Main:z from

Percentage Accurate: 91.5% → 99.6%

Time: 33.8s

Alternatives: 19

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.5% 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

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 2.5 \cdot 10^{+32}:\\ \;\;\;\;\left(\left(t_3 - \sqrt{y}\right) + \left(t_1 - \sqrt{x}\right)\right) + \left(\frac{1}{t_2 + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t_1 + \sqrt{x}} + \frac{1}{t_3 + \sqrt{y}}\right) + \left(t_2 - \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))) (t_2 (sqrt (+ 1.0 z))) (t_3 (sqrt (+ 1.0 y))))
   (if (<= z 2.5e+32)
     (+
      (+ (- t_3 (sqrt y)) (- t_1 (sqrt x)))
      (+ (/ 1.0 (+ t_2 (sqrt z))) (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t)))))
     (+
      (+ (/ 1.0 (+ t_1 (sqrt x))) (/ 1.0 (+ t_3 (sqrt y))))
      (- t_2 (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 + z));
	double t_3 = sqrt((1.0 + y));
	double tmp;
	if (z <= 2.5e+32) {
		tmp = ((t_3 - sqrt(y)) + (t_1 - sqrt(x))) + ((1.0 / (t_2 + sqrt(z))) + (1.0 / (sqrt((1.0 + t)) + sqrt(t))));
	} else {
		tmp = ((1.0 / (t_1 + sqrt(x))) + (1.0 / (t_3 + sqrt(y)))) + (t_2 - 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) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + z))
    t_3 = sqrt((1.0d0 + y))
    if (z <= 2.5d+32) then
        tmp = ((t_3 - sqrt(y)) + (t_1 - sqrt(x))) + ((1.0d0 / (t_2 + sqrt(z))) + (1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t))))
    else
        tmp = ((1.0d0 / (t_1 + sqrt(x))) + (1.0d0 / (t_3 + sqrt(y)))) + (t_2 - 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 + z));
	double t_3 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 2.5e+32) {
		tmp = ((t_3 - Math.sqrt(y)) + (t_1 - Math.sqrt(x))) + ((1.0 / (t_2 + Math.sqrt(z))) + (1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))));
	} else {
		tmp = ((1.0 / (t_1 + Math.sqrt(x))) + (1.0 / (t_3 + Math.sqrt(y)))) + (t_2 - 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 + z))
	t_3 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 2.5e+32:
		tmp = ((t_3 - math.sqrt(y)) + (t_1 - math.sqrt(x))) + ((1.0 / (t_2 + math.sqrt(z))) + (1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))))
	else:
		tmp = ((1.0 / (t_1 + math.sqrt(x))) + (1.0 / (t_3 + math.sqrt(y)))) + (t_2 - 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 + z))
	t_3 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 2.5e+32)
		tmp = Float64(Float64(Float64(t_3 - sqrt(y)) + Float64(t_1 - sqrt(x))) + Float64(Float64(1.0 / Float64(t_2 + sqrt(z))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_1 + sqrt(x))) + Float64(1.0 / Float64(t_3 + sqrt(y)))) + Float64(t_2 - 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 + z));
	t_3 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 2.5e+32)
		tmp = ((t_3 - sqrt(y)) + (t_1 - sqrt(x))) + ((1.0 / (t_2 + sqrt(z))) + (1.0 / (sqrt((1.0 + t)) + sqrt(t))));
	else
		tmp = ((1.0 / (t_1 + sqrt(x))) + (1.0 / (t_3 + sqrt(y)))) + (t_2 - 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 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 2.5e+32], N[(N[(N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$3 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < 2.4999999999999999e32

   1. Initial program 96.4%

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

      1. associate-+l+ 96.4%

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

      2. associate-+l- 80.9%

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

      3. associate-+l- 96.4%

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

      4. sub-neg 96.4%

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

      5. sub-neg 96.4%

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

      6. +-commutative 96.4%

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

      7. +-commutative 96.4%

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

      8. +-commutative 96.4%

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

   3. Simplified 96.4%

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

      1. flip-- 96.6%

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

      2. add-sqr-sqrt 96.0%

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

      3. add-sqr-sqrt 97.0%

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

   5. Applied egg-rr 97.0%

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

      1. associate--l+ 97.8%

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

      2. +-inverses 97.8%

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

      3. metadata-eval 97.8%

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

   7. Simplified 97.8%

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

      1. flip-- 97.8%

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

      2. add-sqr-sqrt 78.5%

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

      3. add-sqr-sqrt 97.9%

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

   9. Applied egg-rr 97.9%

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

       1. associate--l+ 98.4%

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

       2. +-inverses 98.4%

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

       3. metadata-eval 98.4%

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

   11. Simplified 98.4%

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

   if 2.4999999999999999e32 < z

   1. Initial program 89.2%

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

      1. associate-+l+ 89.2%

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

      2. associate-+l- 64.8%

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

      3. associate-+l- 89.2%

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

      4. sub-neg 89.2%

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

      5. sub-neg 89.2%

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

      6. +-commutative 89.2%

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

      7. +-commutative 89.2%

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

      8. +-commutative 89.2%

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

   3. Simplified 89.2%

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

      1. flip-- 89.9%

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

      2. add-sqr-sqrt 67.0%

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

      3. +-commutative 67.0%

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

      4. add-sqr-sqrt 90.1%

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

      5. +-commutative 90.1%

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

   5. Applied egg-rr 90.1%

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

      1. associate--l+ 92.1%

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

      2. +-inverses 92.1%

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

      3. metadata-eval 92.1%

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

   7. Simplified 92.1%

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

      1. flip-- 92.4%

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

      2. add-sqr-sqrt 79.2%

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

      3. add-sqr-sqrt 93.2%

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

   9. Applied egg-rr 93.2%

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

       1. associate--l+ 95.9%

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

       2. +-inverses 95.9%

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

       3. metadata-eval 95.9%

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

   11. Simplified 95.9%

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

   12. Taylor expanded in t around inf 57.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{+32}:\\ \;\;\;\;\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \end{array} \]

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{1 + z}\\ \mathbf{if}\;t_2 - \sqrt{y} \leq 0.9999999999999998:\\ \;\;\;\;\left(\frac{1}{t_1 + \sqrt{x}} + \frac{1}{t_2 + \sqrt{y}}\right) + \left(t_3 - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{t_3 + \sqrt{z}}\right) + \left(1 + \left(t_1 - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))) (t_2 (sqrt (+ 1.0 y))) (t_3 (sqrt (+ 1.0 z))))
   (if (<= (- t_2 (sqrt y)) 0.9999999999999998)
     (+ (+ (/ 1.0 (+ t_1 (sqrt x))) (/ 1.0 (+ t_2 (sqrt y)))) (- t_3 (sqrt z)))
     (+
      (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (/ 1.0 (+ t_3 (sqrt z))))
      (+ 1.0 (- t_1 (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((1.0 + z));
	double tmp;
	if ((t_2 - sqrt(y)) <= 0.9999999999999998) {
		tmp = ((1.0 / (t_1 + sqrt(x))) + (1.0 / (t_2 + sqrt(y)))) + (t_3 - sqrt(z));
	} else {
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 / (t_3 + sqrt(z)))) + (1.0 + (t_1 - sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + y))
    t_3 = sqrt((1.0d0 + z))
    if ((t_2 - sqrt(y)) <= 0.9999999999999998d0) then
        tmp = ((1.0d0 / (t_1 + sqrt(x))) + (1.0d0 / (t_2 + sqrt(y)))) + (t_3 - sqrt(z))
    else
        tmp = ((sqrt((1.0d0 + t)) - sqrt(t)) + (1.0d0 / (t_3 + sqrt(z)))) + (1.0d0 + (t_1 - sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + y));
	double t_3 = Math.sqrt((1.0 + z));
	double tmp;
	if ((t_2 - Math.sqrt(y)) <= 0.9999999999999998) {
		tmp = ((1.0 / (t_1 + Math.sqrt(x))) + (1.0 / (t_2 + Math.sqrt(y)))) + (t_3 - Math.sqrt(z));
	} else {
		tmp = ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (1.0 / (t_3 + Math.sqrt(z)))) + (1.0 + (t_1 - Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + y))
	t_3 = math.sqrt((1.0 + z))
	tmp = 0
	if (t_2 - math.sqrt(y)) <= 0.9999999999999998:
		tmp = ((1.0 / (t_1 + math.sqrt(x))) + (1.0 / (t_2 + math.sqrt(y)))) + (t_3 - math.sqrt(z))
	else:
		tmp = ((math.sqrt((1.0 + t)) - math.sqrt(t)) + (1.0 / (t_3 + math.sqrt(z)))) + (1.0 + (t_1 - math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (Float64(t_2 - sqrt(y)) <= 0.9999999999999998)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_1 + sqrt(x))) + Float64(1.0 / Float64(t_2 + sqrt(y)))) + Float64(t_3 - sqrt(z)));
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(1.0 / Float64(t_3 + sqrt(z)))) + Float64(1.0 + Float64(t_1 - sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((1.0 + y));
	t_3 = sqrt((1.0 + z));
	tmp = 0.0;
	if ((t_2 - sqrt(y)) <= 0.9999999999999998)
		tmp = ((1.0 / (t_1 + sqrt(x))) + (1.0 / (t_2 + sqrt(y)))) + (t_3 - sqrt(z));
	else
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 / (t_3 + sqrt(z)))) + (1.0 + (t_1 - sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 0.9999999999999998], N[(N[(N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$3 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)) < 0.99999999999999978

   1. Initial program 89.4%

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

      1. associate-+l+ 89.4%

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

      2. associate-+l- 81.7%

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

      3. associate-+l- 89.4%

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

      4. sub-neg 89.4%

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

      5. sub-neg 89.4%

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

      6. +-commutative 89.4%

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

      7. +-commutative 89.4%

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

      8. +-commutative 89.4%

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

   3. Simplified 89.4%

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

      1. flip-- 89.9%

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

      2. add-sqr-sqrt 67.4%

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

      3. +-commutative 67.4%

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

      4. add-sqr-sqrt 90.1%

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

      5. +-commutative 90.1%

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

   5. Applied egg-rr 90.1%

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

      1. associate--l+ 91.7%

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

      2. +-inverses 91.7%

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

      3. metadata-eval 91.7%

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

   7. Simplified 91.7%

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

      1. flip-- 92.4%

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

      2. add-sqr-sqrt 62.1%

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

      3. add-sqr-sqrt 93.6%

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

   9. Applied egg-rr 93.6%

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

       1. associate--l+ 96.0%

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

       2. +-inverses 96.0%

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

       3. metadata-eval 96.0%

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

   11. Simplified 96.0%

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

   12. Taylor expanded in t around inf 49.7%

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

   if 0.99999999999999978 < (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))

   1. Initial program 98.2%

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

      1. associate-+l+ 98.2%

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

      2. associate-+l- 65.8%

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

      3. associate-+l- 98.2%

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

      4. sub-neg 98.2%

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

      5. sub-neg 98.2%

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

      6. +-commutative 98.2%

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

      7. +-commutative 98.2%

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

      8. +-commutative 98.2%

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

   3. Simplified 98.2%

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

      1. flip-- 98.5%

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

      2. add-sqr-sqrt 77.5%

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

      3. add-sqr-sqrt 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. associate--l+ 99.5%

