Main:z from

Percentage Accurate: 91.6% → 99.3%

Time: 26.3s

Alternatives: 19

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \mathbf{if}\;t_1 - \sqrt{x} \leq 0.4:\\ \;\;\;\;\left(t_4 + \frac{1}{\sqrt{x} + t_1}\right) + \left(\left(t_2 - \sqrt{z}\right) + t_3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \left(t_4 - \sqrt{x}\right)\right) + \left(t_3 + \frac{1 + \left(z - z\right)}{t_2 + \sqrt{z}}\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))))
   (if (<= (- t_1 (sqrt x)) 0.4)
     (+ (+ t_4 (/ 1.0 (+ (sqrt x) t_1))) (+ (- t_2 (sqrt z)) t_3))
     (+
      (+ t_1 (- t_4 (sqrt x)))
      (+ t_3 (/ (+ 1.0 (- z z)) (+ 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((x + 1.0));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = 1.0 / (sqrt((1.0 + y)) + sqrt(y));
	double tmp;
	if ((t_1 - sqrt(x)) <= 0.4) {
		tmp = (t_4 + (1.0 / (sqrt(x) + t_1))) + ((t_2 - sqrt(z)) + t_3);
	} else {
		tmp = (t_1 + (t_4 - sqrt(x))) + (t_3 + ((1.0 + (z - z)) / (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) :: t_4
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((1.0d0 + z))
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    t_4 = 1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))
    if ((t_1 - sqrt(x)) <= 0.4d0) then
        tmp = (t_4 + (1.0d0 / (sqrt(x) + t_1))) + ((t_2 - sqrt(z)) + t_3)
    else
        tmp = (t_1 + (t_4 - sqrt(x))) + (t_3 + ((1.0d0 + (z - z)) / (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((x + 1.0));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_4 = 1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y));
	double tmp;
	if ((t_1 - Math.sqrt(x)) <= 0.4) {
		tmp = (t_4 + (1.0 / (Math.sqrt(x) + t_1))) + ((t_2 - Math.sqrt(z)) + t_3);
	} else {
		tmp = (t_1 + (t_4 - Math.sqrt(x))) + (t_3 + ((1.0 + (z - z)) / (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((x + 1.0))
	t_2 = math.sqrt((1.0 + z))
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_4 = 1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))
	tmp = 0
	if (t_1 - math.sqrt(x)) <= 0.4:
		tmp = (t_4 + (1.0 / (math.sqrt(x) + t_1))) + ((t_2 - math.sqrt(z)) + t_3)
	else:
		tmp = (t_1 + (t_4 - math.sqrt(x))) + (t_3 + ((1.0 + (z - z)) / (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(x + 1.0))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))
	tmp = 0.0
	if (Float64(t_1 - sqrt(x)) <= 0.4)
		tmp = Float64(Float64(t_4 + Float64(1.0 / Float64(sqrt(x) + t_1))) + Float64(Float64(t_2 - sqrt(z)) + t_3));
	else
		tmp = Float64(Float64(t_1 + Float64(t_4 - sqrt(x))) + Float64(t_3 + Float64(Float64(1.0 + Float64(z - z)) / 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((x + 1.0));
	t_2 = sqrt((1.0 + z));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	t_4 = 1.0 / (sqrt((1.0 + y)) + sqrt(y));
	tmp = 0.0;
	if ((t_1 - sqrt(x)) <= 0.4)
		tmp = (t_4 + (1.0 / (sqrt(x) + t_1))) + ((t_2 - sqrt(z)) + t_3);
	else
		tmp = (t_1 + (t_4 - sqrt(x))) + (t_3 + ((1.0 + (z - z)) / (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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.4], N[(N[(t$95$4 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(N[(1.0 + N[(z - z), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 0.40000000000000002

   1. Initial program 86.2%

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

      1. associate-+l+ 86.2%

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

      2. associate-+l- 51.2%

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

      3. +-commutative 51.2%

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

      4. sub-neg 51.2%

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

      5. sub-neg 51.2%

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

      6. +-commutative 51.2%

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

      7. +-commutative 51.2%

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

   3. Simplified 51.2%

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

      1. flip-- 51.2%

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

      2. add-sqr-sqrt 43.0%

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

      3. add-sqr-sqrt 51.2%

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

   5. Applied egg-rr 51.2%

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

      1. associate--l+ 51.2%

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

      2. +-inverses 51.2%

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

      3. metadata-eval 51.2%

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

   7. Simplified 51.2%

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

      1. add-exp-log 51.2%

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

      2. associate--r- 89.1%

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

   9. Applied egg-rr 89.1%

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

       1. flip-- 89.6%

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

       2. add-sqr-sqrt 54.8%

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

       3. +-commutative 54.8%

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

       4. add-sqr-sqrt 89.9%

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

       5. +-commutative 89.9%

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

   11. Applied egg-rr 89.9%

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

       1. associate--l+ 92.7%

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

       2. +-inverses 92.7%

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

       3. metadata-eval 92.7%

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

   13. Simplified 92.7%

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

       1. add-exp-log 92.9%

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

   15. Applied egg-rr 92.9%

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

       1. +-commutative 92.9%

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

   17. Simplified 92.9%

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

   if 0.40000000000000002 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   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. associate-+l- 97.2%

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

      3. +-commutative 97.2%

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

      4. sub-neg 97.2%

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

      5. sub-neg 97.2%

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

      6. +-commutative 97.2%

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

      7. +-commutative 97.2%

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

   3. Simplified 97.2%

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

      1. flip-- 97.5%

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

      2. add-sqr-sqrt 72.5%

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

      3. +-commutative 72.5%

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

      4. add-sqr-sqrt 98.1%

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

      5. +-commutative 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. associate--l+ 98.1%

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

   7. Applied egg-rr 98.1%

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

      1. associate-+r- 98.1%

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

      2. +-commutative 98.1%

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

      3. associate--l+ 98.6%

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

   9. Simplified 98.6%

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

       1. flip-- 97.4%

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

       2. add-sqr-sqrt 79.4%

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

       3. add-sqr-sqrt 97.9%

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

   11. Applied egg-rr 99.3%

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

       1. associate--l+ 98.1%

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

       2. +-inverses 98.1%

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

       3. metadata-eval 98.1%

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

   13. Simplified 99.5%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.4:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \sqrt{x}\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1 + \left(z - z\right)}{\sqrt{1 + z} + \sqrt{z}}\right)\\ \end{array} \]

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;z \leq 4.4 \cdot 10^{+28}:\\ \;\;\;\;\left(t_3 + \frac{1 + \left(z - z\right)}{t_1 + \sqrt{z}}\right) + \left(1 + \left(t_2 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t_2 + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right) + \left(\left(t_1 - \sqrt{z}\right) + t_3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= z 4.4e+28)
     (+ (+ t_3 (/ (+ 1.0 (- z z)) (+ t_1 (sqrt z)))) (+ 1.0 (- t_2 (sqrt y))))
     (+
      (+ (/ 1.0 (+ t_2 (sqrt y))) (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))))
      (+ (- t_1 (sqrt z)) t_3)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (z <= 4.4e+28) {
		tmp = (t_3 + ((1.0 + (z - z)) / (t_1 + sqrt(z)))) + (1.0 + (t_2 - sqrt(y)));
	} else {
		tmp = ((1.0 / (t_2 + sqrt(y))) + (1.0 / (sqrt(x) + sqrt((x + 1.0))))) + ((t_1 - sqrt(z)) + t_3);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + y))
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    if (z <= 4.4d+28) then
        tmp = (t_3 + ((1.0d0 + (z - z)) / (t_1 + sqrt(z)))) + (1.0d0 + (t_2 - sqrt(y)))
    else
        tmp = ((1.0d0 / (t_2 + sqrt(y))) + (1.0d0 / (sqrt(x) + sqrt((x + 1.0d0))))) + ((t_1 - sqrt(z)) + t_3)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((1.0 + y));
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (z <= 4.4e+28) {
		tmp = (t_3 + ((1.0 + (z - z)) / (t_1 + Math.sqrt(z)))) + (1.0 + (t_2 - Math.sqrt(y)));
	} else {
		tmp = ((1.0 / (t_2 + Math.sqrt(y))) + (1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0))))) + ((t_1 - Math.sqrt(z)) + t_3);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = math.sqrt((1.0 + y))
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if z <= 4.4e+28:
		tmp = (t_3 + ((1.0 + (z - z)) / (t_1 + math.sqrt(z)))) + (1.0 + (t_2 - math.sqrt(y)))
	else:
		tmp = ((1.0 / (t_2 + math.sqrt(y))) + (1.0 / (math.sqrt(x) + math.sqrt((x + 1.0))))) + ((t_1 - math.sqrt(z)) + t_3)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (z <= 4.4e+28)
		tmp = Float64(Float64(t_3 + Float64(Float64(1.0 + Float64(z - z)) / Float64(t_1 + sqrt(z)))) + Float64(1.0 + Float64(t_2 - sqrt(y))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_2 + sqrt(y))) + Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0))))) + Float64(Float64(t_1 - sqrt(z)) + t_3));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((1.0 + y));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (z <= 4.4e+28)
		tmp = (t_3 + ((1.0 + (z - z)) / (t_1 + sqrt(z)))) + (1.0 + (t_2 - sqrt(y)));
	else
		tmp = ((1.0 / (t_2 + sqrt(y))) + (1.0 / (sqrt(x) + sqrt((x + 1.0))))) + ((t_1 - sqrt(z)) + t_3);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 4.4e+28], N[(N[(t$95$3 + N[(N[(1.0 + N[(z - z), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < 4.39999999999999973e28

