Main:z from

Percentage Accurate: 91.7% → 98.2%

Time: 42.8s

Alternatives: 19

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{t + 1}\\ t_3 := \sqrt{z + 1}\\ \mathbf{if}\;t\_1 - \sqrt{y} \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(0.5 \cdot \sqrt{\frac{1}{y}} + \left(t\_2 + \left(\left(t\_3 - \sqrt{z}\right) - \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 + \left(\left(t\_3 - \left(\sqrt{x} + \sqrt{z}\right)\right) + 1\right)\right) - \sqrt{y}\right) + \frac{1}{t\_2 + \sqrt{t}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))) (t_2 (sqrt (+ t 1.0))) (t_3 (sqrt (+ z 1.0))))
   (if (<= (- t_1 (sqrt y)) 3.2e-5)
     (+
      (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
      (+ (* 0.5 (sqrt (/ 1.0 y))) (+ t_2 (- (- t_3 (sqrt z)) (sqrt t)))))
     (+
      (- (+ t_1 (+ (- t_3 (+ (sqrt x) (sqrt z))) 1.0)) (sqrt y))
      (/ 1.0 (+ t_2 (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((t + 1.0));
	double t_3 = sqrt((z + 1.0));
	double tmp;
	if ((t_1 - sqrt(y)) <= 3.2e-5) {
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((0.5 * sqrt((1.0 / y))) + (t_2 + ((t_3 - sqrt(z)) - sqrt(t))));
	} else {
		tmp = ((t_1 + ((t_3 - (sqrt(x) + sqrt(z))) + 1.0)) - sqrt(y)) + (1.0 / (t_2 + sqrt(t)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((t + 1.0d0))
    t_3 = sqrt((z + 1.0d0))
    if ((t_1 - sqrt(y)) <= 3.2d-5) then
        tmp = (1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + ((0.5d0 * sqrt((1.0d0 / y))) + (t_2 + ((t_3 - sqrt(z)) - sqrt(t))))
    else
        tmp = ((t_1 + ((t_3 - (sqrt(x) + sqrt(z))) + 1.0d0)) - sqrt(y)) + (1.0d0 / (t_2 + sqrt(t)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((t + 1.0));
	double t_3 = Math.sqrt((z + 1.0));
	double tmp;
	if ((t_1 - Math.sqrt(y)) <= 3.2e-5) {
		tmp = (1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + ((0.5 * Math.sqrt((1.0 / y))) + (t_2 + ((t_3 - Math.sqrt(z)) - Math.sqrt(t))));
	} else {
		tmp = ((t_1 + ((t_3 - (Math.sqrt(x) + Math.sqrt(z))) + 1.0)) - Math.sqrt(y)) + (1.0 / (t_2 + Math.sqrt(t)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((t + 1.0))
	t_3 = math.sqrt((z + 1.0))
	tmp = 0
	if (t_1 - math.sqrt(y)) <= 3.2e-5:
		tmp = (1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + ((0.5 * math.sqrt((1.0 / y))) + (t_2 + ((t_3 - math.sqrt(z)) - math.sqrt(t))))
	else:
		tmp = ((t_1 + ((t_3 - (math.sqrt(x) + math.sqrt(z))) + 1.0)) - math.sqrt(y)) + (1.0 / (t_2 + math.sqrt(t)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(t + 1.0))
	t_3 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (Float64(t_1 - sqrt(y)) <= 3.2e-5)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(t_2 + Float64(Float64(t_3 - sqrt(z)) - sqrt(t)))));
	else
		tmp = Float64(Float64(Float64(t_1 + Float64(Float64(t_3 - Float64(sqrt(x) + sqrt(z))) + 1.0)) - sqrt(y)) + Float64(1.0 / Float64(t_2 + sqrt(t))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt((t + 1.0));
	t_3 = sqrt((z + 1.0));
	tmp = 0.0;
	if ((t_1 - sqrt(y)) <= 3.2e-5)
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((0.5 * sqrt((1.0 / y))) + (t_2 + ((t_3 - sqrt(z)) - sqrt(t))));
	else
		tmp = ((t_1 + ((t_3 - (sqrt(x) + sqrt(z))) + 1.0)) - sqrt(y)) + (1.0 / (t_2 + sqrt(t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 3.2e-5], N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 + N[(N[(t$95$3 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$2 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)) < 3.19999999999999986e-5

   1. Initial program 83.2%

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

      1. associate-+l+ 83.2%

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

      2. associate-+l+ 83.2%

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

      3. +-commutative 83.2%

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

      4. +-commutative 83.2%

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

      5. associate-+l- 67.4%

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

      6. +-commutative 67.4%

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

      7. +-commutative 67.4%

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

   3. Simplified 67.4%

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

      1. flip-- 67.5%

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

      2. add-sqr-sqrt 55.4%

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

      3. +-commutative 55.4%

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

      4. add-sqr-sqrt 67.6%

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

      5. +-commutative 67.6%

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

   6. Applied egg-rr 67.6%

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

      1. associate--l+ 69.1%

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

      2. +-inverses 69.1%

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

      3. metadata-eval 69.1%

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

   8. Simplified 69.1%

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

   9. Taylor expanded in y around inf 74.6%

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

   if 3.19999999999999986e-5 < (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

      2. associate-+r+ 97.5%

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

      3. associate-+r- 97.4%

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

      4. associate-+l- 97.5%

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

      5. associate-+r- 97.5%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in x around 0 40.5%