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

      2. +-inverses 99.5%

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

      3. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

      1. flip-- 98.4%

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

      2. add-sqr-sqrt 98.4%

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

      3. add-sqr-sqrt 98.4%

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

   9. Applied egg-rr 99.5%

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

       1. associate--l+ 98.4%

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

       2. +-inverses 98.4%

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

       3. metadata-eval 98.4%

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

   11. Simplified 99.5%

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

   12. Taylor expanded in y around 0 99.5%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 0.9999999999999998:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{1 + z}\\ \mathbf{if}\;z \leq 3.1 \cdot 10^{+32}:\\ \;\;\;\;\left(\left(t_2 - \sqrt{y}\right) + \left(t_1 - \sqrt{x}\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{t_3 + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t_1 + \sqrt{x}} + \frac{1}{t_2 + \sqrt{y}}\right) + \left(t_3 - \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))) (t_2 (sqrt (+ 1.0 y))) (t_3 (sqrt (+ 1.0 z))))
   (if (<= z 3.1e+32)
     (+
      (+ (- t_2 (sqrt y)) (- t_1 (sqrt x)))
      (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (/ 1.0 (+ t_3 (sqrt z)))))
     (+
      (+ (/ 1.0 (+ t_1 (sqrt x))) (/ 1.0 (+ t_2 (sqrt y))))
      (- t_3 (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 + z));
	double tmp;
	if (z <= 3.1e+32) {
		tmp = ((t_2 - sqrt(y)) + (t_1 - sqrt(x))) + ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 / (t_3 + sqrt(z))));
	} else {
		tmp = ((1.0 / (t_1 + sqrt(x))) + (1.0 / (t_2 + sqrt(y)))) + (t_3 - 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) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + y))
    t_3 = sqrt((1.0d0 + z))
    if (z <= 3.1d+32) then
        tmp = ((t_2 - sqrt(y)) + (t_1 - sqrt(x))) + ((sqrt((1.0d0 + t)) - sqrt(t)) + (1.0d0 / (t_3 + sqrt(z))))
    else
        tmp = ((1.0d0 / (t_1 + sqrt(x))) + (1.0d0 / (t_2 + sqrt(y)))) + (t_3 - 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 + z));
	double tmp;
	if (z <= 3.1e+32) {
		tmp = ((t_2 - Math.sqrt(y)) + (t_1 - Math.sqrt(x))) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (1.0 / (t_3 + Math.sqrt(z))));
	} else {
		tmp = ((1.0 / (t_1 + Math.sqrt(x))) + (1.0 / (t_2 + Math.sqrt(y)))) + (t_3 - 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 + z))
	tmp = 0
	if z <= 3.1e+32:
		tmp = ((t_2 - math.sqrt(y)) + (t_1 - math.sqrt(x))) + ((math.sqrt((1.0 + t)) - math.sqrt(t)) + (1.0 / (t_3 + math.sqrt(z))))
	else:
		tmp = ((1.0 / (t_1 + math.sqrt(x))) + (1.0 / (t_2 + math.sqrt(y)))) + (t_3 - 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 + z))
	tmp = 0.0
	if (z <= 3.1e+32)
		tmp = Float64(Float64(Float64(t_2 - sqrt(y)) + Float64(t_1 - sqrt(x))) + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(1.0 / Float64(t_3 + sqrt(z)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_1 + sqrt(x))) + Float64(1.0 / Float64(t_2 + sqrt(y)))) + Float64(t_3 - 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 + z));
	tmp = 0.0;
	if (z <= 3.1e+32)
		tmp = ((t_2 - sqrt(y)) + (t_1 - sqrt(x))) + ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 / (t_3 + sqrt(z))));
	else
		tmp = ((1.0 / (t_1 + sqrt(x))) + (1.0 / (t_2 + sqrt(y)))) + (t_3 - 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 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 3.1e+32], N[(N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$3 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < 3.09999999999999993e32

   1. Initial program 96.4%

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

      1. associate-+l+ 96.4%

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

      2. associate-+l- 80.9%

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

      3. associate-+l- 96.4%

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

      4. sub-neg 96.4%

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

      5. sub-neg 96.4%

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

      6. +-commutative 96.4%

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

      7. +-commutative 96.4%

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

      8. +-commutative 96.4%

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

   3. Simplified 96.4%

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

      1. flip-- 96.6%

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

      2. add-sqr-sqrt 96.0%

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

      3. add-sqr-sqrt 97.0%

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

   5. Applied egg-rr 97.0%

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

      1. associate--l+ 97.8%

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

      2. +-inverses 97.8%

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

      3. metadata-eval 97.8%

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

   7. Simplified 97.8%

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

   if 3.09999999999999993e32 < z

   1. Initial program 89.2%

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

      1. associate-+l+ 89.2%

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

      2. associate-+l- 64.8%

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

      3. associate-+l- 89.2%

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

      4. sub-neg 89.2%

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

      5. sub-neg 89.2%

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

      6. +-commutative 89.2%

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

      7. +-commutative 89.2%

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

      8. +-commutative 89.2%

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

   3. Simplified 89.2%

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

      1. flip-- 89.9%

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

      2. add-sqr-sqrt 67.0%

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

      3. +-commutative 67.0%

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

      4. add-sqr-sqrt 90.1%

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

      5. +-commutative 90.1%

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

   5. Applied egg-rr 90.1%

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

      1. associate--l+ 92.1%

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

      2. +-inverses 92.1%

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

      3. metadata-eval 92.1%

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

   7. Simplified 92.1%

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

      1. flip-- 92.4%

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

      2. add-sqr-sqrt 79.2%

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

      3. add-sqr-sqrt 93.2%

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

   9. Applied egg-rr 93.2%

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

       1. associate--l+ 95.9%

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

       2. +-inverses 95.9%

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

       3. metadata-eval 95.9%

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

   11. Simplified 95.9%

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

   12. Taylor expanded in t around inf 57.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.1 \cdot 10^{+32}:\\ \;\;\;\;\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \end{array} \]

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.5%

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

   1. associate-+l+ 93.5%

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

   2. associate-+l- 74.4%

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

   3. associate-+l- 93.5%

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

   4. sub-neg 93.5%

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

   5. sub-neg 93.5%

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

   6. +-commutative 93.5%

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

   7. +-commutative 93.5%

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

   8. +-commutative 93.5%

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

3. Simplified 93.5%

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

   1. flip-- 93.7%

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

   2. add-sqr-sqrt 71.3%

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

   3. +-commutative 71.3%

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

   4. add-sqr-sqrt 93.9%

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

   5. +-commutative 93.9%

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

5. Applied egg-rr 93.9%

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

   1. associate--l+ 94.8%

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

   2. +-inverses 94.8%

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

   3. metadata-eval 94.8%

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

7. Simplified 94.8%

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

   1. flip-- 95.1%

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

   2. add-sqr-sqrt 78.8%

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

   3. add-sqr-sqrt 95.8%

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

9. Applied egg-rr 95.8%

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

    1. associate--l+ 97.1%

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

    2. +-inverses 97.1%

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

    3. metadata-eval 97.1%

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

11. Simplified 97.1%

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

12. Final simplification 97.1%

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

Alternative 5: 96.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 5.8 \cdot 10^{-32}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{t_1 + \sqrt{z}}\right) + \left(1 + \left(t_2 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - \sqrt{z}\right) + \left(\frac{1}{t_2 + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= y 5.8e-32)
     (+
      (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (/ 1.0 (+ t_1 (sqrt z))))
      (+ 1.0 (- t_2 (sqrt x))))
     (+
      (- t_1 (sqrt z))
      (+ (/ 1.0 (+ t_2 (sqrt x))) (- (sqrt (+ 1.0 y)) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (y <= 5.8e-32) {
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 / (t_1 + sqrt(z)))) + (1.0 + (t_2 - sqrt(x)));
	} else {
		tmp = (t_1 - sqrt(z)) + ((1.0 / (t_2 + sqrt(x))) + (sqrt((1.0 + y)) - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + x))
    if (y <= 5.8d-32) then
        tmp = ((sqrt((1.0d0 + t)) - sqrt(t)) + (1.0d0 / (t_1 + sqrt(z)))) + (1.0d0 + (t_2 - sqrt(x)))
    else
        tmp = (t_1 - sqrt(z)) + ((1.0d0 / (t_2 + sqrt(x))) + (sqrt((1.0d0 + y)) - sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 5.8e-32) {
		tmp = ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (1.0 / (t_1 + Math.sqrt(z)))) + (1.0 + (t_2 - Math.sqrt(x)));
	} else {
		tmp = (t_1 - Math.sqrt(z)) + ((1.0 / (t_2 + Math.sqrt(x))) + (Math.sqrt((1.0 + y)) - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 5.8e-32:
		tmp = ((math.sqrt((1.0 + t)) - math.sqrt(t)) + (1.0 / (t_1 + math.sqrt(z)))) + (1.0 + (t_2 - math.sqrt(x)))
	else:
		tmp = (t_1 - math.sqrt(z)) + ((1.0 / (t_2 + math.sqrt(x))) + (math.sqrt((1.0 + y)) - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 5.8e-32)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(1.0 / Float64(t_1 + sqrt(z)))) + Float64(1.0 + Float64(t_2 - sqrt(x))));
	else
		tmp = Float64(Float64(t_1 - sqrt(z)) + Float64(Float64(1.0 / Float64(t_2 + sqrt(x))) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 5.8e-32)
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 / (t_1 + sqrt(z)))) + (1.0 + (t_2 - sqrt(x)));
	else
		tmp = (t_1 - sqrt(z)) + ((1.0 / (t_2 + sqrt(x))) + (sqrt((1.0 + y)) - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 5.8e-32], N[(N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$1 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.79999999999999991e-32

   1. Initial program 98.2%

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

      1. associate-+l+ 98.2%

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

      2. associate-+l- 65.8%

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

      3. associate-+l- 98.2%

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

      4. sub-neg 98.2%

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

      5. sub-neg 98.2%

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

      6. +-commutative 98.2%

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

      7. +-commutative 98.2%

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

      8. +-commutative 98.2%

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

   3. Simplified 98.2%

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

      1. flip-- 98.5%

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

      2. add-sqr-sqrt 77.5%

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

      3. add-sqr-sqrt 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. associate--l+ 99.5%

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

      2. +-inverses 99.5%

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

      3. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

      1. flip-- 98.4%

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

      2. add-sqr-sqrt 98.4%

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

      3. add-sqr-sqrt 98.4%

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

   9. Applied egg-rr 99.5%

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

       1. associate--l+ 98.4%

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

       2. +-inverses 98.4%

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

       3. metadata-eval 98.4%

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

   11. Simplified 99.5%

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

   12. Taylor expanded in y around 0 99.5%

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

   if 5.79999999999999991e-32 < y

   1. Initial program 89.4%

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

      1. associate-+l+ 89.4%

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

      2. associate-+l- 81.7%

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

      3. associate-+l- 89.4%

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

      4. sub-neg 89.4%

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

      5. sub-neg 89.4%

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

      6. +-commutative 89.4%

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

      7. +-commutative 89.4%

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

      8. +-commutative 89.4%

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

   3. Simplified 89.4%

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

      1. flip-- 89.9%

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

      2. add-sqr-sqrt 67.4%

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

      3. +-commutative 67.4%

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

      4. add-sqr-sqrt 90.1%

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

      5. +-commutative 90.1%

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

   5. Applied egg-rr 90.1%

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

      1. associate--l+ 91.7%

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

      2. +-inverses 91.7%

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

      3. metadata-eval 91.7%

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

   7. Simplified 91.7%

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

   8. Taylor expanded in t around inf 46.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{-32}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 6: 95.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;t \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;t_1 + \left(1 + \left(\left(\left(1 + \sqrt{1 + t}\right) - \sqrt{t}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(t_1 - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))))
   (if (<= t 1.3e+15)
     (+
      t_1
      (+ 1.0 (- (- (+ 1.0 (sqrt (+ 1.0 t))) (sqrt t)) (+ (sqrt z) (sqrt y)))))
     (+
      (- (sqrt (+ 1.0 z)) (sqrt z))
      (+ (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))) (- t_1 (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double tmp;
	if (t <= 1.3e+15) {
		tmp = t_1 + (1.0 + (((1.0 + sqrt((1.0 + t))) - sqrt(t)) - (sqrt(z) + sqrt(y))));
	} else {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (t_1 - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    if (t <= 1.3d+15) then
        tmp = t_1 + (1.0d0 + (((1.0d0 + sqrt((1.0d0 + t))) - sqrt(t)) - (sqrt(z) + sqrt(y))))
    else
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + ((1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))) + (t_1 - sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double tmp;
	if (t <= 1.3e+15) {
		tmp = t_1 + (1.0 + (((1.0 + Math.sqrt((1.0 + t))) - Math.sqrt(t)) - (Math.sqrt(z) + Math.sqrt(y))));
	} else {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + ((1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) + (t_1 - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	tmp = 0
	if t <= 1.3e+15:
		tmp = t_1 + (1.0 + (((1.0 + math.sqrt((1.0 + t))) - math.sqrt(t)) - (math.sqrt(z) + math.sqrt(y))))
	else:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + ((1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) + (t_1 - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (t <= 1.3e+15)
		tmp = Float64(t_1 + Float64(1.0 + Float64(Float64(Float64(1.0 + sqrt(Float64(1.0 + t))) - sqrt(t)) - Float64(sqrt(z) + sqrt(y)))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + Float64(t_1 - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	tmp = 0.0;
	if (t <= 1.3e+15)
		tmp = t_1 + (1.0 + (((1.0 + sqrt((1.0 + t))) - sqrt(t)) - (sqrt(z) + sqrt(y))));
	else
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (t_1 - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1.3e+15], N[(t$95$1 + N[(1.0 + N[(N[(N[(1.0 + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.3e15