   1. Initial program 95.4%

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

      1. associate-+l+ 95.4%

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

      2. associate-+l- 78.3%

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

      3. +-commutative 78.3%

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

      4. sub-neg 78.3%

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

      5. sub-neg 78.3%

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

      6. +-commutative 78.3%

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

      7. +-commutative 78.3%

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

   3. Simplified 78.3%

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

      1. flip-- 79.0%

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

      2. add-sqr-sqrt 78.9%

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

      3. +-commutative 78.9%

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

      4. add-sqr-sqrt 79.5%

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

      5. +-commutative 79.5%

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

   5. Applied egg-rr 79.5%

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

      1. associate--l+ 79.5%

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

   7. Applied egg-rr 79.5%

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

      1. associate-+r- 79.5%

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

      2. +-commutative 79.5%

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

      3. associate--l+ 80.0%

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

   9. Simplified 80.0%

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

   10. Taylor expanded in x around 0 55.5%

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

       1. associate--l+ 53.9%

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

   12. Simplified 53.9%

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

   if 4.39999999999999973e28 < z

   1. Initial program 87.3%

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

      1. associate-+l+ 87.3%

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

      2. associate-+l- 69.2%

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

      3. +-commutative 69.2%

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

      4. sub-neg 69.2%

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

      5. sub-neg 69.2%

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

      6. +-commutative 69.2%

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

      7. +-commutative 69.2%

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

   3. Simplified 69.2%

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

      1. flip-- 69.1%

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

      2. add-sqr-sqrt 55.6%

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

      3. add-sqr-sqrt 69.7%

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

   5. Applied egg-rr 69.7%

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

      1. associate--l+ 69.7%

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

      2. +-inverses 69.7%

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

      3. metadata-eval 69.7%

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

   7. Simplified 69.7%

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

      1. add-exp-log 69.7%

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

      2. associate--r- 90.4%

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

   9. Applied egg-rr 90.4%

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

       1. flip-- 90.5%

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

       2. add-sqr-sqrt 71.9%

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

       3. +-commutative 71.9%

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

       4. add-sqr-sqrt 90.5%

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

       5. +-commutative 90.5%

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

   11. Applied egg-rr 90.5%

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

       1. associate--l+ 93.4%

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

       2. +-inverses 93.4%

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

       3. metadata-eval 93.4%

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

   13. Simplified 93.4%

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

       1. add-exp-log 93.6%

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

   15. Applied egg-rr 93.6%

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

       1. +-commutative 93.6%

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

   17. Simplified 93.6%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.4 \cdot 10^{+28}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1 + \left(z - z\right)}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \end{array} \]

Alternative 3: 96.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{+26}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1 + \left(z - z\right)}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \end{array} \]

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 1.45e+26)
   (+
    (+
     (- (sqrt (+ 1.0 t)) (sqrt t))
     (/ (+ 1.0 (- z z)) (+ (sqrt (+ 1.0 z)) (sqrt z))))
    (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y))))
   (/ (+ 1.0 (- x x)) (+ (sqrt x) (sqrt (+ x 1.0))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.45e+26) {
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) + ((1.0 + (z - z)) / (sqrt((1.0 + z)) + sqrt(z)))) + (1.0 + (sqrt((1.0 + y)) - sqrt(y)));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((x + 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 <= 1.45d+26) then
        tmp = ((sqrt((1.0d0 + t)) - sqrt(t)) + ((1.0d0 + (z - z)) / (sqrt((1.0d0 + z)) + sqrt(z)))) + (1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y)))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt(x) + sqrt((x + 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 <= 1.45e+26) {
		tmp = ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + ((1.0 + (z - z)) / (Math.sqrt((1.0 + z)) + Math.sqrt(z)))) + (1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y)));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + Math.sqrt((x + 1.0)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 1.45e+26:
		tmp = ((math.sqrt((1.0 + t)) - math.sqrt(t)) + ((1.0 + (z - z)) / (math.sqrt((1.0 + z)) + math.sqrt(z)))) + (1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y)))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + math.sqrt((x + 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 <= 1.45e+26)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(1.0 + Float64(z - z)) / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))) + Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + sqrt(Float64(x + 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 <= 1.45e+26)
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) + ((1.0 + (z - z)) / (sqrt((1.0 + z)) + sqrt(z)))) + (1.0 + (sqrt((1.0 + y)) - sqrt(y)));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((x + 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, 1.45e+26], N[(N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(z - z), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45e26

   1. Initial program 95.0%

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

      1. associate-+l+ 95.0%

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

      2. associate-+l- 62.0%

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

      3. +-commutative 62.0%

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

      4. sub-neg 62.0%

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

      5. sub-neg 62.0%

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

      6. +-commutative 62.0%

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

      7. +-commutative 62.0%

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

   3. Simplified 62.0%

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

      1. flip-- 62.1%

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

      2. add-sqr-sqrt 49.4%

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

      3. +-commutative 49.4%

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

      4. add-sqr-sqrt 62.6%

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

      5. +-commutative 62.6%

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

   5. Applied egg-rr 62.6%

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

      1. associate--l+ 62.6%

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

   7. Applied egg-rr 62.6%

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

      1. associate-+r- 62.6%

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

      2. +-commutative 62.6%

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

      3. associate--l+ 63.4%

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

   9. Simplified 63.4%

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

   10. Taylor expanded in x around 0 56.5%

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

       1. associate--l+ 56.5%

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

   12. Simplified 56.5%

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

   if 1.45e26 < y

   1. Initial program 88.0%

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

      1. associate-+l+ 88.0%

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

      2. +-commutative 88.0%

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

      3. associate-+r- 88.0%

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

      4. associate-+l- 53.0%

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

      5. +-commutative 53.0%

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

      6. associate--l+ 53.0%

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

      7. +-commutative 53.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in t around inf 3.2%

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

      1. associate--l+ 4.4%

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

      2. +-commutative 4.4%

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

      3. +-commutative 4.4%

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

   6. Simplified 4.4%

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

   7. Taylor expanded in z around inf 4.0%

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

      1. +-commutative 4.0%

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

   9. Simplified 4.0%

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

   10. Taylor expanded in y around inf 21.0%

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

       1. flip-- 21.0%

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

       2. add-sqr-sqrt 21.0%

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

       3. +-commutative 21.0%

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

       4. add-sqr-sqrt 21.0%

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

       5. +-commutative 21.0%

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

   12. Applied egg-rr 21.0%

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

       1. +-commutative 21.0%

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

       2. associate--l+ 24.9%

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

       3. +-commutative 24.9%

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

       4. +-commutative 24.9%

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

   14. Simplified 24.9%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{+26}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1 + \left(z - z\right)}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 4: 96.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+31}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \end{array} \]

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 2.7e+31)
   (+
    (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (- (sqrt (+ 1.0 t)) (sqrt t)))
    (+ 1.0 (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))))
   (/ (+ 1.0 (- x x)) (+ (sqrt x) (sqrt (+ x 1.0))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.7e+31) {
		tmp = ((sqrt((1.0 + z)) - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t))) + (1.0 + (1.0 / (sqrt((1.0 + y)) + sqrt(y))));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((x + 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 <= 2.7d+31) then
        tmp = ((sqrt((1.0d0 + z)) - sqrt(z)) + (sqrt((1.0d0 + t)) - sqrt(t))) + (1.0d0 + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt(x) + sqrt((x + 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 <= 2.7e+31) {
		tmp = ((Math.sqrt((1.0 + z)) - Math.sqrt(z)) + (Math.sqrt((1.0 + t)) - Math.sqrt(t))) + (1.0 + (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + Math.sqrt((x + 1.0)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 2.7e+31:
		tmp = ((math.sqrt((1.0 + z)) - math.sqrt(z)) + (math.sqrt((1.0 + t)) - math.sqrt(t))) + (1.0 + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + math.sqrt((x + 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 <= 2.7e+31)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) + Float64(1.0 + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + sqrt(Float64(x + 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 <= 2.7e+31)
		tmp = ((sqrt((1.0 + z)) - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t))) + (1.0 + (1.0 / (sqrt((1.0 + y)) + sqrt(y))));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((x + 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, 2.7e+31], N[(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] + N[(1.0 + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.69999999999999986e31