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

      1. flip-- 40.5%

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

      2. add-sqr-sqrt 30.8%

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

      3. add-sqr-sqrt 40.5%

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

      4. +-commutative 40.5%

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

      5. +-commutative 40.5%

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

   7. Applied egg-rr 40.5%

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

      1. associate--r+ 40.8%

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

      2. +-inverses 40.8%

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

      3. metadata-eval 40.8%

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

   9. Simplified 40.8%

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

       1. associate-+r- 40.8%

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

       2. associate--l+ 58.7%

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

       3. +-commutative 58.7%

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

       4. +-commutative 58.7%

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

       5. +-commutative 58.7%

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

       6. add-sqr-sqrt 58.7%

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

       7. hypot-1-def 58.7%

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

   11. Applied egg-rr 58.7%

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

       1. associate--r+ 58.7%

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

       2. sub-neg 58.7%

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

       3. associate-+r- 40.8%

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

       4. +-commutative 40.8%

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

       5. associate--l+ 58.7%

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

       6. metadata-eval 58.7%

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

       7. associate-*r/ 58.7%

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

       8. mul-1-neg 58.7%

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

       9. remove-double-neg 58.7%

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

       10. +-commutative 58.7%

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

       11. hypot-undefine 58.7%

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

   13. Simplified 58.7%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{y + 1} - \sqrt{y} \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(0.5 \cdot \sqrt{\frac{1}{y}} + \left(\sqrt{t + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) - \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + \left(\left(\sqrt{z + 1} - \left(\sqrt{x} + \sqrt{z}\right)\right) + 1\right)\right) - \sqrt{y}\right) + \frac{1}{\sqrt{t + 1} + \sqrt{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{y + 1}\\ \mathbf{if}\;y \leq 4.5 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(t\_2 + \left(\left(t\_1 - \left(\sqrt{x} + \sqrt{z}\right)\right) + 1\right)\right) - \sqrt{y}\right) + \frac{1}{\sqrt{t + 1} + \sqrt{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(t\_1 + t\_2\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0))) (t_2 (sqrt (+ y 1.0))))
   (if (<= y 4.5e+19)
     (+
      (- (+ t_2 (+ (- t_1 (+ (sqrt x) (sqrt z))) 1.0)) (sqrt y))
      (/ 1.0 (+ (sqrt (+ t 1.0)) (sqrt t))))
     (+
      (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
      (- (+ t_1 t_2) (+ (sqrt z) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt((y + 1.0));
	double tmp;
	if (y <= 4.5e+19) {
		tmp = ((t_2 + ((t_1 - (sqrt(x) + sqrt(z))) + 1.0)) - sqrt(y)) + (1.0 / (sqrt((t + 1.0)) + sqrt(t)));
	} else {
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((t_1 + t_2) - (sqrt(z) + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = sqrt((y + 1.0d0))
    if (y <= 4.5d+19) then
        tmp = ((t_2 + ((t_1 - (sqrt(x) + sqrt(z))) + 1.0d0)) - sqrt(y)) + (1.0d0 / (sqrt((t + 1.0d0)) + sqrt(t)))
    else
        tmp = (1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + ((t_1 + t_2) - (sqrt(z) + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double t_2 = Math.sqrt((y + 1.0));
	double tmp;
	if (y <= 4.5e+19) {
		tmp = ((t_2 + ((t_1 - (Math.sqrt(x) + Math.sqrt(z))) + 1.0)) - Math.sqrt(y)) + (1.0 / (Math.sqrt((t + 1.0)) + Math.sqrt(t)));
	} else {
		tmp = (1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + ((t_1 + t_2) - (Math.sqrt(z) + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = math.sqrt((y + 1.0))
	tmp = 0
	if y <= 4.5e+19:
		tmp = ((t_2 + ((t_1 - (math.sqrt(x) + math.sqrt(z))) + 1.0)) - math.sqrt(y)) + (1.0 / (math.sqrt((t + 1.0)) + math.sqrt(t)))
	else:
		tmp = (1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + ((t_1 + t_2) - (math.sqrt(z) + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (y <= 4.5e+19)
		tmp = Float64(Float64(Float64(t_2 + Float64(Float64(t_1 - Float64(sqrt(x) + sqrt(z))) + 1.0)) - sqrt(y)) + Float64(1.0 / Float64(sqrt(Float64(t + 1.0)) + sqrt(t))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(Float64(t_1 + t_2) - Float64(sqrt(z) + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	t_2 = sqrt((y + 1.0));
	tmp = 0.0;
	if (y <= 4.5e+19)
		tmp = ((t_2 + ((t_1 - (sqrt(x) + sqrt(z))) + 1.0)) - sqrt(y)) + (1.0 / (sqrt((t + 1.0)) + sqrt(t)));
	else
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((t_1 + t_2) - (sqrt(z) + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.5e+19], N[(N[(N[(t$95$2 + N[(N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 + t$95$2), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5e19

   1. Initial program 96.1%

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

      1. +-commutative 96.1%

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

      2. associate-+r+ 96.1%

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

      3. associate-+r- 96.0%

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

      4. associate-+l- 96.1%

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

      5. associate-+r- 96.1%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in x around 0 39.1%