   1. Initial program 97.7%

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

      1. associate-+l+ 97.7%

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

      2. +-commutative 97.7%

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

      3. associate-+l+ 97.7%

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

      4. +-commutative 97.7%

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

      5. associate-+l+ 97.7%

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

      6. +-commutative 97.7%

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

      7. associate-+r- 50.3%

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

   3. Simplified 49.3%

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

      1. add-cube-cbrt 48.8%

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

      2. pow3 48.6%

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

   5. Applied egg-rr 48.6%

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

   6. Taylor expanded in x around 0 22.3%

      \[\leadsto \color{blue}{\sqrt{1 + y} + {1}^{0.3333333333333333} \cdot \left(\left(1 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 22.3%

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

      2. associate--l+ 28.8%

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

      3. associate--r+ 28.8%

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

      4. +-commutative 28.8%

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

   8. Simplified 28.8%

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

   9. Taylor expanded in z around 0 16.6%

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

   if 1.3e15 < t

   1. Initial program 89.2%

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

      1. associate-+l+ 89.2%

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

      2. associate-+l- 69.1%

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

      3. associate-+l- 89.2%

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

      4. sub-neg 89.2%

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

      5. sub-neg 89.2%

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

      6. +-commutative 89.2%

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

      7. +-commutative 89.2%

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

      8. +-commutative 89.2%

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

   3. Simplified 89.2%

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

      1. flip-- 89.7%

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

      2. add-sqr-sqrt 70.6%

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

      3. +-commutative 70.6%

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

      4. add-sqr-sqrt 89.9%

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

      5. +-commutative 89.9%

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

   5. Applied egg-rr 89.9%

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

      1. associate--l+ 91.7%

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

      2. +-inverses 91.7%

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

      3. metadata-eval 91.7%

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

   7. Simplified 91.7%

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

   8. Taylor expanded in t around inf 91.6%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 + \left(\left(\left(1 + \sqrt{1 + t}\right) - \sqrt{t}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 7: 90.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.75 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{1 + y} + \left(\left(t_1 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{x}\right)\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))))
   (if (<= y 1.75e+19)
     (+
      (sqrt (+ 1.0 y))
      (- (+ t_1 (- (- (sqrt (+ 1.0 z)) (sqrt z)) (sqrt x))) (sqrt y)))
     (/ 1.0 (+ t_1 (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double tmp;
	if (y <= 1.75e+19) {
		tmp = sqrt((1.0 + y)) + ((t_1 + ((sqrt((1.0 + z)) - sqrt(z)) - sqrt(x))) - sqrt(y));
	} else {
		tmp = 1.0 / (t_1 + sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    if (y <= 1.75d+19) then
        tmp = sqrt((1.0d0 + y)) + ((t_1 + ((sqrt((1.0d0 + z)) - sqrt(z)) - sqrt(x))) - sqrt(y))
    else
        tmp = 1.0d0 / (t_1 + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 1.75e+19) {
		tmp = Math.sqrt((1.0 + y)) + ((t_1 + ((Math.sqrt((1.0 + z)) - Math.sqrt(z)) - Math.sqrt(x))) - Math.sqrt(y));
	} else {
		tmp = 1.0 / (t_1 + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 1.75e+19:
		tmp = math.sqrt((1.0 + y)) + ((t_1 + ((math.sqrt((1.0 + z)) - math.sqrt(z)) - math.sqrt(x))) - math.sqrt(y))
	else:
		tmp = 1.0 / (t_1 + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 1.75e+19)
		tmp = Float64(sqrt(Float64(1.0 + y)) + Float64(Float64(t_1 + Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) - sqrt(x))) - sqrt(y)));
	else
		tmp = Float64(1.0 / Float64(t_1 + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 1.75e+19)
		tmp = sqrt((1.0 + y)) + ((t_1 + ((sqrt((1.0 + z)) - sqrt(z)) - sqrt(x))) - sqrt(y));
	else
		tmp = 1.0 / (t_1 + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.75e+19], N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(N[(t$95$1 + N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.75e19

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. +-commutative 96.2%

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

      3. associate-+l+ 96.2%

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

      4. +-commutative 96.2%

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

      5. associate-+l+ 96.2%

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

      6. +-commutative 96.2%

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

      7. associate-+r- 96.2%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in t around inf 44.4%

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

      1. associate--l+ 49.6%

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

      2. +-commutative 49.6%

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

      3. associate--r+ 54.5%

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

   6. Simplified 54.5%

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

   if 1.75e19 < y

   1. Initial program 89.9%

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

      1. associate-+l+ 89.9%

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

      2. +-commutative 89.9%

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

      3. associate-+l+ 89.9%

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

      4. +-commutative 89.9%

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

      5. associate-+l+ 89.9%

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

      6. +-commutative 89.9%

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

      7. associate-+r- 5.3%

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

   3. Simplified 4.1%

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

   4. Taylor expanded in t around inf 3.3%

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

      1. associate--l+ 20.6%

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

      2. +-commutative 20.6%

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

      3. associate--l+ 26.5%

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

      4. +-commutative 26.5%

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

   6. Simplified 26.5%

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

   7. Taylor expanded in y around inf 30.0%

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

      1. associate--r+ 30.0%

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

   9. Simplified 30.0%

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

   10. Taylor expanded in z around inf 20.8%

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

       1. flip-- 21.4%

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

       2. add-sqr-sqrt 22.0%

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

       3. add-sqr-sqrt 21.6%

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

   12. Applied egg-rr 21.6%

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

       1. associate--l+ 24.9%

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

       2. +-inverses 24.9%

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

       3. metadata-eval 24.9%

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

   14. Simplified 24.9%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{1 + y} + \left(\left(\sqrt{1 + x} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{x}\right)\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]

Alternative 8: 89.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;y \leq 6.8 \cdot 10^{-23}:\\ \;\;\;\;1 + \left(t_1 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;t_1 + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))))
   (if (<= y 6.8e-23)
     (+ 1.0 (+ t_1 (- (sqrt (+ 1.0 z)) (+ (sqrt y) (sqrt z)))))
     (if (<= y 5e+15)
       (+ t_1 (- 1.0 (sqrt y)))
       (/ 1.0 (+ (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 t_1 = sqrt((1.0 + y));
	double tmp;
	if (y <= 6.8e-23) {
		tmp = 1.0 + (t_1 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z))));
	} else if (y <= 5e+15) {
		tmp = t_1 + (1.0 - sqrt(y));
	} else {
		tmp = 1.0 / (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) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    if (y <= 6.8d-23) then
        tmp = 1.0d0 + (t_1 + (sqrt((1.0d0 + z)) - (sqrt(y) + sqrt(z))))
    else if (y <= 5d+15) then
        tmp = t_1 + (1.0d0 - sqrt(y))
    else
        tmp = 1.0d0 / (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 t_1 = Math.sqrt((1.0 + y));
	double tmp;
	if (y <= 6.8e-23) {
		tmp = 1.0 + (t_1 + (Math.sqrt((1.0 + z)) - (Math.sqrt(y) + Math.sqrt(z))));
	} else if (y <= 5e+15) {
		tmp = t_1 + (1.0 - Math.sqrt(y));
	} else {
		tmp = 1.0 / (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):
	t_1 = math.sqrt((1.0 + y))
	tmp = 0
	if y <= 6.8e-23:
		tmp = 1.0 + (t_1 + (math.sqrt((1.0 + z)) - (math.sqrt(y) + math.sqrt(z))))
	elif y <= 5e+15:
		tmp = t_1 + (1.0 - math.sqrt(y))
	else:
		tmp = 1.0 / (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)
	t_1 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (y <= 6.8e-23)
		tmp = Float64(1.0 + Float64(t_1 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(y) + sqrt(z)))));
	elseif (y <= 5e+15)
		tmp = Float64(t_1 + Float64(1.0 - sqrt(y)));
	else
		tmp = Float64(1.0 / 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)
	t_1 = sqrt((1.0 + y));
	tmp = 0.0;
	if (y <= 6.8e-23)
		tmp = 1.0 + (t_1 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z))));
	elseif (y <= 5e+15)
		tmp = t_1 + (1.0 - sqrt(y));
	else
		tmp = 1.0 / (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_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 6.8e-23], N[(1.0 + N[(t$95$1 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+15], N[(t$95$1 + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.8000000000000001e-23

   1. Initial program 98.0%

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

      1. associate-+l+ 98.0%

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

      2. +-commutative 98.0%

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

      3. associate-+l+ 98.0%

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

      4. +-commutative 98.0%

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

      5. associate-+l+ 98.0%

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

      6. +-commutative 98.0%

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

      7. associate-+r- 98.0%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in t around inf 27.0%

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

      1. associate--l+ 30.6%

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

      2. +-commutative 30.6%

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

      3. associate--l+ 30.6%

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

      4. +-commutative 30.6%

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

   6. Simplified 30.6%

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

   7. Taylor expanded in x around 0 36.4%

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

      1. associate--l+ 52.0%

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

      2. associate--l+ 58.6%

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

      3. +-commutative 58.6%

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

   9. Simplified 58.6%

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

   if 6.8000000000000001e-23 < y < 5e15

   1. Initial program 87.8%

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

      1. associate-+l+ 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-+l+ 87.8%

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

      4. +-commutative 87.8%

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

      5. associate-+l+ 87.8%

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

      6. +-commutative 87.8%

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

      7. associate-+r- 87.4%

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

   3. Simplified 60.2%

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

   4. Taylor expanded in t around inf 20.2%

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

      1. associate--l+ 23.0%

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

      2. +-commutative 23.0%

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

      3. associate--l+ 23.0%

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

      4. +-commutative 23.0%

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

   6. Simplified 23.0%

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

   7. Taylor expanded in x around 0 32.4%

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

   8. Taylor expanded in z around inf 42.2%

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

      1. +-commutative 42.2%

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

      2. associate--l+ 42.3%

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

   10. Simplified 42.3%

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

   if 5e15 < y

   1. Initial program 89.7%

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

      1. associate-+l+ 89.7%

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

      2. +-commutative 89.7%

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

      3. associate-+l+ 89.7%

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

      4. +-commutative 89.7%

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

      5. associate-+l+ 89.7%

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

      6. +-commutative 89.7%

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

      7. associate-+r- 5.9%

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

   3. Simplified 4.1%

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

   4. Taylor expanded in t around inf 3.3%

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

      1. associate--l+ 20.4%

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

      2. +-commutative 20.4%

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

      3. associate--l+ 26.3%

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

      4. +-commutative 26.3%

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

   6. Simplified 26.3%

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

   7. Taylor expanded in y around inf 29.7%

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

      1. associate--r+ 29.7%

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

   9. Simplified 29.7%

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

   10. Taylor expanded in z around inf 20.7%

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

       1. flip-- 21.3%

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

       2. add-sqr-sqrt 21.8%

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

       3. add-sqr-sqrt 21.4%

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

   12. Applied egg-rr 21.4%

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

       1. associate--l+ 24.7%

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

       2. +-inverses 24.7%

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

       3. metadata-eval 24.7%

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

   14. Simplified 24.7%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-23}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]

