 2e-33], N[(3.0 + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.5], N[(3.0 + N[(z * N[(0.5 - N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 1.2e+122], And[N[Not[LessEqual[z, 2.3e+144]], $MachinePrecision], LessEqual[z, 5.4e+180]]], N[(2.0 + N[(N[(y * 0.5), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < 2.0000000000000001e-33

   1. Initial program 97.8%

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

      1. associate-+l+ 97.8%

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

      2. sub-neg 97.8%

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

      3. sub-neg 97.8%

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

      4. +-commutative 97.8%

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

      5. +-commutative 97.8%

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

      6. +-commutative 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in x around 0 50.1%

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

   6. Taylor expanded in z around 0 50.1%

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

      1. associate--l+ 50.1%

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

   8. Simplified 50.1%

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

   9. Taylor expanded in y around 0 14.1%

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

       1. associate--l+ 26.5%

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

       2. +-commutative 26.5%

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

       3. associate-+r+ 26.5%

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

   11. Simplified 26.5%

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

   12. Taylor expanded in t around inf 37.0%

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

   if 2.0000000000000001e-33 < z < 4.5

   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. sub-neg 96.3%

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

      3. sub-neg 96.3%

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

      4. +-commutative 96.3%

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

      5. +-commutative 96.3%

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

      6. +-commutative 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in x around 0 49.5%

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

   6. Taylor expanded in z around 0 41.9%

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

      1. associate--l+ 41.9%

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

   8. Simplified 41.9%

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

   9. Taylor expanded in y around 0 6.3%

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

       1. associate--l+ 11.7%

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

       2. +-commutative 11.7%

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

       3. associate-+r+ 11.7%

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

   11. Simplified 11.7%

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

   12. Taylor expanded in z around inf 55.0%

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

   if 4.5 < z < 1.2000000000000001e122 or 2.3000000000000001e144 < z < 5.40000000000000033e180

   1. Initial program 90.1%

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

      1. associate-+l+ 90.1%

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

      2. sub-neg 90.1%

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

      3. sub-neg 90.1%

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

      4. +-commutative 90.1%

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

      5. +-commutative 90.1%

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

      6. +-commutative 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in x around 0 52.2%

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

   6. Taylor expanded in t around inf 29.0%

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

   7. Taylor expanded in y around 0 13.4%

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

      1. associate--l+ 13.4%

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

      2. *-commutative 13.4%

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

      3. +-commutative 13.4%

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

   9. Simplified 13.4%

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

   10. Taylor expanded in z around inf 13.2%

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

       1. associate--l+ 13.2%

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

       2. +-commutative 13.2%

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

       3. *-commutative 13.2%

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

   12. Simplified 13.2%

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

   if 1.2000000000000001e122 < z < 2.3000000000000001e144 or 5.40000000000000033e180 < z

   1. Initial program 89.0%

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

      1. associate-+l+ 89.0%

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

      2. sub-neg 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 89.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 57.8%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 32.4%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in y around inf 2.9%

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \sqrt{z}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 32.8%

         \[\leadsto \color{blue}{1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)} \]

   9. Simplified 32.8%

      \[\leadsto \color{blue}{1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-33}:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 4.5:\\ \;\;\;\;3 + z \cdot \left(0.5 - \sqrt{\frac{1}{z}}\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+122} \lor \neg \left(z \leq 2.3 \cdot 10^{+144}\right) \land z \leq 5.4 \cdot 10^{+180}:\\ \;\;\;\;2 + \left(y \cdot 0.5 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 68.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} + \sqrt{y}\\ \mathbf{if}\;z \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 1.92:\\ \;\;\;\;3 + \left(z \cdot 0.5 - \left(\sqrt{z} + t\_1\right)\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+122} \lor \neg \left(z \leq 2.3 \cdot 10^{+144}\right) \land z \leq 6.5 \cdot 10^{+180}:\\ \;\;\;\;2 + \left(y \cdot 0.5 - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt x) (sqrt y))))
   (if (<= z 1.05e-32)
     (+ 3.0 (- (+ (sqrt (+ 1.0 t)) (* z 0.5)) (sqrt t)))
     (if (<= z 1.92)
       (+ 3.0 (- (* z 0.5) (+ (sqrt z) t_1)))
       (if (or (<= z 3.2e+122) (and (not (<= z 2.3e+144)) (<= z 6.5e+180)))
         (+ 2.0 (- (* y 0.5) t_1))
         (+ 1.0 (- (sqrt (+ 1.0 z)) (+ (sqrt x) (sqrt z)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(x) + sqrt(y);
	double tmp;
	if (z <= 1.05e-32) {
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - sqrt(t));
	} else if (z <= 1.92) {
		tmp = 3.0 + ((z * 0.5) - (sqrt(z) + t_1));
	} else if ((z <= 3.2e+122) || (!(z <= 2.3e+144) && (z <= 6.5e+180))) {
		tmp = 2.0 + ((y * 0.5) - t_1);
	} else {
		tmp = 1.0 + (sqrt((1.0 + z)) - (sqrt(x) + 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) :: tmp
    t_1 = sqrt(x) + sqrt(y)
    if (z <= 1.05d-32) then
        tmp = 3.0d0 + ((sqrt((1.0d0 + t)) + (z * 0.5d0)) - sqrt(t))
    else if (z <= 1.92d0) then
        tmp = 3.0d0 + ((z * 0.5d0) - (sqrt(z) + t_1))
    else if ((z <= 3.2d+122) .or. (.not. (z <= 2.3d+144)) .and. (z <= 6.5d+180)) then
        tmp = 2.0d0 + ((y * 0.5d0) - t_1)
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + z)) - (sqrt(x) + 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 = Math.sqrt(x) + Math.sqrt(y);
	double tmp;
	if (z <= 1.05e-32) {
		tmp = 3.0 + ((Math.sqrt((1.0 + t)) + (z * 0.5)) - Math.sqrt(t));
	} else if (z <= 1.92) {
		tmp = 3.0 + ((z * 0.5) - (Math.sqrt(z) + t_1));
	} else if ((z <= 3.2e+122) || (!(z <= 2.3e+144) && (z <= 6.5e+180))) {
		tmp = 2.0 + ((y * 0.5) - t_1);
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(x) + Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(x) + math.sqrt(y)
	tmp = 0
	if z <= 1.05e-32:
		tmp = 3.0 + ((math.sqrt((1.0 + t)) + (z * 0.5)) - math.sqrt(t))
	elif z <= 1.92:
		tmp = 3.0 + ((z * 0.5) - (math.sqrt(z) + t_1))
	elif (z <= 3.2e+122) or (not (z <= 2.3e+144) and (z <= 6.5e+180)):
		tmp = 2.0 + ((y * 0.5) - t_1)
	else:
		tmp = 1.0 + (math.sqrt((1.0 + z)) - (math.sqrt(x) + math.sqrt(z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) + sqrt(y))
	tmp = 0.0
	if (z <= 1.05e-32)
		tmp = Float64(3.0 + Float64(Float64(sqrt(Float64(1.0 + t)) + Float64(z * 0.5)) - sqrt(t)));
	elseif (z <= 1.92)
		tmp = Float64(3.0 + Float64(Float64(z * 0.5) - Float64(sqrt(z) + t_1)));
	elseif ((z <= 3.2e+122) || (!(z <= 2.3e+144) && (z <= 6.5e+180)))
		tmp = Float64(2.0 + Float64(Float64(y * 0.5) - t_1));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(x) + sqrt(z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(x) + sqrt(y);
	tmp = 0.0;
	if (z <= 1.05e-32)
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - sqrt(t));
	elseif (z <= 1.92)
		tmp = 3.0 + ((z * 0.5) - (sqrt(z) + t_1));
	elseif ((z <= 3.2e+122) || (~((z <= 2.3e+144)) && (z <= 6.5e+180)))
		tmp = 2.0 + ((y * 0.5) - t_1);
	else
		tmp = 1.0 + (sqrt((1.0 + z)) - (sqrt(x) + 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[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1.05e-32], N[(3.0 + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.92], N[(3.0 + N[(N[(z * 0.5), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 3.2e+122], And[N[Not[LessEqual[z, 2.3e+144]], $MachinePrecision], LessEqual[z, 6.5e+180]]], N[(2.0 + N[(N[(y * 0.5), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < 1.05e-32