   1. Initial program 95.0%

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

      1. associate-+l+ 95.0%

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

      2. associate-+l- 62.3%

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

      3. +-commutative 62.3%

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

      4. sub-neg 62.3%

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

      5. sub-neg 62.3%

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

      6. +-commutative 62.3%

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

      7. +-commutative 62.3%

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

   3. Simplified 62.3%

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

      1. flip-- 62.4%

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

      2. add-sqr-sqrt 62.7%

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

      3. add-sqr-sqrt 62.9%

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

   5. Applied egg-rr 62.9%

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

      1. associate--l+ 63.1%

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

      2. +-inverses 63.1%

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

      3. metadata-eval 63.1%

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

   7. Simplified 63.1%

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

   8. Taylor expanded in x around 0 56.3%

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

   if 2.69999999999999986e31 < y

   1. Initial program 87.9%

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

      1. associate-+l+ 87.9%

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

      2. +-commutative 87.9%

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

      3. associate-+r- 87.9%

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

      4. associate-+l- 53.5%

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

      5. +-commutative 53.5%

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

      6. associate--l+ 53.5%

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

      7. +-commutative 53.5%

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

   3. Simplified 37.3%

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

   4. Taylor expanded in t around inf 3.2%

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

      1. associate--l+ 4.4%

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

      2. +-commutative 4.4%

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

      3. +-commutative 4.4%

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

   6. Simplified 4.4%

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

   7. Taylor expanded in z around inf 4.0%

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

      1. +-commutative 4.0%

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

   9. Simplified 4.0%

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

   10. Taylor expanded in y around inf 21.2%

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

       1. flip-- 21.2%

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

       2. add-sqr-sqrt 21.2%

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

       3. +-commutative 21.2%

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

       4. add-sqr-sqrt 21.2%

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

       5. +-commutative 21.2%

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

   12. Applied egg-rr 21.2%

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

       1. +-commutative 21.2%

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

       2. associate--l+ 25.1%

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

       3. +-commutative 25.1%

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

       4. +-commutative 25.1%

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

   14. Simplified 25.1%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+31}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 5: 95.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 2.9 \cdot 10^{-63}:\\ \;\;\;\;\left(t_2 + \left(\left(\mathsf{hypot}\left(1, \sqrt{t}\right) - \sqrt{t}\right) - \sqrt{z}\right)\right) + 2\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+26}:\\ \;\;\;\;t_1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(t_2 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))) (t_2 (sqrt (+ 1.0 z))))
   (if (<= y 2.9e-63)
     (+ (+ t_2 (- (- (hypot 1.0 (sqrt t)) (sqrt t)) (sqrt z))) 2.0)
     (if (<= y 1.45e+26)
       (+ t_1 (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (- (- t_2 (sqrt z)) (sqrt x))))
       (/ (+ 1.0 (- x x)) (+ (sqrt x) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((1.0 + z));
	double tmp;
	if (y <= 2.9e-63) {
		tmp = (t_2 + ((hypot(1.0, sqrt(t)) - sqrt(t)) - sqrt(z))) + 2.0;
	} else if (y <= 1.45e+26) {
		tmp = t_1 + ((sqrt((1.0 + y)) - sqrt(y)) + ((t_2 - sqrt(z)) - sqrt(x)));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + t_1);
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double t_2 = Math.sqrt((1.0 + z));
	double tmp;
	if (y <= 2.9e-63) {
		tmp = (t_2 + ((Math.hypot(1.0, Math.sqrt(t)) - Math.sqrt(t)) - Math.sqrt(z))) + 2.0;
	} else if (y <= 1.45e+26) {
		tmp = t_1 + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) + ((t_2 - Math.sqrt(z)) - Math.sqrt(x)));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	t_2 = math.sqrt((1.0 + z))
	tmp = 0
	if y <= 2.9e-63:
		tmp = (t_2 + ((math.hypot(1.0, math.sqrt(t)) - math.sqrt(t)) - math.sqrt(z))) + 2.0
	elif y <= 1.45e+26:
		tmp = t_1 + ((math.sqrt((1.0 + y)) - math.sqrt(y)) + ((t_2 - math.sqrt(z)) - math.sqrt(x)))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (y <= 2.9e-63)
		tmp = Float64(Float64(t_2 + Float64(Float64(hypot(1.0, sqrt(t)) - sqrt(t)) - sqrt(z))) + 2.0);
	elseif (y <= 1.45e+26)
		tmp = Float64(t_1 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(Float64(t_2 - sqrt(z)) - sqrt(x))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + t_1));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	t_2 = sqrt((1.0 + z));
	tmp = 0.0;
	if (y <= 2.9e-63)
		tmp = (t_2 + ((hypot(1.0, sqrt(t)) - sqrt(t)) - sqrt(z))) + 2.0;
	elseif (y <= 1.45e+26)
		tmp = t_1 + ((sqrt((1.0 + y)) - sqrt(y)) + ((t_2 - sqrt(z)) - sqrt(x)));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 2.9e-63], N[(N[(t$95$2 + N[(N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[t], $MachinePrecision] ^ 2], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 1.45e+26], N[(t$95$1 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.89999999999999975e-63

   1. Initial program 96.3%

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

      1. associate-+l+ 96.3%

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

      2. +-commutative 96.3%

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

      3. associate-+r- 60.7%

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

      4. associate-+l- 55.2%

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

      5. +-commutative 55.2%

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

      6. associate--l+ 55.2%

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

      7. +-commutative 55.2%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in y around 0 17.6%

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

      1. associate--l+ 37.9%

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

      2. associate-+r+ 37.8%

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

      3. +-commutative 37.8%

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

      4. +-commutative 37.8%

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

      5. +-commutative 37.8%

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

   6. Simplified 37.8%

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

   7. Taylor expanded in x around 0 17.2%

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

      1. associate--l+ 52.4%

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

      2. +-commutative 52.4%

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

      3. associate--l+ 45.3%

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

      4. +-commutative 45.3%

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

      5. associate--r+ 55.0%

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

      6. unpow1 55.0%

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

      7. sqr-pow 55.0%

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

      8. hypot-1-def 55.0%

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

      9. metadata-eval 55.0%

         \[\leadsto \left(\sqrt{1 + z} + \left(\left(\mathsf{hypot}\left(1, {t}^{\color{blue}{0.5}}\right) - \sqrt{t}\right) - \sqrt{z}\right)\right) + 2 \]

      10. unpow1/2 55.0%

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

   9. Simplified 55.0%

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

   if 2.89999999999999975e-63 < y < 1.45e26

   1. Initial program 90.6%

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

      1. associate-+l+ 90.6%

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

      2. +-commutative 90.6%

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

      3. associate-+r- 66.4%

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

      4. associate-+l- 58.0%

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

      5. +-commutative 58.0%

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

      6. associate--l+ 58.0%

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

      7. +-commutative 58.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in t around inf 36.8%

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

      1. +-commutative 36.8%

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

      2. +-commutative 36.8%

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

      3. associate--l+ 36.5%

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

   6. Simplified 36.5%

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

   if 1.45e26 < y

   1. Initial program 88.0%

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

      1. associate-+l+ 88.0%

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

      2. +-commutative 88.0%

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

      3. associate-+r- 88.0%

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

      4. associate-+l- 53.0%

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

      5. +-commutative 53.0%

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

      6. associate--l+ 53.0%

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

      7. +-commutative 53.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in t around inf 3.2%

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

      1. associate--l+ 4.4%

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

      2. +-commutative 4.4%

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

      3. +-commutative 4.4%

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

   6. Simplified 4.4%

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

   7. Taylor expanded in z around inf 4.0%

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

      1. +-commutative 4.0%

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

   9. Simplified 4.0%

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

   10. Taylor expanded in y around inf 21.0%

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

       1. flip-- 21.0%

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

       2. add-sqr-sqrt 21.0%

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

       3. +-commutative 21.0%

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

       4. add-sqr-sqrt 21.0%

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

       5. +-commutative 21.0%

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

   12. Applied egg-rr 21.0%

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

       1. +-commutative 21.0%

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

       2. associate--l+ 24.9%

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

       3. +-commutative 24.9%

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

       4. +-commutative 24.9%

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

   14. Simplified 24.9%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-63}:\\ \;\;\;\;\left(\sqrt{1 + z} + \left(\left(\mathsf{hypot}\left(1, \sqrt{t}\right) - \sqrt{t}\right) - \sqrt{z}\right)\right) + 2\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 6: 95.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{+26}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \end{array} \]