Alternative 9: 84.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 34000000000000:\\ \;\;\;\;\left(1 + \left(1 + \left(\sqrt{1 + z} + y \cdot 0.5\right)\right)\right) + \frac{y - z}{\sqrt{z} - \sqrt{y}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 34000000000000.0)
   (+
    (+ 1.0 (+ 1.0 (+ (sqrt (+ 1.0 z)) (* y 0.5))))
    (/ (- y z) (- (sqrt z) (sqrt y))))
   (+ (sqrt (+ 1.0 x)) (- (sqrt (+ 1.0 y)) (+ (sqrt x) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 34000000000000.0) {
		tmp = (1.0 + (1.0 + (sqrt((1.0 + z)) + (y * 0.5)))) + ((y - z) / (sqrt(z) - sqrt(y)));
	} else {
		tmp = sqrt((1.0 + x)) + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 34000000000000.0d0) then
        tmp = (1.0d0 + (1.0d0 + (sqrt((1.0d0 + z)) + (y * 0.5d0)))) + ((y - z) / (sqrt(z) - sqrt(y)))
    else
        tmp = sqrt((1.0d0 + x)) + (sqrt((1.0d0 + y)) - (sqrt(x) + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 34000000000000.0) {
		tmp = (1.0 + (1.0 + (Math.sqrt((1.0 + z)) + (y * 0.5)))) + ((y - z) / (Math.sqrt(z) - Math.sqrt(y)));
	} else {
		tmp = Math.sqrt((1.0 + x)) + (Math.sqrt((1.0 + y)) - (Math.sqrt(x) + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 34000000000000.0:
		tmp = (1.0 + (1.0 + (math.sqrt((1.0 + z)) + (y * 0.5)))) + ((y - z) / (math.sqrt(z) - math.sqrt(y)))
	else:
		tmp = math.sqrt((1.0 + x)) + (math.sqrt((1.0 + y)) - (math.sqrt(x) + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 34000000000000.0)
		tmp = Float64(Float64(1.0 + Float64(1.0 + Float64(sqrt(Float64(1.0 + z)) + Float64(y * 0.5)))) + Float64(Float64(y - z) / Float64(sqrt(z) - sqrt(y))));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(sqrt(Float64(1.0 + y)) - Float64(sqrt(x) + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 34000000000000.0)
		tmp = (1.0 + (1.0 + (sqrt((1.0 + z)) + (y * 0.5)))) + ((y - z) / (sqrt(z) - sqrt(y)));
	else
		tmp = sqrt((1.0 + x)) + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 34000000000000.0], N[(N[(1.0 + N[(1.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y - z), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.4e13

   1. Initial program 97.2%

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

      1. associate-+l+ 97.2%

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

      2. +-commutative 97.2%

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

      3. associate-+l+ 97.2%

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

      4. +-commutative 97.2%

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

      5. associate-+l+ 97.2%

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

      6. +-commutative 97.2%

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

      7. associate-+r- 54.6%

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

   3. Simplified 29.2%

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

   4. Taylor expanded in t around inf 25.4%

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

      1. associate--l+ 29.5%

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

      2. +-commutative 29.5%

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

      3. associate--l+ 35.7%

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

      4. +-commutative 35.7%

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

   6. Simplified 35.7%

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

   7. Taylor expanded in x around 0 35.2%

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

   8. Taylor expanded in y around 0 32.4%

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

      1. *-commutative 32.4%

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

   10. Simplified 32.4%

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

       1. +-commutative 32.4%

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

       2. flip-+ 32.4%

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

       3. add-sqr-sqrt 32.4%

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

       4. add-sqr-sqrt 32.4%

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

   12. Applied egg-rr 32.4%

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

   if 3.4e13 < z

   1. Initial program 88.5%

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

      1. associate-+l+ 88.5%

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

      2. +-commutative 88.5%

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

      3. associate-+l+ 88.5%

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

      4. +-commutative 88.5%

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

      5. associate-+l+ 88.5%

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

      6. +-commutative 88.5%

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

      7. associate-+r- 59.7%

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

   3. Simplified 40.6%

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

   4. Taylor expanded in t around inf 3.6%

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

      1. associate--l+ 20.2%

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

      2. +-commutative 20.2%

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

      3. associate--l+ 18.0%

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

      4. +-commutative 18.0%

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

   6. Simplified 18.0%

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

   7. Taylor expanded in z around inf 22.7%

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

      1. associate--l+ 33.3%

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

      2. +-commutative 33.3%

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

   9. Simplified 33.3%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 34000000000000:\\ \;\;\;\;\left(1 + \left(1 + \left(\sqrt{1 + z} + y \cdot 0.5\right)\right)\right) + \frac{y - z}{\sqrt{z} - \sqrt{y}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 10: 72.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.25 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(1 + \left(\sqrt{1 + z} + y \cdot 0.5\right)\right)\right) + \frac{y - z}{\sqrt{z} - \sqrt{y}}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+127} \lor \neg \left(z \leq 3.8 \cdot 10^{+141}\right) \land z \leq 4.25 \cdot 10^{+173}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

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 (<= z 1.25e+15)
   (+
    (+ 1.0 (+ 1.0 (+ (sqrt (+ 1.0 z)) (* y 0.5))))
    (/ (- y z) (- (sqrt z) (sqrt y))))
   (if (or (<= z 4.5e+127) (and (not (<= z 3.8e+141)) (<= z 4.25e+173)))
     (+ (sqrt (+ 1.0 y)) (- 1.0 (sqrt y)))
     (/ 1.0 (+ (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 (z <= 1.25e+15) {
		tmp = (1.0 + (1.0 + (sqrt((1.0 + z)) + (y * 0.5)))) + ((y - z) / (sqrt(z) - sqrt(y)));
	} else if ((z <= 4.5e+127) || (!(z <= 3.8e+141) && (z <= 4.25e+173))) {
		tmp = sqrt((1.0 + y)) + (1.0 - sqrt(y));
	} else {
		tmp = 1.0 / (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 (z <= 1.25d+15) then
        tmp = (1.0d0 + (1.0d0 + (sqrt((1.0d0 + z)) + (y * 0.5d0)))) + ((y - z) / (sqrt(z) - sqrt(y)))
    else if ((z <= 4.5d+127) .or. (.not. (z <= 3.8d+141)) .and. (z <= 4.25d+173)) then
        tmp = sqrt((1.0d0 + y)) + (1.0d0 - sqrt(y))
    else
        tmp = 1.0d0 / (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 (z <= 1.25e+15) {
		tmp = (1.0 + (1.0 + (Math.sqrt((1.0 + z)) + (y * 0.5)))) + ((y - z) / (Math.sqrt(z) - Math.sqrt(y)));
	} else if ((z <= 4.5e+127) || (!(z <= 3.8e+141) && (z <= 4.25e+173))) {
		tmp = Math.sqrt((1.0 + y)) + (1.0 - Math.sqrt(y));
	} else {
		tmp = 1.0 / (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 z <= 1.25e+15:
		tmp = (1.0 + (1.0 + (math.sqrt((1.0 + z)) + (y * 0.5)))) + ((y - z) / (math.sqrt(z) - math.sqrt(y)))
	elif (z <= 4.5e+127) or (not (z <= 3.8e+141) and (z <= 4.25e+173)):
		tmp = math.sqrt((1.0 + y)) + (1.0 - math.sqrt(y))
	else:
		tmp = 1.0 / (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 (z <= 1.25e+15)
		tmp = Float64(Float64(1.0 + Float64(1.0 + Float64(sqrt(Float64(1.0 + z)) + Float64(y * 0.5)))) + Float64(Float64(y - z) / Float64(sqrt(z) - sqrt(y))));
	elseif ((z <= 4.5e+127) || (!(z <= 3.8e+141) && (z <= 4.25e+173)))
		tmp = Float64(sqrt(Float64(1.0 + y)) + Float64(1.0 - sqrt(y)));
	else
		tmp = Float64(1.0 / 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 (z <= 1.25e+15)
		tmp = (1.0 + (1.0 + (sqrt((1.0 + z)) + (y * 0.5)))) + ((y - z) / (sqrt(z) - sqrt(y)));
	elseif ((z <= 4.5e+127) || (~((z <= 3.8e+141)) && (z <= 4.25e+173)))
		tmp = sqrt((1.0 + y)) + (1.0 - sqrt(y));
	else
		tmp = 1.0 / (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[z, 1.25e+15], N[(N[(1.0 + N[(1.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y - z), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 4.5e+127], And[N[Not[LessEqual[z, 3.8e+141]], $MachinePrecision], LessEqual[z, 4.25e+173]]], N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.25e15

   1. Initial program 97.2%

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

      1. associate-+l+ 97.2%

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

      2. +-commutative 97.2%

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

      3. associate-+l+ 97.2%

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

      4. +-commutative 97.2%

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

      5. associate-+l+ 97.2%

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

      6. +-commutative 97.2%

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

      7. associate-+r- 54.6%

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

   3. Simplified 29.2%

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

   4. Taylor expanded in t around inf 25.4%

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

      1. associate--l+ 29.5%

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

      2. +-commutative 29.5%

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

      3. associate--l+ 35.7%

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

      4. +-commutative 35.7%

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

   6. Simplified 35.7%

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

   7. Taylor expanded in x around 0 35.2%

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

   8. Taylor expanded in y around 0 32.4%

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

      1. *-commutative 32.4%

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

   10. Simplified 32.4%

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

       1. +-commutative 32.4%

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

       2. flip-+ 32.4%

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

       3. add-sqr-sqrt 32.4%

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

       4. add-sqr-sqrt 32.4%

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

   12. Applied egg-rr 32.4%

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

   if 1.25e15 < z < 4.50000000000000034e127 or 3.79999999999999976e141 < z < 4.2500000000000001e173

   1. Initial program 84.4%

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

      1. associate-+l+ 84.4%

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

      2. +-commutative 84.4%

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

      3. associate-+l+ 84.4%

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

      4. +-commutative 84.4%

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

      5. associate-+l+ 84.4%

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

      6. +-commutative 84.4%

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

      7. associate-+r- 65.0%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in t around inf 4.0%

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

      1. associate--l+ 15.4%

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

      2. +-commutative 15.4%

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

      3. associate--l+ 13.5%

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

      4. +-commutative 13.5%

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

   6. Simplified 13.5%

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

   7. Taylor expanded in x around 0 4.1%

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

   8. Taylor expanded in z around inf 35.0%

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

      1. +-commutative 35.0%

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

      2. associate--l+ 35.1%

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

   10. Simplified 35.1%

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

   if 4.50000000000000034e127 < z < 3.79999999999999976e141 or 4.2500000000000001e173 < z

   1. Initial program 92.6%

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

      1. associate-+l+ 92.6%

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

      2. +-commutative 92.6%

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

      3. associate-+l+ 92.6%

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

      4. +-commutative 92.6%

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

      5. associate-+l+ 92.6%

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

      6. +-commutative 92.6%

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

      7. associate-+r- 54.4%

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

   3. Simplified 30.7%

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

   4. Taylor expanded in t around inf 3.1%

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

      1. associate--l+ 24.9%

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

      2. +-commutative 24.9%

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

      3. associate--l+ 22.5%

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

      4. +-commutative 22.5%

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

   6. Simplified 22.5%

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

   7. Taylor expanded in y around inf 24.9%

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

      1. associate--r+ 24.9%

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

   9. Simplified 24.9%

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

   10. Taylor expanded in z around inf 24.3%

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

       1. flip-- 25.5%

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

       2. add-sqr-sqrt 25.5%

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

       3. add-sqr-sqrt 25.5%

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

   12. Applied egg-rr 25.5%

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

       1. associate--l+ 28.5%

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

       2. +-inverses 28.5%

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

       3. metadata-eval 28.5%

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

   14. Simplified 28.5%

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

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.25 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(1 + \left(\sqrt{1 + z} + y \cdot 0.5\right)\right)\right) + \frac{y - z}{\sqrt{z} - \sqrt{y}}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+127} \lor \neg \left(z \leq 3.8 \cdot 10^{+141}\right) \land z \leq 4.25 \cdot 10^{+173}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]