   1. Initial program 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 97.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 97.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 97.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 97.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 97.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 97.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 50.1%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in z around 0 50.1%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(\left(1 + 0.5 \cdot z\right) - \sqrt{z}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   7. Step-by-step derivation

      1. associate--l+ 50.1%

         \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   8. Simplified 50.1%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   9. Taylor expanded in y around 0 14.1%

      \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + 0.5 \cdot z\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 26.5%

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + 0.5 \cdot z\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       2. +-commutative 26.5%

          \[\leadsto 3 + \left(\color{blue}{\left(0.5 \cdot z + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       3. associate-+r+ 26.5%

          \[\leadsto 3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   11. Simplified 26.5%

       \[\leadsto \color{blue}{3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   12. Taylor expanded in t around inf 37.0%

       \[\leadsto 3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \color{blue}{\sqrt{t}}\right) \]

   if 1.05e-32 < z < 1.9199999999999999

   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. sub-neg 96.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 96.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 96.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 96.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 96.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 96.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 49.5%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in z around 0 41.9%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(\left(1 + 0.5 \cdot z\right) - \sqrt{z}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   7. Step-by-step derivation

      1. associate--l+ 41.9%

         \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   8. Simplified 41.9%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   9. Taylor expanded in y around 0 6.3%

      \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + 0.5 \cdot z\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 11.7%

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + 0.5 \cdot z\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       2. +-commutative 11.7%

          \[\leadsto 3 + \left(\color{blue}{\left(0.5 \cdot z + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       3. associate-+r+ 11.7%

          \[\leadsto 3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   11. Simplified 11.7%

       \[\leadsto \color{blue}{3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   12. Taylor expanded in t around inf 13.2%

       \[\leadsto 3 + \color{blue}{\left(0.5 \cdot z - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-+r+ 13.2%

          \[\leadsto 3 + \left(0.5 \cdot z - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

       2. +-commutative 13.2%

          \[\leadsto 3 + \left(0.5 \cdot z - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right) \]

   14. Simplified 13.2%

       \[\leadsto 3 + \color{blue}{\left(0.5 \cdot z - \left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)\right)} \]

   if 1.9199999999999999 < z < 3.20000000000000012e122 or 2.3000000000000001e144 < z < 6.5e180

   1. Initial program 90.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 90.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 90.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 90.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 90.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 90.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 90.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 90.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 52.9%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 30.1%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in y around 0 14.7%

      \[\leadsto \color{blue}{\left(\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]
   8. Step-by-step derivation

      1. associate--l+ 14.7%

         \[\leadsto \color{blue}{\left(2 + \left(0.5 \cdot y - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      2. *-commutative 14.7%

         \[\leadsto \left(2 + \left(\color{blue}{y \cdot 0.5} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      3. +-commutative 14.7%

         \[\leadsto \left(2 + \left(y \cdot 0.5 - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   9. Simplified 14.7%

      \[\leadsto \color{blue}{\left(2 + \left(y \cdot 0.5 - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   10. Taylor expanded in z around inf 14.5%

       \[\leadsto \color{blue}{\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   11. Step-by-step derivation

       1. associate--l+ 14.5%

          \[\leadsto \color{blue}{2 + \left(0.5 \cdot y - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       2. +-commutative 14.5%

          \[\leadsto 2 + \left(0.5 \cdot y - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

       3. *-commutative 14.5%

          \[\leadsto 2 + \left(\color{blue}{y \cdot 0.5} - \left(\sqrt{y} + \sqrt{x}\right)\right) \]

   12. Simplified 14.5%

       \[\leadsto \color{blue}{2 + \left(y \cdot 0.5 - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 3.20000000000000012e122 < z < 2.3000000000000001e144 or 6.5e180 < z

   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. associate-+l+ 88.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 88.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 88.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 88.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 88.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 88.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 88.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 57.0%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 31.3%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in y around inf 2.9%

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \sqrt{z}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 33.0%

         \[\leadsto \color{blue}{1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)} \]

