Main:z from

Percentage Accurate: 91.5% → 99.6%

Time: 26.8s

Alternatives: 22

Speedup: 2.0×

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: 91.5% 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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{1 + t}\\ t_4 := \sqrt{z + 1}\\ \mathbf{if}\;z \leq 1.1 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\sqrt{t} + t_3} + \left(\left(\left(t_1 - \sqrt{x}\right) + \left(t_2 - \sqrt{y}\right)\right) + \frac{1}{\sqrt{z} + t_4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_3 - \sqrt{t}\right) + \left(\left(\frac{1}{\sqrt{x} + t_1} + \frac{1}{t_2 + \sqrt{y}}\right) + \left(t_4 - \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

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 x)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (sqrt (+ 1.0 t)))
        (t_4 (sqrt (+ z 1.0))))
   (if (<= z 1.1e+30)
     (+
      (/ 1.0 (+ (sqrt t) t_3))
      (+ (+ (- t_1 (sqrt x)) (- t_2 (sqrt y))) (/ 1.0 (+ (sqrt z) t_4))))
     (+
      (- t_3 (sqrt t))
      (+
       (+ (/ 1.0 (+ (sqrt x) t_1)) (/ 1.0 (+ t_2 (sqrt y))))
       (- t_4 (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 + x));
	double t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((1.0 + t));
	double t_4 = sqrt((z + 1.0));
	double tmp;
	if (z <= 1.1e+30) {
		tmp = (1.0 / (sqrt(t) + t_3)) + (((t_1 - sqrt(x)) + (t_2 - sqrt(y))) + (1.0 / (sqrt(z) + t_4)));
	} else {
		tmp = (t_3 - sqrt(t)) + (((1.0 / (sqrt(x) + t_1)) + (1.0 / (t_2 + sqrt(y)))) + (t_4 - sqrt(z)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + y))
    t_3 = sqrt((1.0d0 + t))
    t_4 = sqrt((z + 1.0d0))
    if (z <= 1.1d+30) then
        tmp = (1.0d0 / (sqrt(t) + t_3)) + (((t_1 - sqrt(x)) + (t_2 - sqrt(y))) + (1.0d0 / (sqrt(z) + t_4)))
    else
        tmp = (t_3 - sqrt(t)) + (((1.0d0 / (sqrt(x) + t_1)) + (1.0d0 / (t_2 + sqrt(y)))) + (t_4 - 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 + x));
	double t_2 = Math.sqrt((1.0 + y));
	double t_3 = Math.sqrt((1.0 + t));
	double t_4 = Math.sqrt((z + 1.0));
	double tmp;
	if (z <= 1.1e+30) {
		tmp = (1.0 / (Math.sqrt(t) + t_3)) + (((t_1 - Math.sqrt(x)) + (t_2 - Math.sqrt(y))) + (1.0 / (Math.sqrt(z) + t_4)));
	} else {
		tmp = (t_3 - Math.sqrt(t)) + (((1.0 / (Math.sqrt(x) + t_1)) + (1.0 / (t_2 + Math.sqrt(y)))) + (t_4 - 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 + x))
	t_2 = math.sqrt((1.0 + y))
	t_3 = math.sqrt((1.0 + t))
	t_4 = math.sqrt((z + 1.0))
	tmp = 0
	if z <= 1.1e+30:
		tmp = (1.0 / (math.sqrt(t) + t_3)) + (((t_1 - math.sqrt(x)) + (t_2 - math.sqrt(y))) + (1.0 / (math.sqrt(z) + t_4)))
	else:
		tmp = (t_3 - math.sqrt(t)) + (((1.0 / (math.sqrt(x) + t_1)) + (1.0 / (t_2 + math.sqrt(y)))) + (t_4 - 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 + x))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = sqrt(Float64(1.0 + t))
	t_4 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (z <= 1.1e+30)
		tmp = Float64(Float64(1.0 / Float64(sqrt(t) + t_3)) + Float64(Float64(Float64(t_1 - sqrt(x)) + Float64(t_2 - sqrt(y))) + Float64(1.0 / Float64(sqrt(z) + t_4))));
	else
		tmp = Float64(Float64(t_3 - sqrt(t)) + Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_1)) + Float64(1.0 / Float64(t_2 + sqrt(y)))) + Float64(t_4 - 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 + x));
	t_2 = sqrt((1.0 + y));
	t_3 = sqrt((1.0 + t));
	t_4 = sqrt((z + 1.0));
	tmp = 0.0;
	if (z <= 1.1e+30)
		tmp = (1.0 / (sqrt(t) + t_3)) + (((t_1 - sqrt(x)) + (t_2 - sqrt(y))) + (1.0 / (sqrt(z) + t_4)));
	else
		tmp = (t_3 - sqrt(t)) + (((1.0 / (sqrt(x) + t_1)) + (1.0 / (t_2 + sqrt(y)))) + (t_4 - 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 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1.1e+30], N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.1e30

   1. Initial program 95.5%

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

      1. +-commutative 95.5%

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

      2. associate-+r- 95.0%

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

      3. +-commutative 95.0%

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

      4. associate-+r- 95.5%

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

      5. associate-+l+ 95.5%

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

   3. Simplified 95.5%

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

      1. flip-- 95.8%

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

      2. add-sqr-sqrt 95.4%

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

      3. add-sqr-sqrt 96.6%

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

   5. Applied egg-rr 96.6%

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

      1. associate--l+ 97.1%

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

      2. +-inverses 97.1%

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

      3. metadata-eval 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

   7. Simplified 97.1%

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

      1. flip-- 97.3%

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

      2. add-sqr-sqrt 79.4%

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

      3. add-sqr-sqrt 97.8%

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

   9. Applied egg-rr 97.8%

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

       1. associate--l+ 98.2%

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

       2. +-inverses 98.2%

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

       3. metadata-eval 98.2%

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

       4. +-commutative 98.2%

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

       5. +-commutative 98.2%

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

   11. Simplified 98.2%

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

   if 1.1e30 < z

   1. Initial program 86.5%

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

      1. +-commutative 86.5%

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

      2. associate-+r- 19.4%

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

      3. +-commutative 19.4%

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

      4. associate-+r- 86.5%

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

      5. associate-+l+ 86.5%

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

   3. Simplified 86.5%

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

      1. flip-- 86.5%

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

      2. add-sqr-sqrt 64.3%

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

      3. add-sqr-sqrt 87.3%

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

   5. Applied egg-rr 87.3%

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

      1. +-commutative 87.3%

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

      2. associate--l+ 90.8%

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

      3. +-inverses 90.8%

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

      4. metadata-eval 90.8%

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

      5. +-commutative 90.8%

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

      6. +-commutative 90.8%

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

   7. Simplified 90.8%

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

      1. flip-- 91.5%

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

      2. add-sqr-sqrt 70.4%

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

      3. add-sqr-sqrt 91.8%

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

   9. Applied egg-rr 91.8%

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

       1. associate--l+ 93.3%

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

       2. +-inverses 93.3%

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

       3. metadata-eval 93.3%

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

   11. Simplified 93.3%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.1 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\sqrt{t} + \sqrt{1 + t}} + \left(\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\\ \end{array} \]

Alternative 2: 92.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{1 + x}\\ t_3 := t_2 - \sqrt{x}\\ t_4 := \sqrt{z + 1}\\ t_5 := \sqrt{z} + t_4\\ t_6 := \sqrt{1 + y}\\ t_7 := t_3 + \left(t_6 - \sqrt{y}\right)\\ \mathbf{if}\;t_7 \leq 0.1:\\ \;\;\;\;\left(t_1 - \sqrt{t}\right) + \left(\left(t_4 - \sqrt{z}\right) + \left(\frac{1}{\sqrt{x} + t_2} + \frac{1}{\sqrt{y} + \left(1 + y \cdot 0.5\right)}\right)\right)\\ \mathbf{elif}\;t_7 \leq 2:\\ \;\;\;\;t_2 + \left(\frac{1}{t_6 + \sqrt{y}} + \left(\frac{1 + \left(z - z\right)}{t_5} - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{t} + t_1} + \left(\frac{1}{t_5} + \left(1 + t_3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 t)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (- t_2 (sqrt x)))
        (t_4 (sqrt (+ z 1.0)))
        (t_5 (+ (sqrt z) t_4))
        (t_6 (sqrt (+ 1.0 y)))
        (t_7 (+ t_3 (- t_6 (sqrt y)))))
   (if (<= t_7 0.1)
     (+
      (- t_1 (sqrt t))
      (+
       (- t_4 (sqrt z))
       (+ (/ 1.0 (+ (sqrt x) t_2)) (/ 1.0 (+ (sqrt y) (+ 1.0 (* y 0.5)))))))
     (if (<= t_7 2.0)
       (+
        t_2
        (+ (/ 1.0 (+ t_6 (sqrt y))) (- (/ (+ 1.0 (- z z)) t_5) (sqrt x))))
       (+ (/ 1.0 (+ (sqrt t) t_1)) (+ (/ 1.0 t_5) (+ 1.0 t_3)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t));
	double t_2 = sqrt((1.0 + x));
	double t_3 = t_2 - sqrt(x);
	double t_4 = sqrt((z + 1.0));
	double t_5 = sqrt(z) + t_4;
	double t_6 = sqrt((1.0 + y));
	double t_7 = t_3 + (t_6 - sqrt(y));
	double tmp;
	if (t_7 <= 0.1) {
		tmp = (t_1 - sqrt(t)) + ((t_4 - sqrt(z)) + ((1.0 / (sqrt(x) + t_2)) + (1.0 / (sqrt(y) + (1.0 + (y * 0.5))))));
	} else if (t_7 <= 2.0) {
		tmp = t_2 + ((1.0 / (t_6 + sqrt(y))) + (((1.0 + (z - z)) / t_5) - sqrt(x)));
	} else {
		tmp = (1.0 / (sqrt(t) + t_1)) + ((1.0 / t_5) + (1.0 + t_3));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + t))
    t_2 = sqrt((1.0d0 + x))
    t_3 = t_2 - sqrt(x)
    t_4 = sqrt((z + 1.0d0))
    t_5 = sqrt(z) + t_4
    t_6 = sqrt((1.0d0 + y))
    t_7 = t_3 + (t_6 - sqrt(y))
    if (t_7 <= 0.1d0) then
        tmp = (t_1 - sqrt(t)) + ((t_4 - sqrt(z)) + ((1.0d0 / (sqrt(x) + t_2)) + (1.0d0 / (sqrt(y) + (1.0d0 + (y * 0.5d0))))))
    else if (t_7 <= 2.0d0) then
        tmp = t_2 + ((1.0d0 / (t_6 + sqrt(y))) + (((1.0d0 + (z - z)) / t_5) - sqrt(x)))
    else
        tmp = (1.0d0 / (sqrt(t) + t_1)) + ((1.0d0 / t_5) + (1.0d0 + t_3))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + t));
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = t_2 - Math.sqrt(x);
	double t_4 = Math.sqrt((z + 1.0));
	double t_5 = Math.sqrt(z) + t_4;
	double t_6 = Math.sqrt((1.0 + y));
	double t_7 = t_3 + (t_6 - Math.sqrt(y));
	double tmp;
	if (t_7 <= 0.1) {
		tmp = (t_1 - Math.sqrt(t)) + ((t_4 - Math.sqrt(z)) + ((1.0 / (Math.sqrt(x) + t_2)) + (1.0 / (Math.sqrt(y) + (1.0 + (y * 0.5))))));
	} else if (t_7 <= 2.0) {
		tmp = t_2 + ((1.0 / (t_6 + Math.sqrt(y))) + (((1.0 + (z - z)) / t_5) - Math.sqrt(x)));
	} else {
		tmp = (1.0 / (Math.sqrt(t) + t_1)) + ((1.0 / t_5) + (1.0 + t_3));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + t))
	t_2 = math.sqrt((1.0 + x))
	t_3 = t_2 - math.sqrt(x)
	t_4 = math.sqrt((z + 1.0))
	t_5 = math.sqrt(z) + t_4
	t_6 = math.sqrt((1.0 + y))
	t_7 = t_3 + (t_6 - math.sqrt(y))
	tmp = 0
	if t_7 <= 0.1:
		tmp = (t_1 - math.sqrt(t)) + ((t_4 - math.sqrt(z)) + ((1.0 / (math.sqrt(x) + t_2)) + (1.0 / (math.sqrt(y) + (1.0 + (y * 0.5))))))
	elif t_7 <= 2.0:
		tmp = t_2 + ((1.0 / (t_6 + math.sqrt(y))) + (((1.0 + (z - z)) / t_5) - math.sqrt(x)))
	else:
		tmp = (1.0 / (math.sqrt(t) + t_1)) + ((1.0 / t_5) + (1.0 + t_3))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + t))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = Float64(t_2 - sqrt(x))
	t_4 = sqrt(Float64(z + 1.0))
	t_5 = Float64(sqrt(z) + t_4)
	t_6 = sqrt(Float64(1.0 + y))
	t_7 = Float64(t_3 + Float64(t_6 - sqrt(y)))
	tmp = 0.0
	if (t_7 <= 0.1)
		tmp = Float64(Float64(t_1 - sqrt(t)) + Float64(Float64(t_4 - sqrt(z)) + Float64(Float64(1.0 / Float64(sqrt(x) + t_2)) + Float64(1.0 / Float64(sqrt(y) + Float64(1.0 + Float64(y * 0.5)))))));
	elseif (t_7 <= 2.0)
		tmp = Float64(t_2 + Float64(Float64(1.0 / Float64(t_6 + sqrt(y))) + Float64(Float64(Float64(1.0 + Float64(z - z)) / t_5) - sqrt(x))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(t) + t_1)) + Float64(Float64(1.0 / t_5) + Float64(1.0 + t_3)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + t));
	t_2 = sqrt((1.0 + x));
	t_3 = t_2 - sqrt(x);
	t_4 = sqrt((z + 1.0));
	t_5 = sqrt(z) + t_4;
	t_6 = sqrt((1.0 + y));
	t_7 = t_3 + (t_6 - sqrt(y));
	tmp = 0.0;
	if (t_7 <= 0.1)
		tmp = (t_1 - sqrt(t)) + ((t_4 - sqrt(z)) + ((1.0 / (sqrt(x) + t_2)) + (1.0 / (sqrt(y) + (1.0 + (y * 0.5))))));
	elseif (t_7 <= 2.0)
		tmp = t_2 + ((1.0 / (t_6 + sqrt(y))) + (((1.0 + (z - z)) / t_5) - sqrt(x)));
	else
		tmp = (1.0 / (sqrt(t) + t_1)) + ((1.0 / t_5) + (1.0 + t_3));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[z], $MachinePrecision] + t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(t$95$3 + N[(t$95$6 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 0.1], N[(N[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[(1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 2.0], N[(t$95$2 + N[(N[(1.0 / N[(t$95$6 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(z - z), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / t$95$5), $MachinePrecision] + N[(1.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) < 0.10000000000000001