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

      1. flip-- 39.1%

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

      2. add-sqr-sqrt 29.9%

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

      3. add-sqr-sqrt 39.1%

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

      4. +-commutative 39.1%

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

      5. +-commutative 39.1%

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

   7. Applied egg-rr 39.1%

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

      1. associate--r+ 39.4%

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

      2. +-inverses 39.4%

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

      3. metadata-eval 39.4%

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

   9. Simplified 39.4%

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

       1. associate-+r- 39.4%

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

       2. associate--l+ 57.8%

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

       3. +-commutative 57.8%

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

       4. +-commutative 57.8%

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

       5. +-commutative 57.8%

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

       6. add-sqr-sqrt 57.8%

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

       7. hypot-1-def 57.8%

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

   11. Applied egg-rr 57.8%

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

       1. associate--r+ 57.8%

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

       2. sub-neg 57.8%

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

       3. associate-+r- 39.4%

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

       4. +-commutative 39.4%

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

       5. associate--l+ 57.8%

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

       6. metadata-eval 57.8%

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

       7. associate-*r/ 57.8%

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

       8. mul-1-neg 57.8%

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

       9. remove-double-neg 57.8%

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

       10. +-commutative 57.8%

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

       11. hypot-undefine 57.8%

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

   13. Simplified 57.8%

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

   if 4.5e19 < y

   1. Initial program 83.9%

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

      1. associate-+l+ 83.9%

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

      2. associate-+l+ 83.9%

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

      3. +-commutative 83.9%

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

      4. +-commutative 83.9%

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

      5. associate-+l- 68.5%

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

      6. +-commutative 68.5%

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

      7. +-commutative 68.5%

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

   3. Simplified 68.5%

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

      1. flip-- 68.7%

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

      2. add-sqr-sqrt 56.4%

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

      3. +-commutative 56.4%

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

      4. add-sqr-sqrt 68.8%

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

      5. +-commutative 68.8%

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

   6. Applied egg-rr 68.8%

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

      1. associate--l+ 70.3%

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

      2. +-inverses 70.3%

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

      3. metadata-eval 70.3%

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

   8. Simplified 70.3%

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

   9. Taylor expanded in t around inf 24.5%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + \left(\left(\sqrt{z + 1} - \left(\sqrt{x} + \sqrt{z}\right)\right) + 1\right)\right) - \sqrt{y}\right) + \frac{1}{\sqrt{t + 1} + \sqrt{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z} + \sqrt{y}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 39000:\\ \;\;\;\;\left(\frac{1}{\sqrt{t + 1} + \sqrt{t}} + \left(t\_2 + 2\right)\right) - \left(\sqrt{x} + t\_1\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+219}:\\ \;\;\;\;\left(\left(t\_3 + \left(-0.125 \cdot \sqrt{\frac{1}{{z}^{3}}} + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(t\_2 + t\_3\right) - t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt z) (sqrt y)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (sqrt (+ y 1.0))))
   (if (<= z 39000.0)
     (- (+ (/ 1.0 (+ (sqrt (+ t 1.0)) (sqrt t))) (+ t_2 2.0)) (+ (sqrt x) t_1))
     (if (<= z 7.5e+219)
       (+
        (-
         (+
          t_3
          (+ (* -0.125 (sqrt (/ 1.0 (pow z 3.0)))) (* 0.5 (sqrt (/ 1.0 z)))))
         (+ (sqrt x) (sqrt y)))
        1.0)
       (+ (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))) (- (+ t_2 t_3) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(z) + sqrt(y);
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((y + 1.0));
	double tmp;
	if (z <= 39000.0) {
		tmp = ((1.0 / (sqrt((t + 1.0)) + sqrt(t))) + (t_2 + 2.0)) - (sqrt(x) + t_1);
	} else if (z <= 7.5e+219) {
		tmp = ((t_3 + ((-0.125 * sqrt((1.0 / pow(z, 3.0)))) + (0.5 * sqrt((1.0 / z))))) - (sqrt(x) + sqrt(y))) + 1.0;
	} else {
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((t_2 + t_3) - t_1);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt(z) + sqrt(y)
    t_2 = sqrt((z + 1.0d0))
    t_3 = sqrt((y + 1.0d0))
    if (z <= 39000.0d0) then
        tmp = ((1.0d0 / (sqrt((t + 1.0d0)) + sqrt(t))) + (t_2 + 2.0d0)) - (sqrt(x) + t_1)
    else if (z <= 7.5d+219) then
        tmp = ((t_3 + (((-0.125d0) * sqrt((1.0d0 / (z ** 3.0d0)))) + (0.5d0 * sqrt((1.0d0 / z))))) - (sqrt(x) + sqrt(y))) + 1.0d0
    else
        tmp = (1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + ((t_2 + t_3) - t_1)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(z) + Math.sqrt(y);
	double t_2 = Math.sqrt((z + 1.0));
	double t_3 = Math.sqrt((y + 1.0));
	double tmp;
	if (z <= 39000.0) {
		tmp = ((1.0 / (Math.sqrt((t + 1.0)) + Math.sqrt(t))) + (t_2 + 2.0)) - (Math.sqrt(x) + t_1);
	} else if (z <= 7.5e+219) {
		tmp = ((t_3 + ((-0.125 * Math.sqrt((1.0 / Math.pow(z, 3.0)))) + (0.5 * Math.sqrt((1.0 / z))))) - (Math.sqrt(x) + Math.sqrt(y))) + 1.0;
	} else {
		tmp = (1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + ((t_2 + t_3) - t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(z) + math.sqrt(y)
	t_2 = math.sqrt((z + 1.0))
	t_3 = math.sqrt((y + 1.0))
	tmp = 0
	if z <= 39000.0:
		tmp = ((1.0 / (math.sqrt((t + 1.0)) + math.sqrt(t))) + (t_2 + 2.0)) - (math.sqrt(x) + t_1)
	elif z <= 7.5e+219:
		tmp = ((t_3 + ((-0.125 * math.sqrt((1.0 / math.pow(z, 3.0)))) + (0.5 * math.sqrt((1.0 / z))))) - (math.sqrt(x) + math.sqrt(y))) + 1.0
	else:
		tmp = (1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + ((t_2 + t_3) - t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(z) + sqrt(y))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 39000.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(t + 1.0)) + sqrt(t))) + Float64(t_2 + 2.0)) - Float64(sqrt(x) + t_1));
	elseif (z <= 7.5e+219)
		tmp = Float64(Float64(Float64(t_3 + Float64(Float64(-0.125 * sqrt(Float64(1.0 / (z ^ 3.0)))) + Float64(0.5 * sqrt(Float64(1.0 / z))))) - Float64(sqrt(x) + sqrt(y))) + 1.0);
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(Float64(t_2 + t_3) - 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(z) + sqrt(y);
	t_2 = sqrt((z + 1.0));
	t_3 = sqrt((y + 1.0));
	tmp = 0.0;
	if (z <= 39000.0)
		tmp = ((1.0 / (sqrt((t + 1.0)) + sqrt(t))) + (t_2 + 2.0)) - (sqrt(x) + t_1);
	elseif (z <= 7.5e+219)
		tmp = ((t_3 + ((-0.125 * sqrt((1.0 / (z ^ 3.0)))) + (0.5 * sqrt((1.0 / z))))) - (sqrt(x) + sqrt(y))) + 1.0;
	else
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((t_2 + t_3) - t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 39000.0], N[(N[(N[(1.0 / N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.5e+219], N[(N[(N[(t$95$3 + N[(N[(-0.125 * N[Sqrt[N[(1.0 / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + t$95$3), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 39000

   1. Initial program 96.7%

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

      1. +-commutative 96.7%

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

      2. associate-+r+ 96.7%

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

      3. associate-+r- 76.9%

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

      4. associate-+l- 70.2%

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

      5. associate-+r- 56.8%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in x around 0 32.2%

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

      1. flip-- 32.2%

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

      2. add-sqr-sqrt 22.7%

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

      3. add-sqr-sqrt 32.2%

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

      4. +-commutative 32.2%

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

      5. +-commutative 32.2%

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

   7. Applied egg-rr 32.2%

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

      1. associate--r+ 32.5%

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

      2. +-inverses 32.5%

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

      3. metadata-eval 32.5%

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

   9. Simplified 32.5%

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

   10. Taylor expanded in y around 0 26.8%

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

       1. associate-+r+ 26.9%

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

       2. +-commutative 26.9%

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

   12. Simplified 26.9%

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

   if 39000 < z < 7.5000000000000006e219

   1. Initial program 84.0%

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

      1. +-commutative 84.0%

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

      2. associate-+r+ 84.0%

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

      3. associate-+r- 62.4%

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

      4. associate-+l- 50.0%

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

      5. associate-+r- 49.8%

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

   3. Simplified 28.3%

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

   5. Taylor expanded in t around inf 4.8%

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

      1. associate--l+ 21.7%

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

      2. associate--l+ 28.8%

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

      3. associate-+r+ 28.8%

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

   7. Simplified 28.8%

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

   8. Taylor expanded in x around 0 4.3%

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

      1. associate--l+ 24.5%

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

      2. associate--l+ 33.4%

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

      3. associate-+r+ 33.4%

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

      4. +-commutative 33.4%

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

      5. associate--l- 33.4%

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

      6. associate--l- 33.4%

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

      7. +-commutative 33.4%

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

      8. associate-+r+ 33.4%

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

      9. associate--r+ 33.4%

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

   10. Simplified 33.4%

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

   11. Taylor expanded in z around inf 35.5%

       \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \left(-0.125 \cdot \sqrt{\frac{1}{{z}^{3}}} + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 7.5000000000000006e219 < z