   9. Simplified 33.0%

      \[\leadsto \color{blue}{1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 1.92:\\ \;\;\;\;3 + \left(z \cdot 0.5 - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+122} \lor \neg \left(z \leq 2.3 \cdot 10^{+144}\right) \land z \leq 6.5 \cdot 10^{+180}:\\ \;\;\;\;2 + \left(y \cdot 0.5 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 70.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} + \sqrt{y}\\ \mathbf{if}\;z \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 0.74:\\ \;\;\;\;3 + \left(z \cdot 0.5 - \left(\sqrt{z} + t\_1\right)\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+122}:\\ \;\;\;\;2 + \left(0.5 \cdot \left(y + \sqrt{\frac{1}{z}}\right) - t\_1\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+144} \lor \neg \left(z \leq 5.4 \cdot 10^{+180}\right):\\ \;\;\;\;1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(y \cdot 0.5 - t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt x) (sqrt y))))
   (if (<= z 1.05e-32)
     (+ 3.0 (- (+ (sqrt (+ 1.0 t)) (* z 0.5)) (sqrt t)))
     (if (<= z 0.74)
       (+ 3.0 (- (* z 0.5) (+ (sqrt z) t_1)))
       (if (<= z 2.9e+122)
         (+ 2.0 (- (* 0.5 (+ y (sqrt (/ 1.0 z)))) t_1))
         (if (or (<= z 2.3e+144) (not (<= z 5.4e+180)))
           (+ 1.0 (- (sqrt (+ 1.0 z)) (+ (sqrt x) (sqrt z))))
           (+ 2.0 (- (* y 0.5) t_1))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(x) + sqrt(y);
	double tmp;
	if (z <= 1.05e-32) {
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - sqrt(t));
	} else if (z <= 0.74) {
		tmp = 3.0 + ((z * 0.5) - (sqrt(z) + t_1));
	} else if (z <= 2.9e+122) {
		tmp = 2.0 + ((0.5 * (y + sqrt((1.0 / z)))) - t_1);
	} else if ((z <= 2.3e+144) || !(z <= 5.4e+180)) {
		tmp = 1.0 + (sqrt((1.0 + z)) - (sqrt(x) + sqrt(z)));
	} else {
		tmp = 2.0 + ((y * 0.5) - t_1);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(x) + sqrt(y)
    if (z <= 1.05d-32) then
        tmp = 3.0d0 + ((sqrt((1.0d0 + t)) + (z * 0.5d0)) - sqrt(t))
    else if (z <= 0.74d0) then
        tmp = 3.0d0 + ((z * 0.5d0) - (sqrt(z) + t_1))
    else if (z <= 2.9d+122) then
        tmp = 2.0d0 + ((0.5d0 * (y + sqrt((1.0d0 / z)))) - t_1)
    else if ((z <= 2.3d+144) .or. (.not. (z <= 5.4d+180))) then
        tmp = 1.0d0 + (sqrt((1.0d0 + z)) - (sqrt(x) + sqrt(z)))
    else
        tmp = 2.0d0 + ((y * 0.5d0) - 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(x) + Math.sqrt(y);
	double tmp;
	if (z <= 1.05e-32) {
		tmp = 3.0 + ((Math.sqrt((1.0 + t)) + (z * 0.5)) - Math.sqrt(t));
	} else if (z <= 0.74) {
		tmp = 3.0 + ((z * 0.5) - (Math.sqrt(z) + t_1));
	} else if (z <= 2.9e+122) {
		tmp = 2.0 + ((0.5 * (y + Math.sqrt((1.0 / z)))) - t_1);
	} else if ((z <= 2.3e+144) || !(z <= 5.4e+180)) {
		tmp = 1.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(x) + Math.sqrt(z)));
	} else {
		tmp = 2.0 + ((y * 0.5) - t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(x) + math.sqrt(y)
	tmp = 0
	if z <= 1.05e-32:
		tmp = 3.0 + ((math.sqrt((1.0 + t)) + (z * 0.5)) - math.sqrt(t))
	elif z <= 0.74:
		tmp = 3.0 + ((z * 0.5) - (math.sqrt(z) + t_1))
	elif z <= 2.9e+122:
		tmp = 2.0 + ((0.5 * (y + math.sqrt((1.0 / z)))) - t_1)
	elif (z <= 2.3e+144) or not (z <= 5.4e+180):
		tmp = 1.0 + (math.sqrt((1.0 + z)) - (math.sqrt(x) + math.sqrt(z)))
	else:
		tmp = 2.0 + ((y * 0.5) - t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) + sqrt(y))
	tmp = 0.0
	if (z <= 1.05e-32)
		tmp = Float64(3.0 + Float64(Float64(sqrt(Float64(1.0 + t)) + Float64(z * 0.5)) - sqrt(t)));
	elseif (z <= 0.74)
		tmp = Float64(3.0 + Float64(Float64(z * 0.5) - Float64(sqrt(z) + t_1)));
	elseif (z <= 2.9e+122)
		tmp = Float64(2.0 + Float64(Float64(0.5 * Float64(y + sqrt(Float64(1.0 / z)))) - t_1));
	elseif ((z <= 2.3e+144) || !(z <= 5.4e+180))
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(x) + sqrt(z))));
	else
		tmp = Float64(2.0 + Float64(Float64(y * 0.5) - 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(x) + sqrt(y);
	tmp = 0.0;
	if (z <= 1.05e-32)
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - sqrt(t));
	elseif (z <= 0.74)
		tmp = 3.0 + ((z * 0.5) - (sqrt(z) + t_1));
	elseif (z <= 2.9e+122)
		tmp = 2.0 + ((0.5 * (y + sqrt((1.0 / z)))) - t_1);
	elseif ((z <= 2.3e+144) || ~((z <= 5.4e+180)))
		tmp = 1.0 + (sqrt((1.0 + z)) - (sqrt(x) + sqrt(z)));
	else
		tmp = 2.0 + ((y * 0.5) - t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1.05e-32], N[(3.0 + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.74], N[(3.0 + N[(N[(z * 0.5), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.9e+122], N[(2.0 + N[(N[(0.5 * N[(y + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 2.3e+144], N[Not[LessEqual[z, 5.4e+180]], $MachinePrecision]], N[(1.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(N[(y * 0.5), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < 1.05e-32

   1. Initial program 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 97.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 97.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 97.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 97.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 97.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 97.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 50.1%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in z around 0 50.1%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(\left(1 + 0.5 \cdot z\right) - \sqrt{z}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   7. Step-by-step derivation

      1. associate--l+ 50.1%

         \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   8. Simplified 50.1%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   9. Taylor expanded in y around 0 14.1%

      \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + 0.5 \cdot z\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 26.5%

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + 0.5 \cdot z\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       2. +-commutative 26.5%

          \[\leadsto 3 + \left(\color{blue}{\left(0.5 \cdot z + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       3. associate-+r+ 26.5%

          \[\leadsto 3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   11. Simplified 26.5%

       \[\leadsto \color{blue}{3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   12. Taylor expanded in t around inf 37.0%

       \[\leadsto 3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \color{blue}{\sqrt{t}}\right) \]

   if 1.05e-32 < z < 0.73999999999999999

   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. sub-neg 96.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 96.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 96.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 96.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 96.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 96.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 49.5%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in z around 0 41.9%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(\left(1 + 0.5 \cdot z\right) - \sqrt{z}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   7. Step-by-step derivation

      1. associate--l+ 41.9%

         \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   8. Simplified 41.9%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   9. Taylor expanded in y around 0 6.3%

      \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + 0.5 \cdot z\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 11.7%

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + 0.5 \cdot z\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       2. +-commutative 11.7%

          \[\leadsto 3 + \left(\color{blue}{\left(0.5 \cdot z + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       3. associate-+r+ 11.7%

          \[\leadsto 3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   11. Simplified 11.7%

       \[\leadsto \color{blue}{3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   12. Taylor expanded in t around inf 13.2%

       \[\leadsto 3 + \color{blue}{\left(0.5 \cdot z - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-+r+ 13.2%

          \[\leadsto 3 + \left(0.5 \cdot z - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

       2. +-commutative 13.2%

          \[\leadsto 3 + \left(0.5 \cdot z - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right) \]

   14. Simplified 13.2%

       \[\leadsto 3 + \color{blue}{\left(0.5 \cdot z - \left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)\right)} \]

   if 0.73999999999999999 < z < 2.9000000000000001e122

   1. Initial program 90.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 90.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 90.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 90.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 90.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 90.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 90.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 90.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 55.7%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 29.8%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in y around 0 13.8%

      \[\leadsto \color{blue}{\left(\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]
   8. Step-by-step derivation

      1. associate--l+ 13.8%

         \[\leadsto \color{blue}{\left(2 + \left(0.5 \cdot y - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      2. *-commutative 13.8%

         \[\leadsto \left(2 + \left(\color{blue}{y \cdot 0.5} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      3. +-commutative 13.8%

         \[\leadsto \left(2 + \left(y \cdot 0.5 - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   9. Simplified 13.8%

      \[\leadsto \color{blue}{\left(2 + \left(y \cdot 0.5 - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   10. Taylor expanded in z around inf 14.3%

       \[\leadsto \color{blue}{\left(2 + \left(0.5 \cdot y + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   11. Step-by-step derivation

       1. associate--l+ 14.3%

          \[\leadsto \color{blue}{2 + \left(\left(0.5 \cdot y + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       2. distribute-lft-out 14.3%

          \[\leadsto 2 + \left(\color{blue}{0.5 \cdot \left(y + \sqrt{\frac{1}{z}}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       3. +-commutative 14.3%

          \[\leadsto 2 + \left(0.5 \cdot \left(y + \sqrt{\frac{1}{z}}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   12. Simplified 14.3%

       \[\leadsto \color{blue}{2 + \left(0.5 \cdot \left(y + \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 2.9000000000000001e122 < z < 2.3000000000000001e144 or 5.40000000000000033e180 < z

   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. associate-+l+ 88.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 88.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 88.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 88.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 88.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 88.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 88.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 57.0%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 31.3%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in y around inf 2.9%

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \sqrt{z}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 33.0%

         \[\leadsto \color{blue}{1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)} \]