   1. Initial program 76.1%

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

      1. +-commutative 76.1%

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

      2. associate-+r- 72.2%

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

      3. +-commutative 72.2%

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

      4. associate-+r- 76.1%

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

      5. associate-+l+ 76.1%

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

   3. Simplified 76.1%

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

      1. flip-- 76.1%

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

      2. add-sqr-sqrt 38.5%

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

      3. add-sqr-sqrt 77.5%

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

   5. Applied egg-rr 77.5%

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

      1. +-commutative 77.5%

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

      2. associate--l+ 83.9%

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

      3. +-inverses 83.9%

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

      4. metadata-eval 83.9%

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

      5. +-commutative 83.9%

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

      6. +-commutative 83.9%

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

   7. Simplified 83.9%

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

      1. flip-- 85.2%

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

      2. add-sqr-sqrt 39.7%

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

      3. add-sqr-sqrt 85.4%

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

   9. Applied egg-rr 85.4%

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

       1. associate--l+ 89.1%

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

       2. +-inverses 89.1%

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

       3. metadata-eval 89.1%

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

   11. Simplified 89.1%

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

   12. Taylor expanded in y around 0 81.6%

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

       1. *-commutative 81.6%

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

   14. Simplified 81.6%

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

   if 0.10000000000000001 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) < 2

   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. +-commutative 96.6%

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

      3. associate-+r- 67.7%

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

      4. associate-+l- 63.5%

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

      5. +-commutative 63.5%

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

      6. +-commutative 63.5%

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

      7. associate--l+ 63.4%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in t around inf 37.5%

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

      1. associate--l+ 38.2%

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

      2. +-commutative 38.2%

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

   6. Simplified 38.2%

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

      1. flip-- 97.2%

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

      2. add-sqr-sqrt 79.9%

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

      3. add-sqr-sqrt 97.4%

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

   8. Applied egg-rr 38.4%

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

      1. associate--l+ 97.5%

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

      2. +-inverses 97.5%

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

      3. metadata-eval 97.5%

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

   10. Simplified 38.4%

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

       1. flip-- 38.6%

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

       2. add-sqr-sqrt 31.0%

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

       3. add-sqr-sqrt 38.7%

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

       4. +-commutative 38.7%

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

       5. +-commutative 38.7%

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

   12. Applied egg-rr 38.7%

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

       1. +-commutative 38.7%

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

       2. associate--r+ 38.8%

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

       3. +-commutative 38.8%

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

   14. Simplified 38.8%

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

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

   1. Initial program 91.4%

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

      1. +-commutative 91.4%

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

      2. associate-+r- 60.7%

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

      3. +-commutative 60.7%

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

      4. associate-+r- 91.4%

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

      5. associate-+l+ 91.4%

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

   3. Simplified 91.4%

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

      1. flip-- 91.6%

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

      2. add-sqr-sqrt 72.4%

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

      3. add-sqr-sqrt 92.0%

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

   5. Applied egg-rr 92.0%

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

      1. associate--l+ 92.8%

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

      2. +-inverses 92.8%

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

      3. metadata-eval 92.8%

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

      4. +-commutative 92.8%

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

      5. +-commutative 92.8%

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

   7. Simplified 92.8%

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

   8. Taylor expanded in y around 0 55.3%

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

      1. flip-- 93.1%

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

      2. add-sqr-sqrt 75.4%

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

      3. add-sqr-sqrt 93.5%

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

   10. Applied egg-rr 55.4%

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

       1. associate--l+ 95.5%

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

       2. +-inverses 95.5%

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

       3. metadata-eval 95.5%

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

       4. +-commutative 95.5%

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

       5. +-commutative 95.5%

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

   12. Simplified 55.7%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 0.1:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \left(1 + y \cdot 0.5\right)}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 2:\\ \;\;\;\;\sqrt{1 + x} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1 + \left(z - z\right)}{\sqrt{z} + \sqrt{z + 1}} - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{t} + \sqrt{1 + t}} + \left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \end{array} \]

Alternative 3: 92.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{1 + x}\\ t_3 := t_2 - \sqrt{x}\\ t_4 := \sqrt{z + 1}\\ t_5 := \sqrt{1 + y}\\ t_6 := t_5 - \sqrt{y}\\ t_7 := t_3 + t_6\\ t_8 := \sqrt{z} + t_4\\ \mathbf{if}\;t_7 \leq 0.1:\\ \;\;\;\;\left(t_1 - \sqrt{t}\right) + \left(\left(t_4 - \sqrt{z}\right) + \left(t_6 + \frac{1}{\sqrt{x} + t_2}\right)\right)\\ \mathbf{elif}\;t_7 \leq 2:\\ \;\;\;\;t_2 + \left(\frac{1}{t_5 + \sqrt{y}} + \left(\frac{1 + \left(z - z\right)}{t_8} - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{t} + t_1} + \left(\frac{1}{t_8} + \left(1 + t_3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 t)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (- t_2 (sqrt x)))
        (t_4 (sqrt (+ z 1.0)))
        (t_5 (sqrt (+ 1.0 y)))
        (t_6 (- t_5 (sqrt y)))
        (t_7 (+ t_3 t_6))
        (t_8 (+ (sqrt z) t_4)))
   (if (<= t_7 0.1)
     (+ (- t_1 (sqrt t)) (+ (- t_4 (sqrt z)) (+ t_6 (/ 1.0 (+ (sqrt x) t_2)))))
     (if (<= t_7 2.0)
       (+
        t_2
        (+ (/ 1.0 (+ t_5 (sqrt y))) (- (/ (+ 1.0 (- z z)) t_8) (sqrt x))))
       (+ (/ 1.0 (+ (sqrt t) t_1)) (+ (/ 1.0 t_8) (+ 1.0 t_3)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t));
	double t_2 = sqrt((1.0 + x));
	double t_3 = t_2 - sqrt(x);
	double t_4 = sqrt((z + 1.0));
	double t_5 = sqrt((1.0 + y));
	double t_6 = t_5 - sqrt(y);
	double t_7 = t_3 + t_6;
	double t_8 = sqrt(z) + t_4;
	double tmp;
	if (t_7 <= 0.1) {
		tmp = (t_1 - sqrt(t)) + ((t_4 - sqrt(z)) + (t_6 + (1.0 / (sqrt(x) + t_2))));
	} else if (t_7 <= 2.0) {
		tmp = t_2 + ((1.0 / (t_5 + sqrt(y))) + (((1.0 + (z - z)) / t_8) - sqrt(x)));
	} else {
		tmp = (1.0 / (sqrt(t) + t_1)) + ((1.0 / t_8) + (1.0 + t_3));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + t))
    t_2 = sqrt((1.0d0 + x))
    t_3 = t_2 - sqrt(x)
    t_4 = sqrt((z + 1.0d0))
    t_5 = sqrt((1.0d0 + y))
    t_6 = t_5 - sqrt(y)
    t_7 = t_3 + t_6
    t_8 = sqrt(z) + t_4
    if (t_7 <= 0.1d0) then
        tmp = (t_1 - sqrt(t)) + ((t_4 - sqrt(z)) + (t_6 + (1.0d0 / (sqrt(x) + t_2))))
    else if (t_7 <= 2.0d0) then
        tmp = t_2 + ((1.0d0 / (t_5 + sqrt(y))) + (((1.0d0 + (z - z)) / t_8) - sqrt(x)))
    else
        tmp = (1.0d0 / (sqrt(t) + t_1)) + ((1.0d0 / t_8) + (1.0d0 + t_3))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + t));
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = t_2 - Math.sqrt(x);
	double t_4 = Math.sqrt((z + 1.0));
	double t_5 = Math.sqrt((1.0 + y));
	double t_6 = t_5 - Math.sqrt(y);
	double t_7 = t_3 + t_6;
	double t_8 = Math.sqrt(z) + t_4;
	double tmp;
	if (t_7 <= 0.1) {
		tmp = (t_1 - Math.sqrt(t)) + ((t_4 - Math.sqrt(z)) + (t_6 + (1.0 / (Math.sqrt(x) + t_2))));
	} else if (t_7 <= 2.0) {
		tmp = t_2 + ((1.0 / (t_5 + Math.sqrt(y))) + (((1.0 + (z - z)) / t_8) - Math.sqrt(x)));
	} else {
		tmp = (1.0 / (Math.sqrt(t) + t_1)) + ((1.0 / t_8) + (1.0 + t_3));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + t))
	t_2 = math.sqrt((1.0 + x))
	t_3 = t_2 - math.sqrt(x)
	t_4 = math.sqrt((z + 1.0))
	t_5 = math.sqrt((1.0 + y))
	t_6 = t_5 - math.sqrt(y)
	t_7 = t_3 + t_6
	t_8 = math.sqrt(z) + t_4
	tmp = 0
	if t_7 <= 0.1:
		tmp = (t_1 - math.sqrt(t)) + ((t_4 - math.sqrt(z)) + (t_6 + (1.0 / (math.sqrt(x) + t_2))))
	elif t_7 <= 2.0:
		tmp = t_2 + ((1.0 / (t_5 + math.sqrt(y))) + (((1.0 + (z - z)) / t_8) - math.sqrt(x)))
	else:
		tmp = (1.0 / (math.sqrt(t) + t_1)) + ((1.0 / t_8) + (1.0 + t_3))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + t))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = Float64(t_2 - sqrt(x))
	t_4 = sqrt(Float64(z + 1.0))
	t_5 = sqrt(Float64(1.0 + y))
	t_6 = Float64(t_5 - sqrt(y))
	t_7 = Float64(t_3 + t_6)
	t_8 = Float64(sqrt(z) + t_4)
	tmp = 0.0
	if (t_7 <= 0.1)
		tmp = Float64(Float64(t_1 - sqrt(t)) + Float64(Float64(t_4 - sqrt(z)) + Float64(t_6 + Float64(1.0 / Float64(sqrt(x) + t_2)))));
	elseif (t_7 <= 2.0)
		tmp = Float64(t_2 + Float64(Float64(1.0 / Float64(t_5 + sqrt(y))) + Float64(Float64(Float64(1.0 + Float64(z - z)) / t_8) - sqrt(x))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(t) + t_1)) + Float64(Float64(1.0 / t_8) + Float64(1.0 + t_3)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + t));
	t_2 = sqrt((1.0 + x));
	t_3 = t_2 - sqrt(x);
	t_4 = sqrt((z + 1.0));
	t_5 = sqrt((1.0 + y));
	t_6 = t_5 - sqrt(y);
	t_7 = t_3 + t_6;
	t_8 = sqrt(z) + t_4;
	tmp = 0.0;
	if (t_7 <= 0.1)
		tmp = (t_1 - sqrt(t)) + ((t_4 - sqrt(z)) + (t_6 + (1.0 / (sqrt(x) + t_2))));
	elseif (t_7 <= 2.0)
		tmp = t_2 + ((1.0 / (t_5 + sqrt(y))) + (((1.0 + (z - z)) / t_8) - sqrt(x)));
	else
		tmp = (1.0 / (sqrt(t) + t_1)) + ((1.0 / t_8) + (1.0 + t_3));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$3 + t$95$6), $MachinePrecision]}, Block[{t$95$8 = N[(N[Sqrt[z], $MachinePrecision] + t$95$4), $MachinePrecision]}, If[LessEqual[t$95$7, 0.1], N[(N[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$6 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 2.0], N[(t$95$2 + N[(N[(1.0 / N[(t$95$5 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(z - z), $MachinePrecision]), $MachinePrecision] / t$95$8), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / t$95$8), $MachinePrecision] + N[(1.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) < 0.10000000000000001

   1. Initial program 76.1%

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

      1. +-commutative 76.1%

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

      2. associate-+r- 72.2%

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

      3. +-commutative 72.2%

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

      4. associate-+r- 76.1%

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

      5. associate-+l+ 76.1%

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

   3. Simplified 76.1%

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

      1. flip-- 76.1%

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

      2. add-sqr-sqrt 38.5%

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

      3. add-sqr-sqrt 77.5%

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

   5. Applied egg-rr 77.5%

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

      1. +-commutative 77.5%

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

      2. associate--l+ 83.9%

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

      3. +-inverses 83.9%

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

      4. metadata-eval 83.9%

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

      5. +-commutative 83.9%

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

      6. +-commutative 83.9%

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

   7. Simplified 83.9%

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

   if 0.10000000000000001 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) < 2

   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. +-commutative 96.6%

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

      3. associate-+r- 67.7%

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

      4. associate-+l- 63.5%

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

      5. +-commutative 63.5%

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

      6. +-commutative 63.5%

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

      7. associate--l+ 63.4%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in t around inf 37.5%