Alternative 11: 72.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.5 \cdot 10^{+17}:\\ \;\;\;\;\left(1 + \left(1 + \left(\sqrt{1 + z} + y \cdot 0.5\right)\right)\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+128} \lor \neg \left(z \leq 3.8 \cdot 10^{+141}\right) \land z \leq 4.25 \cdot 10^{+173}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

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 (<= z 1.5e+17)
   (- (+ 1.0 (+ 1.0 (+ (sqrt (+ 1.0 z)) (* y 0.5)))) (+ (sqrt z) (sqrt y)))
   (if (or (<= z 1.25e+128) (and (not (<= z 3.8e+141)) (<= z 4.25e+173)))
     (+ (sqrt (+ 1.0 y)) (- 1.0 (sqrt y)))
     (/ 1.0 (+ (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 (z <= 1.5e+17) {
		tmp = (1.0 + (1.0 + (sqrt((1.0 + z)) + (y * 0.5)))) - (sqrt(z) + sqrt(y));
	} else if ((z <= 1.25e+128) || (!(z <= 3.8e+141) && (z <= 4.25e+173))) {
		tmp = sqrt((1.0 + y)) + (1.0 - sqrt(y));
	} else {
		tmp = 1.0 / (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 (z <= 1.5d+17) then
        tmp = (1.0d0 + (1.0d0 + (sqrt((1.0d0 + z)) + (y * 0.5d0)))) - (sqrt(z) + sqrt(y))
    else if ((z <= 1.25d+128) .or. (.not. (z <= 3.8d+141)) .and. (z <= 4.25d+173)) then
        tmp = sqrt((1.0d0 + y)) + (1.0d0 - sqrt(y))
    else
        tmp = 1.0d0 / (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 (z <= 1.5e+17) {
		tmp = (1.0 + (1.0 + (Math.sqrt((1.0 + z)) + (y * 0.5)))) - (Math.sqrt(z) + Math.sqrt(y));
	} else if ((z <= 1.25e+128) || (!(z <= 3.8e+141) && (z <= 4.25e+173))) {
		tmp = Math.sqrt((1.0 + y)) + (1.0 - Math.sqrt(y));
	} else {
		tmp = 1.0 / (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 z <= 1.5e+17:
		tmp = (1.0 + (1.0 + (math.sqrt((1.0 + z)) + (y * 0.5)))) - (math.sqrt(z) + math.sqrt(y))
	elif (z <= 1.25e+128) or (not (z <= 3.8e+141) and (z <= 4.25e+173)):
		tmp = math.sqrt((1.0 + y)) + (1.0 - math.sqrt(y))
	else:
		tmp = 1.0 / (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 (z <= 1.5e+17)
		tmp = Float64(Float64(1.0 + Float64(1.0 + Float64(sqrt(Float64(1.0 + z)) + Float64(y * 0.5)))) - Float64(sqrt(z) + sqrt(y)));
	elseif ((z <= 1.25e+128) || (!(z <= 3.8e+141) && (z <= 4.25e+173)))
		tmp = Float64(sqrt(Float64(1.0 + y)) + Float64(1.0 - sqrt(y)));
	else
		tmp = Float64(1.0 / 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 (z <= 1.5e+17)
		tmp = (1.0 + (1.0 + (sqrt((1.0 + z)) + (y * 0.5)))) - (sqrt(z) + sqrt(y));
	elseif ((z <= 1.25e+128) || (~((z <= 3.8e+141)) && (z <= 4.25e+173)))
		tmp = sqrt((1.0 + y)) + (1.0 - sqrt(y));
	else
		tmp = 1.0 / (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[z, 1.5e+17], N[(N[(1.0 + N[(1.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 1.25e+128], And[N[Not[LessEqual[z, 3.8e+141]], $MachinePrecision], LessEqual[z, 4.25e+173]]], N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.5e17

   1. Initial program 97.2%

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

      1. associate-+l+ 97.2%

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

      2. +-commutative 97.2%

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

      3. associate-+l+ 97.2%

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

      4. +-commutative 97.2%

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

      5. associate-+l+ 97.2%

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

      6. +-commutative 97.2%

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

      7. associate-+r- 54.6%

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

   3. Simplified 29.2%

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

   4. Taylor expanded in t around inf 25.4%

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

      1. associate--l+ 29.5%

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

      2. +-commutative 29.5%

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

      3. associate--l+ 35.7%

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

      4. +-commutative 35.7%

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

   6. Simplified 35.7%

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

   7. Taylor expanded in x around 0 35.2%

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

   8. Taylor expanded in y around 0 32.4%

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

      1. *-commutative 32.4%

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

   10. Simplified 32.4%

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

   if 1.5e17 < z < 1.25e128 or 3.79999999999999976e141 < z < 4.2500000000000001e173

   1. Initial program 84.4%

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

      1. associate-+l+ 84.4%

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

      2. +-commutative 84.4%

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

      3. associate-+l+ 84.4%

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

      4. +-commutative 84.4%

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

      5. associate-+l+ 84.4%

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

      6. +-commutative 84.4%

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

      7. associate-+r- 65.0%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in t around inf 4.0%

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

      1. associate--l+ 15.4%

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

      2. +-commutative 15.4%

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

      3. associate--l+ 13.5%

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

      4. +-commutative 13.5%

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

   6. Simplified 13.5%

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

   7. Taylor expanded in x around 0 4.1%

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

   8. Taylor expanded in z around inf 35.0%

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

      1. +-commutative 35.0%

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

      2. associate--l+ 35.1%

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

   10. Simplified 35.1%

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

   if 1.25e128 < z < 3.79999999999999976e141 or 4.2500000000000001e173 < z

   1. Initial program 92.6%

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

      1. associate-+l+ 92.6%

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

      2. +-commutative 92.6%

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

      3. associate-+l+ 92.6%

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

      4. +-commutative 92.6%

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

      5. associate-+l+ 92.6%

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

      6. +-commutative 92.6%

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

      7. associate-+r- 54.4%

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

   3. Simplified 30.7%

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

   4. Taylor expanded in t around inf 3.1%

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

      1. associate--l+ 24.9%

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

      2. +-commutative 24.9%

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

      3. associate--l+ 22.5%

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

      4. +-commutative 22.5%

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

   6. Simplified 22.5%

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

   7. Taylor expanded in y around inf 24.9%

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

      1. associate--r+ 24.9%

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

   9. Simplified 24.9%

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

   10. Taylor expanded in z around inf 24.3%

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

       1. flip-- 25.5%

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

       2. add-sqr-sqrt 25.5%

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

       3. add-sqr-sqrt 25.5%

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

   12. Applied egg-rr 25.5%

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

       1. associate--l+ 28.5%

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

       2. +-inverses 28.5%

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

       3. metadata-eval 28.5%

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

   14. Simplified 28.5%

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

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.5 \cdot 10^{+17}:\\ \;\;\;\;\left(1 + \left(1 + \left(\sqrt{1 + z} + y \cdot 0.5\right)\right)\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+128} \lor \neg \left(z \leq 3.8 \cdot 10^{+141}\right) \land z \leq 4.25 \cdot 10^{+173}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]

Alternative 12: 72.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 4.4 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(1 + \sqrt{1 + z}\right)\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+128} \lor \neg \left(z \leq 1.1 \cdot 10^{+142}\right) \land z \leq 4.25 \cdot 10^{+173}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

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 (<= z 4.4e+15)
   (- (+ 1.0 (+ 1.0 (sqrt (+ 1.0 z)))) (+ (sqrt z) (sqrt y)))
   (if (or (<= z 1.1e+128) (and (not (<= z 1.1e+142)) (<= z 4.25e+173)))
     (+ (sqrt (+ 1.0 y)) (- 1.0 (sqrt y)))
     (/ 1.0 (+ (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 (z <= 4.4e+15) {
		tmp = (1.0 + (1.0 + sqrt((1.0 + z)))) - (sqrt(z) + sqrt(y));
	} else if ((z <= 1.1e+128) || (!(z <= 1.1e+142) && (z <= 4.25e+173))) {
		tmp = sqrt((1.0 + y)) + (1.0 - sqrt(y));
	} else {
		tmp = 1.0 / (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 (z <= 4.4d+15) then
        tmp = (1.0d0 + (1.0d0 + sqrt((1.0d0 + z)))) - (sqrt(z) + sqrt(y))
    else if ((z <= 1.1d+128) .or. (.not. (z <= 1.1d+142)) .and. (z <= 4.25d+173)) then
        tmp = sqrt((1.0d0 + y)) + (1.0d0 - sqrt(y))
    else
        tmp = 1.0d0 / (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 (z <= 4.4e+15) {
		tmp = (1.0 + (1.0 + Math.sqrt((1.0 + z)))) - (Math.sqrt(z) + Math.sqrt(y));
	} else if ((z <= 1.1e+128) || (!(z <= 1.1e+142) && (z <= 4.25e+173))) {
		tmp = Math.sqrt((1.0 + y)) + (1.0 - Math.sqrt(y));
	} else {
		tmp = 1.0 / (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 z <= 4.4e+15:
		tmp = (1.0 + (1.0 + math.sqrt((1.0 + z)))) - (math.sqrt(z) + math.sqrt(y))
	elif (z <= 1.1e+128) or (not (z <= 1.1e+142) and (z <= 4.25e+173)):
		tmp = math.sqrt((1.0 + y)) + (1.0 - math.sqrt(y))
	else:
		tmp = 1.0 / (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 (z <= 4.4e+15)
		tmp = Float64(Float64(1.0 + Float64(1.0 + sqrt(Float64(1.0 + z)))) - Float64(sqrt(z) + sqrt(y)));
	elseif ((z <= 1.1e+128) || (!(z <= 1.1e+142) && (z <= 4.25e+173)))
		tmp = Float64(sqrt(Float64(1.0 + y)) + Float64(1.0 - sqrt(y)));
	else
		tmp = Float64(1.0 / 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 (z <= 4.4e+15)
		tmp = (1.0 + (1.0 + sqrt((1.0 + z)))) - (sqrt(z) + sqrt(y));
	elseif ((z <= 1.1e+128) || (~((z <= 1.1e+142)) && (z <= 4.25e+173)))
		tmp = sqrt((1.0 + y)) + (1.0 - sqrt(y));
	else
		tmp = 1.0 / (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[z, 4.4e+15], N[(N[(1.0 + N[(1.0 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 1.1e+128], And[N[Not[LessEqual[z, 1.1e+142]], $MachinePrecision], LessEqual[z, 4.25e+173]]], N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 4.4e15

   1. Initial program 97.2%

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

      1. associate-+l+ 97.2%

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

      2. +-commutative 97.2%

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

      3. associate-+l+ 97.2%

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

      4. +-commutative 97.2%

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

      5. associate-+l+ 97.2%

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

      6. +-commutative 97.2%

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

      7. associate-+r- 54.6%

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

   3. Simplified 29.2%

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

   4. Taylor expanded in t around inf 25.4%

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

      1. associate--l+ 29.5%

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

      2. +-commutative 29.5%

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

      3. associate--l+ 35.7%

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

      4. +-commutative 35.7%

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

   6. Simplified 35.7%

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

   7. Taylor expanded in x around 0 35.2%

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

   8. Taylor expanded in y around 0 30.0%

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

   if 4.4e15 < z < 1.10000000000000008e128 or 1.09999999999999993e142 < z < 4.2500000000000001e173

   1. Initial program 84.4%

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

      1. associate-+l+ 84.4%

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

      2. +-commutative 84.4%

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

      3. associate-+l+ 84.4%

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

      4. +-commutative 84.4%

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

      5. associate-+l+ 84.4%

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

      6. +-commutative 84.4%

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

      7. associate-+r- 65.0%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in t around inf 4.0%

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

      1. associate--l+ 15.4%

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

      2. +-commutative 15.4%

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

      3. associate--l+ 13.5%

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

      4. +-commutative 13.5%

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

   6. Simplified 13.5%

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

   7. Taylor expanded in x around 0 4.1%

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

   8. Taylor expanded in z around inf 35.0%

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

      1. +-commutative 35.0%

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

      2. associate--l+ 35.1%

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

   10. Simplified 35.1%

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

   if 1.10000000000000008e128 < z < 1.09999999999999993e142 or 4.2500000000000001e173 < z