   9. Simplified 33.0%

      \[\leadsto \color{blue}{1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)} \]

   if 2.3000000000000001e144 < z < 5.40000000000000033e180

   1. Initial program 91.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 91.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 91.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 91.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 91.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 91.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 91.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 91.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 42.5%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 31.1%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in y around 0 18.3%

      \[\leadsto \color{blue}{\left(\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]
   8. Step-by-step derivation

      1. associate--l+ 18.3%

         \[\leadsto \color{blue}{\left(2 + \left(0.5 \cdot y - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      2. *-commutative 18.3%

         \[\leadsto \left(2 + \left(\color{blue}{y \cdot 0.5} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      3. +-commutative 18.3%

         \[\leadsto \left(2 + \left(y \cdot 0.5 - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   9. Simplified 18.3%

      \[\leadsto \color{blue}{\left(2 + \left(y \cdot 0.5 - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   10. Taylor expanded in z around inf 18.3%

       \[\leadsto \color{blue}{\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   11. Step-by-step derivation

       1. associate--l+ 18.3%

          \[\leadsto \color{blue}{2 + \left(0.5 \cdot y - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       2. +-commutative 18.3%

          \[\leadsto 2 + \left(0.5 \cdot y - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

       3. *-commutative 18.3%

          \[\leadsto 2 + \left(\color{blue}{y \cdot 0.5} - \left(\sqrt{y} + \sqrt{x}\right)\right) \]

   12. Simplified 18.3%

       \[\leadsto \color{blue}{2 + \left(y \cdot 0.5 - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 0.74:\\ \;\;\;\;3 + \left(z \cdot 0.5 - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+122}:\\ \;\;\;\;2 + \left(0.5 \cdot \left(y + \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+144} \lor \neg \left(z \leq 5.4 \cdot 10^{+180}\right):\\ \;\;\;\;1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(y \cdot 0.5 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 84.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 2.65 \cdot 10^{-5}:\\ \;\;\;\;\left(t\_1 - \sqrt{z}\right) + \left(2 + \left(y \cdot 0.5 - \sqrt{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(\frac{\sqrt{1 + y}}{x} + \left(\frac{1}{x} - \left(\sqrt{\frac{1}{x}} + \frac{\sqrt{y}}{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t\_1 - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))))
   (if (<= y 2.65e-5)
     (+ (- t_1 (sqrt z)) (+ 2.0 (- (* y 0.5) (sqrt y))))
     (if (<= y 1.35e+14)
       (*
        x
        (+
         (/ (sqrt (+ 1.0 y)) x)
         (- (/ 1.0 x) (+ (sqrt (/ 1.0 x)) (/ (sqrt y) x)))))
       (+ 1.0 (- t_1 (+ (sqrt x) (sqrt z))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double tmp;
	if (y <= 2.65e-5) {
		tmp = (t_1 - sqrt(z)) + (2.0 + ((y * 0.5) - sqrt(y)));
	} else if (y <= 1.35e+14) {
		tmp = x * ((sqrt((1.0 + y)) / x) + ((1.0 / x) - (sqrt((1.0 / x)) + (sqrt(y) / x))));
	} else {
		tmp = 1.0 + (t_1 - (sqrt(x) + 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) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if (y <= 2.65d-5) then
        tmp = (t_1 - sqrt(z)) + (2.0d0 + ((y * 0.5d0) - sqrt(y)))
    else if (y <= 1.35d+14) then
        tmp = x * ((sqrt((1.0d0 + y)) / x) + ((1.0d0 / x) - (sqrt((1.0d0 / x)) + (sqrt(y) / x))))
    else
        tmp = 1.0d0 + (t_1 - (sqrt(x) + 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 = Math.sqrt((1.0 + z));
	double tmp;
	if (y <= 2.65e-5) {
		tmp = (t_1 - Math.sqrt(z)) + (2.0 + ((y * 0.5) - Math.sqrt(y)));
	} else if (y <= 1.35e+14) {
		tmp = x * ((Math.sqrt((1.0 + y)) / x) + ((1.0 / x) - (Math.sqrt((1.0 / x)) + (Math.sqrt(y) / x))));
	} else {
		tmp = 1.0 + (t_1 - (Math.sqrt(x) + Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	tmp = 0
	if y <= 2.65e-5:
		tmp = (t_1 - math.sqrt(z)) + (2.0 + ((y * 0.5) - math.sqrt(y)))
	elif y <= 1.35e+14:
		tmp = x * ((math.sqrt((1.0 + y)) / x) + ((1.0 / x) - (math.sqrt((1.0 / x)) + (math.sqrt(y) / x))))
	else:
		tmp = 1.0 + (t_1 - (math.sqrt(x) + math.sqrt(z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (y <= 2.65e-5)
		tmp = Float64(Float64(t_1 - sqrt(z)) + Float64(2.0 + Float64(Float64(y * 0.5) - sqrt(y))));
	elseif (y <= 1.35e+14)
		tmp = Float64(x * Float64(Float64(sqrt(Float64(1.0 + y)) / x) + Float64(Float64(1.0 / x) - Float64(sqrt(Float64(1.0 / x)) + Float64(sqrt(y) / x)))));
	else
		tmp = Float64(1.0 + Float64(t_1 - Float64(sqrt(x) + sqrt(z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	tmp = 0.0;
	if (y <= 2.65e-5)
		tmp = (t_1 - sqrt(z)) + (2.0 + ((y * 0.5) - sqrt(y)));
	elseif (y <= 1.35e+14)
		tmp = x * ((sqrt((1.0 + y)) / x) + ((1.0 / x) - (sqrt((1.0 / x)) + (sqrt(y) / x))));
	else
		tmp = 1.0 + (t_1 - (sqrt(x) + sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 2.65e-5], N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(N[(y * 0.5), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+14], N[(x * N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.65e-5

   1. Initial program 98.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 98.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 50.0%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 29.0%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in y around 0 29.0%

      \[\leadsto \color{blue}{\left(\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]
   8. Step-by-step derivation

      1. associate--l+ 29.0%

         \[\leadsto \color{blue}{\left(2 + \left(0.5 \cdot y - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      2. *-commutative 29.0%

         \[\leadsto \left(2 + \left(\color{blue}{y \cdot 0.5} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      3. +-commutative 29.0%

         \[\leadsto \left(2 + \left(y \cdot 0.5 - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   9. Simplified 29.0%

      \[\leadsto \color{blue}{\left(2 + \left(y \cdot 0.5 - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   10. Taylor expanded in y around inf 56.4%

       \[\leadsto \left(2 + \left(y \cdot 0.5 - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   if 2.65e-5 < y < 1.35e14

   1. Initial program 89.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 89.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 89.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 89.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 89.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 89.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 89.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 89.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 46.1%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 28.9%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in x around inf 3.3%

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{x} \cdot \sqrt{1 + y} + \left(\frac{1}{x} \cdot \sqrt{1 + z} + \frac{1}{x}\right)\right) - \left(\sqrt{\frac{1}{x}} + \left(\frac{1}{x} \cdot \sqrt{y} + \frac{1}{x} \cdot \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in z around inf 21.4%

      \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{x} \cdot \sqrt{1 + y} + \frac{1}{x}\right) - \left(\sqrt{\frac{1}{x}} + \frac{1}{x} \cdot \sqrt{y}\right)\right)} \]
   9. Step-by-step derivation