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

      1. associate--l+ 38.2%

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

      2. +-commutative 38.2%

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

   6. Simplified 38.2%

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

      1. flip-- 97.2%

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

      2. add-sqr-sqrt 79.9%

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

      3. add-sqr-sqrt 97.4%

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

   8. Applied egg-rr 38.4%

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

      1. associate--l+ 97.5%

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

      2. +-inverses 97.5%

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

      3. metadata-eval 97.5%

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

   10. Simplified 38.4%

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

       1. flip-- 38.6%

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

       2. add-sqr-sqrt 31.0%

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

       3. add-sqr-sqrt 38.7%

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

       4. +-commutative 38.7%

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

       5. +-commutative 38.7%

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

   12. Applied egg-rr 38.7%

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

       1. +-commutative 38.7%

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

       2. associate--r+ 38.8%

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

       3. +-commutative 38.8%

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

   14. Simplified 38.8%

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

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

   1. Initial program 91.4%

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

      1. +-commutative 91.4%

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

      2. associate-+r- 60.7%

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

      3. +-commutative 60.7%

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

      4. associate-+r- 91.4%

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

      5. associate-+l+ 91.4%

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

   3. Simplified 91.4%

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

      1. flip-- 91.6%

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

      2. add-sqr-sqrt 72.4%

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

      3. add-sqr-sqrt 92.0%

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

   5. Applied egg-rr 92.0%

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

      1. associate--l+ 92.8%

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

      2. +-inverses 92.8%

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

      3. metadata-eval 92.8%

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

      4. +-commutative 92.8%

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

      5. +-commutative 92.8%

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

   7. Simplified 92.8%

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

   8. Taylor expanded in y around 0 55.3%

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

      1. flip-- 93.1%

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

      2. add-sqr-sqrt 75.4%

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

      3. add-sqr-sqrt 93.5%

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

   10. Applied egg-rr 55.4%

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

       1. associate--l+ 95.5%

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

       2. +-inverses 95.5%

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

       3. metadata-eval 95.5%

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

       4. +-commutative 95.5%

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

       5. +-commutative 95.5%

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

   12. Simplified 55.7%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 0.1:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 2:\\ \;\;\;\;\sqrt{1 + x} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1 + \left(z - z\right)}{\sqrt{z} + \sqrt{z + 1}} - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{t} + \sqrt{1 + t}} + \left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \end{array} \]

Alternative 4: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{1 + x}\\ t_3 := t_2 - \sqrt{x}\\ t_4 := \sqrt{1 + y}\\ t_5 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t_3 \leq 0.1:\\ \;\;\;\;t_5 + \left(\left(t_1 - \sqrt{z}\right) + \left(\left(t_4 - \sqrt{y}\right) + \frac{1}{\sqrt{x} + t_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_5 + \left(\frac{1}{\sqrt{z} + t_1} + \left(t_3 + \frac{1}{t_4 + \sqrt{y}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (- t_2 (sqrt x)))
        (t_4 (sqrt (+ 1.0 y)))
        (t_5 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_3 0.1)
     (+ t_5 (+ (- t_1 (sqrt z)) (+ (- t_4 (sqrt y)) (/ 1.0 (+ (sqrt x) t_2)))))
     (+ t_5 (+ (/ 1.0 (+ (sqrt z) t_1)) (+ t_3 (/ 1.0 (+ t_4 (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt((1.0 + x));
	double t_3 = t_2 - sqrt(x);
	double t_4 = sqrt((1.0 + y));
	double t_5 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_5 + ((t_1 - sqrt(z)) + ((t_4 - sqrt(y)) + (1.0 / (sqrt(x) + t_2))));
	} else {
		tmp = t_5 + ((1.0 / (sqrt(z) + t_1)) + (t_3 + (1.0 / (t_4 + sqrt(y)))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = sqrt((1.0d0 + x))
    t_3 = t_2 - sqrt(x)
    t_4 = sqrt((1.0d0 + y))
    t_5 = sqrt((1.0d0 + t)) - sqrt(t)
    if (t_3 <= 0.1d0) then
        tmp = t_5 + ((t_1 - sqrt(z)) + ((t_4 - sqrt(y)) + (1.0d0 / (sqrt(x) + t_2))))
    else
        tmp = t_5 + ((1.0d0 / (sqrt(z) + t_1)) + (t_3 + (1.0d0 / (t_4 + sqrt(y)))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = t_2 - Math.sqrt(x);
	double t_4 = Math.sqrt((1.0 + y));
	double t_5 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_5 + ((t_1 - Math.sqrt(z)) + ((t_4 - Math.sqrt(y)) + (1.0 / (Math.sqrt(x) + t_2))));
	} else {
		tmp = t_5 + ((1.0 / (Math.sqrt(z) + t_1)) + (t_3 + (1.0 / (t_4 + Math.sqrt(y)))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = math.sqrt((1.0 + x))
	t_3 = t_2 - math.sqrt(x)
	t_4 = math.sqrt((1.0 + y))
	t_5 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if t_3 <= 0.1:
		tmp = t_5 + ((t_1 - math.sqrt(z)) + ((t_4 - math.sqrt(y)) + (1.0 / (math.sqrt(x) + t_2))))
	else:
		tmp = t_5 + ((1.0 / (math.sqrt(z) + t_1)) + (t_3 + (1.0 / (t_4 + math.sqrt(y)))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = Float64(t_2 - sqrt(x))
	t_4 = sqrt(Float64(1.0 + y))
	t_5 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_3 <= 0.1)
		tmp = Float64(t_5 + Float64(Float64(t_1 - sqrt(z)) + Float64(Float64(t_4 - sqrt(y)) + Float64(1.0 / Float64(sqrt(x) + t_2)))));
	else
		tmp = Float64(t_5 + Float64(Float64(1.0 / Float64(sqrt(z) + t_1)) + Float64(t_3 + Float64(1.0 / Float64(t_4 + sqrt(y))))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	t_2 = sqrt((1.0 + x));
	t_3 = t_2 - sqrt(x);
	t_4 = sqrt((1.0 + y));
	t_5 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (t_3 <= 0.1)
		tmp = t_5 + ((t_1 - sqrt(z)) + ((t_4 - sqrt(y)) + (1.0 / (sqrt(x) + t_2))));
	else
		tmp = t_5 + ((1.0 / (sqrt(z) + t_1)) + (t_3 + (1.0 / (t_4 + sqrt(y)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.1], N[(t$95$5 + N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$5 + N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(1.0 / N[(t$95$4 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 0.10000000000000001

   1. Initial program 86.8%

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

      1. +-commutative 86.8%

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

      2. associate-+r- 62.3%

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

      3. +-commutative 62.3%

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

      4. associate-+r- 86.8%

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

      5. associate-+l+ 86.8%

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

   3. Simplified 86.8%

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

      1. flip-- 87.0%

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

      2. add-sqr-sqrt 47.9%

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

      3. add-sqr-sqrt 88.0%

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

   5. Applied egg-rr 88.0%

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

      1. +-commutative 88.0%

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

      2. associate--l+ 91.3%

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

      3. +-inverses 91.3%

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

      4. metadata-eval 91.3%

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

      5. +-commutative 91.3%

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

      6. +-commutative 91.3%

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

   7. Simplified 91.3%

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

   if 0.10000000000000001 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   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. +-commutative 96.9%

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

      2. associate-+r- 58.8%

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

      3. +-commutative 58.8%

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

      4. associate-+r- 96.9%

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

      5. associate-+l+ 96.9%

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

   3. Simplified 96.9%

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

      1. flip-- 97.1%

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

      2. add-sqr-sqrt 76.6%

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

      3. add-sqr-sqrt 97.4%

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

   5. Applied egg-rr 97.4%

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

      1. associate--l+ 97.6%

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

      2. +-inverses 97.6%

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

      3. metadata-eval 97.6%

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

      4. +-commutative 97.6%

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

      5. +-commutative 97.6%

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

   7. Simplified 97.6%

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

      1. flip-- 96.9%

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

      2. add-sqr-sqrt 68.5%

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

      3. add-sqr-sqrt 97.2%

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

   9. Applied egg-rr 97.9%

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

       1. associate--l+ 97.3%

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

       2. +-inverses 97.3%

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

       3. metadata-eval 97.3%

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

   11. Simplified 98.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.1:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)\\ \end{array} \]

Alternative 5: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. +-commutative 91.4%

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

   2. associate-+r- 60.7%

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

   3. +-commutative 60.7%

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

   4. associate-+r- 91.4%

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

   5. associate-+l+ 91.4%

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

3. Simplified 91.4%

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

   1. flip-- 91.5%

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

   2. add-sqr-sqrt 70.3%

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

   3. add-sqr-sqrt 92.1%

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

5. Applied egg-rr 92.1%

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

   1. +-commutative 92.1%

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

   2. associate--l+ 93.9%

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

   3. +-inverses 93.9%

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

   4. metadata-eval 93.9%

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

   5. +-commutative 93.9%

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

   6. +-commutative 93.9%

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

7. Simplified 93.9%

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

   1. flip-- 94.2%

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

   2. add-sqr-sqrt 69.7%

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

   3. add-sqr-sqrt 94.4%

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

9. Applied egg-rr 94.4%

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

    1. associate--l+ 95.4%

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

    2. +-inverses 95.4%

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

    3. metadata-eval 95.4%

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

11. Simplified 95.4%

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

12. Final simplification 95.4%

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

Alternative 6: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z} + \sqrt{z + 1}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 6.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\sqrt{t} + \sqrt{1 + t}} + \left(\frac{1}{t_1} + \left(1 + \left(t_2 - \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+37}:\\ \;\;\;\;t_2 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1 + \left(z - z\right)}{t_1} - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt z) (sqrt (+ z 1.0)))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= y 6.2e-34)
     (+
      (/ 1.0 (+ (sqrt t) (sqrt (+ 1.0 t))))
      (+ (/ 1.0 t_1) (+ 1.0 (- t_2 (sqrt x)))))
     (if (<= y 2.8e+37)
       (+
        t_2
        (+
         (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))
         (- (/ (+ 1.0 (- z z)) t_1) (sqrt x))))
       (/ 1.0 (+ (sqrt x) t_2))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(z) + sqrt((z + 1.0));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (y <= 6.2e-34) {
		tmp = (1.0 / (sqrt(t) + sqrt((1.0 + t)))) + ((1.0 / t_1) + (1.0 + (t_2 - sqrt(x))));
	} else if (y <= 2.8e+37) {
		tmp = t_2 + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (((1.0 + (z - z)) / t_1) - sqrt(x)));
	} else {
		tmp = 1.0 / (sqrt(x) + t_2);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt(z) + sqrt((z + 1.0d0))
    t_2 = sqrt((1.0d0 + x))
    if (y <= 6.2d-34) then
        tmp = (1.0d0 / (sqrt(t) + sqrt((1.0d0 + t)))) + ((1.0d0 / t_1) + (1.0d0 + (t_2 - sqrt(x))))
    else if (y <= 2.8d+37) then
        tmp = t_2 + ((1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))) + (((1.0d0 + (z - z)) / t_1) - sqrt(x)))
    else
        tmp = 1.0d0 / (sqrt(x) + t_2)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(z) + Math.sqrt((z + 1.0));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 6.2e-34) {
		tmp = (1.0 / (Math.sqrt(t) + Math.sqrt((1.0 + t)))) + ((1.0 / t_1) + (1.0 + (t_2 - Math.sqrt(x))));
	} else if (y <= 2.8e+37) {
		tmp = t_2 + ((1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) + (((1.0 + (z - z)) / t_1) - Math.sqrt(x)));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_2);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(z) + math.sqrt((z + 1.0))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 6.2e-34:
		tmp = (1.0 / (math.sqrt(t) + math.sqrt((1.0 + t)))) + ((1.0 / t_1) + (1.0 + (t_2 - math.sqrt(x))))
	elif y <= 2.8e+37:
		tmp = t_2 + ((1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) + (((1.0 + (z - z)) / t_1) - math.sqrt(x)))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_2)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(z) + sqrt(Float64(z + 1.0)))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 6.2e-34)
		tmp = Float64(Float64(1.0 / Float64(sqrt(t) + sqrt(Float64(1.0 + t)))) + Float64(Float64(1.0 / t_1) + Float64(1.0 + Float64(t_2 - sqrt(x)))));
	elseif (y <= 2.8e+37)
		tmp = Float64(t_2 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))) + Float64(Float64(Float64(1.0 + Float64(z - z)) / t_1) - sqrt(x))));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_2));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(z) + sqrt((z + 1.0));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 6.2e-34)
		tmp = (1.0 / (sqrt(t) + sqrt((1.0 + t)))) + ((1.0 / t_1) + (1.0 + (t_2 - sqrt(x))));
	elseif (y <= 2.8e+37)
		tmp = t_2 + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (((1.0 + (z - z)) / t_1) - sqrt(x)));
	else
		tmp = 1.0 / (sqrt(x) + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 6.2e-34], N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / t$95$1), $MachinePrecision] + N[(1.0 + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e+37], N[(t$95$2 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(z - z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.1999999999999996e-34