   1. Initial program 92.6%

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

      1. associate-+l+ 92.6%

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

      2. +-commutative 92.6%

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

      3. associate-+l+ 92.6%

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

      4. +-commutative 92.6%

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

      5. associate-+l+ 92.6%

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

      6. +-commutative 92.6%

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

      7. associate-+r- 54.4%

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

   3. Simplified 30.7%

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

   4. Taylor expanded in t around inf 3.1%

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

      1. associate--l+ 24.9%

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

      2. +-commutative 24.9%

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

      3. associate--l+ 22.5%

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

      4. +-commutative 22.5%

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

   6. Simplified 22.5%

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

   7. Taylor expanded in y around inf 24.9%

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

      1. associate--r+ 24.9%

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

   9. Simplified 24.9%

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

   10. Taylor expanded in z around inf 24.3%

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

       1. flip-- 25.5%

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

       2. add-sqr-sqrt 25.5%

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

       3. add-sqr-sqrt 25.5%

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

   12. Applied egg-rr 25.5%

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

       1. associate--l+ 28.5%

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

       2. +-inverses 28.5%

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

       3. metadata-eval 28.5%

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

   14. Simplified 28.5%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.4 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(1 + \sqrt{1 + z}\right)\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+128} \lor \neg \left(z \leq 1.1 \cdot 10^{+142}\right) \land z \leq 4.25 \cdot 10^{+173}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]

Alternative 13: 69.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + z} + \left(2 - \sqrt{z}\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+128} \lor \neg \left(z \leq 5.2 \cdot 10^{+136}\right) \land z \leq 4.25 \cdot 10^{+173}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \end{array} \]

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 (<= z 5.8e+15)
   (+ (sqrt (+ 1.0 z)) (- 2.0 (sqrt z)))
   (if (or (<= z 1.1e+128) (and (not (<= z 5.2e+136)) (<= z 4.25e+173)))
     (+ (sqrt (+ 1.0 y)) (- 1.0 (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 (z <= 5.8e+15) {
		tmp = sqrt((1.0 + z)) + (2.0 - sqrt(z));
	} else if ((z <= 1.1e+128) || (!(z <= 5.2e+136) && (z <= 4.25e+173))) {
		tmp = sqrt((1.0 + y)) + (1.0 - 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 (z <= 5.8d+15) then
        tmp = sqrt((1.0d0 + z)) + (2.0d0 - sqrt(z))
    else if ((z <= 1.1d+128) .or. (.not. (z <= 5.2d+136)) .and. (z <= 4.25d+173)) then
        tmp = sqrt((1.0d0 + y)) + (1.0d0 - 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 (z <= 5.8e+15) {
		tmp = Math.sqrt((1.0 + z)) + (2.0 - Math.sqrt(z));
	} else if ((z <= 1.1e+128) || (!(z <= 5.2e+136) && (z <= 4.25e+173))) {
		tmp = Math.sqrt((1.0 + y)) + (1.0 - 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 z <= 5.8e+15:
		tmp = math.sqrt((1.0 + z)) + (2.0 - math.sqrt(z))
	elif (z <= 1.1e+128) or (not (z <= 5.2e+136) and (z <= 4.25e+173)):
		tmp = math.sqrt((1.0 + y)) + (1.0 - 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 (z <= 5.8e+15)
		tmp = Float64(sqrt(Float64(1.0 + z)) + Float64(2.0 - sqrt(z)));
	elseif ((z <= 1.1e+128) || (!(z <= 5.2e+136) && (z <= 4.25e+173)))
		tmp = Float64(sqrt(Float64(1.0 + y)) + Float64(1.0 - 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 (z <= 5.8e+15)
		tmp = sqrt((1.0 + z)) + (2.0 - sqrt(z));
	elseif ((z <= 1.1e+128) || (~((z <= 5.2e+136)) && (z <= 4.25e+173)))
		tmp = sqrt((1.0 + y)) + (1.0 - 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[z, 5.8e+15], N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(2.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 1.1e+128], And[N[Not[LessEqual[z, 5.2e+136]], $MachinePrecision], LessEqual[z, 4.25e+173]]], N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(1.0 - 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 z < 5.8e15

   1. Initial program 97.2%

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

      1. associate-+l+ 97.2%

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

      2. +-commutative 97.2%

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

      3. associate-+l+ 97.2%

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

      4. +-commutative 97.2%

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

      5. associate-+l+ 97.2%

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

      6. +-commutative 97.2%

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

      7. associate-+r- 54.6%

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

   3. Simplified 29.2%

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

   4. Taylor expanded in t around inf 25.4%

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

      1. associate--l+ 29.5%

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

      2. +-commutative 29.5%

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

      3. associate--l+ 35.7%

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

      4. +-commutative 35.7%

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

   6. Simplified 35.7%

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

   7. Taylor expanded in x around 0 35.2%

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

   8. Taylor expanded in y around 0 51.6%

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

      1. +-commutative 51.6%

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

      2. associate--l+ 51.5%

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

   10. Simplified 51.5%

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

   if 5.8e15 < z < 1.10000000000000008e128 or 5.2000000000000003e136 < z < 4.2500000000000001e173

   1. Initial program 84.6%

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

      1. associate-+l+ 84.6%

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

      2. +-commutative 84.6%

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

      3. associate-+l+ 84.6%

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

      4. +-commutative 84.6%

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

      5. associate-+l+ 84.6%

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

      6. +-commutative 84.6%

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

      7. associate-+r- 63.9%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in t around inf 4.0%

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

      1. associate--l+ 15.5%

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

      2. +-commutative 15.5%

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

      3. associate--l+ 13.4%

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

      4. +-commutative 13.4%

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

   6. Simplified 13.4%

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

   7. Taylor expanded in x around 0 4.0%

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

   8. Taylor expanded in z around inf 34.4%

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

      1. +-commutative 34.4%

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

      2. associate--l+ 34.5%

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

   10. Simplified 34.5%

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

   if 1.10000000000000008e128 < z < 5.2000000000000003e136 or 4.2500000000000001e173 < z

   1. Initial program 92.4%

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

      1. associate-+l+ 92.4%

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

      2. +-commutative 92.4%

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

      3. associate-+l+ 92.4%

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

      4. +-commutative 92.4%

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

      5. associate-+l+ 92.4%

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

      6. +-commutative 92.4%

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

      7. associate-+r- 55.3%

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

   3. Simplified 31.2%

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

   4. Taylor expanded in t around inf 3.1%

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

      1. associate--l+ 25.0%

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

      2. +-commutative 25.0%

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

      3. associate--l+ 22.7%

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

      4. +-commutative 22.7%

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

   6. Simplified 22.7%

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

   7. Taylor expanded in y around inf 25.0%

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

      1. associate--r+ 25.0%

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

   9. Simplified 25.0%

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

   10. Taylor expanded in z around inf 24.4%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + z} + \left(2 - \sqrt{z}\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+128} \lor \neg \left(z \leq 5.2 \cdot 10^{+136}\right) \land z \leq 4.25 \cdot 10^{+173}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 14: 72.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{1 + z} + \left(2 - \sqrt{z}\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+128} \lor \neg \left(z \leq 3.8 \cdot 10^{+141}\right) \land z \leq 4.25 \cdot 10^{+173}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

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 (<= z 6.2e+16)
   (+ (sqrt (+ 1.0 z)) (- 2.0 (sqrt z)))
   (if (or (<= z 1.25e+128) (and (not (<= z 3.8e+141)) (<= z 4.25e+173)))
     (+ (sqrt (+ 1.0 y)) (- 1.0 (sqrt y)))
     (/ 1.0 (+ (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 (z <= 6.2e+16) {
		tmp = sqrt((1.0 + z)) + (2.0 - sqrt(z));
	} else if ((z <= 1.25e+128) || (!(z <= 3.8e+141) && (z <= 4.25e+173))) {
		tmp = sqrt((1.0 + y)) + (1.0 - sqrt(y));
	} else {
		tmp = 1.0 / (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 (z <= 6.2d+16) then
        tmp = sqrt((1.0d0 + z)) + (2.0d0 - sqrt(z))
    else if ((z <= 1.25d+128) .or. (.not. (z <= 3.8d+141)) .and. (z <= 4.25d+173)) then
        tmp = sqrt((1.0d0 + y)) + (1.0d0 - sqrt(y))
    else
        tmp = 1.0d0 / (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 (z <= 6.2e+16) {
		tmp = Math.sqrt((1.0 + z)) + (2.0 - Math.sqrt(z));
	} else if ((z <= 1.25e+128) || (!(z <= 3.8e+141) && (z <= 4.25e+173))) {
		tmp = Math.sqrt((1.0 + y)) + (1.0 - Math.sqrt(y));
	} else {
		tmp = 1.0 / (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 z <= 6.2e+16:
		tmp = math.sqrt((1.0 + z)) + (2.0 - math.sqrt(z))
	elif (z <= 1.25e+128) or (not (z <= 3.8e+141) and (z <= 4.25e+173)):
		tmp = math.sqrt((1.0 + y)) + (1.0 - math.sqrt(y))
	else:
		tmp = 1.0 / (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 (z <= 6.2e+16)
		tmp = Float64(sqrt(Float64(1.0 + z)) + Float64(2.0 - sqrt(z)));
	elseif ((z <= 1.25e+128) || (!(z <= 3.8e+141) && (z <= 4.25e+173)))
		tmp = Float64(sqrt(Float64(1.0 + y)) + Float64(1.0 - sqrt(y)));
	else
		tmp = Float64(1.0 / 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 (z <= 6.2e+16)
		tmp = sqrt((1.0 + z)) + (2.0 - sqrt(z));
	elseif ((z <= 1.25e+128) || (~((z <= 3.8e+141)) && (z <= 4.25e+173)))
		tmp = sqrt((1.0 + y)) + (1.0 - sqrt(y));
	else
		tmp = 1.0 / (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[z, 6.2e+16], N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(2.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 1.25e+128], And[N[Not[LessEqual[z, 3.8e+141]], $MachinePrecision], LessEqual[z, 4.25e+173]]], N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 6.2e16

   1. Initial program 97.2%

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

      1. associate-+l+ 97.2%

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

      2. +-commutative 97.2%

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

      3. associate-+l+ 97.2%

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

      4. +-commutative 97.2%

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

      5. associate-+l+ 97.2%

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

      6. +-commutative 97.2%

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

      7. associate-+r- 54.6%

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

   3. Simplified 29.2%

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

   4. Taylor expanded in t around inf 25.4%

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

      1. associate--l+ 29.5%

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

      2. +-commutative 29.5%

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

      3. associate--l+ 35.7%

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

      4. +-commutative 35.7%

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

   6. Simplified 35.7%

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

   7. Taylor expanded in x around 0 35.2%

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

   8. Taylor expanded in y around 0 51.6%

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

      1. +-commutative 51.6%

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

      2. associate--l+ 51.5%

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

   10. Simplified 51.5%

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

   if 6.2e16 < z < 1.25e128 or 3.79999999999999976e141 < z < 4.2500000000000001e173

   1. Initial program 84.4%

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

      1. associate-+l+ 84.4%

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

      2. +-commutative 84.4%

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

      3. associate-+l+ 84.4%

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

      4. +-commutative 84.4%

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

      5. associate-+l+ 84.4%

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

      6. +-commutative 84.4%

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

      7. associate-+r- 65.0%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in t around inf 4.0%