      1. associate--l+ 21.0%

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{x} \cdot \sqrt{1 + y} + \left(\frac{1}{x} - \left(\sqrt{\frac{1}{x}} + \frac{1}{x} \cdot \sqrt{y}\right)\right)\right)} \]

      2. associate-*l/ 21.0%

         \[\leadsto x \cdot \left(\color{blue}{\frac{1 \cdot \sqrt{1 + y}}{x}} + \left(\frac{1}{x} - \left(\sqrt{\frac{1}{x}} + \frac{1}{x} \cdot \sqrt{y}\right)\right)\right) \]

      3. +-commutative 21.0%

         \[\leadsto x \cdot \left(\frac{1 \cdot \sqrt{\color{blue}{y + 1}}}{x} + \left(\frac{1}{x} - \left(\sqrt{\frac{1}{x}} + \frac{1}{x} \cdot \sqrt{y}\right)\right)\right) \]

      4. *-lft-identity 21.0%

         \[\leadsto x \cdot \left(\frac{\color{blue}{\sqrt{y + 1}}}{x} + \left(\frac{1}{x} - \left(\sqrt{\frac{1}{x}} + \frac{1}{x} \cdot \sqrt{y}\right)\right)\right) \]

      5. +-commutative 21.0%

         \[\leadsto x \cdot \left(\frac{\sqrt{\color{blue}{1 + y}}}{x} + \left(\frac{1}{x} - \left(\sqrt{\frac{1}{x}} + \frac{1}{x} \cdot \sqrt{y}\right)\right)\right) \]

      6. associate-*l/ 21.0%

         \[\leadsto x \cdot \left(\frac{\sqrt{1 + y}}{x} + \left(\frac{1}{x} - \left(\sqrt{\frac{1}{x}} + \color{blue}{\frac{1 \cdot \sqrt{y}}{x}}\right)\right)\right) \]

      7. *-lft-identity 21.0%

         \[\leadsto x \cdot \left(\frac{\sqrt{1 + y}}{x} + \left(\frac{1}{x} - \left(\sqrt{\frac{1}{x}} + \frac{\color{blue}{\sqrt{y}}}{x}\right)\right)\right) \]

   10. Simplified 21.0%

       \[\leadsto x \cdot \color{blue}{\left(\frac{\sqrt{1 + y}}{x} + \left(\frac{1}{x} - \left(\sqrt{\frac{1}{x}} + \frac{\sqrt{y}}{x}\right)\right)\right)} \]

   if 1.35e14 < y

   1. Initial program 89.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 89.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 89.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 89.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 89.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 89.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 89.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 89.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 55.3%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 32.0%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in y around inf 18.2%

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \sqrt{z}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 35.5%

         \[\leadsto \color{blue}{1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)} \]

   9. Simplified 35.5%

      \[\leadsto \color{blue}{1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-5}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(2 + \left(y \cdot 0.5 - \sqrt{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(\frac{\sqrt{1 + y}}{x} + \left(\frac{1}{x} - \left(\sqrt{\frac{1}{x}} + \frac{\sqrt{y}}{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 82.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 2.2:\\ \;\;\;\;\left(t\_1 - \sqrt{z}\right) + \left(2 + \left(y \cdot 0.5 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t\_1 - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))))
   (if (<= y 2.2)
     (+ (- t_1 (sqrt z)) (+ 2.0 (- (* y 0.5) (sqrt y))))
     (+ 1.0 (- t_1 (+ (sqrt x) (sqrt z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double tmp;
	if (y <= 2.2) {
		tmp = (t_1 - sqrt(z)) + (2.0 + ((y * 0.5) - sqrt(y)));
	} else {
		tmp = 1.0 + (t_1 - (sqrt(x) + 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) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if (y <= 2.2d0) then
        tmp = (t_1 - sqrt(z)) + (2.0d0 + ((y * 0.5d0) - sqrt(y)))
    else
        tmp = 1.0d0 + (t_1 - (sqrt(x) + 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 = Math.sqrt((1.0 + z));
	double tmp;
	if (y <= 2.2) {
		tmp = (t_1 - Math.sqrt(z)) + (2.0 + ((y * 0.5) - Math.sqrt(y)));
	} else {
		tmp = 1.0 + (t_1 - (Math.sqrt(x) + Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	tmp = 0
	if y <= 2.2:
		tmp = (t_1 - math.sqrt(z)) + (2.0 + ((y * 0.5) - math.sqrt(y)))
	else:
		tmp = 1.0 + (t_1 - (math.sqrt(x) + math.sqrt(z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (y <= 2.2)
		tmp = Float64(Float64(t_1 - sqrt(z)) + Float64(2.0 + Float64(Float64(y * 0.5) - sqrt(y))));
	else
		tmp = Float64(1.0 + Float64(t_1 - Float64(sqrt(x) + sqrt(z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	tmp = 0.0;
	if (y <= 2.2)
		tmp = (t_1 - sqrt(z)) + (2.0 + ((y * 0.5) - sqrt(y)));
	else
		tmp = 1.0 + (t_1 - (sqrt(x) + sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 2.2], N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(N[(y * 0.5), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.2000000000000002

   1. Initial program 98.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 98.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 49.2%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 28.2%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in y around 0 28.2%

      \[\leadsto \color{blue}{\left(\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]
   8. Step-by-step derivation

      1. associate--l+ 28.2%

         \[\leadsto \color{blue}{\left(2 + \left(0.5 \cdot y - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      2. *-commutative 28.2%

         \[\leadsto \left(2 + \left(\color{blue}{y \cdot 0.5} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      3. +-commutative 28.2%

         \[\leadsto \left(2 + \left(y \cdot 0.5 - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   9. Simplified 28.2%

      \[\leadsto \color{blue}{\left(2 + \left(y \cdot 0.5 - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   10. Taylor expanded in y around inf 55.6%

       \[\leadsto \left(2 + \left(y \cdot 0.5 - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   if 2.2000000000000002 < y

   1. Initial program 89.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 89.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 89.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 89.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 89.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 89.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 89.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 89.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 55.4%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 32.5%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in y around inf 17.8%

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \sqrt{z}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 35.1%

         \[\leadsto \color{blue}{1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)} \]