   1. Initial program 97.6%

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

      1. +-commutative 97.6%

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

      2. associate-+r- 58.0%

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

      3. +-commutative 58.0%

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

      4. associate-+r- 97.6%

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

      5. associate-+l+ 97.6%

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

   3. Simplified 97.6%

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

      1. flip-- 97.5%

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

      2. add-sqr-sqrt 72.5%

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

      3. add-sqr-sqrt 97.9%

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

   5. Applied egg-rr 97.9%

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

      1. associate--l+ 98.4%

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

      2. +-inverses 98.4%

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

      3. metadata-eval 98.4%

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

      4. +-commutative 98.4%

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

      5. +-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in y around 0 98.4%

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

      1. flip-- 98.4%

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

      2. add-sqr-sqrt 77.6%

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

      3. add-sqr-sqrt 98.7%

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

   10. Applied egg-rr 98.7%

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

       1. associate--l+ 99.2%

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

       2. +-inverses 99.2%

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

       3. metadata-eval 99.2%

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

       4. +-commutative 99.2%

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

       5. +-commutative 99.2%

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

   12. Simplified 99.2%

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

   if 6.1999999999999996e-34 < y < 2.7999999999999998e37

   1. Initial program 83.9%

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

      1. associate-+l+ 83.9%

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

      2. +-commutative 83.9%

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

      3. associate-+r- 54.4%

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

      4. associate-+l- 36.0%

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

      5. +-commutative 36.0%

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

      6. +-commutative 36.0%

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

      7. associate--l+ 36.0%

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

   3. Simplified 26.2%

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

   4. Taylor expanded in t around inf 20.7%

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

      1. associate--l+ 20.3%

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

      2. +-commutative 20.3%

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

   6. Simplified 20.3%

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

      1. flip-- 88.2%

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

      2. add-sqr-sqrt 79.8%

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

      3. add-sqr-sqrt 89.8%

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

   8. Applied egg-rr 21.5%

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

      1. associate--l+ 97.3%

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

      2. +-inverses 97.3%

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

      3. metadata-eval 97.3%

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

   10. Simplified 21.5%

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

       1. flip-- 21.5%

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

       2. add-sqr-sqrt 20.5%

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

       3. add-sqr-sqrt 21.5%

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

       4. +-commutative 21.5%

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

       5. +-commutative 21.5%

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

   12. Applied egg-rr 21.5%

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

       1. +-commutative 21.5%

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

       2. associate--r+ 21.5%

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

       3. +-commutative 21.5%

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

   14. Simplified 21.5%

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

   if 2.7999999999999998e37 < y

   1. Initial program 87.5%

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

      1. associate-+l+ 87.5%

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

      2. +-commutative 87.5%

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

      3. associate-+r- 87.5%

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

      4. associate-+l- 53.8%

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

      5. +-commutative 53.8%

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

      6. +-commutative 53.8%

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

      7. associate--l+ 53.8%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in t around inf 34.8%

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

      1. associate--l+ 36.6%

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

      2. +-commutative 36.6%

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

   6. Simplified 36.6%

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

   7. Taylor expanded in z around inf 3.2%

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

      1. associate--l+ 22.0%

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

      2. +-commutative 22.0%

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

   9. Simplified 22.0%

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

   10. Taylor expanded in y around inf 23.3%

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

       1. flip-- 23.3%

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

       2. add-sqr-sqrt 23.4%

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

       3. add-sqr-sqrt 24.0%

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

       4. +-commutative 24.0%

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

   12. Applied egg-rr 24.0%

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

       1. associate--l+ 28.5%

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

       2. +-inverses 28.5%

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

       3. metadata-eval 28.5%

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

   14. Simplified 28.5%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\sqrt{t} + \sqrt{1 + t}} + \left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{1 + x} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1 + \left(z - z\right)}{\sqrt{z} + \sqrt{z + 1}} - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 7: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 10^{-34}:\\ \;\;\;\;\frac{1}{\sqrt{t} + \sqrt{1 + t}} + \left(\frac{1}{\sqrt{z} + t_1} + \left(1 + \left(t_2 - \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+37}:\\ \;\;\;\;t_2 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(t_1 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= y 1e-34)
     (+
      (/ 1.0 (+ (sqrt t) (sqrt (+ 1.0 t))))
      (+ (/ 1.0 (+ (sqrt z) t_1)) (+ 1.0 (- t_2 (sqrt x)))))
     (if (<= y 3.7e+37)
       (+
        t_2
        (+
         (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))
         (- (- t_1 (sqrt z)) (sqrt x))))
       (/ 1.0 (+ (sqrt x) t_2))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (y <= 1e-34) {
		tmp = (1.0 / (sqrt(t) + sqrt((1.0 + t)))) + ((1.0 / (sqrt(z) + t_1)) + (1.0 + (t_2 - sqrt(x))));
	} else if (y <= 3.7e+37) {
		tmp = t_2 + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + ((t_1 - sqrt(z)) - sqrt(x)));
	} else {
		tmp = 1.0 / (sqrt(x) + t_2);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = sqrt((1.0d0 + x))
    if (y <= 1d-34) then
        tmp = (1.0d0 / (sqrt(t) + sqrt((1.0d0 + t)))) + ((1.0d0 / (sqrt(z) + t_1)) + (1.0d0 + (t_2 - sqrt(x))))
    else if (y <= 3.7d+37) then
        tmp = t_2 + ((1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))) + ((t_1 - sqrt(z)) - sqrt(x)))
    else
        tmp = 1.0d0 / (sqrt(x) + t_2)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 1e-34) {
		tmp = (1.0 / (Math.sqrt(t) + Math.sqrt((1.0 + t)))) + ((1.0 / (Math.sqrt(z) + t_1)) + (1.0 + (t_2 - Math.sqrt(x))));
	} else if (y <= 3.7e+37) {
		tmp = t_2 + ((1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) + ((t_1 - Math.sqrt(z)) - Math.sqrt(x)));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_2);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 1e-34:
		tmp = (1.0 / (math.sqrt(t) + math.sqrt((1.0 + t)))) + ((1.0 / (math.sqrt(z) + t_1)) + (1.0 + (t_2 - math.sqrt(x))))
	elif y <= 3.7e+37:
		tmp = t_2 + ((1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) + ((t_1 - math.sqrt(z)) - math.sqrt(x)))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_2)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 1e-34)
		tmp = Float64(Float64(1.0 / Float64(sqrt(t) + sqrt(Float64(1.0 + t)))) + Float64(Float64(1.0 / Float64(sqrt(z) + t_1)) + Float64(1.0 + Float64(t_2 - sqrt(x)))));
	elseif (y <= 3.7e+37)
		tmp = Float64(t_2 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))) + Float64(Float64(t_1 - sqrt(z)) - sqrt(x))));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_2));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 1e-34)
		tmp = (1.0 / (sqrt(t) + sqrt((1.0 + t)))) + ((1.0 / (sqrt(z) + t_1)) + (1.0 + (t_2 - sqrt(x))));
	elseif (y <= 3.7e+37)
		tmp = t_2 + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + ((t_1 - sqrt(z)) - sqrt(x)));
	else
		tmp = 1.0 / (sqrt(x) + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1e-34], N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e+37], N[(t$95$2 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.99999999999999928e-35

   1. Initial program 97.6%

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

      1. +-commutative 97.6%

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

      2. associate-+r- 58.0%

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

      3. +-commutative 58.0%

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

      4. associate-+r- 97.6%

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

      5. associate-+l+ 97.6%

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

   3. Simplified 97.6%

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

      1. flip-- 97.5%

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

      2. add-sqr-sqrt 72.5%

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

      3. add-sqr-sqrt 97.9%

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

   5. Applied egg-rr 97.9%

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

      1. associate--l+ 98.4%

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

      2. +-inverses 98.4%

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

      3. metadata-eval 98.4%

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

      4. +-commutative 98.4%

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

      5. +-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in y around 0 98.4%

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

      1. flip-- 98.4%

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

      2. add-sqr-sqrt 77.6%

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

      3. add-sqr-sqrt 98.7%

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

   10. Applied egg-rr 98.7%

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

       1. associate--l+ 99.2%

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

       2. +-inverses 99.2%

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

       3. metadata-eval 99.2%

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

       4. +-commutative 99.2%

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

       5. +-commutative 99.2%

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

   12. Simplified 99.2%

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

   if 9.99999999999999928e-35 < y < 3.6999999999999999e37

   1. Initial program 83.9%

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

      1. associate-+l+ 83.9%

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

      2. +-commutative 83.9%

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

      3. associate-+r- 54.4%

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

      4. associate-+l- 36.0%

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

      5. +-commutative 36.0%

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

      6. +-commutative 36.0%

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

      7. associate--l+ 36.0%

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

   3. Simplified 26.2%

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

   4. Taylor expanded in t around inf 20.7%

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

      1. associate--l+ 20.3%

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

      2. +-commutative 20.3%

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

   6. Simplified 20.3%

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

      1. flip-- 88.2%

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

      2. add-sqr-sqrt 79.8%

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

      3. add-sqr-sqrt 89.8%

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

   8. Applied egg-rr 21.5%

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

      1. associate--l+ 97.3%

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

      2. +-inverses 97.3%

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

      3. metadata-eval 97.3%

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

   10. Simplified 21.5%

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

   if 3.6999999999999999e37 < y

   1. Initial program 87.5%

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

      1. associate-+l+ 87.5%

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

      2. +-commutative 87.5%

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

      3. associate-+r- 87.5%

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

      4. associate-+l- 53.8%

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

      5. +-commutative 53.8%

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

      6. +-commutative 53.8%

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

      7. associate--l+ 53.8%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in t around inf 34.8%

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

      1. associate--l+ 36.6%

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

      2. +-commutative 36.6%

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

   6. Simplified 36.6%

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

   7. Taylor expanded in z around inf 3.2%

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

      1. associate--l+ 22.0%

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

      2. +-commutative 22.0%

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

   9. Simplified 22.0%

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

   10. Taylor expanded in y around inf 23.3%

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

       1. flip-- 23.3%

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

       2. add-sqr-sqrt 23.4%

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

       3. add-sqr-sqrt 24.0%

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

       4. +-commutative 24.0%

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

   12. Applied egg-rr 24.0%

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

       1. associate--l+ 28.5%

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

       2. +-inverses 28.5%

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

       3. metadata-eval 28.5%

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

   14. Simplified 28.5%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-34}:\\ \;\;\;\;\frac{1}{\sqrt{t} + \sqrt{1 + t}} + \left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{1 + x} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 8: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 4.8 \cdot 10^{-34}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{z} + t_1} + 2\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+37}:\\ \;\;\;\;t_2 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(t_1 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= y 4.8e-34)
     (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (+ (/ 1.0 (+ (sqrt z) t_1)) 2.0))
     (if (<= y 3.7e+37)
       (+
        t_2
        (+
         (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))
         (- (- t_1 (sqrt z)) (sqrt x))))
       (/ 1.0 (+ (sqrt x) t_2))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (y <= 4.8e-34) {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((1.0 / (sqrt(z) + t_1)) + 2.0);
	} else if (y <= 3.7e+37) {
		tmp = t_2 + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + ((t_1 - sqrt(z)) - sqrt(x)));
	} else {
		tmp = 1.0 / (sqrt(x) + t_2);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = sqrt((1.0d0 + x))
    if (y <= 4.8d-34) then
        tmp = (sqrt((1.0d0 + t)) - sqrt(t)) + ((1.0d0 / (sqrt(z) + t_1)) + 2.0d0)
    else if (y <= 3.7d+37) then
        tmp = t_2 + ((1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))) + ((t_1 - sqrt(z)) - sqrt(x)))
    else
        tmp = 1.0d0 / (sqrt(x) + t_2)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 4.8e-34) {
		tmp = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) + ((1.0 / (Math.sqrt(z) + t_1)) + 2.0);
	} else if (y <= 3.7e+37) {
		tmp = t_2 + ((1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) + ((t_1 - Math.sqrt(z)) - Math.sqrt(x)));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_2);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 4.8e-34:
		tmp = (math.sqrt((1.0 + t)) - math.sqrt(t)) + ((1.0 / (math.sqrt(z) + t_1)) + 2.0)
	elif y <= 3.7e+37:
		tmp = t_2 + ((1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) + ((t_1 - math.sqrt(z)) - math.sqrt(x)))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_2)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 4.8e-34)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(1.0 / Float64(sqrt(z) + t_1)) + 2.0));
	elseif (y <= 3.7e+37)
		tmp = Float64(t_2 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))) + Float64(Float64(t_1 - sqrt(z)) - sqrt(x))));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_2));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 4.8e-34)
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((1.0 / (sqrt(z) + t_1)) + 2.0);
	elseif (y <= 3.7e+37)
		tmp = t_2 + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + ((t_1 - sqrt(z)) - sqrt(x)));
	else
		tmp = 1.0 / (sqrt(x) + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.8e-34], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e+37], N[(t$95$2 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.79999999999999982e-34

   1. Initial program 97.6%

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

      1. +-commutative 97.6%

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

      2. associate-+r- 58.0%

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

      3. +-commutative 58.0%

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

      4. associate-+r- 97.6%

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

      5. associate-+l+ 97.6%

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

   3. Simplified 97.6%

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

      1. flip-- 97.5%

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

      2. add-sqr-sqrt 72.5%

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

      3. add-sqr-sqrt 97.9%

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

   5. Applied egg-rr 97.9%

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

      1. associate--l+ 98.4%

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

      2. +-inverses 98.4%

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

      3. metadata-eval 98.4%

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

      4. +-commutative 98.4%

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

      5. +-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in y around 0 98.4%

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

   9. Taylor expanded in x around 0 53.7%

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

       1. +-commutative 53.7%

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

   11. Simplified 53.7%

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

   if 4.79999999999999982e-34 < y < 3.6999999999999999e37

   1. Initial program 83.9%

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

      1. associate-+l+ 83.9%

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

      2. +-commutative 83.9%

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

      3. associate-+r- 54.4%

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

      4. associate-+l- 36.0%

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

      5. +-commutative 36.0%

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

      6. +-commutative 36.0%

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

      7. associate--l+ 36.0%

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

   3. Simplified 26.2%

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

   4. Taylor expanded in t around inf 20.7%

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

      1. associate--l+ 20.3%

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

      2. +-commutative 20.3%

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

   6. Simplified 20.3%

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

      1. flip-- 88.2%

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

      2. add-sqr-sqrt 79.8%

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

      3. add-sqr-sqrt 89.8%

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

   8. Applied egg-rr 21.5%

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

      1. associate--l+ 97.3%

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

      2. +-inverses 97.3%

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

      3. metadata-eval 97.3%

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

   10. Simplified 21.5%

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

   if 3.6999999999999999e37 < y

   1. Initial program 87.5%

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

      1. associate-+l+ 87.5%

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

      2. +-commutative 87.5%

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

      3. associate-+r- 87.5%

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

      4. associate-+l- 53.8%

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

      5. +-commutative 53.8%

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

      6. +-commutative 53.8%

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

      7. associate--l+ 53.8%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in t around inf 34.8%