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 1.45e+26)
   (+
    (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (- (sqrt (+ 1.0 t)) (sqrt t)))
    (- (+ 1.0 (sqrt (+ 1.0 y))) (sqrt y)))
   (/ (+ 1.0 (- x x)) (+ (sqrt x) (sqrt (+ x 1.0))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.45e+26) {
		tmp = ((sqrt((1.0 + z)) - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t))) + ((1.0 + sqrt((1.0 + y))) - sqrt(y));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((x + 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 <= 1.45d+26) then
        tmp = ((sqrt((1.0d0 + z)) - sqrt(z)) + (sqrt((1.0d0 + t)) - sqrt(t))) + ((1.0d0 + sqrt((1.0d0 + y))) - sqrt(y))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt(x) + sqrt((x + 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 <= 1.45e+26) {
		tmp = ((Math.sqrt((1.0 + z)) - Math.sqrt(z)) + (Math.sqrt((1.0 + t)) - Math.sqrt(t))) + ((1.0 + Math.sqrt((1.0 + y))) - Math.sqrt(y));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + Math.sqrt((x + 1.0)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 1.45e+26:
		tmp = ((math.sqrt((1.0 + z)) - math.sqrt(z)) + (math.sqrt((1.0 + t)) - math.sqrt(t))) + ((1.0 + math.sqrt((1.0 + y))) - math.sqrt(y))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + math.sqrt((x + 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 <= 1.45e+26)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) + Float64(Float64(1.0 + sqrt(Float64(1.0 + y))) - sqrt(y)));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + sqrt(Float64(x + 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 <= 1.45e+26)
		tmp = ((sqrt((1.0 + z)) - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t))) + ((1.0 + sqrt((1.0 + y))) - sqrt(y));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((x + 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, 1.45e+26], N[(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] + N[(N[(1.0 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45e26

   1. Initial program 95.0%

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

      1. associate-+l+ 95.0%

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

      2. associate-+l- 62.0%

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

      3. +-commutative 62.0%

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

      4. sub-neg 62.0%

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

      5. sub-neg 62.0%

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

      6. +-commutative 62.0%

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

      7. +-commutative 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in x around 0 55.7%

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

   if 1.45e26 < y

   1. Initial program 88.0%

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

      1. associate-+l+ 88.0%

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

      2. +-commutative 88.0%

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

      3. associate-+r- 88.0%

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

      4. associate-+l- 53.0%

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

      5. +-commutative 53.0%

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

      6. associate--l+ 53.0%

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

      7. +-commutative 53.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in t around inf 3.2%

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

      1. associate--l+ 4.4%

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

      2. +-commutative 4.4%

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

      3. +-commutative 4.4%

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

   6. Simplified 4.4%

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

   7. Taylor expanded in z around inf 4.0%

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

      1. +-commutative 4.0%

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

   9. Simplified 4.0%

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

   10. Taylor expanded in y around inf 21.0%

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

       1. flip-- 21.0%

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

       2. add-sqr-sqrt 21.0%

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

       3. +-commutative 21.0%

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

       4. add-sqr-sqrt 21.0%

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

       5. +-commutative 21.0%

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

   12. Applied egg-rr 21.0%

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

       1. +-commutative 21.0%

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

       2. associate--l+ 24.9%

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

       3. +-commutative 24.9%

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

       4. +-commutative 24.9%

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

   14. Simplified 24.9%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{+26}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 7: 95.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 1.9 \cdot 10^{-20}:\\ \;\;\;\;\left(\sqrt{1 + z} + \left(\left(\mathsf{hypot}\left(1, \sqrt{t}\right) - \sqrt{t}\right) - \sqrt{z}\right)\right) + 2\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{1 + y} + t_1\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))))
   (if (<= y 1.9e-20)
     (+
      (+ (sqrt (+ 1.0 z)) (- (- (hypot 1.0 (sqrt t)) (sqrt t)) (sqrt z)))
      2.0)
     (if (<= y 5e+14)
       (- (+ (sqrt (+ 1.0 y)) t_1) (+ (sqrt x) (sqrt y)))
       (/ (+ 1.0 (- x x)) (+ (sqrt x) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double tmp;
	if (y <= 1.9e-20) {
		tmp = (sqrt((1.0 + z)) + ((hypot(1.0, sqrt(t)) - sqrt(t)) - sqrt(z))) + 2.0;
	} else if (y <= 5e+14) {
		tmp = (sqrt((1.0 + y)) + t_1) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + t_1);
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double tmp;
	if (y <= 1.9e-20) {
		tmp = (Math.sqrt((1.0 + z)) + ((Math.hypot(1.0, Math.sqrt(t)) - Math.sqrt(t)) - Math.sqrt(z))) + 2.0;
	} else if (y <= 5e+14) {
		tmp = (Math.sqrt((1.0 + y)) + t_1) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	tmp = 0
	if y <= 1.9e-20:
		tmp = (math.sqrt((1.0 + z)) + ((math.hypot(1.0, math.sqrt(t)) - math.sqrt(t)) - math.sqrt(z))) + 2.0
	elif y <= 5e+14:
		tmp = (math.sqrt((1.0 + y)) + t_1) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (y <= 1.9e-20)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) + Float64(Float64(hypot(1.0, sqrt(t)) - sqrt(t)) - sqrt(z))) + 2.0);
	elseif (y <= 5e+14)
		tmp = Float64(Float64(sqrt(Float64(1.0 + y)) + t_1) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + t_1));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	tmp = 0.0;
	if (y <= 1.9e-20)
		tmp = (sqrt((1.0 + z)) + ((hypot(1.0, sqrt(t)) - sqrt(t)) - sqrt(z))) + 2.0;
	elseif (y <= 5e+14)
		tmp = (sqrt((1.0 + y)) + t_1) - (sqrt(x) + sqrt(y));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.9e-20], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[t], $MachinePrecision] ^ 2], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 5e+14], N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.8999999999999999e-20

   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. +-commutative 96.4%

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

      3. associate-+r- 62.1%

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

      4. associate-+l- 56.7%

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

      5. +-commutative 56.7%

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

      6. associate--l+ 56.7%

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

      7. +-commutative 56.7%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in y around 0 17.1%

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

      1. associate--l+ 36.7%

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

      2. associate-+r+ 36.7%

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

      3. +-commutative 36.7%

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

      4. +-commutative 36.7%

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

      5. +-commutative 36.7%

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

   6. Simplified 36.7%

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

   7. Taylor expanded in x around 0 16.8%

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

      1. associate--l+ 52.3%

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

      2. +-commutative 52.3%

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

      3. associate--l+ 45.5%

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

      4. +-commutative 45.5%

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

      5. associate--r+ 55.4%

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

      6. unpow1 55.4%

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

      7. sqr-pow 55.4%

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

      8. hypot-1-def 55.4%

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

      9. metadata-eval 55.4%

         \[\leadsto \left(\sqrt{1 + z} + \left(\left(\mathsf{hypot}\left(1, {t}^{\color{blue}{0.5}}\right) - \sqrt{t}\right) - \sqrt{z}\right)\right) + 2 \]

      10. unpow1/2 55.4%

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

   9. Simplified 55.4%

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

   if 1.8999999999999999e-20 < y < 5e14

   1. Initial program 89.6%

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

      1. associate-+l+ 89.6%

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

      2. +-commutative 89.6%

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

      3. associate-+r- 59.5%

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

      4. associate-+l- 55.2%

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

      5. +-commutative 55.2%

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

      6. associate--l+ 55.2%

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

      7. +-commutative 55.2%

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

   3. Simplified 41.6%

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

   4. Taylor expanded in t around inf 18.1%

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

      1. associate--l+ 30.0%

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

      2. +-commutative 30.0%

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

      3. +-commutative 30.0%

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

   6. Simplified 30.0%

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

   7. Taylor expanded in z around inf 25.8%

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

   if 5e14 < y

   1. Initial program 87.6%

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

      1. associate-+l+ 87.6%

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

      2. +-commutative 87.6%

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

      3. associate-+r- 87.6%

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

      4. associate-+l- 52.4%

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

      5. +-commutative 52.4%

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

      6. associate--l+ 52.4%

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

      7. +-commutative 52.4%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in t around inf 3.7%

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

      1. associate--l+ 5.1%

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

      2. +-commutative 5.1%

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

      3. +-commutative 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in z around inf 4.3%