   9. Simplified 35.1%

      \[\leadsto \color{blue}{1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(2 + \left(y \cdot 0.5 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 66.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 4.3:\\ \;\;\;\;3 + z \cdot \left(0.5 - \sqrt{\frac{1}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(y \cdot 0.5 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 4e-34)
   (+ 3.0 (- (+ (sqrt (+ 1.0 t)) (* z 0.5)) (sqrt t)))
   (if (<= z 4.3)
     (+ 3.0 (* z (- 0.5 (sqrt (/ 1.0 z)))))
     (+ 2.0 (- (* y 0.5) (+ (sqrt x) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4e-34) {
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - sqrt(t));
	} else if (z <= 4.3) {
		tmp = 3.0 + (z * (0.5 - sqrt((1.0 / z))));
	} else {
		tmp = 2.0 + ((y * 0.5) - (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 4d-34) then
        tmp = 3.0d0 + ((sqrt((1.0d0 + t)) + (z * 0.5d0)) - sqrt(t))
    else if (z <= 4.3d0) then
        tmp = 3.0d0 + (z * (0.5d0 - sqrt((1.0d0 / z))))
    else
        tmp = 2.0d0 + ((y * 0.5d0) - (sqrt(x) + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4e-34) {
		tmp = 3.0 + ((Math.sqrt((1.0 + t)) + (z * 0.5)) - Math.sqrt(t));
	} else if (z <= 4.3) {
		tmp = 3.0 + (z * (0.5 - Math.sqrt((1.0 / z))));
	} else {
		tmp = 2.0 + ((y * 0.5) - (Math.sqrt(x) + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 4e-34:
		tmp = 3.0 + ((math.sqrt((1.0 + t)) + (z * 0.5)) - math.sqrt(t))
	elif z <= 4.3:
		tmp = 3.0 + (z * (0.5 - math.sqrt((1.0 / z))))
	else:
		tmp = 2.0 + ((y * 0.5) - (math.sqrt(x) + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 4e-34)
		tmp = Float64(3.0 + Float64(Float64(sqrt(Float64(1.0 + t)) + Float64(z * 0.5)) - sqrt(t)));
	elseif (z <= 4.3)
		tmp = Float64(3.0 + Float64(z * Float64(0.5 - sqrt(Float64(1.0 / z)))));
	else
		tmp = Float64(2.0 + Float64(Float64(y * 0.5) - Float64(sqrt(x) + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 4e-34)
		tmp = 3.0 + ((sqrt((1.0 + t)) + (z * 0.5)) - sqrt(t));
	elseif (z <= 4.3)
		tmp = 3.0 + (z * (0.5 - sqrt((1.0 / z))));
	else
		tmp = 2.0 + ((y * 0.5) - (sqrt(x) + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 4e-34], N[(3.0 + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.3], N[(3.0 + N[(z * N[(0.5 - N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(N[(y * 0.5), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.99999999999999971e-34

   1. Initial program 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 97.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 97.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 97.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 97.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 97.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 97.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 50.1%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in z around 0 50.1%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(\left(1 + 0.5 \cdot z\right) - \sqrt{z}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   7. Step-by-step derivation

      1. associate--l+ 50.1%

         \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   8. Simplified 50.1%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   9. Taylor expanded in y around 0 14.1%

      \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + 0.5 \cdot z\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 26.5%

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + 0.5 \cdot z\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       2. +-commutative 26.5%

          \[\leadsto 3 + \left(\color{blue}{\left(0.5 \cdot z + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       3. associate-+r+ 26.5%

          \[\leadsto 3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   11. Simplified 26.5%

       \[\leadsto \color{blue}{3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   12. Taylor expanded in t around inf 37.0%

       \[\leadsto 3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \color{blue}{\sqrt{t}}\right) \]

   if 3.99999999999999971e-34 < z < 4.29999999999999982

   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. sub-neg 96.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 96.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 96.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 96.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 96.3%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 96.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 49.5%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in z around 0 41.9%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(\left(1 + 0.5 \cdot z\right) - \sqrt{z}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   7. Step-by-step derivation

      1. associate--l+ 41.9%

         \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   8. Simplified 41.9%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   9. Taylor expanded in y around 0 6.3%

      \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + 0.5 \cdot z\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 11.7%

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + 0.5 \cdot z\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       2. +-commutative 11.7%

          \[\leadsto 3 + \left(\color{blue}{\left(0.5 \cdot z + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       3. associate-+r+ 11.7%

          \[\leadsto 3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   11. Simplified 11.7%

       \[\leadsto \color{blue}{3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   12. Taylor expanded in z around inf 55.0%

       \[\leadsto 3 + \color{blue}{z \cdot \left(0.5 - \sqrt{\frac{1}{z}}\right)} \]

   if 4.29999999999999982 < z

   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. associate-+l+ 89.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 89.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 89.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 89.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 89.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 89.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 54.9%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 30.7%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in y around 0 14.8%

      \[\leadsto \color{blue}{\left(\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]
   8. Step-by-step derivation

      1. associate--l+ 14.8%

         \[\leadsto \color{blue}{\left(2 + \left(0.5 \cdot y - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      2. *-commutative 14.8%

         \[\leadsto \left(2 + \left(\color{blue}{y \cdot 0.5} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      3. +-commutative 14.8%

         \[\leadsto \left(2 + \left(y \cdot 0.5 - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   9. Simplified 14.8%

      \[\leadsto \color{blue}{\left(2 + \left(y \cdot 0.5 - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   10. Taylor expanded in z around inf 14.7%

       \[\leadsto \color{blue}{\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   11. Step-by-step derivation

       1. associate--l+ 14.7%

          \[\leadsto \color{blue}{2 + \left(0.5 \cdot y - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       2. +-commutative 14.7%

          \[\leadsto 2 + \left(0.5 \cdot y - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

       3. *-commutative 14.7%

          \[\leadsto 2 + \left(\color{blue}{y \cdot 0.5} - \left(\sqrt{y} + \sqrt{x}\right)\right) \]

   12. Simplified 14.7%

       \[\leadsto \color{blue}{2 + \left(y \cdot 0.5 - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(\left(\sqrt{1 + t} + z \cdot 0.5\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 4.3:\\ \;\;\;\;3 + z \cdot \left(0.5 - \sqrt{\frac{1}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(y \cdot 0.5 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 60.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3.3:\\ \;\;\;\;3 + z \cdot \left(0.5 - \sqrt{\frac{1}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(y \cdot 0.5 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.3)
   (+ 3.0 (* z (- 0.5 (sqrt (/ 1.0 z)))))
   (+ 2.0 (- (* y 0.5) (+ (sqrt x) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.3) {
		tmp = 3.0 + (z * (0.5 - sqrt((1.0 / z))));
	} else {
		tmp = 2.0 + ((y * 0.5) - (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 3.3d0) then
        tmp = 3.0d0 + (z * (0.5d0 - sqrt((1.0d0 / z))))
    else
        tmp = 2.0d0 + ((y * 0.5d0) - (sqrt(x) + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.3) {
		tmp = 3.0 + (z * (0.5 - Math.sqrt((1.0 / z))));
	} else {
		tmp = 2.0 + ((y * 0.5) - (Math.sqrt(x) + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 3.3:
		tmp = 3.0 + (z * (0.5 - math.sqrt((1.0 / z))))
	else:
		tmp = 2.0 + ((y * 0.5) - (math.sqrt(x) + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.3)
		tmp = Float64(3.0 + Float64(z * Float64(0.5 - sqrt(Float64(1.0 / z)))));
	else
		tmp = Float64(2.0 + Float64(Float64(y * 0.5) - Float64(sqrt(x) + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3.3)
		tmp = 3.0 + (z * (0.5 - sqrt((1.0 / z))));
	else
		tmp = 2.0 + ((y * 0.5) - (sqrt(x) + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 3.3], N[(3.0 + N[(z * N[(0.5 - N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(N[(y * 0.5), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.2999999999999998

   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. sub-neg 97.7%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 97.7%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 97.7%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 97.7%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 97.7%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 97.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 50.1%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in z around 0 49.2%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(\left(1 + 0.5 \cdot z\right) - \sqrt{z}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   7. Step-by-step derivation

      1. associate--l+ 49.2%

         \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   8. Simplified 49.2%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   9. Taylor expanded in y around 0 13.2%

      \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + 0.5 \cdot z\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 24.9%

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + 0.5 \cdot z\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       2. +-commutative 24.9%

          \[\leadsto 3 + \left(\color{blue}{\left(0.5 \cdot z + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       3. associate-+r+ 24.9%

          \[\leadsto 3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   11. Simplified 24.9%

       \[\leadsto \color{blue}{3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   12. Taylor expanded in z around inf 47.6%

       \[\leadsto 3 + \color{blue}{z \cdot \left(0.5 - \sqrt{\frac{1}{z}}\right)} \]

   if 3.2999999999999998 < z

   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. associate-+l+ 89.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 89.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 89.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 89.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 89.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 89.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 54.9%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 30.7%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in y around 0 14.8%

      \[\leadsto \color{blue}{\left(\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]
   8. Step-by-step derivation