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

      1. associate--l+ 36.6%

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

      2. +-commutative 36.6%

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

   6. Simplified 36.6%

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

   7. Taylor expanded in z around inf 3.2%

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

      1. associate--l+ 22.0%

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

      2. +-commutative 22.0%

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

   9. Simplified 22.0%

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

   10. Taylor expanded in y around inf 23.3%

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

       1. flip-- 23.3%

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

       2. add-sqr-sqrt 23.4%

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

       3. add-sqr-sqrt 24.0%

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

       4. +-commutative 24.0%

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

   12. Applied egg-rr 24.0%

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

       1. associate--l+ 28.5%

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

       2. +-inverses 28.5%

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

       3. metadata-eval 28.5%

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

   14. Simplified 28.5%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-34}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + 2\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{1 + x} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 9: 91.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\sqrt{z} + \sqrt{z + 1}} + 2\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\left(t_1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))))
   (if (<= y 1.8e-26)
     (+ (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0)))) 2.0)
     (if (<= y 5e+29)
       (- (+ t_1 (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))) (sqrt x))
       (/ 1.0 (+ (sqrt x) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double tmp;
	if (y <= 1.8e-26) {
		tmp = (1.0 / (sqrt(z) + sqrt((z + 1.0)))) + 2.0;
	} else if (y <= 5e+29) {
		tmp = (t_1 + (1.0 / (sqrt((1.0 + y)) + sqrt(y)))) - sqrt(x);
	} else {
		tmp = 1.0 / (sqrt(x) + t_1);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    if (y <= 1.8d-26) then
        tmp = (1.0d0 / (sqrt(z) + sqrt((z + 1.0d0)))) + 2.0d0
    else if (y <= 5d+29) then
        tmp = (t_1 + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y)))) - sqrt(x)
    else
        tmp = 1.0d0 / (sqrt(x) + t_1)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 1.8e-26) {
		tmp = (1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0)))) + 2.0;
	} else if (y <= 5e+29) {
		tmp = (t_1 + (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y)))) - Math.sqrt(x);
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 1.8e-26:
		tmp = (1.0 / (math.sqrt(z) + math.sqrt((z + 1.0)))) + 2.0
	elif y <= 5e+29:
		tmp = (t_1 + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y)))) - math.sqrt(x)
	else:
		tmp = 1.0 / (math.sqrt(x) + t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 1.8e-26)
		tmp = Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0)))) + 2.0);
	elseif (y <= 5e+29)
		tmp = Float64(Float64(t_1 + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))) - sqrt(x));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_1));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 1.8e-26)
		tmp = (1.0 / (sqrt(z) + sqrt((z + 1.0)))) + 2.0;
	elseif (y <= 5e+29)
		tmp = (t_1 + (1.0 / (sqrt((1.0 + y)) + sqrt(y)))) - sqrt(x);
	else
		tmp = 1.0 / (sqrt(x) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.8e-26], N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 5e+29], N[(N[(t$95$1 + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.8000000000000001e-26

   1. Initial program 97.6%

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

      1. associate-+l+ 97.6%

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

      2. +-commutative 97.6%

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

      3. associate-+r- 55.6%

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

      4. associate-+l- 49.4%

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

      5. +-commutative 49.4%

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

      6. +-commutative 49.4%

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

      7. associate--l+ 49.3%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in t around inf 27.5%

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

      1. associate--l+ 27.7%

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

      2. +-commutative 27.7%

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

   6. Simplified 27.7%

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

   7. Taylor expanded in x around 0 29.4%

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

   8. Taylor expanded in y around 0 29.4%

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

      1. associate--l+ 54.2%

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

      2. rem-square-sqrt 54.2%

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

      3. hypot-1-def 54.2%

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

   10. Simplified 54.2%

       \[\leadsto \color{blue}{2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)} \]
   11. Step-by-step derivation

       1. flip-- 54.1%

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

       2. hypot-udef 54.1%

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

       3. metadata-eval 54.1%

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

       4. add-sqr-sqrt 54.1%

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

       5. hypot-udef 54.1%

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

       6. metadata-eval 54.1%

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

       7. add-sqr-sqrt 54.1%

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

       8. add-sqr-sqrt 41.7%

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

       9. add-sqr-sqrt 54.2%

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

       10. hypot-udef 54.2%

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

       11. metadata-eval 54.2%

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

       12. add-sqr-sqrt 54.2%

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

   12. Applied egg-rr 54.2%

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

       1. associate--l+ 54.6%

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

       2. +-inverses 54.6%

          \[\leadsto 2 + \frac{1 + \color{blue}{0}}{\sqrt{1 + z} + \sqrt{z}} \]

       3. metadata-eval 54.6%

          \[\leadsto 2 + \frac{\color{blue}{1}}{\sqrt{1 + z} + \sqrt{z}} \]

       4. +-commutative 54.6%

          \[\leadsto 2 + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}} \]

   14. Simplified 54.6%

       \[\leadsto 2 + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}} \]

   if 1.8000000000000001e-26 < y < 5.0000000000000001e29

   1. Initial program 84.9%

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

      1. associate-+l+ 84.9%

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

      2. +-commutative 84.9%

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

      3. associate-+r- 53.0%

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

      4. associate-+l- 31.8%

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

      5. +-commutative 31.8%

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

      6. +-commutative 31.8%

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

      7. associate--l+ 31.8%

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

   3. Simplified 26.0%

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

   4. Taylor expanded in t around inf 16.6%

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

      1. associate--l+ 16.3%

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

      2. +-commutative 16.3%

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

   6. Simplified 16.3%

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

      1. flip-- 89.5%

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

      2. add-sqr-sqrt 79.6%

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

      3. add-sqr-sqrt 91.4%

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

   8. Applied egg-rr 17.6%

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

      1. associate--l+ 96.8%

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

      2. +-inverses 96.8%

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

      3. metadata-eval 96.8%

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

   10. Simplified 17.7%

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

   11. Taylor expanded in z around inf 11.7%

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

   if 5.0000000000000001e29 < y

   1. Initial program 86.8%

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

      1. associate-+l+ 86.8%

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

      2. +-commutative 86.8%

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

      3. associate-+r- 86.8%

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

      4. associate-+l- 53.6%

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

      5. +-commutative 53.6%

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

      6. +-commutative 53.6%

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

      7. associate--l+ 53.6%

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

   3. Simplified 31.7%

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

   4. Taylor expanded in t around inf 35.1%

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

      1. associate--l+ 36.8%

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

      2. +-commutative 36.8%

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

   6. Simplified 36.8%

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

   7. Taylor expanded in z around inf 3.2%

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

      1. associate--l+ 21.7%

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

      2. +-commutative 21.7%

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

   9. Simplified 21.7%

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

   10. Taylor expanded in y around inf 23.0%

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

       1. flip-- 23.0%

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

       2. add-sqr-sqrt 23.2%

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

       3. add-sqr-sqrt 23.7%

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

       4. +-commutative 23.7%

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

   12. Applied egg-rr 23.7%

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

       1. associate--l+ 28.2%

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

       2. +-inverses 28.2%

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

       3. metadata-eval 28.2%

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

   14. Simplified 28.2%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\sqrt{z} + \sqrt{z + 1}} + 2\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\left(\sqrt{1 + x} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 10: 97.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + 2\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\left(t_1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))))
   (if (<= y 2.15e-26)
     (+
      (- (sqrt (+ 1.0 t)) (sqrt t))
      (+ (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0)))) 2.0))
     (if (<= y 5e+33)
       (- (+ t_1 (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))) (sqrt x))
       (/ 1.0 (+ (sqrt x) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double tmp;
	if (y <= 2.15e-26) {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((1.0 / (sqrt(z) + sqrt((z + 1.0)))) + 2.0);
	} else if (y <= 5e+33) {
		tmp = (t_1 + (1.0 / (sqrt((1.0 + y)) + sqrt(y)))) - sqrt(x);
	} else {
		tmp = 1.0 / (sqrt(x) + t_1);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    if (y <= 2.15d-26) then
        tmp = (sqrt((1.0d0 + t)) - sqrt(t)) + ((1.0d0 / (sqrt(z) + sqrt((z + 1.0d0)))) + 2.0d0)
    else if (y <= 5d+33) then
        tmp = (t_1 + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y)))) - sqrt(x)
    else
        tmp = 1.0d0 / (sqrt(x) + t_1)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 2.15e-26) {
		tmp = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) + ((1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0)))) + 2.0);
	} else if (y <= 5e+33) {
		tmp = (t_1 + (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y)))) - Math.sqrt(x);
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 2.15e-26:
		tmp = (math.sqrt((1.0 + t)) - math.sqrt(t)) + ((1.0 / (math.sqrt(z) + math.sqrt((z + 1.0)))) + 2.0)
	elif y <= 5e+33:
		tmp = (t_1 + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y)))) - math.sqrt(x)
	else:
		tmp = 1.0 / (math.sqrt(x) + t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 2.15e-26)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0)))) + 2.0));
	elseif (y <= 5e+33)
		tmp = Float64(Float64(t_1 + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))) - sqrt(x));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_1));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 2.15e-26)
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((1.0 / (sqrt(z) + sqrt((z + 1.0)))) + 2.0);
	elseif (y <= 5e+33)
		tmp = (t_1 + (1.0 / (sqrt((1.0 + y)) + sqrt(y)))) - sqrt(x);
	else
		tmp = 1.0 / (sqrt(x) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 2.15e-26], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+33], N[(N[(t$95$1 + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.14999999999999994e-26

   1. Initial program 97.6%

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

      1. +-commutative 97.6%

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

      2. associate-+r- 58.4%

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

      3. +-commutative 58.4%

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

      4. associate-+r- 97.6%

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

      5. associate-+l+ 97.6%

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

   3. Simplified 97.6%

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

      1. flip-- 97.6%

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

      2. add-sqr-sqrt 73.2%

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

      3. add-sqr-sqrt 97.9%

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

   5. Applied egg-rr 97.9%

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

      1. associate--l+ 98.4%

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

      2. +-inverses 98.4%

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

      3. metadata-eval 98.4%

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

      4. +-commutative 98.4%

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

      5. +-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in y around 0 98.3%

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

   9. Taylor expanded in x around 0 54.1%

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

       1. +-commutative 54.1%

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

   11. Simplified 54.1%

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

   if 2.14999999999999994e-26 < y < 4.99999999999999973e33

   1. Initial program 85.1%

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

      1. associate-+l+ 85.1%

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

      2. +-commutative 85.1%

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

      3. associate-+r- 54.3%

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

      4. associate-+l- 33.9%

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

      5. +-commutative 33.9%

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

      6. +-commutative 33.9%

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

      7. associate--l+ 33.9%

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

   3. Simplified 25.5%

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

   4. Taylor expanded in t around inf 19.1%

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

      1. associate--l+ 18.8%

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

      2. +-commutative 18.8%

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

   6. Simplified 18.8%

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

      1. flip-- 89.8%

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

      2. add-sqr-sqrt 80.3%

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

      3. add-sqr-sqrt 91.7%

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

   8. Applied egg-rr 20.1%

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

      1. associate--l+ 96.9%

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

      2. +-inverses 96.9%

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

      3. metadata-eval 96.9%

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

   10. Simplified 20.2%

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

   11. Taylor expanded in z around inf 11.4%

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

   if 4.99999999999999973e33 < y

   1. Initial program 86.8%

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

      1. associate-+l+ 86.8%

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

      2. +-commutative 86.8%

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

      3. associate-+r- 86.8%

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

      4. associate-+l- 53.3%

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

      5. +-commutative 53.3%

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

      6. +-commutative 53.3%

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

      7. associate--l+ 53.3%

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

   3. Simplified 31.9%

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

   4. Taylor expanded in t around inf 34.6%

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

      1. associate--l+ 36.3%

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

      2. +-commutative 36.3%

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

   6. Simplified 36.3%

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

   7. Taylor expanded in z around inf 3.2%

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

      1. associate--l+ 21.9%

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

      2. +-commutative 21.9%

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

   9. Simplified 21.9%

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

   10. Taylor expanded in y around inf 23.1%

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

       1. flip-- 23.1%

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

       2. add-sqr-sqrt 23.2%

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

       3. add-sqr-sqrt 23.8%

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

       4. +-commutative 23.8%

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

   12. Applied egg-rr 23.8%

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

       1. associate--l+ 28.3%

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

       2. +-inverses 28.3%

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

       3. metadata-eval 28.3%

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

   14. Simplified 28.3%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + 2\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\left(\sqrt{1 + x} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 11: 90.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\sqrt{z} + \sqrt{z + 1}} + 2\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+15}:\\ \;\;\;\;t_1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))))
   (if (<= y 2e-26)
     (+ (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0)))) 2.0)
     (if (<= y 5.8e+15)
       (+ t_1 (- (sqrt (+ 1.0 y)) (+ (sqrt x) (sqrt y))))
       (/ 1.0 (+ (sqrt x) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double tmp;
	if (y <= 2e-26) {
		tmp = (1.0 / (sqrt(z) + sqrt((z + 1.0)))) + 2.0;
	} else if (y <= 5.8e+15) {
		tmp = t_1 + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 / (sqrt(x) + t_1);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    if (y <= 2d-26) then
        tmp = (1.0d0 / (sqrt(z) + sqrt((z + 1.0d0)))) + 2.0d0
    else if (y <= 5.8d+15) then
        tmp = t_1 + (sqrt((1.0d0 + y)) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 / (sqrt(x) + t_1)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 2e-26) {
		tmp = (1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0)))) + 2.0;
	} else if (y <= 5.8e+15) {
		tmp = t_1 + (Math.sqrt((1.0 + y)) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 2e-26:
		tmp = (1.0 / (math.sqrt(z) + math.sqrt((z + 1.0)))) + 2.0
	elif y <= 5.8e+15:
		tmp = t_1 + (math.sqrt((1.0 + y)) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 2e-26)
		tmp = Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0)))) + 2.0);
	elseif (y <= 5.8e+15)
		tmp = Float64(t_1 + Float64(sqrt(Float64(1.0 + y)) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_1));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 2e-26)
		tmp = (1.0 / (sqrt(z) + sqrt((z + 1.0)))) + 2.0;
	elseif (y <= 5.8e+15)
		tmp = t_1 + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 / (sqrt(x) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 2e-26], N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 5.8e+15], N[(t$95$1 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.0000000000000001e-26