   1. Initial program 83.8%

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

      1. associate-+l+ 83.8%

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

      2. associate-+l+ 83.8%

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

      3. +-commutative 83.8%

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

      4. +-commutative 83.8%

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

      5. associate-+l- 83.8%

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

      6. +-commutative 83.8%

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

      7. +-commutative 83.8%

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

   3. Simplified 83.8%

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

      1. flip-- 83.8%

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

      2. add-sqr-sqrt 68.0%

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

      3. +-commutative 68.0%

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

      4. add-sqr-sqrt 83.8%

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

      5. +-commutative 83.8%

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

   6. Applied egg-rr 83.8%

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

      1. associate--l+ 85.6%

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

      2. +-inverses 85.6%

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

      3. metadata-eval 85.6%

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

   8. Simplified 85.6%

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

   9. Taylor expanded in t around inf 27.3%

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

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 39000:\\ \;\;\;\;\left(\frac{1}{\sqrt{t + 1} + \sqrt{t}} + \left(\sqrt{z + 1} + 2\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+219}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + \left(-0.125 \cdot \sqrt{\frac{1}{{z}^{3}}} + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z} + \sqrt{y}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{z + 1}\\ \mathbf{if}\;y \leq 4.8 \cdot 10^{-19}:\\ \;\;\;\;2 + \left(\left(t\_3 - \sqrt{x}\right) - t\_1\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(t\_2 + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + 1\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(t\_3 + t\_2\right) - t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt z) (sqrt y)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (+ z 1.0))))
   (if (<= y 4.8e-19)
     (+ 2.0 (- (- t_3 (sqrt x)) t_1))
     (if (<= y 1.15e+15)
       (- (+ (+ t_2 (* 0.5 (sqrt (/ 1.0 z)))) 1.0) (+ (sqrt x) (sqrt y)))
       (+ (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))) (- (+ t_3 t_2) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(z) + sqrt(y);
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((z + 1.0));
	double tmp;
	if (y <= 4.8e-19) {
		tmp = 2.0 + ((t_3 - sqrt(x)) - t_1);
	} else if (y <= 1.15e+15) {
		tmp = ((t_2 + (0.5 * sqrt((1.0 / z)))) + 1.0) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((t_3 + t_2) - t_1);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt(z) + sqrt(y)
    t_2 = sqrt((y + 1.0d0))
    t_3 = sqrt((z + 1.0d0))
    if (y <= 4.8d-19) then
        tmp = 2.0d0 + ((t_3 - sqrt(x)) - t_1)
    else if (y <= 1.15d+15) then
        tmp = ((t_2 + (0.5d0 * sqrt((1.0d0 / z)))) + 1.0d0) - (sqrt(x) + sqrt(y))
    else
        tmp = (1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + ((t_3 + t_2) - t_1)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(z) + Math.sqrt(y);
	double t_2 = Math.sqrt((y + 1.0));
	double t_3 = Math.sqrt((z + 1.0));
	double tmp;
	if (y <= 4.8e-19) {
		tmp = 2.0 + ((t_3 - Math.sqrt(x)) - t_1);
	} else if (y <= 1.15e+15) {
		tmp = ((t_2 + (0.5 * Math.sqrt((1.0 / z)))) + 1.0) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = (1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + ((t_3 + t_2) - t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(z) + math.sqrt(y)
	t_2 = math.sqrt((y + 1.0))
	t_3 = math.sqrt((z + 1.0))
	tmp = 0
	if y <= 4.8e-19:
		tmp = 2.0 + ((t_3 - math.sqrt(x)) - t_1)
	elif y <= 1.15e+15:
		tmp = ((t_2 + (0.5 * math.sqrt((1.0 / z)))) + 1.0) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = (1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + ((t_3 + t_2) - t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(z) + sqrt(y))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (y <= 4.8e-19)
		tmp = Float64(2.0 + Float64(Float64(t_3 - sqrt(x)) - t_1));
	elseif (y <= 1.15e+15)
		tmp = Float64(Float64(Float64(t_2 + Float64(0.5 * sqrt(Float64(1.0 / z)))) + 1.0) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(Float64(t_3 + t_2) - t_1));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(z) + sqrt(y);
	t_2 = sqrt((y + 1.0));
	t_3 = sqrt((z + 1.0));
	tmp = 0.0;
	if (y <= 4.8e-19)
		tmp = 2.0 + ((t_3 - sqrt(x)) - t_1);
	elseif (y <= 1.15e+15)
		tmp = ((t_2 + (0.5 * sqrt((1.0 / z)))) + 1.0) - (sqrt(x) + sqrt(y));
	else
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((t_3 + t_2) - t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.8e-19], N[(2.0 + N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+15], N[(N[(N[(t$95$2 + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 + t$95$2), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.80000000000000046e-19

   1. Initial program 97.6%

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

      1. +-commutative 97.6%

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

      2. associate-+r+ 97.6%

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

      3. associate-+r- 97.6%

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

      4. associate-+l- 97.6%

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

      5. associate-+r- 97.6%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in t around inf 24.1%

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

      1. associate--l+ 28.3%

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

      2. associate--l+ 37.5%

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

      3. associate-+r+ 37.5%

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

   7. Simplified 37.5%

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

   8. Taylor expanded in x around 0 22.3%

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

      1. associate--l+ 27.9%

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

      2. associate--l+ 39.1%

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

      3. associate-+r+ 39.1%

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

      4. +-commutative 39.1%

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

      5. associate--l- 39.1%

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

      6. associate--l- 39.1%

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

      7. +-commutative 39.1%

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

      8. associate-+r+ 39.1%

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

      9. associate--r+ 39.1%

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

   10. Simplified 39.1%

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

   11. Taylor expanded in y around 0 22.3%

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

       1. associate--l+ 39.1%

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

       2. associate--r+ 39.1%

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

       3. +-commutative 39.1%

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

   13. Simplified 39.1%

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

   if 4.80000000000000046e-19 < y < 1.15e15

   1. Initial program 88.6%

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

      1. +-commutative 88.6%

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

      2. associate-+r+ 88.6%

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

      3. associate-+r- 88.4%

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

      4. associate-+l- 88.5%

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

      5. associate-+r- 88.4%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in t around inf 26.0%