      1. associate--l+ 14.8%

         \[\leadsto \color{blue}{\left(2 + \left(0.5 \cdot y - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      2. *-commutative 14.8%

         \[\leadsto \left(2 + \left(\color{blue}{y \cdot 0.5} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      3. +-commutative 14.8%

         \[\leadsto \left(2 + \left(y \cdot 0.5 - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   9. Simplified 14.8%

      \[\leadsto \color{blue}{\left(2 + \left(y \cdot 0.5 - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   10. Taylor expanded in z around inf 14.7%

       \[\leadsto \color{blue}{\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   11. Step-by-step derivation

       1. associate--l+ 14.7%

          \[\leadsto \color{blue}{2 + \left(0.5 \cdot y - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       2. +-commutative 14.7%

          \[\leadsto 2 + \left(0.5 \cdot y - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

       3. *-commutative 14.7%

          \[\leadsto 2 + \left(\color{blue}{y \cdot 0.5} - \left(\sqrt{y} + \sqrt{x}\right)\right) \]

   12. Simplified 14.7%

       \[\leadsto \color{blue}{2 + \left(y \cdot 0.5 - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.3:\\ \;\;\;\;3 + z \cdot \left(0.5 - \sqrt{\frac{1}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(y \cdot 0.5 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 32.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 4.5:\\ \;\;\;\;3 + z \cdot \left(0.5 - \sqrt{\frac{1}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \sqrt{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 4.5) (+ 3.0 (* z (- 0.5 (sqrt (/ 1.0 z))))) (- 3.0 (sqrt y))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.5) {
		tmp = 3.0 + (z * (0.5 - sqrt((1.0 / z))));
	} else {
		tmp = 3.0 - sqrt(y);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 4.5d0) then
        tmp = 3.0d0 + (z * (0.5d0 - sqrt((1.0d0 / z))))
    else
        tmp = 3.0d0 - sqrt(y)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.5) {
		tmp = 3.0 + (z * (0.5 - Math.sqrt((1.0 / z))));
	} else {
		tmp = 3.0 - Math.sqrt(y);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 4.5:
		tmp = 3.0 + (z * (0.5 - math.sqrt((1.0 / z))))
	else:
		tmp = 3.0 - math.sqrt(y)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 4.5)
		tmp = Float64(3.0 + Float64(z * Float64(0.5 - sqrt(Float64(1.0 / z)))));
	else
		tmp = Float64(3.0 - sqrt(y));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 4.5)
		tmp = 3.0 + (z * (0.5 - sqrt((1.0 / z))));
	else
		tmp = 3.0 - sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 4.5], N[(3.0 + N[(z * N[(0.5 - N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.5

   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. sub-neg 97.7%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 97.7%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 97.7%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 97.7%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 97.7%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 97.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 50.1%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in z around 0 49.2%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(\left(1 + 0.5 \cdot z\right) - \sqrt{z}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   7. Step-by-step derivation

      1. associate--l+ 49.2%

         \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   8. Simplified 49.2%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   9. Taylor expanded in y around 0 13.2%

      \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + 0.5 \cdot z\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 24.9%

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + 0.5 \cdot z\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       2. +-commutative 24.9%

          \[\leadsto 3 + \left(\color{blue}{\left(0.5 \cdot z + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       3. associate-+r+ 24.9%

          \[\leadsto 3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   11. Simplified 24.9%

       \[\leadsto \color{blue}{3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   12. Taylor expanded in z around inf 47.6%

       \[\leadsto 3 + \color{blue}{z \cdot \left(0.5 - \sqrt{\frac{1}{z}}\right)} \]

   if 4.5 < z

   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. associate-+l+ 89.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 89.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 89.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 89.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 89.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 89.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 54.9%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in z around 0 4.6%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(\left(1 + 0.5 \cdot z\right) - \sqrt{z}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   7. Step-by-step derivation

      1. associate--l+ 4.6%

         \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   8. Simplified 4.6%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   9. Taylor expanded in y around 0 3.8%

      \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + 0.5 \cdot z\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 4.3%

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + 0.5 \cdot z\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       2. +-commutative 4.3%

          \[\leadsto 3 + \left(\color{blue}{\left(0.5 \cdot z + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       3. associate-+r+ 4.3%

          \[\leadsto 3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   11. Simplified 4.3%

       \[\leadsto \color{blue}{3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   12. Taylor expanded in y around inf 20.9%

       \[\leadsto 3 + \color{blue}{-1 \cdot \sqrt{y}} \]
   13. Step-by-step derivation

       1. mul-1-neg 20.9%

          \[\leadsto 3 + \color{blue}{\left(-\sqrt{y}\right)} \]

   14. Simplified 20.9%

       \[\leadsto 3 + \color{blue}{\left(-\sqrt{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.5:\\ \;\;\;\;3 + z \cdot \left(0.5 - \sqrt{\frac{1}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \sqrt{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 32.7% accurate, 7.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6:\\ \;\;\;\;3 - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 - \sqrt{\frac{1}{y}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 6.0) (- 3.0 (sqrt y)) (* y (- 0.5 (sqrt (/ 1.0 y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.0) {
		tmp = 3.0 - sqrt(y);
	} else {
		tmp = y * (0.5 - sqrt((1.0 / y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 6.0d0) then
        tmp = 3.0d0 - sqrt(y)
    else
        tmp = y * (0.5d0 - sqrt((1.0d0 / y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.0) {
		tmp = 3.0 - Math.sqrt(y);
	} else {
		tmp = y * (0.5 - Math.sqrt((1.0 / y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 6.0:
		tmp = 3.0 - math.sqrt(y)
	else:
		tmp = y * (0.5 - math.sqrt((1.0 / y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 6.0)
		tmp = Float64(3.0 - sqrt(y));
	else
		tmp = Float64(y * Float64(0.5 - sqrt(Float64(1.0 / y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 6.0)
		tmp = 3.0 - sqrt(y);
	else
		tmp = y * (0.5 - sqrt((1.0 / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 6.0], N[(3.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(y * N[(0.5 - N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6

   1. Initial program 98.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 98.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 49.2%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in z around 0 25.7%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(\left(1 + 0.5 \cdot z\right) - \sqrt{z}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   7. Step-by-step derivation

      1. associate--l+ 25.7%

         \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   8. Simplified 25.7%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   9. Taylor expanded in y around 0 15.2%

      \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + 0.5 \cdot z\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 26.1%

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + 0.5 \cdot z\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       2. +-commutative 26.1%

          \[\leadsto 3 + \left(\color{blue}{\left(0.5 \cdot z + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       3. associate-+r+ 26.1%

          \[\leadsto 3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   11. Simplified 26.1%

       \[\leadsto \color{blue}{3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   12. Taylor expanded in y around inf 48.1%

       \[\leadsto 3 + \color{blue}{-1 \cdot \sqrt{y}} \]
   13. Step-by-step derivation

       1. mul-1-neg 48.1%

          \[\leadsto 3 + \color{blue}{\left(-\sqrt{y}\right)} \]

   14. Simplified 48.1%

       \[\leadsto 3 + \color{blue}{\left(-\sqrt{y}\right)} \]

   if 6 < y

   1. Initial program 89.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 89.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 89.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 89.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 89.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 89.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 89.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 89.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 55.4%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 32.5%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in y around 0 4.7%

      \[\leadsto \color{blue}{\left(\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]
   8. Step-by-step derivation