   1. Initial program 97.6%

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

      1. associate-+l+ 97.6%

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

      2. +-commutative 97.6%

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

      3. associate-+r- 55.6%

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

      4. associate-+l- 49.4%

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

      5. +-commutative 49.4%

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

      6. +-commutative 49.4%

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

      7. associate--l+ 49.3%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in t around inf 27.5%

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

      1. associate--l+ 27.7%

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

      2. +-commutative 27.7%

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

   6. Simplified 27.7%

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

   7. Taylor expanded in x around 0 29.4%

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

   8. Taylor expanded in y around 0 29.4%

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

      1. associate--l+ 54.2%

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

      2. rem-square-sqrt 54.2%

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

      3. hypot-1-def 54.2%

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

   10. Simplified 54.2%

       \[\leadsto \color{blue}{2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)} \]
   11. Step-by-step derivation

       1. flip-- 54.1%

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

       2. hypot-udef 54.1%

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

       3. metadata-eval 54.1%

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

       4. add-sqr-sqrt 54.1%

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

       5. hypot-udef 54.1%

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

       6. metadata-eval 54.1%

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

       7. add-sqr-sqrt 54.1%

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

       8. add-sqr-sqrt 41.7%

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

       9. add-sqr-sqrt 54.2%

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

       10. hypot-udef 54.2%

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

       11. metadata-eval 54.2%

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

       12. add-sqr-sqrt 54.2%

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

   12. Applied egg-rr 54.2%

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

       1. associate--l+ 54.6%

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

       2. +-inverses 54.6%

          \[\leadsto 2 + \frac{1 + \color{blue}{0}}{\sqrt{1 + z} + \sqrt{z}} \]

       3. metadata-eval 54.6%

          \[\leadsto 2 + \frac{\color{blue}{1}}{\sqrt{1 + z} + \sqrt{z}} \]

       4. +-commutative 54.6%

          \[\leadsto 2 + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}} \]

   14. Simplified 54.6%

       \[\leadsto 2 + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}} \]

   if 2.0000000000000001e-26 < y < 5.8e15

   1. Initial program 89.5%

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

      1. associate-+l+ 89.5%

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

      2. +-commutative 89.5%

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

      3. associate-+r- 44.7%

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

      4. associate-+l- 33.5%

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

      5. +-commutative 33.5%

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

      6. +-commutative 33.5%

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

      7. associate--l+ 33.5%

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

   3. Simplified 29.9%

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

   4. Taylor expanded in t around inf 16.0%

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

      1. associate--l+ 15.2%

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

      2. +-commutative 15.2%

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

   6. Simplified 15.2%

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

   7. Taylor expanded in z around inf 11.3%

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

      1. associate--l+ 11.3%

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

      2. +-commutative 11.3%

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

   9. Simplified 11.3%

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

   if 5.8e15 < y

   1. Initial program 86.0%

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

      1. associate-+l+ 85.9%

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

      2. +-commutative 85.9%

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

      3. associate-+r- 86.0%

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

      4. associate-+l- 51.9%

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

      5. +-commutative 51.9%

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

      6. +-commutative 51.9%

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

      7. associate--l+ 51.9%

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

   3. Simplified 30.7%

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

   4. Taylor expanded in t around inf 33.9%

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

      1. associate--l+ 35.6%

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

      2. +-commutative 35.6%

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

   6. Simplified 35.6%

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

   7. Taylor expanded in z around inf 3.4%

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

      1. associate--l+ 20.7%

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

      2. +-commutative 20.7%

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

   9. Simplified 20.7%

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

   10. Taylor expanded in y around inf 22.0%

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

       1. flip-- 22.0%

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

       2. add-sqr-sqrt 22.3%

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

       3. add-sqr-sqrt 22.8%

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

       4. +-commutative 22.8%

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

   12. Applied egg-rr 22.8%

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

       1. associate--l+ 27.1%

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

       2. +-inverses 27.1%

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

       3. metadata-eval 27.1%

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

   14. Simplified 27.1%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\sqrt{z} + \sqrt{z + 1}} + 2\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 12: 90.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\sqrt{z} + \sqrt{z + 1}} + 2\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+15}:\\ \;\;\;\;t_1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))))
   (if (<= y 2.15e-26)
     (+ (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0)))) 2.0)
     (if (<= y 7.6e+15)
       (+ t_1 (- (- (sqrt (+ 1.0 y)) (sqrt y)) (sqrt x)))
       (/ 1.0 (+ (sqrt x) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double tmp;
	if (y <= 2.15e-26) {
		tmp = (1.0 / (sqrt(z) + sqrt((z + 1.0)))) + 2.0;
	} else if (y <= 7.6e+15) {
		tmp = t_1 + ((sqrt((1.0 + y)) - sqrt(y)) - sqrt(x));
	} else {
		tmp = 1.0 / (sqrt(x) + t_1);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    if (y <= 2.15d-26) then
        tmp = (1.0d0 / (sqrt(z) + sqrt((z + 1.0d0)))) + 2.0d0
    else if (y <= 7.6d+15) then
        tmp = t_1 + ((sqrt((1.0d0 + y)) - sqrt(y)) - sqrt(x))
    else
        tmp = 1.0d0 / (sqrt(x) + t_1)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 2.15e-26) {
		tmp = (1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0)))) + 2.0;
	} else if (y <= 7.6e+15) {
		tmp = t_1 + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) - Math.sqrt(x));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 2.15e-26:
		tmp = (1.0 / (math.sqrt(z) + math.sqrt((z + 1.0)))) + 2.0
	elif y <= 7.6e+15:
		tmp = t_1 + ((math.sqrt((1.0 + y)) - math.sqrt(y)) - math.sqrt(x))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 2.15e-26)
		tmp = Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0)))) + 2.0);
	elseif (y <= 7.6e+15)
		tmp = Float64(t_1 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) - sqrt(x)));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_1));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 2.15e-26)
		tmp = (1.0 / (sqrt(z) + sqrt((z + 1.0)))) + 2.0;
	elseif (y <= 7.6e+15)
		tmp = t_1 + ((sqrt((1.0 + y)) - sqrt(y)) - sqrt(x));
	else
		tmp = 1.0 / (sqrt(x) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 2.15e-26], N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 7.6e+15], N[(t$95$1 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.14999999999999994e-26

   1. Initial program 97.6%

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

      1. associate-+l+ 97.6%

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

      2. +-commutative 97.6%

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

      3. associate-+r- 55.6%

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

      4. associate-+l- 49.4%

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

      5. +-commutative 49.4%

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

      6. +-commutative 49.4%

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

      7. associate--l+ 49.3%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in t around inf 27.5%

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

      1. associate--l+ 27.7%

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

      2. +-commutative 27.7%

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

   6. Simplified 27.7%

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

   7. Taylor expanded in x around 0 29.4%

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

   8. Taylor expanded in y around 0 29.4%

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

      1. associate--l+ 54.2%

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

      2. rem-square-sqrt 54.2%

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

      3. hypot-1-def 54.2%

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

   10. Simplified 54.2%

       \[\leadsto \color{blue}{2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)} \]
   11. Step-by-step derivation

       1. flip-- 54.1%

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

       2. hypot-udef 54.1%

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

       3. metadata-eval 54.1%

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

       4. add-sqr-sqrt 54.1%

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

       5. hypot-udef 54.1%

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

       6. metadata-eval 54.1%

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

       7. add-sqr-sqrt 54.1%

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

       8. add-sqr-sqrt 41.7%

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

       9. add-sqr-sqrt 54.2%

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

       10. hypot-udef 54.2%

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

       11. metadata-eval 54.2%

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

       12. add-sqr-sqrt 54.2%

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

   12. Applied egg-rr 54.2%

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

       1. associate--l+ 54.6%

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

       2. +-inverses 54.6%

          \[\leadsto 2 + \frac{1 + \color{blue}{0}}{\sqrt{1 + z} + \sqrt{z}} \]

       3. metadata-eval 54.6%

          \[\leadsto 2 + \frac{\color{blue}{1}}{\sqrt{1 + z} + \sqrt{z}} \]

       4. +-commutative 54.6%

          \[\leadsto 2 + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}} \]

   14. Simplified 54.6%

       \[\leadsto 2 + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}} \]

   if 2.14999999999999994e-26 < y < 7.6e15

   1. Initial program 89.5%

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

      1. associate-+l+ 89.5%

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

      2. +-commutative 89.5%

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

      3. associate-+r- 44.7%

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

      4. associate-+l- 33.5%

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

      5. +-commutative 33.5%

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

      6. +-commutative 33.5%

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

      7. associate--l+ 33.5%

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

   3. Simplified 29.9%

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

   4. Taylor expanded in t around inf 16.0%

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

      1. associate--l+ 15.2%

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

      2. +-commutative 15.2%

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

   6. Simplified 15.2%

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

   7. Taylor expanded in z around inf 11.3%

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

   if 7.6e15 < y

   1. Initial program 86.0%

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

      1. associate-+l+ 85.9%

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

      2. +-commutative 85.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 86.0%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 51.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 51.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 51.9%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 51.9%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 30.7%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 33.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 35.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 35.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 35.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 3.4%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 20.7%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 20.7%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 20.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 22.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 22.0%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 22.3%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. add-sqr-sqrt 22.8%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 22.8%

          \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   12. Applied egg-rr 22.8%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{1 + x}}} \]
   13. Step-by-step derivation

       1. associate--l+ 27.1%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{x} + \sqrt{1 + x}} \]

       2. +-inverses 27.1%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 27.1%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{x} + \sqrt{1 + x}} \]

   14. Simplified 27.1%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\sqrt{z} + \sqrt{z + 1}} + 2\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 13: 90.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\sqrt{z} + \sqrt{z + 1}} + 2\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\mathsf{hypot}\left(1, \sqrt{y}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \end{array} \]

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 2.15e-26)
   (+ (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0)))) 2.0)
   (if (<= y 1.05e+15)
     (+ 1.0 (- (hypot 1.0 (sqrt y)) (sqrt y)))
     (/ 1.0 (+ (sqrt 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 <= 2.15e-26) {
		tmp = (1.0 / (sqrt(z) + sqrt((z + 1.0)))) + 2.0;
	} else if (y <= 1.05e+15) {
		tmp = 1.0 + (hypot(1.0, sqrt(y)) - sqrt(y));
	} else {
		tmp = 1.0 / (sqrt(x) + sqrt((1.0 + x)));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.15e-26) {
		tmp = (1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0)))) + 2.0;
	} else if (y <= 1.05e+15) {
		tmp = 1.0 + (Math.hypot(1.0, Math.sqrt(y)) - Math.sqrt(y));
	} else {
		tmp = 1.0 / (Math.sqrt(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 <= 2.15e-26:
		tmp = (1.0 / (math.sqrt(z) + math.sqrt((z + 1.0)))) + 2.0
	elif y <= 1.05e+15:
		tmp = 1.0 + (math.hypot(1.0, math.sqrt(y)) - math.sqrt(y))
	else:
		tmp = 1.0 / (math.sqrt(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 <= 2.15e-26)
		tmp = Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0)))) + 2.0);
	elseif (y <= 1.05e+15)
		tmp = Float64(1.0 + Float64(hypot(1.0, sqrt(y)) - sqrt(y)));
	else
		tmp = Float64(1.0 / Float64(sqrt(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 <= 2.15e-26)
		tmp = (1.0 / (sqrt(z) + sqrt((z + 1.0)))) + 2.0;
	elseif (y <= 1.05e+15)
		tmp = 1.0 + (hypot(1.0, sqrt(y)) - sqrt(y));
	else
		tmp = 1.0 / (sqrt(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, 2.15e-26], N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 1.05e+15], N[(1.0 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[y], $MachinePrecision] ^ 2], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.14999999999999994e-26

   1. Initial program 97.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 97.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 55.6%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 49.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 49.4%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 49.4%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 49.3%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 37.0%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 27.5%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 27.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 27.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 27.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in x around 0 29.4%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   8. Taylor expanded in y around 0 29.4%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \sqrt{z}} \]
   9. Step-by-step derivation

      1. associate--l+ 54.2%

         \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

      2. rem-square-sqrt 54.2%

         \[\leadsto 2 + \left(\sqrt{1 + \color{blue}{\sqrt{z} \cdot \sqrt{z}}} - \sqrt{z}\right) \]

      3. hypot-1-def 54.2%

         \[\leadsto 2 + \left(\color{blue}{\mathsf{hypot}\left(1, \sqrt{z}\right)} - \sqrt{z}\right) \]

   10. Simplified 54.2%

       \[\leadsto \color{blue}{2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)} \]
   11. Step-by-step derivation

       1. flip-- 54.1%

          \[\leadsto 2 + \color{blue}{\frac{\mathsf{hypot}\left(1, \sqrt{z}\right) \cdot \mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}}} \]

       2. hypot-udef 54.1%

          \[\leadsto 2 + \frac{\color{blue}{\sqrt{1 \cdot 1 + \sqrt{z} \cdot \sqrt{z}}} \cdot \mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       3. metadata-eval 54.1%

          \[\leadsto 2 + \frac{\sqrt{\color{blue}{1} + \sqrt{z} \cdot \sqrt{z}} \cdot \mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       4. add-sqr-sqrt 54.1%

          \[\leadsto 2 + \frac{\sqrt{1 + \color{blue}{z}} \cdot \mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       5. hypot-udef 54.1%

          \[\leadsto 2 + \frac{\sqrt{1 + z} \cdot \color{blue}{\sqrt{1 \cdot 1 + \sqrt{z} \cdot \sqrt{z}}} - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       6. metadata-eval 54.1%

          \[\leadsto 2 + \frac{\sqrt{1 + z} \cdot \sqrt{\color{blue}{1} + \sqrt{z} \cdot \sqrt{z}} - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       7. add-sqr-sqrt 54.1%

          \[\leadsto 2 + \frac{\sqrt{1 + z} \cdot \sqrt{1 + \color{blue}{z}} - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       8. add-sqr-sqrt 41.7%

          \[\leadsto 2 + \frac{\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       9. add-sqr-sqrt 54.2%

          \[\leadsto 2 + \frac{\left(1 + z\right) - \color{blue}{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       10. hypot-udef 54.2%

           \[\leadsto 2 + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{1 \cdot 1 + \sqrt{z} \cdot \sqrt{z}}} + \sqrt{z}} \]