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

      1. associate--l+ 15.4%

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

      2. +-commutative 15.4%

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

      3. associate--l+ 13.5%

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

      4. +-commutative 13.5%

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

   6. Simplified 13.5%

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

   7. Taylor expanded in x around 0 4.1%

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

   8. Taylor expanded in z around inf 35.0%

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

      1. +-commutative 35.0%

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

      2. associate--l+ 35.1%

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

   10. Simplified 35.1%

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

   if 1.25e128 < z < 3.79999999999999976e141 or 4.2500000000000001e173 < z

   1. Initial program 92.6%

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

      1. associate-+l+ 92.6%

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

      2. +-commutative 92.6%

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

      3. associate-+l+ 92.6%

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

      4. +-commutative 92.6%

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

      5. associate-+l+ 92.6%

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

      6. +-commutative 92.6%

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

      7. associate-+r- 54.4%

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

   3. Simplified 30.7%

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

   4. Taylor expanded in t around inf 3.1%

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

      1. associate--l+ 24.9%

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

      2. +-commutative 24.9%

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

      3. associate--l+ 22.5%

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

      4. +-commutative 22.5%

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

   6. Simplified 22.5%

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

   7. Taylor expanded in y around inf 24.9%

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

      1. associate--r+ 24.9%

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

   9. Simplified 24.9%

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

   10. Taylor expanded in z around inf 24.3%

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

       1. flip-- 25.5%

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

       2. add-sqr-sqrt 25.5%

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

       3. add-sqr-sqrt 25.5%

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

   12. Applied egg-rr 25.5%

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

       1. associate--l+ 28.5%

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

       2. +-inverses 28.5%

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

       3. metadata-eval 28.5%

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

   14. Simplified 28.5%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{1 + z} + \left(2 - \sqrt{z}\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+128} \lor \neg \left(z \leq 3.8 \cdot 10^{+141}\right) \land z \leq 4.25 \cdot 10^{+173}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]

Alternative 15: 67.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-155}:\\ \;\;\;\;\left(3 + y \cdot 0.5\right) - \sqrt{y}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \end{array} \]

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 3.3e-155)
   (- (+ 3.0 (* y 0.5)) (sqrt y))
   (if (<= y 1.5e+17)
     (+ (sqrt (+ 1.0 y)) (- 1.0 (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 <= 3.3e-155) {
		tmp = (3.0 + (y * 0.5)) - sqrt(y);
	} else if (y <= 1.5e+17) {
		tmp = sqrt((1.0 + y)) + (1.0 - 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 <= 3.3d-155) then
        tmp = (3.0d0 + (y * 0.5d0)) - sqrt(y)
    else if (y <= 1.5d+17) then
        tmp = sqrt((1.0d0 + y)) + (1.0d0 - 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 <= 3.3e-155) {
		tmp = (3.0 + (y * 0.5)) - Math.sqrt(y);
	} else if (y <= 1.5e+17) {
		tmp = Math.sqrt((1.0 + y)) + (1.0 - 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 <= 3.3e-155:
		tmp = (3.0 + (y * 0.5)) - math.sqrt(y)
	elif y <= 1.5e+17:
		tmp = math.sqrt((1.0 + y)) + (1.0 - 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 <= 3.3e-155)
		tmp = Float64(Float64(3.0 + Float64(y * 0.5)) - sqrt(y));
	elseif (y <= 1.5e+17)
		tmp = Float64(sqrt(Float64(1.0 + y)) + Float64(1.0 - 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 <= 3.3e-155)
		tmp = (3.0 + (y * 0.5)) - sqrt(y);
	elseif (y <= 1.5e+17)
		tmp = sqrt((1.0 + y)) + (1.0 - 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, 3.3e-155], N[(N[(3.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+17], N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(1.0 - 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 < 3.29999999999999986e-155

   1. Initial program 98.2%

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

      1. associate-+l+ 98.2%

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

      2. +-commutative 98.2%

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

      3. associate-+l+ 98.2%

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

      4. +-commutative 98.2%

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

      5. associate-+l+ 98.2%

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

      6. +-commutative 98.2%

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

      7. associate-+r- 98.2%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in t around inf 27.7%

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

      1. associate--l+ 31.7%

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

      2. +-commutative 31.7%

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

      3. associate--l+ 31.7%

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

      4. +-commutative 31.7%

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

   6. Simplified 31.7%

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

   7. Taylor expanded in x around 0 36.2%

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

   8. Taylor expanded in y around 0 36.2%

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

      1. *-commutative 36.2%

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

   10. Simplified 36.2%

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

   11. Taylor expanded in z around 0 47.1%

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

   if 3.29999999999999986e-155 < y < 1.5e17

   1. Initial program 95.2%

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

      1. associate-+l+ 95.2%

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

      2. +-commutative 95.2%

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

      3. associate-+l+ 95.2%

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

      4. +-commutative 95.2%

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

      5. associate-+l+ 95.2%

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

      6. +-commutative 95.2%

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

      7. associate-+r- 95.1%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in t around inf 24.7%

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

      1. associate--l+ 27.9%

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

      2. +-commutative 27.9%

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. associate--l+ 27.9%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative 27.9%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   6. Simplified 27.9%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)} \]

   7. Taylor expanded in x around 0 35.5%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   8. Taylor expanded in z around inf 44.5%

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   9. Step-by-step derivation

      1. +-commutative 44.5%

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + 1\right)} - \sqrt{y} \]

      2. associate--l+ 44.5%

         \[\leadsto \color{blue}{\sqrt{1 + y} + \left(1 - \sqrt{y}\right)} \]

   10. Simplified 44.5%

       \[\leadsto \color{blue}{\sqrt{1 + y} + \left(1 - \sqrt{y}\right)} \]

   if 1.5e17 < y

   1. Initial program 89.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 89.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 89.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+l+ 89.7%

         \[\leadsto \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      4. +-commutative 89.7%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      5. associate-+l+ 89.7%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)} \]

      6. +-commutative 89.7%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)} \]

      7. associate-+r- 5.9%

         \[\leadsto \color{blue}{\left(\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \sqrt{y + 1}\right) - \sqrt{y}} \]

   3. Simplified 4.1%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} - \sqrt{x + 1}\right) + \sqrt{x}\right)\right) - \sqrt{y}\right)} \]

   4. Taylor expanded in t around inf 3.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 20.4%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative 20.4%

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. associate--l+ 26.3%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative 26.3%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   6. Simplified 26.3%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)} \]

   7. Taylor expanded in y around inf 29.7%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)} \]
   8. Step-by-step derivation

      1. associate--r+ 29.7%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + z} - \sqrt{x}\right) - \sqrt{z}\right)} \]

   9. Simplified 29.7%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + z} - \sqrt{x}\right) - \sqrt{z}\right)} \]

   10. Taylor expanded in z around inf 20.7%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-155}:\\ \;\;\;\;\left(3 + y \cdot 0.5\right) - \sqrt{y}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 16: 65.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-155}:\\ \;\;\;\;\left(3 + y \cdot 0.5\right) - \sqrt{y}\\ \mathbf{elif}\;y \leq 6:\\ \;\;\;\;\left(2 + y \cdot 0.5\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \end{array} \]

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 2.2e-155)
   (- (+ 3.0 (* y 0.5)) (sqrt y))
   (if (<= y 6.0)
     (- (+ 2.0 (* y 0.5)) (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 <= 2.2e-155) {
		tmp = (3.0 + (y * 0.5)) - sqrt(y);
	} else if (y <= 6.0) {
		tmp = (2.0 + (y * 0.5)) - 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 <= 2.2d-155) then
        tmp = (3.0d0 + (y * 0.5d0)) - sqrt(y)
    else if (y <= 6.0d0) then
        tmp = (2.0d0 + (y * 0.5d0)) - 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 <= 2.2e-155) {
		tmp = (3.0 + (y * 0.5)) - Math.sqrt(y);
	} else if (y <= 6.0) {
		tmp = (2.0 + (y * 0.5)) - 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 <= 2.2e-155:
		tmp = (3.0 + (y * 0.5)) - math.sqrt(y)
	elif y <= 6.0:
		tmp = (2.0 + (y * 0.5)) - 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 <= 2.2e-155)
		tmp = Float64(Float64(3.0 + Float64(y * 0.5)) - sqrt(y));
	elseif (y <= 6.0)
		tmp = Float64(Float64(2.0 + Float64(y * 0.5)) - 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 <= 2.2e-155)
		tmp = (3.0 + (y * 0.5)) - sqrt(y);
	elseif (y <= 6.0)
		tmp = (2.0 + (y * 0.5)) - 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, 2.2e-155], N[(N[(3.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.0], N[(N[(2.0 + N[(y * 0.5), $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 < 2.1999999999999999e-155

   1. Initial program 98.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 98.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 98.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+l+ 98.2%

         \[\leadsto \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      4. +-commutative 98.2%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      5. associate-+l+ 98.2%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)} \]

      6. +-commutative 98.2%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)} \]

      7. associate-+r- 98.2%

         \[\leadsto \color{blue}{\left(\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \sqrt{y + 1}\right) - \sqrt{y}} \]

   3. Simplified 57.1%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} - \sqrt{x + 1}\right) + \sqrt{x}\right)\right) - \sqrt{y}\right)} \]

   4. Taylor expanded in t around inf 27.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 31.7%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative 31.7%

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. associate--l+ 31.7%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative 31.7%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   6. Simplified 31.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)} \]

   7. Taylor expanded in x around 0 36.2%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   8. Taylor expanded in y around 0 36.2%

      \[\leadsto \left(1 + \color{blue}{\left(1 + \left(\sqrt{1 + z} + 0.5 \cdot y\right)\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
   9. Step-by-step derivation

      1. *-commutative 36.2%

         \[\leadsto \left(1 + \left(1 + \left(\sqrt{1 + z} + \color{blue}{y \cdot 0.5}\right)\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   10. Simplified 36.2%

       \[\leadsto \left(1 + \color{blue}{\left(1 + \left(\sqrt{1 + z} + y \cdot 0.5\right)\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   11. Taylor expanded in z around 0 47.1%

       \[\leadsto \color{blue}{\left(3 + 0.5 \cdot y\right) - \sqrt{y}} \]

   if 2.1999999999999999e-155 < y < 6

   1. Initial program 98.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 98.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 98.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+l+ 98.0%

         \[\leadsto \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      4. +-commutative 98.0%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      5. associate-+l+ 98.0%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)} \]

      6. +-commutative 98.0%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)} \]

      7. associate-+r- 97.9%

         \[\leadsto \color{blue}{\left(\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \sqrt{y + 1}\right) - \sqrt{y}} \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} - \sqrt{x + 1}\right) + \sqrt{x}\right)\right) - \sqrt{y}\right)} \]

   4. Taylor expanded in t around inf 23.6%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 27.2%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative 27.2%

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. associate--l+ 27.2%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative 27.2%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   6. Simplified 27.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)} \]

   7. Taylor expanded in x around 0 34.2%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   8. Taylor expanded in y around 0 33.1%

      \[\leadsto \left(1 + \color{blue}{\left(1 + \left(\sqrt{1 + z} + 0.5 \cdot y\right)\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
   9. Step-by-step derivation

      1. *-commutative 33.1%

         \[\leadsto \left(1 + \left(1 + \left(\sqrt{1 + z} + \color{blue}{y \cdot 0.5}\right)\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   10. Simplified 33.1%

       \[\leadsto \left(1 + \color{blue}{\left(1 + \left(\sqrt{1 + z} + y \cdot 0.5\right)\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   11. Taylor expanded in z around inf 47.0%

       \[\leadsto \color{blue}{\left(2 + 0.5 \cdot y\right) - \sqrt{y}} \]

   if 6 < y

   1. Initial program 88.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 88.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 88.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+l+ 88.6%

         \[\leadsto \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      4. +-commutative 88.6%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      5. associate-+l+ 88.6%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)} \]

      6. +-commutative 88.6%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)} \]

      7. associate-+r- 13.5%

         \[\leadsto \color{blue}{\left(\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \sqrt{y + 1}\right) - \sqrt{y}} \]

   3. Simplified 8.5%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} - \sqrt{x + 1}\right) + \sqrt{x}\right)\right) - \sqrt{y}\right)} \]

   4. Taylor expanded in t around inf 6.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 21.6%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative 21.6%

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. associate--l+ 26.8%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative 26.8%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   6. Simplified 26.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)} \]

   7. Taylor expanded in y around inf 28.6%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)} \]
   8. Step-by-step derivation