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

      1. associate--l+ 33.9%

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

      2. associate--l+ 37.2%

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

      3. associate-+r+ 37.2%

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

   7. Simplified 37.2%

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

   8. Taylor expanded in x around 0 21.0%

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

      1. associate--l+ 28.9%

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

      2. associate--l+ 32.2%

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

      3. associate-+r+ 32.2%

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

      4. +-commutative 32.2%

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

      5. associate--l- 32.2%

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

      6. associate--l- 32.2%

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

      7. +-commutative 32.2%

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

      8. associate-+r+ 32.2%

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

      9. associate--r+ 32.2%

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

   10. Simplified 32.2%

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

   11. Taylor expanded in z around inf 25.0%

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

   if 1.15e15 < y

   1. Initial program 83.6%

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

      1. associate-+l+ 83.6%

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

      2. associate-+l+ 83.6%

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

      3. +-commutative 83.6%

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

      4. +-commutative 83.6%

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

      5. associate-+l- 68.5%

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

      6. +-commutative 68.5%

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

      7. +-commutative 68.5%

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

   3. Simplified 68.5%

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

      1. flip-- 68.7%

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

      2. add-sqr-sqrt 56.6%

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

      3. +-commutative 56.6%

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

      4. add-sqr-sqrt 68.8%

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

      5. +-commutative 68.8%

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

   6. Applied egg-rr 68.8%

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

      1. associate--l+ 70.3%

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

      2. +-inverses 70.3%

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

      3. metadata-eval 70.3%

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

   8. Simplified 70.3%

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

   9. Taylor expanded in t around inf 25.2%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-19}:\\ \;\;\;\;2 + \left(\left(\sqrt{z + 1} - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + 1\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 4.3 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(\left(\sqrt{z + 1} + t\_1\right) \cdot \frac{1}{x} + \left(\left(\sqrt{z} + \sqrt{y}\right) \cdot \frac{-1}{x} - \sqrt{\frac{1}{x}}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(t\_1 - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))))
   (if (<= z 4.3e+14)
     (+
      (*
       x
       (+
        (* (+ (sqrt (+ z 1.0)) t_1) (/ 1.0 x))
        (- (* (+ (sqrt z) (sqrt y)) (/ -1.0 x)) (sqrt (/ 1.0 x)))))
      1.0)
     (+
      (- (sqrt (+ x 1.0)) (sqrt x))
      (+ (- t_1 (sqrt y)) (- (sqrt (+ t 1.0)) (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double tmp;
	if (z <= 4.3e+14) {
		tmp = (x * (((sqrt((z + 1.0)) + t_1) * (1.0 / x)) + (((sqrt(z) + sqrt(y)) * (-1.0 / x)) - sqrt((1.0 / x))))) + 1.0;
	} else {
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((t_1 - sqrt(y)) + (sqrt((t + 1.0)) - sqrt(t)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    if (z <= 4.3d+14) then
        tmp = (x * (((sqrt((z + 1.0d0)) + t_1) * (1.0d0 / x)) + (((sqrt(z) + sqrt(y)) * ((-1.0d0) / x)) - sqrt((1.0d0 / x))))) + 1.0d0
    else
        tmp = (sqrt((x + 1.0d0)) - sqrt(x)) + ((t_1 - sqrt(y)) + (sqrt((t + 1.0d0)) - sqrt(t)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double tmp;
	if (z <= 4.3e+14) {
		tmp = (x * (((Math.sqrt((z + 1.0)) + t_1) * (1.0 / x)) + (((Math.sqrt(z) + Math.sqrt(y)) * (-1.0 / x)) - Math.sqrt((1.0 / x))))) + 1.0;
	} else {
		tmp = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + ((t_1 - Math.sqrt(y)) + (Math.sqrt((t + 1.0)) - Math.sqrt(t)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	tmp = 0
	if z <= 4.3e+14:
		tmp = (x * (((math.sqrt((z + 1.0)) + t_1) * (1.0 / x)) + (((math.sqrt(z) + math.sqrt(y)) * (-1.0 / x)) - math.sqrt((1.0 / x))))) + 1.0
	else:
		tmp = (math.sqrt((x + 1.0)) - math.sqrt(x)) + ((t_1 - math.sqrt(y)) + (math.sqrt((t + 1.0)) - math.sqrt(t)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 4.3e+14)
		tmp = Float64(Float64(x * Float64(Float64(Float64(sqrt(Float64(z + 1.0)) + t_1) * Float64(1.0 / x)) + Float64(Float64(Float64(sqrt(z) + sqrt(y)) * Float64(-1.0 / x)) - sqrt(Float64(1.0 / x))))) + 1.0);
	else
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(Float64(t_1 - sqrt(y)) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	tmp = 0.0;
	if (z <= 4.3e+14)
		tmp = (x * (((sqrt((z + 1.0)) + t_1) * (1.0 / x)) + (((sqrt(z) + sqrt(y)) * (-1.0 / x)) - sqrt((1.0 / x))))) + 1.0;
	else
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((t_1 - sqrt(y)) + (sqrt((t + 1.0)) - sqrt(t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 4.3e+14], N[(N[(x * N[(N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 4.3e14

   1. Initial program 96.1%

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

      1. +-commutative 96.1%

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

      2. associate-+r+ 96.1%

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

      3. associate-+r- 76.4%

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

      4. associate-+l- 69.4%

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

      5. associate-+r- 56.3%

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

   3. Simplified 56.3%

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

   5. Taylor expanded in t around inf 25.1%

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

      1. associate--l+ 29.3%

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

      2. associate--l+ 29.3%

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

      3. associate-+r+ 29.3%

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

   7. Simplified 29.3%

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

   8. Taylor expanded in x around 0 23.0%

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

      1. associate--l+ 29.5%

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

      2. associate--l+ 29.6%

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

      3. associate-+r+ 29.6%

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

      4. +-commutative 29.6%

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

      5. associate--l- 29.6%

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

      6. associate--l- 29.6%

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

      7. +-commutative 29.6%

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

      8. associate-+r+ 29.6%

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

      9. associate--r+ 29.6%

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

   10. Simplified 29.6%

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

   11. Taylor expanded in x around inf 26.7%

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

       1. distribute-lft-out 26.7%

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

       2. +-commutative 26.7%

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

       3. distribute-lft-out 26.7%

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

       4. +-commutative 26.7%

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

   13. Simplified 26.7%

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

   if 4.3e14 < z

   1. Initial program 84.2%

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

      1. associate-+l+ 84.2%

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

      2. associate-+l+ 84.2%

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

      3. +-commutative 84.2%

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

      4. +-commutative 84.2%

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

      5. associate-+l- 84.2%

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

      6. +-commutative 84.2%

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

      7. +-commutative 84.2%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in z around inf 84.2%

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

      1. +-commutative 84.2%

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

   7. Simplified 84.2%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.3 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) \cdot \frac{1}{x} + \left(\left(\sqrt{z} + \sqrt{y}\right) \cdot \frac{-1}{x} - \sqrt{\frac{1}{x}}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 2e+15)
		tmp = Float64(Float64(x * Float64(Float64(Float64(sqrt(Float64(z + 1.0)) + t_1) * Float64(1.0 / x)) + Float64(Float64(Float64(sqrt(z) + sqrt(y)) * Float64(-1.0 / x)) - sqrt(Float64(1.0 / x))))) + 1.0);
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - Float64(Float64(sqrt(x) + sqrt(y)) - 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((y + 1.0));
	tmp = 0.0;
	if (z <= 2e+15)
		tmp = (x * (((sqrt((z + 1.0)) + t_1) * (1.0 / x)) + (((sqrt(z) + sqrt(y)) * (-1.0 / x)) - sqrt((1.0 / x))))) + 1.0;
	else
		tmp = sqrt((x + 1.0)) - ((sqrt(x) + sqrt(y)) - 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[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 2e+15], N[(N[(x * N[(N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 2e15