       11. metadata-eval 54.2%

           \[\leadsto 2 + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1} + \sqrt{z} \cdot \sqrt{z}} + \sqrt{z}} \]

       12. add-sqr-sqrt 54.2%

           \[\leadsto 2 + \frac{\left(1 + z\right) - z}{\sqrt{1 + \color{blue}{z}} + \sqrt{z}} \]

   12. Applied egg-rr 54.2%

       \[\leadsto 2 + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}} \]
   13. Step-by-step derivation

       1. associate--l+ 54.6%

          \[\leadsto 2 + \frac{\color{blue}{1 + \left(z - z\right)}}{\sqrt{1 + z} + \sqrt{z}} \]

       2. +-inverses 54.6%

          \[\leadsto 2 + \frac{1 + \color{blue}{0}}{\sqrt{1 + z} + \sqrt{z}} \]

       3. metadata-eval 54.6%

          \[\leadsto 2 + \frac{\color{blue}{1}}{\sqrt{1 + z} + \sqrt{z}} \]

       4. +-commutative 54.6%

          \[\leadsto 2 + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}} \]

   14. Simplified 54.6%

       \[\leadsto 2 + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}} \]

   if 2.14999999999999994e-26 < y < 1.05e15

   1. Initial program 89.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 89.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 89.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 44.7%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 33.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 33.5%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 33.5%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 33.5%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 29.9%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 16.0%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 15.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 15.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 15.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 11.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 11.3%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 11.3%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 11.3%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in x around 0 50.9%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   11. Step-by-step derivation

       1. associate--l+ 50.9%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

       2. rem-square-sqrt 51.0%

          \[\leadsto 1 + \left(\sqrt{1 + \color{blue}{\sqrt{y} \cdot \sqrt{y}}} - \sqrt{y}\right) \]

       3. hypot-1-def 51.0%

          \[\leadsto 1 + \left(\color{blue}{\mathsf{hypot}\left(1, \sqrt{y}\right)} - \sqrt{y}\right) \]

   12. Simplified 51.0%

       \[\leadsto \color{blue}{1 + \left(\mathsf{hypot}\left(1, \sqrt{y}\right) - \sqrt{y}\right)} \]

   if 1.05e15 < y

   1. Initial program 86.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 85.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 85.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 86.0%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 51.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 51.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 51.9%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 51.9%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 30.7%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 33.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 35.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 35.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 35.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 3.4%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 20.7%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 20.7%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 20.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 22.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 22.0%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 22.3%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. add-sqr-sqrt 22.8%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 22.8%

          \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   12. Applied egg-rr 22.8%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{1 + x}}} \]
   13. Step-by-step derivation

       1. associate--l+ 27.1%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{x} + \sqrt{1 + x}} \]

       2. +-inverses 27.1%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 27.1%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{x} + \sqrt{1 + x}} \]

   14. Simplified 27.1%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\sqrt{z} + \sqrt{z + 1}} + 2\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\mathsf{hypot}\left(1, \sqrt{y}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 14: 85.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-26}:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.65e-26)
   (+ (- (sqrt (+ z 1.0)) (sqrt z)) 2.0)
   (if (<= y 1.05e+15)
     (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))
     (- (sqrt (+ 1.0 x)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.65e-26) {
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + 2.0;
	} else if (y <= 1.05e+15) {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	} else {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.65d-26) then
        tmp = (sqrt((z + 1.0d0)) - sqrt(z)) + 2.0d0
    else if (y <= 1.05d+15) then
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    else
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.65e-26) {
		tmp = (Math.sqrt((z + 1.0)) - Math.sqrt(z)) + 2.0;
	} else if (y <= 1.05e+15) {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	} else {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 1.65e-26:
		tmp = (math.sqrt((z + 1.0)) - math.sqrt(z)) + 2.0
	elif y <= 1.05e+15:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	else:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.65e-26)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) + 2.0);
	elseif (y <= 1.05e+15)
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.65e-26)
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + 2.0;
	elseif (y <= 1.05e+15)
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	else
		tmp = sqrt((1.0 + x)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 1.65e-26], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 1.05e+15], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.6499999999999999e-26

   1. Initial program 97.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 97.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 55.6%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 49.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 49.4%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 49.4%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 49.3%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 37.0%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 27.5%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 27.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 27.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 27.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in x around 0 29.4%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   8. Taylor expanded in y around 0 29.4%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \sqrt{z}} \]
   9. Step-by-step derivation

      1. associate--l+ 54.2%

         \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   10. Simplified 54.2%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   if 1.6499999999999999e-26 < y < 1.05e15

   1. Initial program 89.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 89.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 89.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 44.7%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 33.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 33.5%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 33.5%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 33.5%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 29.9%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 16.0%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 15.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 15.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 15.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 11.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 11.3%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 11.3%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 11.3%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in x around 0 50.9%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   11. Step-by-step derivation

       1. associate--l+ 50.9%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   12. Simplified 50.9%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   if 1.05e15 < y

   1. Initial program 86.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 85.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 85.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 86.0%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 51.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 51.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 51.9%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 51.9%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 30.7%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 33.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 35.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 35.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 35.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 3.4%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 20.7%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 20.7%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 20.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 22.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-26}:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 15: 85.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+14}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.15e-26)
   (+ (- (sqrt (+ z 1.0)) (sqrt z)) 2.0)
   (if (<= y 9.6e+14)
     (- (+ 1.0 (sqrt (+ 1.0 y))) (sqrt y))
     (- (sqrt (+ 1.0 x)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.15e-26) {
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + 2.0;
	} else if (y <= 9.6e+14) {
		tmp = (1.0 + sqrt((1.0 + y))) - sqrt(y);
	} else {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 2.15d-26) then
        tmp = (sqrt((z + 1.0d0)) - sqrt(z)) + 2.0d0
    else if (y <= 9.6d+14) then
        tmp = (1.0d0 + sqrt((1.0d0 + y))) - sqrt(y)
    else
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.15e-26) {
		tmp = (Math.sqrt((z + 1.0)) - Math.sqrt(z)) + 2.0;
	} else if (y <= 9.6e+14) {
		tmp = (1.0 + Math.sqrt((1.0 + y))) - Math.sqrt(y);
	} else {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 2.15e-26:
		tmp = (math.sqrt((z + 1.0)) - math.sqrt(z)) + 2.0
	elif y <= 9.6e+14:
		tmp = (1.0 + math.sqrt((1.0 + y))) - math.sqrt(y)
	else:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.15e-26)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) + 2.0);
	elseif (y <= 9.6e+14)
		tmp = Float64(Float64(1.0 + sqrt(Float64(1.0 + y))) - sqrt(y));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.15e-26)
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + 2.0;
	elseif (y <= 9.6e+14)
		tmp = (1.0 + sqrt((1.0 + y))) - sqrt(y);
	else
		tmp = sqrt((1.0 + x)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 2.15e-26], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 9.6e+14], N[(N[(1.0 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.14999999999999994e-26

   1. Initial program 97.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 97.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 55.6%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 49.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 49.4%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 49.4%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 49.3%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 37.0%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 27.5%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 27.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 27.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 27.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in x around 0 29.4%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   8. Taylor expanded in y around 0 29.4%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \sqrt{z}} \]
   9. Step-by-step derivation

      1. associate--l+ 54.2%

         \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   10. Simplified 54.2%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   if 2.14999999999999994e-26 < y < 9.6e14

   1. Initial program 89.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 89.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 89.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 44.7%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 33.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 33.5%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 33.5%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 33.5%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 29.9%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 16.0%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 15.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 15.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 15.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 11.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 11.3%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 11.3%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 11.3%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in x around 0 50.9%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]

   if 9.6e14 < y

   1. Initial program 86.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 85.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 85.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 86.0%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 51.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 51.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 51.9%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 51.9%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 30.7%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 33.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 35.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 35.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 35.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 3.4%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 20.7%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 20.7%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 20.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 22.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+14}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 16: 89.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-26}:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \end{array} \]

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 1.65e-26)
   (+ (- (sqrt (+ z 1.0)) (sqrt z)) 2.0)
   (if (<= y 1.7e+15)
     (- (+ 1.0 (sqrt (+ 1.0 y))) (sqrt y))
     (/ 1.0 (+ (sqrt 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 <= 1.65e-26) {
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + 2.0;
	} else if (y <= 1.7e+15) {
		tmp = (1.0 + sqrt((1.0 + y))) - sqrt(y);
	} else {
		tmp = 1.0 / (sqrt(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 <= 1.65d-26) then
        tmp = (sqrt((z + 1.0d0)) - sqrt(z)) + 2.0d0
    else if (y <= 1.7d+15) then
        tmp = (1.0d0 + sqrt((1.0d0 + y))) - sqrt(y)
    else
        tmp = 1.0d0 / (sqrt(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 <= 1.65e-26) {
		tmp = (Math.sqrt((z + 1.0)) - Math.sqrt(z)) + 2.0;
	} else if (y <= 1.7e+15) {
		tmp = (1.0 + Math.sqrt((1.0 + y))) - Math.sqrt(y);
	} else {
		tmp = 1.0 / (Math.sqrt(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 <= 1.65e-26:
		tmp = (math.sqrt((z + 1.0)) - math.sqrt(z)) + 2.0
	elif y <= 1.7e+15:
		tmp = (1.0 + math.sqrt((1.0 + y))) - math.sqrt(y)
	else:
		tmp = 1.0 / (math.sqrt(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 <= 1.65e-26)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) + 2.0);
	elseif (y <= 1.7e+15)
		tmp = Float64(Float64(1.0 + sqrt(Float64(1.0 + y))) - sqrt(y));
	else
		tmp = Float64(1.0 / Float64(sqrt(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 <= 1.65e-26)
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + 2.0;
	elseif (y <= 1.7e+15)
		tmp = (1.0 + sqrt((1.0 + y))) - sqrt(y);
	else
		tmp = 1.0 / (sqrt(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, 1.65e-26], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 1.7e+15], N[(N[(1.0 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.6499999999999999e-26

   1. Initial program 97.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 97.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 55.6%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 49.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 49.4%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 49.4%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 49.3%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 37.0%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 27.5%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 27.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 27.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 27.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in x around 0 29.4%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   8. Taylor expanded in y around 0 29.4%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \sqrt{z}} \]
   9. Step-by-step derivation

      1. associate--l+ 54.2%

         \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   10. Simplified 54.2%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   if 1.6499999999999999e-26 < y < 1.7e15

   1. Initial program 89.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 89.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 89.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 44.7%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 33.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 33.5%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 33.5%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 33.5%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 29.9%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 16.0%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 15.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 15.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 15.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 11.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 11.3%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 11.3%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 11.3%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in x around 0 50.9%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]

   if 1.7e15 < y

   1. Initial program 86.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 85.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 85.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 86.0%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 51.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 51.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 51.9%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 51.9%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 30.7%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 33.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 35.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 35.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 35.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 3.4%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 20.7%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 20.7%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 20.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 22.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 22.0%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 22.3%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. add-sqr-sqrt 22.8%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 22.8%

          \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   12. Applied egg-rr 22.8%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{1 + x}}} \]
   13. Step-by-step derivation

       1. associate--l+ 27.1%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{x} + \sqrt{1 + x}} \]

       2. +-inverses 27.1%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 27.1%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{x} + \sqrt{1 + x}} \]

   14. Simplified 27.1%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-26}:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 17: 90.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\sqrt{z} + \sqrt{z + 1}} + 2\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \end{array} \]

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 2.05e-26)
   (+ (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0)))) 2.0)
   (if (<= y 1.55e+15)
     (- (+ 1.0 (sqrt (+ 1.0 y))) (sqrt y))
     (/ 1.0 (+ (sqrt 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 <= 2.05e-26) {
		tmp = (1.0 / (sqrt(z) + sqrt((z + 1.0)))) + 2.0;
	} else if (y <= 1.55e+15) {
		tmp = (1.0 + sqrt((1.0 + y))) - sqrt(y);
	} else {
		tmp = 1.0 / (sqrt(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 <= 2.05d-26) then
        tmp = (1.0d0 / (sqrt(z) + sqrt((z + 1.0d0)))) + 2.0d0
    else if (y <= 1.55d+15) then
        tmp = (1.0d0 + sqrt((1.0d0 + y))) - sqrt(y)
    else
        tmp = 1.0d0 / (sqrt(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 <= 2.05e-26) {
		tmp = (1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0)))) + 2.0;
	} else if (y <= 1.55e+15) {
		tmp = (1.0 + Math.sqrt((1.0 + y))) - Math.sqrt(y);
	} else {
		tmp = 1.0 / (Math.sqrt(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 <= 2.05e-26:
		tmp = (1.0 / (math.sqrt(z) + math.sqrt((z + 1.0)))) + 2.0
	elif y <= 1.55e+15:
		tmp = (1.0 + math.sqrt((1.0 + y))) - math.sqrt(y)
	else:
		tmp = 1.0 / (math.sqrt(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 <= 2.05e-26)
		tmp = Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0)))) + 2.0);
	elseif (y <= 1.55e+15)
		tmp = Float64(Float64(1.0 + sqrt(Float64(1.0 + y))) - sqrt(y));
	else
		tmp = Float64(1.0 / Float64(sqrt(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 <= 2.05e-26)
		tmp = (1.0 / (sqrt(z) + sqrt((z + 1.0)))) + 2.0;
	elseif (y <= 1.55e+15)
		tmp = (1.0 + sqrt((1.0 + y))) - sqrt(y);
	else
		tmp = 1.0 / (sqrt(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, 2.05e-26], N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 1.55e+15], N[(N[(1.0 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.0499999999999999e-26