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

      1. +-commutative 4.3%

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

   9. Simplified 4.3%

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

   10. Taylor expanded in y around inf 20.7%

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

       1. flip-- 20.7%

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

       2. add-sqr-sqrt 20.7%

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

       3. +-commutative 20.7%

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

       4. add-sqr-sqrt 20.7%

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

       5. +-commutative 20.7%

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

   12. Applied egg-rr 20.7%

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

       1. +-commutative 20.7%

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

       2. associate--l+ 24.6%

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

       3. +-commutative 24.6%

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

       4. +-commutative 24.6%

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

   14. Simplified 24.6%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-20}:\\ \;\;\;\;\left(\sqrt{1 + z} + \left(\left(\mathsf{hypot}\left(1, \sqrt{t}\right) - \sqrt{t}\right) - \sqrt{z}\right)\right) + 2\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{1 + y} + \sqrt{x + 1}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 8: 89.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 1.04 \cdot 10^{-19}:\\ \;\;\;\;2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{1 + y} + \left(t_1 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))))
   (if (<= y 1.04e-19)
     (+ 2.0 (- (hypot 1.0 (sqrt z)) (sqrt z)))
     (if (<= y 2e+16)
       (+ (sqrt (+ 1.0 y)) (- t_1 (+ (sqrt x) (sqrt y))))
       (/ (+ 1.0 (- x x)) (+ (sqrt x) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double tmp;
	if (y <= 1.04e-19) {
		tmp = 2.0 + (hypot(1.0, sqrt(z)) - sqrt(z));
	} else if (y <= 2e+16) {
		tmp = sqrt((1.0 + y)) + (t_1 - (sqrt(x) + sqrt(y)));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + t_1);
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double tmp;
	if (y <= 1.04e-19) {
		tmp = 2.0 + (Math.hypot(1.0, Math.sqrt(z)) - Math.sqrt(z));
	} else if (y <= 2e+16) {
		tmp = Math.sqrt((1.0 + y)) + (t_1 - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	tmp = 0
	if y <= 1.04e-19:
		tmp = 2.0 + (math.hypot(1.0, math.sqrt(z)) - math.sqrt(z))
	elif y <= 2e+16:
		tmp = math.sqrt((1.0 + y)) + (t_1 - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (y <= 1.04e-19)
		tmp = Float64(2.0 + Float64(hypot(1.0, sqrt(z)) - sqrt(z)));
	elseif (y <= 2e+16)
		tmp = Float64(sqrt(Float64(1.0 + y)) + Float64(t_1 - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + t_1));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	tmp = 0.0;
	if (y <= 1.04e-19)
		tmp = 2.0 + (hypot(1.0, sqrt(z)) - sqrt(z));
	elseif (y <= 2e+16)
		tmp = sqrt((1.0 + y)) + (t_1 - (sqrt(x) + sqrt(y)));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.04e-19], N[(2.0 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[z], $MachinePrecision] ^ 2], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+16], N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.03999999999999998e-19

   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. +-commutative 96.4%

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

      3. associate-+r- 62.1%

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

      4. associate-+l- 56.7%

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

      5. +-commutative 56.7%

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

      6. associate--l+ 56.7%

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

      7. +-commutative 56.7%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in t around inf 24.6%

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

      1. associate--l+ 41.1%

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

      2. +-commutative 41.1%

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

      3. +-commutative 41.1%

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

   6. Simplified 41.1%

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

   7. Taylor expanded in x around 0 36.7%

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

   8. Taylor expanded in y around 0 36.5%

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

      1. associate--l+ 59.0%

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

      2. rem-square-sqrt 59.1%

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

      3. hypot-1-def 59.1%

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

   10. Simplified 59.1%

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

   if 1.03999999999999998e-19 < y < 2e16

   1. Initial program 89.6%

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

      1. associate-+l+ 89.6%

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

      2. +-commutative 89.6%

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

      3. associate-+r- 59.5%

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

      4. associate-+l- 55.2%

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

      5. +-commutative 55.2%

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

      6. associate--l+ 55.2%

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

      7. +-commutative 55.2%

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

   3. Simplified 41.6%

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

   4. Taylor expanded in t around inf 18.1%

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

      1. associate--l+ 30.0%

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

      2. +-commutative 30.0%

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

      3. +-commutative 30.0%

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

   6. Simplified 30.0%

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

   7. Taylor expanded in z around inf 33.3%

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

      1. +-commutative 33.3%

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

   9. Simplified 33.3%

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

   if 2e16 < y

   1. Initial program 87.6%

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

      1. associate-+l+ 87.6%

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

      2. +-commutative 87.6%

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

      3. associate-+r- 87.6%

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

      4. associate-+l- 52.4%

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

      5. +-commutative 52.4%

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

      6. associate--l+ 52.4%

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

      7. +-commutative 52.4%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in t around inf 3.7%

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

      1. associate--l+ 5.1%

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

      2. +-commutative 5.1%

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

      3. +-commutative 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in z around inf 4.3%

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

      1. +-commutative 4.3%

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

   9. Simplified 4.3%

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

   10. Taylor expanded in y around inf 20.7%

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

       1. flip-- 20.7%

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

       2. add-sqr-sqrt 20.7%

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

       3. +-commutative 20.7%

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

       4. add-sqr-sqrt 20.7%

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

       5. +-commutative 20.7%

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

   12. Applied egg-rr 20.7%

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

       1. +-commutative 20.7%

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

       2. associate--l+ 24.6%

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

       3. +-commutative 24.6%

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

       4. +-commutative 24.6%

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

   14. Simplified 24.6%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.04 \cdot 10^{-19}:\\ \;\;\;\;2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{1 + y} + \left(\sqrt{x + 1} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 9: 89.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 2 \cdot 10^{-20}:\\ \;\;\;\;2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + y} + t_1\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))))
   (if (<= y 2e-20)
     (+ 2.0 (- (hypot 1.0 (sqrt z)) (sqrt z)))
     (if (<= y 5e+15)
       (- (+ (sqrt (+ 1.0 y)) t_1) (+ (sqrt x) (sqrt y)))
       (/ (+ 1.0 (- x x)) (+ (sqrt x) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double tmp;
	if (y <= 2e-20) {
		tmp = 2.0 + (hypot(1.0, sqrt(z)) - sqrt(z));
	} else if (y <= 5e+15) {
		tmp = (sqrt((1.0 + y)) + t_1) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + t_1);
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double tmp;
	if (y <= 2e-20) {
		tmp = 2.0 + (Math.hypot(1.0, Math.sqrt(z)) - Math.sqrt(z));
	} else if (y <= 5e+15) {
		tmp = (Math.sqrt((1.0 + y)) + t_1) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	tmp = 0
	if y <= 2e-20:
		tmp = 2.0 + (math.hypot(1.0, math.sqrt(z)) - math.sqrt(z))
	elif y <= 5e+15:
		tmp = (math.sqrt((1.0 + y)) + t_1) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (y <= 2e-20)
		tmp = Float64(2.0 + Float64(hypot(1.0, sqrt(z)) - sqrt(z)));
	elseif (y <= 5e+15)
		tmp = Float64(Float64(sqrt(Float64(1.0 + y)) + t_1) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + t_1));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	tmp = 0.0;
	if (y <= 2e-20)
		tmp = 2.0 + (hypot(1.0, sqrt(z)) - sqrt(z));
	elseif (y <= 5e+15)
		tmp = (sqrt((1.0 + y)) + t_1) - (sqrt(x) + sqrt(y));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 2e-20], N[(2.0 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[z], $MachinePrecision] ^ 2], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+15], N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.99999999999999989e-20

   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. +-commutative 96.4%

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

      3. associate-+r- 62.1%

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

      4. associate-+l- 56.7%

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

      5. +-commutative 56.7%

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

      6. associate--l+ 56.7%

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

      7. +-commutative 56.7%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in t around inf 24.6%

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

      1. associate--l+ 41.1%

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

      2. +-commutative 41.1%

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

      3. +-commutative 41.1%

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

   6. Simplified 41.1%

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

   7. Taylor expanded in x around 0 36.7%

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

   8. Taylor expanded in y around 0 36.5%

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

      1. associate--l+ 59.0%

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

      2. rem-square-sqrt 59.1%

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

      3. hypot-1-def 59.1%

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

   10. Simplified 59.1%

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

   if 1.99999999999999989e-20 < y < 5e15

   1. Initial program 89.6%

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

      1. associate-+l+ 89.6%

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

      2. +-commutative 89.6%

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

      3. associate-+r- 59.5%

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

      4. associate-+l- 55.2%

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

      5. +-commutative 55.2%

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

      6. associate--l+ 55.2%

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

      7. +-commutative 55.2%

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

   3. Simplified 41.6%

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

   4. Taylor expanded in t around inf 18.1%

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

      1. associate--l+ 30.0%

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

      2. +-commutative 30.0%

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

      3. +-commutative 30.0%

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

   6. Simplified 30.0%

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

   7. Taylor expanded in z around inf 25.8%

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

   if 5e15 < y

   1. Initial program 87.6%

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

      1. associate-+l+ 87.6%

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

      2. +-commutative 87.6%

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

      3. associate-+r- 87.6%

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

      4. associate-+l- 52.4%

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

      5. +-commutative 52.4%

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

      6. associate--l+ 52.4%

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

      7. +-commutative 52.4%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in t around inf 3.7%

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

      1. associate--l+ 5.1%

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

      2. +-commutative 5.1%

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

      3. +-commutative 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in z around inf 4.3%

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

      1. +-commutative 4.3%

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

   9. Simplified 4.3%

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

   10. Taylor expanded in y around inf 20.7%

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

       1. flip-- 20.7%

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

       2. add-sqr-sqrt 20.7%

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

       3. +-commutative 20.7%

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

       4. add-sqr-sqrt 20.7%

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

       5. +-commutative 20.7%

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

   12. Applied egg-rr 20.7%

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

       1. +-commutative 20.7%

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

       2. associate--l+ 24.6%

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

       3. +-commutative 24.6%

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

       4. +-commutative 24.6%

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

   14. Simplified 24.6%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-20}:\\ \;\;\;\;2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + y} + \sqrt{x + 1}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 10: 89.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{-20}:\\ \;\;\;\;2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+17}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \end{array} \]