   1. Initial program 96.1%

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

      1. +-commutative 96.1%

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

      2. associate-+r+ 96.1%

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

      3. associate-+r- 76.4%

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

      4. associate-+l- 69.4%

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

      5. associate-+r- 56.3%

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

   3. Simplified 56.3%

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

   5. Taylor expanded in t around inf 25.1%

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

      1. associate--l+ 29.3%

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

      2. associate--l+ 29.3%

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

      3. associate-+r+ 29.3%

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

   7. Simplified 29.3%

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

   8. Taylor expanded in x around 0 23.0%

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

      1. associate--l+ 29.5%

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

      2. associate--l+ 29.6%

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

      3. associate-+r+ 29.6%

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

      4. +-commutative 29.6%

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

      5. associate--l- 29.6%

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

      6. associate--l- 29.6%

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

      7. +-commutative 29.6%

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

      8. associate-+r+ 29.6%

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

      9. associate--r+ 29.6%

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

   10. Simplified 29.6%

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

   11. Taylor expanded in x around inf 26.7%

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

       1. distribute-lft-out 26.7%

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

       2. +-commutative 26.7%

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

       3. distribute-lft-out 26.7%

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

       4. +-commutative 26.7%

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

   13. Simplified 26.7%

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

   if 2e15 < z

   1. Initial program 84.2%

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

      1. +-commutative 84.2%

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

      2. associate-+r+ 84.2%

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

      3. associate-+r- 62.3%

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

      4. associate-+l- 47.7%

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

      5. associate-+r- 47.7%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in t around inf 4.0%

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

      1. associate--l+ 22.1%

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

      2. associate--l+ 24.7%

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

      3. associate-+r+ 24.7%

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

   7. Simplified 24.7%

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

   8. Taylor expanded in z around inf 31.8%

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

      1. +-commutative 31.8%

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

   10. Simplified 31.8%

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

4. Final simplification 29.2%

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

Alternative 7: 85.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 1.4 \cdot 10^{+19}:\\ \;\;\;\;\left(t\_1 + \left(\left(\sqrt{z + 1} - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \left(\left(\sqrt{x} + \sqrt{y}\right) - t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))))
   (if (<= z 1.4e+19)
     (+ (+ t_1 (- (- (sqrt (+ z 1.0)) (sqrt x)) (+ (sqrt z) (sqrt y)))) 1.0)
     (- (sqrt (+ x 1.0)) (- (+ (sqrt x) (sqrt y)) t_1)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double tmp;
	if (z <= 1.4e+19) {
		tmp = (t_1 + ((sqrt((z + 1.0)) - sqrt(x)) - (sqrt(z) + sqrt(y)))) + 1.0;
	} else {
		tmp = sqrt((x + 1.0)) - ((sqrt(x) + sqrt(y)) - t_1);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    if (z <= 1.4d+19) then
        tmp = (t_1 + ((sqrt((z + 1.0d0)) - sqrt(x)) - (sqrt(z) + sqrt(y)))) + 1.0d0
    else
        tmp = sqrt((x + 1.0d0)) - ((sqrt(x) + sqrt(y)) - t_1)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double tmp;
	if (z <= 1.4e+19) {
		tmp = (t_1 + ((Math.sqrt((z + 1.0)) - Math.sqrt(x)) - (Math.sqrt(z) + Math.sqrt(y)))) + 1.0;
	} else {
		tmp = Math.sqrt((x + 1.0)) - ((Math.sqrt(x) + Math.sqrt(y)) - t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	tmp = 0
	if z <= 1.4e+19:
		tmp = (t_1 + ((math.sqrt((z + 1.0)) - math.sqrt(x)) - (math.sqrt(z) + math.sqrt(y)))) + 1.0
	else:
		tmp = math.sqrt((x + 1.0)) - ((math.sqrt(x) + math.sqrt(y)) - t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 1.4e+19)
		tmp = Float64(Float64(t_1 + Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(x)) - Float64(sqrt(z) + sqrt(y)))) + 1.0);
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - Float64(Float64(sqrt(x) + sqrt(y)) - 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((y + 1.0));
	tmp = 0.0;
	if (z <= 1.4e+19)
		tmp = (t_1 + ((sqrt((z + 1.0)) - sqrt(x)) - (sqrt(z) + sqrt(y)))) + 1.0;
	else
		tmp = sqrt((x + 1.0)) - ((sqrt(x) + sqrt(y)) - 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[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1.4e+19], N[(N[(t$95$1 + N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.4e19

   1. Initial program 96.1%

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

      1. +-commutative 96.1%

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

      2. associate-+r+ 96.1%

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

      3. associate-+r- 76.4%

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

      4. associate-+l- 69.4%

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

      5. associate-+r- 56.3%

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

   3. Simplified 56.3%

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

   5. Taylor expanded in t around inf 25.1%

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

      1. associate--l+ 29.3%

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

      2. associate--l+ 29.3%

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

      3. associate-+r+ 29.3%

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

   7. Simplified 29.3%

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

   8. Taylor expanded in x around 0 23.0%

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

      1. associate--l+ 29.5%

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

      2. associate--l+ 29.6%

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

      3. associate-+r+ 29.6%

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

      4. +-commutative 29.6%

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

      5. associate--l- 29.6%

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

      6. associate--l- 29.6%

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

      7. +-commutative 29.6%

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

      8. associate-+r+ 29.6%

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

      9. associate--r+ 29.6%

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

   10. Simplified 29.6%

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

   if 1.4e19 < z

   1. Initial program 84.2%

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

      1. +-commutative 84.2%

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

      2. associate-+r+ 84.2%

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

      3. associate-+r- 62.3%

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

      4. associate-+l- 47.7%

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

      5. associate-+r- 47.7%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in t around inf 4.0%