   1. Initial program 97.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 97.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 55.6%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 49.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 49.4%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 49.4%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 49.3%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 37.0%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 27.5%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 27.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 27.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 27.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in x around 0 29.4%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   8. Taylor expanded in y around 0 29.4%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \sqrt{z}} \]
   9. Step-by-step derivation

      1. associate--l+ 54.2%

         \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

      2. rem-square-sqrt 54.2%

         \[\leadsto 2 + \left(\sqrt{1 + \color{blue}{\sqrt{z} \cdot \sqrt{z}}} - \sqrt{z}\right) \]

      3. hypot-1-def 54.2%

         \[\leadsto 2 + \left(\color{blue}{\mathsf{hypot}\left(1, \sqrt{z}\right)} - \sqrt{z}\right) \]

   10. Simplified 54.2%

       \[\leadsto \color{blue}{2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)} \]
   11. Step-by-step derivation

       1. flip-- 54.1%

          \[\leadsto 2 + \color{blue}{\frac{\mathsf{hypot}\left(1, \sqrt{z}\right) \cdot \mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}}} \]

       2. hypot-udef 54.1%

          \[\leadsto 2 + \frac{\color{blue}{\sqrt{1 \cdot 1 + \sqrt{z} \cdot \sqrt{z}}} \cdot \mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       3. metadata-eval 54.1%

          \[\leadsto 2 + \frac{\sqrt{\color{blue}{1} + \sqrt{z} \cdot \sqrt{z}} \cdot \mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       4. add-sqr-sqrt 54.1%

          \[\leadsto 2 + \frac{\sqrt{1 + \color{blue}{z}} \cdot \mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       5. hypot-udef 54.1%

          \[\leadsto 2 + \frac{\sqrt{1 + z} \cdot \color{blue}{\sqrt{1 \cdot 1 + \sqrt{z} \cdot \sqrt{z}}} - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       6. metadata-eval 54.1%

          \[\leadsto 2 + \frac{\sqrt{1 + z} \cdot \sqrt{\color{blue}{1} + \sqrt{z} \cdot \sqrt{z}} - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       7. add-sqr-sqrt 54.1%

          \[\leadsto 2 + \frac{\sqrt{1 + z} \cdot \sqrt{1 + \color{blue}{z}} - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       8. add-sqr-sqrt 41.7%

          \[\leadsto 2 + \frac{\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       9. add-sqr-sqrt 54.2%

          \[\leadsto 2 + \frac{\left(1 + z\right) - \color{blue}{z}}{\mathsf{hypot}\left(1, \sqrt{z}\right) + \sqrt{z}} \]

       10. hypot-udef 54.2%

           \[\leadsto 2 + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{1 \cdot 1 + \sqrt{z} \cdot \sqrt{z}}} + \sqrt{z}} \]

       11. metadata-eval 54.2%

           \[\leadsto 2 + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1} + \sqrt{z} \cdot \sqrt{z}} + \sqrt{z}} \]

       12. add-sqr-sqrt 54.2%

           \[\leadsto 2 + \frac{\left(1 + z\right) - z}{\sqrt{1 + \color{blue}{z}} + \sqrt{z}} \]

   12. Applied egg-rr 54.2%

       \[\leadsto 2 + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}} \]
   13. Step-by-step derivation

       1. associate--l+ 54.6%

          \[\leadsto 2 + \frac{\color{blue}{1 + \left(z - z\right)}}{\sqrt{1 + z} + \sqrt{z}} \]

       2. +-inverses 54.6%

          \[\leadsto 2 + \frac{1 + \color{blue}{0}}{\sqrt{1 + z} + \sqrt{z}} \]

       3. metadata-eval 54.6%

          \[\leadsto 2 + \frac{\color{blue}{1}}{\sqrt{1 + z} + \sqrt{z}} \]

       4. +-commutative 54.6%

          \[\leadsto 2 + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}} \]

   14. Simplified 54.6%

       \[\leadsto 2 + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}} \]

   if 2.0499999999999999e-26 < y < 1.55e15

   1. Initial program 89.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 89.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 89.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 44.7%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 33.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 33.5%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 33.5%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 33.5%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 29.9%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 16.0%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 15.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 15.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 15.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 11.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 11.3%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 11.3%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 11.3%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in x around 0 50.9%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]

   if 1.55e15 < y

   1. Initial program 86.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 85.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 85.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 86.0%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 51.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 51.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 51.9%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 51.9%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 30.7%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 33.9%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 35.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 35.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 35.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 3.4%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 20.7%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 20.7%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 20.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 22.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 22.0%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 22.3%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. add-sqr-sqrt 22.8%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 22.8%

          \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   12. Applied egg-rr 22.8%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{1 + x}}} \]
   13. Step-by-step derivation

       1. associate--l+ 27.1%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{x} + \sqrt{1 + x}} \]

       2. +-inverses 27.1%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 27.1%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{x} + \sqrt{1 + x}} \]

   14. Simplified 27.1%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\sqrt{z} + \sqrt{z + 1}} + 2\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 18: 80.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.242:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 0.242) 3.0 (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.242) {
		tmp = 3.0;
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 0.242d0) then
        tmp = 3.0d0
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.242) {
		tmp = 3.0;
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 0.242:
		tmp = 3.0
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 0.242)
		tmp = 3.0;
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 0.242)
		tmp = 3.0;
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 0.242], 3.0, N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.242

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 79.2%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 52.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 52.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 52.9%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 52.9%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 35.1%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 31.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 31.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 31.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 31.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in x around 0 29.2%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   8. Taylor expanded in y around 0 40.9%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \sqrt{z}} \]
   9. Step-by-step derivation

      1. associate--l+ 40.9%

         \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

      2. rem-square-sqrt 40.9%

         \[\leadsto 2 + \left(\sqrt{1 + \color{blue}{\sqrt{z} \cdot \sqrt{z}}} - \sqrt{z}\right) \]

      3. hypot-1-def 40.9%

         \[\leadsto 2 + \left(\color{blue}{\mathsf{hypot}\left(1, \sqrt{z}\right)} - \sqrt{z}\right) \]

   10. Simplified 40.9%

       \[\leadsto \color{blue}{2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)} \]

   11. Taylor expanded in z around 0 40.3%

       \[\leadsto \color{blue}{3} \]

   if 0.242 < z

   1. Initial program 85.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 85.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 85.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 59.3%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 45.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 45.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 45.9%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 45.9%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 31.8%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 27.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 29.4%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 29.4%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 29.4%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 14.8%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 27.2%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 27.2%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 27.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in x around 0 33.8%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   11. Step-by-step derivation

       1. associate--l+ 51.9%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   12. Simplified 51.9%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.242:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 19: 53.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.215:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 0.215) 3.0 (- (sqrt (+ 1.0 x)) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.215) {
		tmp = 3.0;
	} else {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 0.215d0) then
        tmp = 3.0d0
    else
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.215) {
		tmp = 3.0;
	} else {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 0.215:
		tmp = 3.0
	else:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 0.215)
		tmp = 3.0;
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 0.215)
		tmp = 3.0;
	else
		tmp = sqrt((1.0 + x)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 0.215], 3.0, N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.214999999999999997

   1. Initial program 97.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 97.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 55.0%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 48.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 48.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 48.9%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 48.9%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 37.6%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 26.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 26.3%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 26.3%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 26.3%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in x around 0 28.1%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   8. Taylor expanded in y around 0 28.1%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \sqrt{z}} \]
   9. Step-by-step derivation

      1. associate--l+ 51.9%

         \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

      2. rem-square-sqrt 51.9%

         \[\leadsto 2 + \left(\sqrt{1 + \color{blue}{\sqrt{z} \cdot \sqrt{z}}} - \sqrt{z}\right) \]

      3. hypot-1-def 51.9%

         \[\leadsto 2 + \left(\color{blue}{\mathsf{hypot}\left(1, \sqrt{z}\right)} - \sqrt{z}\right) \]

   10. Simplified 51.9%

       \[\leadsto \color{blue}{2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)} \]

   11. Taylor expanded in z around 0 41.7%

       \[\leadsto \color{blue}{3} \]

   if 0.214999999999999997 < y

   1. Initial program 85.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 85.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 85.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 82.7%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 49.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 49.8%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 49.8%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 49.8%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 29.5%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 33.0%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 34.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 34.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 34.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 4.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 20.1%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 20.1%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 20.1%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 21.2%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.215:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 20: 52.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.215:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 0.215) 3.0 (- (+ 1.0 (* x 0.5)) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.215) {
		tmp = 3.0;
	} else {
		tmp = (1.0 + (x * 0.5)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 0.215d0) then
        tmp = 3.0d0
    else
        tmp = (1.0d0 + (x * 0.5d0)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.215) {
		tmp = 3.0;
	} else {
		tmp = (1.0 + (x * 0.5)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 0.215:
		tmp = 3.0
	else:
		tmp = (1.0 + (x * 0.5)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 0.215)
		tmp = 3.0;
	else
		tmp = Float64(Float64(1.0 + Float64(x * 0.5)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 0.215)
		tmp = 3.0;
	else
		tmp = (1.0 + (x * 0.5)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 0.215], 3.0, N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.214999999999999997

   1. Initial program 97.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 97.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 55.0%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 48.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 48.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 48.9%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 48.9%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 37.6%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 26.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 26.3%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 26.3%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 26.3%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in x around 0 28.1%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   8. Taylor expanded in y around 0 28.1%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \sqrt{z}} \]
   9. Step-by-step derivation

      1. associate--l+ 51.9%

         \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

      2. rem-square-sqrt 51.9%

         \[\leadsto 2 + \left(\sqrt{1 + \color{blue}{\sqrt{z} \cdot \sqrt{z}}} - \sqrt{z}\right) \]

      3. hypot-1-def 51.9%

         \[\leadsto 2 + \left(\color{blue}{\mathsf{hypot}\left(1, \sqrt{z}\right)} - \sqrt{z}\right) \]

   10. Simplified 51.9%

       \[\leadsto \color{blue}{2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)} \]

   11. Taylor expanded in z around 0 41.7%

       \[\leadsto \color{blue}{3} \]

   if 0.214999999999999997 < y

   1. Initial program 85.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 85.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 85.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 82.7%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 49.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 49.8%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 49.8%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 49.8%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 29.5%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 33.0%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 34.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 34.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 34.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 4.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 20.1%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 20.1%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 20.1%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 21.2%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   11. Taylor expanded in x around 0 20.8%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot x\right)} - \sqrt{x} \]
   12. Step-by-step derivation

       1. *-commutative 20.8%

          \[\leadsto \left(1 + \color{blue}{x \cdot 0.5}\right) - \sqrt{x} \]

   13. Simplified 20.8%

       \[\leadsto \color{blue}{\left(1 + x \cdot 0.5\right)} - \sqrt{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.215:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \sqrt{x}\\ \end{array} \]

Alternative 21: 52.0% accurate, 268.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.122:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (if (<= y 0.122) 3.0 1.0))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.122) {
		tmp = 3.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 0.122d0) then
        tmp = 3.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.122) {
		tmp = 3.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 0.122:
		tmp = 3.0
	else:
		tmp = 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 0.122)
		tmp = 3.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 0.122)
		tmp = 3.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 0.122], 3.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if y < 0.122

   1. Initial program 97.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 97.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 55.0%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 48.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 48.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 48.9%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 48.9%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 37.6%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 26.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 26.3%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 26.3%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 26.3%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in x around 0 28.1%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   8. Taylor expanded in y around 0 28.1%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \sqrt{z}} \]
   9. Step-by-step derivation

      1. associate--l+ 51.9%

         \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

      2. rem-square-sqrt 51.9%

         \[\leadsto 2 + \left(\sqrt{1 + \color{blue}{\sqrt{z} \cdot \sqrt{z}}} - \sqrt{z}\right) \]

      3. hypot-1-def 51.9%

         \[\leadsto 2 + \left(\color{blue}{\mathsf{hypot}\left(1, \sqrt{z}\right)} - \sqrt{z}\right) \]

   10. Simplified 51.9%

       \[\leadsto \color{blue}{2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)} \]

   11. Taylor expanded in z around 0 41.7%

       \[\leadsto \color{blue}{3} \]

   if 0.122 < y

   1. Initial program 85.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 85.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 85.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 82.7%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 49.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 49.8%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 49.8%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 49.8%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 29.5%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 33.0%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. associate--l+ 34.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

      2. +-commutative 34.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

   6. Simplified 34.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 4.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 20.1%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-commutative 20.1%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 20.1%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 21.2%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   11. Taylor expanded in x around 0 40.0%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.122:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 22: 34.5% accurate, 823.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 1.0)

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0
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;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return 1.0
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 91.4%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 91.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. +-commutative 91.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. associate-+r- 69.2%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 49.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. +-commutative 49.4%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

   6. +-commutative 49.4%

      \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

   7. associate--l+ 49.3%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

3. Simplified 33.4%

   \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)} \]

4. Taylor expanded in t around inf 29.7%

   \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{1 + z}\right)}\right) \]
5. Step-by-step derivation

   1. associate--l+ 30.5%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

   2. +-commutative 30.5%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{\color{blue}{z + 1}}\right)\right)\right) \]

6. Simplified 30.5%

   \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

7. Taylor expanded in z around inf 11.2%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
8. Step-by-step derivation

   1. associate--l+ 19.5%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   2. +-commutative 19.5%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

9. Simplified 19.5%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

10. Taylor expanded in y around inf 15.7%

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

11. Taylor expanded in x around 0 34.1%

    \[\leadsto \color{blue}{1} \]

12. Final simplification 34.1%

    \[\leadsto 1 \]

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 2023290 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :herbie-target
  (+ (+ (+ (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))) (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y)))) (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z)))) (- (sqrt (+ t 1.0)) (sqrt t)))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))