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 2.55e-20)
   (+ 2.0 (- (hypot 1.0 (sqrt z)) (sqrt z)))
   (if (<= y 3.9e+17)
     (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))
     (/ (+ 1.0 (- x x)) (+ (sqrt x) (sqrt (+ x 1.0)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.55e-20) {
		tmp = 2.0 + (hypot(1.0, sqrt(z)) - sqrt(z));
	} else if (y <= 3.9e+17) {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((x + 1.0)));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.55e-20) {
		tmp = 2.0 + (Math.hypot(1.0, Math.sqrt(z)) - Math.sqrt(z));
	} else if (y <= 3.9e+17) {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + Math.sqrt((x + 1.0)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 2.55e-20:
		tmp = 2.0 + (math.hypot(1.0, math.sqrt(z)) - math.sqrt(z))
	elif y <= 3.9e+17:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + math.sqrt((x + 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 <= 2.55e-20)
		tmp = Float64(2.0 + Float64(hypot(1.0, sqrt(z)) - sqrt(z)));
	elseif (y <= 3.9e+17)
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + sqrt(Float64(x + 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 <= 2.55e-20)
		tmp = 2.0 + (hypot(1.0, sqrt(z)) - sqrt(z));
	elseif (y <= 3.9e+17)
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((x + 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, 2.55e-20], N[(2.0 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[z], $MachinePrecision] ^ 2], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e+17], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.55000000000000009e-20

   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. +-commutative 96.4%

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

      3. associate-+r- 62.1%

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

      4. associate-+l- 56.7%

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

      5. +-commutative 56.7%

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

      6. associate--l+ 56.7%

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

      7. +-commutative 56.7%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in t around inf 24.6%

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

      1. associate--l+ 41.1%

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

      2. +-commutative 41.1%

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

      3. +-commutative 41.1%

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

   6. Simplified 41.1%

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

   7. Taylor expanded in x around 0 36.7%

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

   8. Taylor expanded in y around 0 36.5%

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

      1. associate--l+ 59.0%

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

      2. rem-square-sqrt 59.1%

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

      3. hypot-1-def 59.1%

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

   10. Simplified 59.1%

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

   if 2.55000000000000009e-20 < y < 3.9e17

   1. Initial program 89.6%

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

      1. associate-+l+ 89.6%

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

      2. +-commutative 89.6%

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

      3. associate-+r- 59.5%

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

      4. associate-+l- 55.2%

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

      5. +-commutative 55.2%

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

      6. associate--l+ 55.2%

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

      7. +-commutative 55.2%

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

   3. Simplified 41.6%

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

   4. Taylor expanded in t around inf 18.1%

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

      1. associate--l+ 30.0%

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

      2. +-commutative 30.0%

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

      3. +-commutative 30.0%

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

   6. Simplified 30.0%

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

   7. Taylor expanded in z around inf 33.3%

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

      1. +-commutative 33.3%

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

   9. Simplified 33.3%

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

   10. Taylor expanded in x around 0 44.0%

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

       1. associate--l+ 44.1%

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

   12. Simplified 44.1%

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

   if 3.9e17 < y

   1. Initial program 87.6%

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

      1. associate-+l+ 87.6%

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

      2. +-commutative 87.6%

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

      3. associate-+r- 87.6%

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

      4. associate-+l- 52.4%

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

      5. +-commutative 52.4%

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

      6. associate--l+ 52.4%

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

      7. +-commutative 52.4%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in t around inf 3.7%

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

      1. associate--l+ 5.1%

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

      2. +-commutative 5.1%

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

      3. +-commutative 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in z around inf 4.3%

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

      1. +-commutative 4.3%

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

   9. Simplified 4.3%

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

   10. Taylor expanded in y around inf 20.7%

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

       1. flip-- 20.7%

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

       2. add-sqr-sqrt 20.7%

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

       3. +-commutative 20.7%

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

       4. add-sqr-sqrt 20.7%

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

       5. +-commutative 20.7%

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

   12. Applied egg-rr 20.7%

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

       1. +-commutative 20.7%

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

       2. associate--l+ 24.6%

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

       3. +-commutative 24.6%

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

       4. +-commutative 24.6%

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

   14. Simplified 24.6%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{-20}:\\ \;\;\;\;2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+17}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 11: 89.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-19}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+16}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \end{array} \]

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 1.02e-19)
   (+ (- (sqrt (+ 1.0 z)) (sqrt z)) 2.0)
   (if (<= y 2e+16)
     (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))
     (/ (+ 1.0 (- x x)) (+ (sqrt x) (sqrt (+ x 1.0)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.02e-19) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	} else if (y <= 2e+16) {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((x + 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 <= 1.02d-19) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + 2.0d0
    else if (y <= 2d+16) then
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt(x) + sqrt((x + 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 <= 1.02e-19) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + 2.0;
	} else if (y <= 2e+16) {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + Math.sqrt((x + 1.0)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 1.02e-19:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + 2.0
	elif y <= 2e+16:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + math.sqrt((x + 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 <= 1.02e-19)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + 2.0);
	elseif (y <= 2e+16)
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + sqrt(Float64(x + 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 <= 1.02e-19)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	elseif (y <= 2e+16)
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((x + 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, 1.02e-19], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 2e+16], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.02000000000000004e-19

   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. +-commutative 96.4%

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

      3. associate-+r- 62.1%

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

      4. associate-+l- 56.7%

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

      5. +-commutative 56.7%

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

      6. associate--l+ 56.7%

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

      7. +-commutative 56.7%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in t around inf 24.6%

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

      1. associate--l+ 41.1%

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

      2. +-commutative 41.1%

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

      3. +-commutative 41.1%

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

   6. Simplified 41.1%

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

   7. Taylor expanded in x around 0 36.7%

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

   8. Taylor expanded in y around 0 36.5%

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

      1. associate--l+ 59.0%

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

   10. Simplified 59.0%

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

   if 1.02000000000000004e-19 < y < 2e16

   1. Initial program 89.6%

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

      1. associate-+l+ 89.6%

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

      2. +-commutative 89.6%

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

      3. associate-+r- 59.5%

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

      4. associate-+l- 55.2%

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

      5. +-commutative 55.2%

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

      6. associate--l+ 55.2%

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

      7. +-commutative 55.2%

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

   3. Simplified 41.6%

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

   4. Taylor expanded in t around inf 18.1%

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

      1. associate--l+ 30.0%

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

      2. +-commutative 30.0%

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

      3. +-commutative 30.0%

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

   6. Simplified 30.0%

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

   7. Taylor expanded in z around inf 33.3%

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

      1. +-commutative 33.3%

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

   9. Simplified 33.3%

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

   10. Taylor expanded in x around 0 44.0%

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

       1. associate--l+ 44.1%

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

   12. Simplified 44.1%

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

   if 2e16 < y

   1. Initial program 87.6%

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

      1. associate-+l+ 87.6%

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

      2. +-commutative 87.6%

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

      3. associate-+r- 87.6%

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

      4. associate-+l- 52.4%

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

      5. +-commutative 52.4%

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

      6. associate--l+ 52.4%

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

      7. +-commutative 52.4%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in t around inf 3.7%

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

      1. associate--l+ 5.1%

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

      2. +-commutative 5.1%

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

      3. +-commutative 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in z around inf 4.3%

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

      1. +-commutative 4.3%

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

   9. Simplified 4.3%

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

   10. Taylor expanded in y around inf 20.7%

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

       1. flip-- 20.7%

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

       2. add-sqr-sqrt 20.7%

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

       3. +-commutative 20.7%

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

       4. add-sqr-sqrt 20.7%

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

       5. +-commutative 20.7%

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

   12. Applied egg-rr 20.7%

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

       1. +-commutative 20.7%

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

       2. associate--l+ 24.6%

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

       3. +-commutative 24.6%

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

       4. +-commutative 24.6%

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

   14. Simplified 24.6%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-19}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+16}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 12: 82.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.8:\\ \;\;\;\;\left(3 + z \cdot 0.5\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.8)
   (- (+ 3.0 (* z 0.5)) (sqrt z))
   (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.8) {
		tmp = (3.0 + (z * 0.5)) - sqrt(z);
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.8d0) then
        tmp = (3.0d0 + (z * 0.5d0)) - sqrt(z)
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.8) {
		tmp = (3.0 + (z * 0.5)) - Math.sqrt(z);
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.8:
		tmp = (3.0 + (z * 0.5)) - math.sqrt(z)
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.8)
		tmp = Float64(Float64(3.0 + Float64(z * 0.5)) - sqrt(z));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.8)
		tmp = (3.0 + (z * 0.5)) - sqrt(z);
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 1.8], N[(N[(3.0 + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.80000000000000004

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. +-commutative 97.1%

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

      3. associate-+r- 78.9%

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

      4. associate-+l- 53.6%

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

      5. +-commutative 53.6%

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

      6. associate--l+ 53.6%

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

      7. +-commutative 53.6%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in t around inf 21.6%