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

      1. associate--l+ 22.1%

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

      2. associate--l+ 24.7%

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

      3. associate-+r+ 24.7%

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

   7. Simplified 24.7%

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

   8. Taylor expanded in z around inf 31.8%

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

      1. +-commutative 31.8%

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

   10. Simplified 31.8%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{+19}:\\ \;\;\;\;\left(\sqrt{y + 1} + \left(\left(\sqrt{z + 1} - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \left(\left(\sqrt{x} + \sqrt{y}\right) - \sqrt{y + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 5.6 \cdot 10^{+14}:\\ \;\;\;\;\left(\left(\sqrt{z + 1} + t\_1\right) + 1\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \left(\left(\sqrt{x} + \sqrt{y}\right) - t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))))
   (if (<= z 5.6e+14)
     (- (+ (+ (sqrt (+ z 1.0)) t_1) 1.0) (+ (sqrt x) (+ (sqrt z) (sqrt y))))
     (- (sqrt (+ x 1.0)) (- (+ (sqrt x) (sqrt y)) t_1)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double tmp;
	if (z <= 5.6e+14) {
		tmp = ((sqrt((z + 1.0)) + t_1) + 1.0) - (sqrt(x) + (sqrt(z) + sqrt(y)));
	} else {
		tmp = sqrt((x + 1.0)) - ((sqrt(x) + sqrt(y)) - t_1);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    if (z <= 5.6d+14) then
        tmp = ((sqrt((z + 1.0d0)) + t_1) + 1.0d0) - (sqrt(x) + (sqrt(z) + sqrt(y)))
    else
        tmp = sqrt((x + 1.0d0)) - ((sqrt(x) + sqrt(y)) - t_1)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double tmp;
	if (z <= 5.6e+14) {
		tmp = ((Math.sqrt((z + 1.0)) + t_1) + 1.0) - (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y)));
	} else {
		tmp = Math.sqrt((x + 1.0)) - ((Math.sqrt(x) + Math.sqrt(y)) - t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	tmp = 0
	if z <= 5.6e+14:
		tmp = ((math.sqrt((z + 1.0)) + t_1) + 1.0) - (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y)))
	else:
		tmp = math.sqrt((x + 1.0)) - ((math.sqrt(x) + math.sqrt(y)) - t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 5.6e+14)
		tmp = Float64(Float64(Float64(sqrt(Float64(z + 1.0)) + t_1) + 1.0) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - Float64(Float64(sqrt(x) + sqrt(y)) - 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((y + 1.0));
	tmp = 0.0;
	if (z <= 5.6e+14)
		tmp = ((sqrt((z + 1.0)) + t_1) + 1.0) - (sqrt(x) + (sqrt(z) + sqrt(y)));
	else
		tmp = sqrt((x + 1.0)) - ((sqrt(x) + sqrt(y)) - 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[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 5.6e+14], N[(N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 5.6e14

   1. Initial program 96.1%

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

      1. +-commutative 96.1%

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

      2. associate-+r+ 96.1%

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

      3. associate-+r- 76.4%

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

      4. associate-+l- 69.4%

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

      5. associate-+r- 56.3%

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

   3. Simplified 56.3%

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

   5. Taylor expanded in t around inf 25.1%

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

      1. associate--l+ 29.3%

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

      2. associate--l+ 29.3%

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

      3. associate-+r+ 29.3%

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

   7. Simplified 29.3%

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

   8. Taylor expanded in x around 0 23.0%

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

   if 5.6e14 < z

   1. Initial program 84.2%

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

      1. +-commutative 84.2%

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

      2. associate-+r+ 84.2%

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

      3. associate-+r- 62.3%

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

      4. associate-+l- 47.7%

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

      5. associate-+r- 47.7%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in t around inf 4.0%

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

      1. associate--l+ 22.1%

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

      2. associate--l+ 24.7%

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

      3. associate-+r+ 24.7%

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

   7. Simplified 24.7%

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

   8. Taylor expanded in z around inf 31.8%

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

      1. +-commutative 31.8%

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

   10. Simplified 31.8%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.6 \cdot 10^{+14}:\\ \;\;\;\;\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + 1\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \left(\left(\sqrt{x} + \sqrt{y}\right) - \sqrt{y + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 480000000000:\\ \;\;\;\;2 + \left(\left(\sqrt{z + 1} - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 480000000000.0)
   (+ 2.0 (- (- (sqrt (+ z 1.0)) (sqrt x)) (+ (sqrt z) (sqrt y))))
   (+ (sqrt (+ x 1.0)) (- (sqrt (+ y 1.0)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 480000000000.0) {
		tmp = 2.0 + ((sqrt((z + 1.0)) - sqrt(x)) - (sqrt(z) + sqrt(y)));
	} else {
		tmp = sqrt((x + 1.0)) + (sqrt((y + 1.0)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 480000000000.0d0) then
        tmp = 2.0d0 + ((sqrt((z + 1.0d0)) - sqrt(x)) - (sqrt(z) + sqrt(y)))
    else
        tmp = sqrt((x + 1.0d0)) + (sqrt((y + 1.0d0)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 480000000000.0) {
		tmp = 2.0 + ((Math.sqrt((z + 1.0)) - Math.sqrt(x)) - (Math.sqrt(z) + Math.sqrt(y)));
	} else {
		tmp = Math.sqrt((x + 1.0)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 480000000000.0:
		tmp = 2.0 + ((math.sqrt((z + 1.0)) - math.sqrt(x)) - (math.sqrt(z) + math.sqrt(y)))
	else:
		tmp = math.sqrt((x + 1.0)) + (math.sqrt((y + 1.0)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 480000000000.0)
		tmp = Float64(2.0 + Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(x)) - Float64(sqrt(z) + sqrt(y))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 480000000000.0)
		tmp = 2.0 + ((sqrt((z + 1.0)) - sqrt(x)) - (sqrt(z) + sqrt(y)));
	else
		tmp = sqrt((x + 1.0)) + (sqrt((y + 1.0)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 480000000000.0], N[(2.0 + N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.8e11

   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. +-commutative 96.3%

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

      2. associate-+r+ 96.3%

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

      3. associate-+r- 76.5%

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

      4. associate-+l- 69.4%

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

      5. associate-+r- 56.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in t around inf 25.0%