      1. associate--r+ 28.6%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + z} - \sqrt{x}\right) - \sqrt{z}\right)} \]

   9. Simplified 28.6%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + z} - \sqrt{x}\right) - \sqrt{z}\right)} \]

   10. Taylor expanded in z around inf 19.6%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-155}:\\ \;\;\;\;\left(3 + y \cdot 0.5\right) - \sqrt{y}\\ \mathbf{elif}\;y \leq 6:\\ \;\;\;\;\left(2 + y \cdot 0.5\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 17: 64.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-155}:\\ \;\;\;\;\left(3 + y \cdot 0.5\right) - \sqrt{y}\\ \mathbf{elif}\;y \leq 5.2:\\ \;\;\;\;\left(2 + y \cdot 0.5\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 6.8e-155)
   (- (+ 3.0 (* y 0.5)) (sqrt y))
   (if (<= y 5.2) (- (+ 2.0 (* y 0.5)) (sqrt y)) 1.0)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.8e-155) {
		tmp = (3.0 + (y * 0.5)) - sqrt(y);
	} else if (y <= 5.2) {
		tmp = (2.0 + (y * 0.5)) - sqrt(y);
	} 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 <= 6.8d-155) then
        tmp = (3.0d0 + (y * 0.5d0)) - sqrt(y)
    else if (y <= 5.2d0) then
        tmp = (2.0d0 + (y * 0.5d0)) - sqrt(y)
    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 <= 6.8e-155) {
		tmp = (3.0 + (y * 0.5)) - Math.sqrt(y);
	} else if (y <= 5.2) {
		tmp = (2.0 + (y * 0.5)) - Math.sqrt(y);
	} 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 <= 6.8e-155:
		tmp = (3.0 + (y * 0.5)) - math.sqrt(y)
	elif y <= 5.2:
		tmp = (2.0 + (y * 0.5)) - math.sqrt(y)
	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 <= 6.8e-155)
		tmp = Float64(Float64(3.0 + Float64(y * 0.5)) - sqrt(y));
	elseif (y <= 5.2)
		tmp = Float64(Float64(2.0 + Float64(y * 0.5)) - sqrt(y));
	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 <= 6.8e-155)
		tmp = (3.0 + (y * 0.5)) - sqrt(y);
	elseif (y <= 5.2)
		tmp = (2.0 + (y * 0.5)) - sqrt(y);
	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, 6.8e-155], N[(N[(3.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2], N[(N[(2.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 6.8e-155

   1. Initial program 98.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 98.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 98.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+l+ 98.2%

         \[\leadsto \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      4. +-commutative 98.2%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      5. associate-+l+ 98.2%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)} \]

      6. +-commutative 98.2%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)} \]

      7. associate-+r- 98.2%

         \[\leadsto \color{blue}{\left(\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \sqrt{y + 1}\right) - \sqrt{y}} \]

   3. Simplified 57.1%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} - \sqrt{x + 1}\right) + \sqrt{x}\right)\right) - \sqrt{y}\right)} \]

   4. Taylor expanded in t around inf 27.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 31.7%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative 31.7%

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. associate--l+ 31.7%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative 31.7%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   6. Simplified 31.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)} \]

   7. Taylor expanded in x around 0 36.2%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   8. Taylor expanded in y around 0 36.2%

      \[\leadsto \left(1 + \color{blue}{\left(1 + \left(\sqrt{1 + z} + 0.5 \cdot y\right)\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
   9. Step-by-step derivation

      1. *-commutative 36.2%

         \[\leadsto \left(1 + \left(1 + \left(\sqrt{1 + z} + \color{blue}{y \cdot 0.5}\right)\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   10. Simplified 36.2%

       \[\leadsto \left(1 + \color{blue}{\left(1 + \left(\sqrt{1 + z} + y \cdot 0.5\right)\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   11. Taylor expanded in z around 0 47.1%

       \[\leadsto \color{blue}{\left(3 + 0.5 \cdot y\right) - \sqrt{y}} \]

   if 6.8e-155 < y < 5.20000000000000018

   1. Initial program 98.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 98.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 98.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+l+ 98.0%

         \[\leadsto \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      4. +-commutative 98.0%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      5. associate-+l+ 98.0%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)} \]

      6. +-commutative 98.0%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)} \]

      7. associate-+r- 97.9%

         \[\leadsto \color{blue}{\left(\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \sqrt{y + 1}\right) - \sqrt{y}} \]

   3. Simplified 59.1%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} - \sqrt{x + 1}\right) + \sqrt{x}\right)\right) - \sqrt{y}\right)} \]

   4. Taylor expanded in t around inf 23.9%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 27.3%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative 27.3%

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. associate--l+ 27.3%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative 27.3%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   6. Simplified 27.3%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)} \]

   7. Taylor expanded in x around 0 34.7%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   8. Taylor expanded in y around 0 33.6%

      \[\leadsto \left(1 + \color{blue}{\left(1 + \left(\sqrt{1 + z} + 0.5 \cdot y\right)\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
   9. Step-by-step derivation

      1. *-commutative 33.6%

         \[\leadsto \left(1 + \left(1 + \left(\sqrt{1 + z} + \color{blue}{y \cdot 0.5}\right)\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   10. Simplified 33.6%

       \[\leadsto \left(1 + \color{blue}{\left(1 + \left(\sqrt{1 + z} + y \cdot 0.5\right)\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   11. Taylor expanded in z around inf 47.3%

       \[\leadsto \color{blue}{\left(2 + 0.5 \cdot y\right) - \sqrt{y}} \]

   if 5.20000000000000018 < y

   1. Initial program 88.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 88.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 88.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+l+ 88.7%

         \[\leadsto \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      4. +-commutative 88.7%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      5. associate-+l+ 88.7%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)} \]

      6. +-commutative 88.7%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)} \]

      7. associate-+r- 14.2%

         \[\leadsto \color{blue}{\left(\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \sqrt{y + 1}\right) - \sqrt{y}} \]

   3. Simplified 9.3%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} - \sqrt{x + 1}\right) + \sqrt{x}\right)\right) - \sqrt{y}\right)} \]

   4. Taylor expanded in t around inf 6.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 21.5%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative 21.5%

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. associate--l+ 26.8%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative 26.8%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   6. Simplified 26.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)} \]

   7. Taylor expanded in y around inf 28.5%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)} \]
   8. Step-by-step derivation

      1. associate--r+ 28.5%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + z} - \sqrt{x}\right) - \sqrt{z}\right)} \]

   9. Simplified 28.5%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + z} - \sqrt{x}\right) - \sqrt{z}\right)} \]

   10. Taylor expanded in z around inf 19.6%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   11. Taylor expanded in x around 0 39.5%

       \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-155}:\\ \;\;\;\;\left(3 + y \cdot 0.5\right) - \sqrt{y}\\ \mathbf{elif}\;y \leq 5.2:\\ \;\;\;\;\left(2 + y \cdot 0.5\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18: 62.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.3:\\ \;\;\;\;\left(2 + y \cdot 0.5\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 5.3) (- (+ 2.0 (* y 0.5)) (sqrt y)) 1.0))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.3) {
		tmp = (2.0 + (y * 0.5)) - sqrt(y);
	} 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 <= 5.3d0) then
        tmp = (2.0d0 + (y * 0.5d0)) - sqrt(y)
    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 <= 5.3) {
		tmp = (2.0 + (y * 0.5)) - Math.sqrt(y);
	} 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 <= 5.3:
		tmp = (2.0 + (y * 0.5)) - math.sqrt(y)
	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 <= 5.3)
		tmp = Float64(Float64(2.0 + Float64(y * 0.5)) - sqrt(y));
	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 <= 5.3)
		tmp = (2.0 + (y * 0.5)) - sqrt(y);
	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, 5.3], N[(N[(2.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if y < 5.29999999999999982

   1. Initial program 98.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 98.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 98.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+l+ 98.1%

         \[\leadsto \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      4. +-commutative 98.1%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      5. associate-+l+ 98.1%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)} \]

      6. +-commutative 98.1%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)} \]

      7. associate-+r- 98.1%

         \[\leadsto \color{blue}{\left(\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \sqrt{y + 1}\right) - \sqrt{y}} \]

   3. Simplified 58.5%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} - \sqrt{x + 1}\right) + \sqrt{x}\right)\right) - \sqrt{y}\right)} \]

   4. Taylor expanded in t around inf 25.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 29.3%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative 29.3%

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. associate--l+ 29.3%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative 29.3%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   6. Simplified 29.3%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)} \]

   7. Taylor expanded in x around 0 35.1%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

   8. Taylor expanded in y around 0 34.5%

      \[\leadsto \left(1 + \color{blue}{\left(1 + \left(\sqrt{1 + z} + 0.5 \cdot y\right)\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
   9. Step-by-step derivation

      1. *-commutative 34.5%

         \[\leadsto \left(1 + \left(1 + \left(\sqrt{1 + z} + \color{blue}{y \cdot 0.5}\right)\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   10. Simplified 34.5%

       \[\leadsto \left(1 + \color{blue}{\left(1 + \left(\sqrt{1 + z} + y \cdot 0.5\right)\right)}\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   11. Taylor expanded in z around inf 44.7%

       \[\leadsto \color{blue}{\left(2 + 0.5 \cdot y\right) - \sqrt{y}} \]

   if 5.29999999999999982 < y

   1. Initial program 88.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 88.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 88.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+l+ 88.6%

         \[\leadsto \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      4. +-commutative 88.6%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      5. associate-+l+ 88.6%

         \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)} \]

      6. +-commutative 88.6%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)} \]

      7. associate-+r- 13.5%

         \[\leadsto \color{blue}{\left(\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \sqrt{y + 1}\right) - \sqrt{y}} \]

   3. Simplified 8.5%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} - \sqrt{x + 1}\right) + \sqrt{x}\right)\right) - \sqrt{y}\right)} \]

   4. Taylor expanded in t around inf 6.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 21.6%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative 21.6%

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. associate--l+ 26.8%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative 26.8%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   6. Simplified 26.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)} \]

   7. Taylor expanded in y around inf 28.6%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)} \]
   8. Step-by-step derivation

      1. associate--r+ 28.6%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + z} - \sqrt{x}\right) - \sqrt{z}\right)} \]

   9. Simplified 28.6%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + z} - \sqrt{x}\right) - \sqrt{z}\right)} \]

   10. Taylor expanded in z around inf 19.6%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   11. Taylor expanded in x around 0 39.7%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.3:\\ \;\;\;\;\left(2 + y \cdot 0.5\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19: 35.2% 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 93.5%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 93.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. +-commutative 93.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. associate-+l+ 93.5%

      \[\leadsto \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   4. +-commutative 93.5%

      \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

   5. associate-+l+ 93.5%

      \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)} \]

   6. +-commutative 93.5%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)} \]

   7. associate-+r- 56.8%

      \[\leadsto \color{blue}{\left(\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \sqrt{y + 1}\right) - \sqrt{y}} \]

3. Simplified 34.1%

   \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \sqrt{1 + t}\right) - \left(\left(\sqrt{t} - \sqrt{x + 1}\right) + \sqrt{x}\right)\right) - \sqrt{y}\right)} \]

4. Taylor expanded in t around inf 16.0%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
5. Step-by-step derivation

   1. associate--l+ 25.5%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. +-commutative 25.5%

      \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   3. associate--l+ 28.1%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   4. +-commutative 28.1%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

6. Simplified 28.1%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)} \]

7. Taylor expanded in y around inf 20.4%

   \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)} \]
8. Step-by-step derivation

   1. associate--r+ 20.4%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + z} - \sqrt{x}\right) - \sqrt{z}\right)} \]

9. Simplified 20.4%

   \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + z} - \sqrt{x}\right) - \sqrt{z}\right)} \]

10. Taylor expanded in z around inf 15.6%

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

11. Taylor expanded in x around 0 33.2%

    \[\leadsto \color{blue}{1} \]

12. Final simplification 33.2%

    \[\leadsto 1 \]

Developer target: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
    (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y)))) + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + (1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y)))) + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023308 
(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))))