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

      1. associate--l+ 25.5%

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

      2. +-commutative 25.5%

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

      3. +-commutative 25.5%

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

   6. Simplified 25.5%

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

   7. Taylor expanded in x around 0 33.6%

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

   8. Taylor expanded in z around 0 33.4%

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

   9. Taylor expanded in y around 0 45.9%

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

   if 1.80000000000000004 < z

   1. Initial program 86.3%

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

      1. associate-+l+ 86.3%

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

      2. +-commutative 86.3%

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

      3. associate-+r- 69.5%

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

      4. associate-+l- 55.4%

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

      5. +-commutative 55.4%

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

      6. associate--l+ 55.5%

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

      7. +-commutative 55.5%

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

   3. Simplified 24.3%

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

   4. Taylor expanded in t around inf 6.7%

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

      1. associate--l+ 20.6%

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

      2. +-commutative 20.6%

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

      3. +-commutative 20.6%

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

   6. Simplified 20.6%

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

   7. Taylor expanded in z around inf 30.6%

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

      1. +-commutative 30.6%

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

   9. Simplified 30.6%

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

   10. Taylor expanded in x around 0 31.3%

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

       1. associate--l+ 57.0%

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

   12. Simplified 57.0%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.8:\\ \;\;\;\;\left(3 + z \cdot 0.5\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 13: 84.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 7e+14)
   (+ (- (sqrt (+ 1.0 z)) (sqrt z)) 2.0)
   (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 7e+14) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 7d+14) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + 2.0d0
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 7e+14) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + 2.0;
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 7e+14:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + 2.0
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 7e+14)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + 2.0);
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 7e+14)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 7e+14], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7e14

   1. Initial program 95.9%

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

      1. associate-+l+ 95.9%

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

      2. +-commutative 95.9%

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

      3. associate-+r- 78.4%

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

      4. associate-+l- 54.4%

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

      5. +-commutative 54.4%

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

      6. associate--l+ 54.4%

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

      7. +-commutative 54.4%

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

   3. Simplified 51.6%

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

   4. Taylor expanded in t around inf 23.0%

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

      1. associate--l+ 26.8%

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

      2. +-commutative 26.8%

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

      3. +-commutative 26.8%

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

   6. Simplified 26.8%

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

   7. Taylor expanded in x around 0 34.9%

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

   8. Taylor expanded in y around 0 46.9%

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

      1. associate--l+ 46.9%

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

   10. Simplified 46.9%

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

   if 7e14 < z

   1. Initial program 86.9%

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

      1. associate-+l+ 86.9%

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

      2. +-commutative 86.9%

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

      3. associate-+r- 69.4%

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

      4. associate-+l- 54.7%

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

      5. +-commutative 54.7%

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

      6. associate--l+ 54.7%

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

      7. +-commutative 54.7%

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

   3. Simplified 23.6%

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

   4. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 18.8%

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

      2. +-commutative 18.8%

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

      3. +-commutative 18.8%

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

   6. Simplified 18.8%

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

   7. Taylor expanded in z around inf 30.9%

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

      1. +-commutative 30.9%

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

   9. Simplified 30.9%

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

   10. Taylor expanded in x around 0 31.6%

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

       1. associate--l+ 59.0%

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

   12. Simplified 59.0%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 14: 84.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 7e+14)
   (- (+ (sqrt (+ 1.0 z)) 2.0) (sqrt z))
   (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 7e+14) {
		tmp = (sqrt((1.0 + z)) + 2.0) - sqrt(z);
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 7d+14) then
        tmp = (sqrt((1.0d0 + z)) + 2.0d0) - sqrt(z)
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 7e+14) {
		tmp = (Math.sqrt((1.0 + z)) + 2.0) - Math.sqrt(z);
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 7e+14:
		tmp = (math.sqrt((1.0 + z)) + 2.0) - math.sqrt(z)
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 7e+14)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) + 2.0) - sqrt(z));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 7e+14)
		tmp = (sqrt((1.0 + z)) + 2.0) - sqrt(z);
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 7e+14], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7e14

   1. Initial program 95.9%

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

      1. associate-+l+ 95.9%

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

      2. +-commutative 95.9%

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

      3. associate-+r- 78.4%

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

      4. associate-+l- 54.4%

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

      5. +-commutative 54.4%

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

      6. associate--l+ 54.4%

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

      7. +-commutative 54.4%

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

   3. Simplified 51.6%

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

   4. Taylor expanded in t around inf 23.0%

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

      1. associate--l+ 26.8%

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

      2. +-commutative 26.8%

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

      3. +-commutative 26.8%

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

   6. Simplified 26.8%

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

   7. Taylor expanded in x around 0 34.9%

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

   8. Taylor expanded in y around 0 46.9%

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

   if 7e14 < z

   1. Initial program 86.9%

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

      1. associate-+l+ 86.9%

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

      2. +-commutative 86.9%

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

      3. associate-+r- 69.4%

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

      4. associate-+l- 54.7%

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

      5. +-commutative 54.7%

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

      6. associate--l+ 54.7%

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

      7. +-commutative 54.7%

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

   3. Simplified 23.6%

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

   4. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 18.8%

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

      2. +-commutative 18.8%

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

      3. +-commutative 18.8%

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

   6. Simplified 18.8%

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

   7. Taylor expanded in z around inf 30.9%

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

      1. +-commutative 30.9%

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

   9. Simplified 30.9%

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

   10. Taylor expanded in x around 0 31.6%

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

       1. associate--l+ 59.0%

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

   12. Simplified 59.0%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 15: 54.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 7.4:\\ \;\;\;\;\left(3 + z \cdot 0.5\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 7.4) (- (+ 3.0 (* z 0.5)) (sqrt z)) (- (sqrt (+ x 1.0)) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 7.4) {
		tmp = (3.0 + (z * 0.5)) - sqrt(z);
	} else {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 7.4d0) then
        tmp = (3.0d0 + (z * 0.5d0)) - sqrt(z)
    else
        tmp = sqrt((x + 1.0d0)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 7.4) {
		tmp = (3.0 + (z * 0.5)) - Math.sqrt(z);
	} else {
		tmp = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 7.4:
		tmp = (3.0 + (z * 0.5)) - math.sqrt(z)
	else:
		tmp = math.sqrt((x + 1.0)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 7.4)
		tmp = Float64(Float64(3.0 + Float64(z * 0.5)) - sqrt(z));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 7.4)
		tmp = (3.0 + (z * 0.5)) - sqrt(z);
	else
		tmp = sqrt((x + 1.0)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 7.4], N[(N[(3.0 + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.4000000000000004

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. +-commutative 97.1%

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

      3. associate-+r- 78.9%

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

      4. associate-+l- 53.6%

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

      5. +-commutative 53.6%

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

      6. associate--l+ 53.6%

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

      7. +-commutative 53.6%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in t around inf 21.6%

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

      1. associate--l+ 25.5%

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

      2. +-commutative 25.5%

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

      3. +-commutative 25.5%

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

   6. Simplified 25.5%

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

   7. Taylor expanded in x around 0 33.6%

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

   8. Taylor expanded in z around 0 33.4%

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

   9. Taylor expanded in y around 0 45.9%

      \[\leadsto \color{blue}{\left(3 + 0.5 \cdot z\right) - \sqrt{z}} \]

   if 7.4000000000000004 < z

   1. Initial program 86.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 86.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 86.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 69.5%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 55.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 55.4%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. associate--l+ 55.5%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

      7. +-commutative 55.5%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Simplified 24.3%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 6.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \sqrt{1 + x}\right)\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 20.6%

         \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \sqrt{1 + x}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative 20.6%

         \[\leadsto \sqrt{1 + y} + \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + z}\right)} - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) \]

      3. +-commutative 20.6%

         \[\leadsto \sqrt{1 + y} + \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{z + 1}}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) \]

   6. Simplified 20.6%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + x} + \sqrt{z + 1}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

   7. Taylor expanded in z around inf 30.6%

      \[\leadsto \sqrt{1 + y} + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 30.6%

         \[\leadsto \sqrt{1 + y} + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 30.6%

      \[\leadsto \sqrt{1 + y} + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 21.8%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.4:\\ \;\;\;\;\left(3 + z \cdot 0.5\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]

Alternative 16: 53.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 11:\\ \;\;\;\;\left(3 + z \cdot 0.5\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 11.0) (- (+ 3.0 (* z 0.5)) (sqrt z)) (- 1.0 (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 11.0) {
		tmp = (3.0 + (z * 0.5)) - sqrt(z);
	} else {
		tmp = 1.0 - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 11.0d0) then
        tmp = (3.0d0 + (z * 0.5d0)) - sqrt(z)
    else
        tmp = 1.0d0 - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 11.0) {
		tmp = (3.0 + (z * 0.5)) - Math.sqrt(z);
	} else {
		tmp = 1.0 - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 11.0:
		tmp = (3.0 + (z * 0.5)) - math.sqrt(z)
	else:
		tmp = 1.0 - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 11.0)
		tmp = Float64(Float64(3.0 + Float64(z * 0.5)) - sqrt(z));
	else
		tmp = Float64(1.0 - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 11.0)
		tmp = (3.0 + (z * 0.5)) - sqrt(z);
	else
		tmp = 1.0 - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 11.0], N[(N[(3.0 + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 11