      1. associate--l+ 4.7%

         \[\leadsto \color{blue}{\left(2 + \left(0.5 \cdot y - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      2. *-commutative 4.7%

         \[\leadsto \left(2 + \left(\color{blue}{y \cdot 0.5} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

      3. +-commutative 4.7%

         \[\leadsto \left(2 + \left(y \cdot 0.5 - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   9. Simplified 4.7%

      \[\leadsto \color{blue}{\left(2 + \left(y \cdot 0.5 - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   10. Taylor expanded in y around inf 5.6%

       \[\leadsto \color{blue}{y \cdot \left(0.5 - \sqrt{\frac{1}{y}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6:\\ \;\;\;\;3 - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 - \sqrt{\frac{1}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 32.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9:\\ \;\;\;\;3 - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 9.0) (- 3.0 (sqrt y)) (* x (sqrt (/ 1.0 x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9.0) {
		tmp = 3.0 - sqrt(y);
	} else {
		tmp = x * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 9.0d0) then
        tmp = 3.0d0 - sqrt(y)
    else
        tmp = x * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9.0) {
		tmp = 3.0 - Math.sqrt(y);
	} else {
		tmp = x * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 9.0:
		tmp = 3.0 - math.sqrt(y)
	else:
		tmp = x * math.sqrt((1.0 / x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 9.0)
		tmp = Float64(3.0 - sqrt(y));
	else
		tmp = Float64(x * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 9.0)
		tmp = 3.0 - sqrt(y);
	else
		tmp = x * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 9.0], N[(3.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9

   1. Initial program 98.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 98.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 98.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 49.2%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in z around 0 25.7%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(\left(1 + 0.5 \cdot z\right) - \sqrt{z}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   7. Step-by-step derivation

      1. associate--l+ 25.7%

         \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   8. Simplified 25.7%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   9. Taylor expanded in y around 0 15.2%

      \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + 0.5 \cdot z\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 26.1%

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + 0.5 \cdot z\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       2. +-commutative 26.1%

          \[\leadsto 3 + \left(\color{blue}{\left(0.5 \cdot z + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       3. associate-+r+ 26.1%

          \[\leadsto 3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   11. Simplified 26.1%

       \[\leadsto \color{blue}{3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   12. Taylor expanded in y around inf 48.1%

       \[\leadsto 3 + \color{blue}{-1 \cdot \sqrt{y}} \]
   13. Step-by-step derivation

       1. mul-1-neg 48.1%

          \[\leadsto 3 + \color{blue}{\left(-\sqrt{y}\right)} \]

   14. Simplified 48.1%

       \[\leadsto 3 + \color{blue}{\left(-\sqrt{y}\right)} \]

   if 9 < y

   1. Initial program 89.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 89.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 89.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 89.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 89.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 89.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 89.4%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 89.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 55.4%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 32.5%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in x around inf 2.2%

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{x} \cdot \sqrt{1 + y} + \left(\frac{1}{x} \cdot \sqrt{1 + z} + \frac{1}{x}\right)\right) - \left(\sqrt{\frac{1}{x}} + \left(\frac{1}{x} \cdot \sqrt{y} + \frac{1}{x} \cdot \sqrt{z}\right)\right)\right)} \]

   8. Taylor expanded in x around -inf 0.0%

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 0.0%

         \[\leadsto x \cdot \color{blue}{\left(-\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow2 0.0%

         \[\leadsto x \cdot \left(-\sqrt{\frac{1}{x}} \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \]

      3. rem-square-sqrt 6.8%

         \[\leadsto x \cdot \left(-\sqrt{\frac{1}{x}} \cdot \color{blue}{-1}\right) \]

      4. distribute-rgt-neg-in 6.8%

         \[\leadsto x \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(--1\right)\right)} \]

      5. metadata-eval 6.8%

         \[\leadsto x \cdot \left(\sqrt{\frac{1}{x}} \cdot \color{blue}{1}\right) \]

      6. *-rgt-identity 6.8%

         \[\leadsto x \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

   10. Simplified 6.8%

       \[\leadsto x \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9:\\ \;\;\;\;3 - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 31.0% accurate, 8.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 3 - \sqrt{y} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- 3.0 (sqrt y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 3.0 - sqrt(y);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 3.0d0 - sqrt(y)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 3.0 - Math.sqrt(y);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 3.0 - math.sqrt(y)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(3.0 - sqrt(y))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 3.0 - sqrt(y);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(3.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 93.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. sub-neg 93.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. sub-neg 93.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. +-commutative 93.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   5. +-commutative 93.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   6. +-commutative 93.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

3. Simplified 93.7%

   \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 52.4%

   \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

6. Taylor expanded in z around 0 27.3%

   \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(\left(1 + 0.5 \cdot z\right) - \sqrt{z}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
7. Step-by-step derivation

   1. associate--l+ 27.3%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

8. Simplified 27.3%

   \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(0.5 \cdot z - \sqrt{z}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

9. Taylor expanded in y around 0 8.6%

   \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + 0.5 \cdot z\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
10. Step-by-step derivation

    1. associate--l+ 14.8%

       \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + 0.5 \cdot z\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

    2. +-commutative 14.8%

       \[\leadsto 3 + \left(\color{blue}{\left(0.5 \cdot z + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

    3. associate-+r+ 14.8%

       \[\leadsto 3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

11. Simplified 14.8%

    \[\leadsto \color{blue}{3 + \left(\left(0.5 \cdot z + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

12. Taylor expanded in y around inf 24.0%

    \[\leadsto 3 + \color{blue}{-1 \cdot \sqrt{y}} \]
13. Step-by-step derivation

    1. mul-1-neg 24.0%

       \[\leadsto 3 + \color{blue}{\left(-\sqrt{y}\right)} \]

14. Simplified 24.0%

    \[\leadsto 3 + \color{blue}{\left(-\sqrt{y}\right)} \]

15. Final simplification 24.0%

    \[\leadsto 3 - \sqrt{y} \]
16. Add Preprocessing

Alternative 27: 1.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ -\sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return -sqrt(x);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -sqrt(x)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return -Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return -math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(-sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = -sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := (-N[Sqrt[x], $MachinePrecision])

Derivation

1. Initial program 93.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 93.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. sub-neg 93.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. sub-neg 93.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. +-commutative 93.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   5. +-commutative 93.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   6. +-commutative 93.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

3. Simplified 93.7%

   \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 52.4%

   \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

6. Taylor expanded in t around inf 30.4%

   \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

7. Taylor expanded in x around inf 6.0%

   \[\leadsto \color{blue}{-1 \cdot \sqrt{x}} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]
8. Step-by-step derivation

   1. mul-1-neg 6.0%

      \[\leadsto \color{blue}{\left(-\sqrt{x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

9. Simplified 6.0%

   \[\leadsto \color{blue}{\left(-\sqrt{x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

10. Taylor expanded in x around inf 1.6%

    \[\leadsto \color{blue}{-1 \cdot \sqrt{x}} \]
11. Step-by-step derivation

    1. neg-mul-1 1.6%

       \[\leadsto \color{blue}{-\sqrt{x}} \]

12. Simplified 1.6%

    \[\leadsto \color{blue}{-\sqrt{x}} \]

13. Final simplification 1.6%

    \[\leadsto -\sqrt{x} \]
14. Add Preprocessing

Developer target: 99.3% 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 2024095 
(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))))