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 78.9%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 53.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 53.6%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. associate--l+ 53.6%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

      7. +-commutative 53.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Simplified 52.9%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 21.6%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \sqrt{1 + x}\right)\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 25.5%

         \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \sqrt{1 + x}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative 25.5%

         \[\leadsto \sqrt{1 + y} + \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + z}\right)} - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) \]

      3. +-commutative 25.5%

         \[\leadsto \sqrt{1 + y} + \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{z + 1}}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) \]

   6. Simplified 25.5%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + x} + \sqrt{z + 1}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 33.6%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{z} + \sqrt{y}\right)} \]

   8. Taylor expanded in z around 0 33.4%

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \color{blue}{\left(1 + 0.5 \cdot z\right)}\right)\right) - \left(\sqrt{z} + \sqrt{y}\right) \]

   9. Taylor expanded in y around 0 45.9%

      \[\leadsto \color{blue}{\left(3 + 0.5 \cdot z\right) - \sqrt{z}} \]

   if 11 < z

   1. Initial program 86.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 86.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 86.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 69.5%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 55.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 55.4%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. associate--l+ 55.5%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

      7. +-commutative 55.5%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Simplified 24.3%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 6.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \sqrt{1 + x}\right)\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 20.6%

         \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \sqrt{1 + x}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative 20.6%

         \[\leadsto \sqrt{1 + y} + \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + z}\right)} - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) \]

      3. +-commutative 20.6%

         \[\leadsto \sqrt{1 + y} + \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{z + 1}}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) \]

   6. Simplified 20.6%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + x} + \sqrt{z + 1}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

   7. Taylor expanded in z around inf 30.6%

      \[\leadsto \sqrt{1 + y} + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 30.6%

         \[\leadsto \sqrt{1 + y} + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 30.6%

      \[\leadsto \sqrt{1 + y} + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 21.8%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   11. Taylor expanded in x around 0 20.7%

       \[\leadsto \color{blue}{1} - \sqrt{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 11:\\ \;\;\;\;\left(3 + z \cdot 0.5\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{x}\\ \end{array} \]

Alternative 17: 53.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 10.5:\\ \;\;\;\;\left(3 + z \cdot 0.5\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 10.5)
   (- (+ 3.0 (* z 0.5)) (sqrt z))
   (- (+ 1.0 (* x 0.5)) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 10.5) {
		tmp = (3.0 + (z * 0.5)) - sqrt(z);
	} else {
		tmp = (1.0 + (x * 0.5)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 10.5d0) then
        tmp = (3.0d0 + (z * 0.5d0)) - sqrt(z)
    else
        tmp = (1.0d0 + (x * 0.5d0)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 10.5) {
		tmp = (3.0 + (z * 0.5)) - Math.sqrt(z);
	} else {
		tmp = (1.0 + (x * 0.5)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 10.5:
		tmp = (3.0 + (z * 0.5)) - math.sqrt(z)
	else:
		tmp = (1.0 + (x * 0.5)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 10.5)
		tmp = Float64(Float64(3.0 + Float64(z * 0.5)) - sqrt(z));
	else
		tmp = Float64(Float64(1.0 + Float64(x * 0.5)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 10.5)
		tmp = (3.0 + (z * 0.5)) - sqrt(z);
	else
		tmp = (1.0 + (x * 0.5)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 10.5], N[(N[(3.0 + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 10.5

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 78.9%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 53.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 53.6%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. associate--l+ 53.6%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

      7. +-commutative 53.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Simplified 52.9%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 21.6%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \sqrt{1 + x}\right)\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 25.5%

         \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \sqrt{1 + x}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative 25.5%

         \[\leadsto \sqrt{1 + y} + \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + z}\right)} - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) \]

      3. +-commutative 25.5%

         \[\leadsto \sqrt{1 + y} + \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{z + 1}}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) \]

   6. Simplified 25.5%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + x} + \sqrt{z + 1}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 33.6%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{z} + \sqrt{y}\right)} \]

   8. Taylor expanded in z around 0 33.4%

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \color{blue}{\left(1 + 0.5 \cdot z\right)}\right)\right) - \left(\sqrt{z} + \sqrt{y}\right) \]

   9. Taylor expanded in y around 0 45.9%

      \[\leadsto \color{blue}{\left(3 + 0.5 \cdot z\right) - \sqrt{z}} \]

   if 10.5 < z

   1. Initial program 86.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 86.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 86.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 69.5%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 55.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 55.4%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. associate--l+ 55.5%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

      7. +-commutative 55.5%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Simplified 24.3%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 6.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \sqrt{1 + x}\right)\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 20.6%

         \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \sqrt{1 + x}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative 20.6%

         \[\leadsto \sqrt{1 + y} + \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + z}\right)} - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) \]

      3. +-commutative 20.6%

         \[\leadsto \sqrt{1 + y} + \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{z + 1}}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) \]

   6. Simplified 20.6%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + x} + \sqrt{z + 1}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

   7. Taylor expanded in z around inf 30.6%

      \[\leadsto \sqrt{1 + y} + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 30.6%

         \[\leadsto \sqrt{1 + y} + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 30.6%

      \[\leadsto \sqrt{1 + y} + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 21.8%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   11. Taylor expanded in x around 0 22.5%

       \[\leadsto \color{blue}{\left(0.5 \cdot x + 1\right)} - \sqrt{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10.5:\\ \;\;\;\;\left(3 + z \cdot 0.5\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \sqrt{x}\\ \end{array} \]

Alternative 18: 34.4% accurate, 8.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 - \sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- 1.0 (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 - sqrt(x);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 - sqrt(x)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 - Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 - math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 - sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 - sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 91.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. +-commutative 91.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. associate-+r- 74.2%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 54.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. +-commutative 54.5%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

   6. associate--l+ 54.5%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

   7. +-commutative 54.5%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

3. Simplified 38.6%

   \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

4. Taylor expanded in t around inf 14.1%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \sqrt{1 + x}\right)\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
5. Step-by-step derivation

   1. associate--l+ 23.1%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \sqrt{1 + x}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

   2. +-commutative 23.1%

      \[\leadsto \sqrt{1 + y} + \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + z}\right)} - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) \]

   3. +-commutative 23.1%

      \[\leadsto \sqrt{1 + y} + \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{z + 1}}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) \]

6. Simplified 23.1%

   \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + x} + \sqrt{z + 1}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

7. Taylor expanded in z around inf 21.2%

   \[\leadsto \sqrt{1 + y} + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative 21.2%

      \[\leadsto \sqrt{1 + y} + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

9. Simplified 21.2%

   \[\leadsto \sqrt{1 + y} + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

10. Taylor expanded in y around inf 16.1%

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

11. Taylor expanded in x around 0 14.9%

    \[\leadsto \color{blue}{1} - \sqrt{x} \]

12. Final simplification 14.9%

    \[\leadsto 1 - \sqrt{x} \]

Alternative 19: 6.1% accurate, 274.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ z \cdot 0.5 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (* z 0.5))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return z * 0.5;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = z * 0.5d0
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return z * 0.5;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return z * 0.5

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(z * 0.5)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = z * 0.5;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(z * 0.5), $MachinePrecision]

Derivation

1. Initial program 91.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 91.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. +-commutative 91.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. associate-+r- 74.2%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 54.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. +-commutative 54.5%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

   6. associate--l+ 54.5%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

   7. +-commutative 54.5%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

3. Simplified 38.6%

   \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

4. Taylor expanded in t around inf 14.1%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \sqrt{1 + x}\right)\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
5. Step-by-step derivation

   1. associate--l+ 23.1%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \sqrt{1 + x}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

   2. +-commutative 23.1%

      \[\leadsto \sqrt{1 + y} + \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + z}\right)} - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) \]

   3. +-commutative 23.1%

      \[\leadsto \sqrt{1 + y} + \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{z + 1}}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) \]

6. Simplified 23.1%

   \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + x} + \sqrt{z + 1}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

7. Taylor expanded in x around 0 20.9%

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{z} + \sqrt{y}\right)} \]

8. Taylor expanded in z around 0 19.2%

   \[\leadsto \left(1 + \left(\sqrt{1 + y} + \color{blue}{\left(1 + 0.5 \cdot z\right)}\right)\right) - \left(\sqrt{z} + \sqrt{y}\right) \]

9. Taylor expanded in z around inf 5.2%

   \[\leadsto \color{blue}{0.5 \cdot z} \]

10. Final simplification 5.2%

    \[\leadsto z \cdot 0.5 \]

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 2023257 
(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))))