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

      1. associate--l+ 29.1%

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

      2. associate--l+ 29.2%

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

      3. associate-+r+ 29.2%

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

   7. Simplified 29.2%

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

   8. Taylor expanded in x around 0 22.8%

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

      1. associate--l+ 29.4%

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

      2. associate--l+ 29.4%

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

      3. associate-+r+ 29.4%

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

      4. +-commutative 29.4%

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

      5. associate--l- 29.4%

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

      6. associate--l- 29.4%

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

      7. +-commutative 29.4%

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

      8. associate-+r+ 29.4%

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

      9. associate--r+ 29.4%

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

   10. Simplified 29.4%

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

   11. Taylor expanded in y around 0 18.4%

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

       1. associate--l+ 18.4%

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

       2. associate--r+ 18.4%

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

       3. +-commutative 18.4%

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

   13. Simplified 18.4%

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

   if 4.8e11 < z

   1. Initial program 84.1%

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

      1. +-commutative 84.1%

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

      2. associate-+r+ 84.1%

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

      3. associate-+r- 62.4%

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

      4. associate-+l- 47.9%

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

      5. associate-+r- 47.9%

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

   3. Simplified 25.2%

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

   5. Taylor expanded in t around inf 4.3%

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

      1. associate--l+ 22.3%

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

      2. associate--l+ 24.9%

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

      3. associate-+r+ 24.9%

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

   7. Simplified 24.9%

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

   8. Taylor expanded in y around inf 32.5%

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

      1. mul-1-neg 32.5%

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

   10. Simplified 32.5%

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

4. Final simplification 25.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 480000000000:\\ \;\;\;\;2 + \left(\left(\sqrt{z + 1} - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 58000000000:\\ \;\;\;\;2 + \left(\left(\sqrt{z + 1} - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \left(\left(\sqrt{x} + \sqrt{y}\right) - \sqrt{y + 1}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 58000000000.0)
   (+ 2.0 (- (- (sqrt (+ z 1.0)) (sqrt x)) (+ (sqrt z) (sqrt y))))
   (- (sqrt (+ x 1.0)) (- (+ (sqrt x) (sqrt y)) (sqrt (+ y 1.0))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 58000000000.0) {
		tmp = 2.0 + ((sqrt((z + 1.0)) - sqrt(x)) - (sqrt(z) + sqrt(y)));
	} else {
		tmp = sqrt((x + 1.0)) - ((sqrt(x) + sqrt(y)) - sqrt((y + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 58000000000.0d0) then
        tmp = 2.0d0 + ((sqrt((z + 1.0d0)) - sqrt(x)) - (sqrt(z) + sqrt(y)))
    else
        tmp = sqrt((x + 1.0d0)) - ((sqrt(x) + sqrt(y)) - sqrt((y + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 58000000000.0) {
		tmp = 2.0 + ((Math.sqrt((z + 1.0)) - Math.sqrt(x)) - (Math.sqrt(z) + Math.sqrt(y)));
	} else {
		tmp = Math.sqrt((x + 1.0)) - ((Math.sqrt(x) + Math.sqrt(y)) - Math.sqrt((y + 1.0)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 58000000000.0:
		tmp = 2.0 + ((math.sqrt((z + 1.0)) - math.sqrt(x)) - (math.sqrt(z) + math.sqrt(y)))
	else:
		tmp = math.sqrt((x + 1.0)) - ((math.sqrt(x) + math.sqrt(y)) - math.sqrt((y + 1.0)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 58000000000.0)
		tmp = Float64(2.0 + Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(x)) - Float64(sqrt(z) + sqrt(y))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - Float64(Float64(sqrt(x) + sqrt(y)) - sqrt(Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 58000000000.0)
		tmp = 2.0 + ((sqrt((z + 1.0)) - sqrt(x)) - (sqrt(z) + sqrt(y)));
	else
		tmp = sqrt((x + 1.0)) - ((sqrt(x) + sqrt(y)) - sqrt((y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 58000000000.0], N[(2.0 + N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.8e10

   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. +-commutative 96.3%

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

      2. associate-+r+ 96.3%

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

      3. associate-+r- 76.5%

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

      4. associate-+l- 69.4%

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

      5. associate-+r- 56.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in t around inf 25.0%

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

      1. associate--l+ 29.1%

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

      2. associate--l+ 29.2%

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

      3. associate-+r+ 29.2%

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

   7. Simplified 29.2%

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

   8. Taylor expanded in x around 0 22.8%

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

      1. associate--l+ 29.4%

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

      2. associate--l+ 29.4%

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

      3. associate-+r+ 29.4%

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

      4. +-commutative 29.4%

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

      5. associate--l- 29.4%

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

      6. associate--l- 29.4%

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

      7. +-commutative 29.4%

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

      8. associate-+r+ 29.4%

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

      9. associate--r+ 29.4%

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

   10. Simplified 29.4%

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

   11. Taylor expanded in y around 0 18.4%

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

       1. associate--l+ 18.4%

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

       2. associate--r+ 18.4%

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

       3. +-commutative 18.4%

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

   13. Simplified 18.4%

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

   if 5.8e10 < z

   1. Initial program 84.1%

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

      1. +-commutative 84.1%

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

      2. associate-+r+ 84.1%

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

      3. associate-+r- 62.4%

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

      4. associate-+l- 47.9%

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

      5. associate-+r- 47.9%

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

   3. Simplified 25.2%

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

   5. Taylor expanded in t around inf 4.3%

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

      1. associate--l+ 22.3%

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

      2. associate--l+ 24.9%

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

      3. associate-+r+ 24.9%

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

   7. Simplified 24.9%

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

   8. Taylor expanded in z around inf 31.9%

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

      1. +-commutative 31.9%

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

   10. Simplified 31.9%

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

4. Final simplification 25.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 58000000000:\\ \;\;\;\;2 + \left(\left(\sqrt{z + 1} - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \left(\left(\sqrt{x} + \sqrt{y}\right) - \sqrt{y + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.58 \cdot 10^{-24}:\\ \;\;\;\;\left(2 + \left(\sqrt{z + 1} + 1\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\left(\sqrt{x} + \sqrt{y}\right) - \sqrt{y + 1}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.58e-24)
   (- (+ 2.0 (+ (sqrt (+ z 1.0)) 1.0)) (sqrt x))
   (- 1.0 (- (+ (sqrt x) (sqrt y)) (sqrt (+ y 1.0))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.58e-24) {
		tmp = (2.0 + (sqrt((z + 1.0)) + 1.0)) - sqrt(x);
	} else {
		tmp = 1.0 - ((sqrt(x) + sqrt(y)) - sqrt((y + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.58d-24) then
        tmp = (2.0d0 + (sqrt((z + 1.0d0)) + 1.0d0)) - sqrt(x)
    else
        tmp = 1.0d0 - ((sqrt(x) + sqrt(y)) - sqrt((y + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.58e-24) {
		tmp = (2.0 + (Math.sqrt((z + 1.0)) + 1.0)) - Math.sqrt(x);
	} else {
		tmp = 1.0 - ((Math.sqrt(x) + Math.sqrt(y)) - Math.sqrt((y + 1.0)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.58e-24:
		tmp = (2.0 + (math.sqrt((z + 1.0)) + 1.0)) - math.sqrt(x)
	else:
		tmp = 1.0 - ((math.sqrt(x) + math.sqrt(y)) - math.sqrt((y + 1.0)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.58e-24)
		tmp = Float64(Float64(2.0 + Float64(sqrt(Float64(z + 1.0)) + 1.0)) - sqrt(x));
	else
		tmp = Float64(1.0 - Float64(Float64(sqrt(x) + sqrt(y)) - sqrt(Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.58e-24)
		tmp = (2.0 + (sqrt((z + 1.0)) + 1.0)) - sqrt(x);
	else
		tmp = 1.0 - ((sqrt(x) + sqrt(y)) - sqrt((y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 1.58e-24], N[(N[(2.0 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.5799999999999999e-24

   1. Initial program 96.7%

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

      1. +-commutative 96.7%

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

      2. associate-+r+ 96.7%

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

      3. associate-+r- 77.9%

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

      4. associate-+l- 70.8%

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

      5. associate-+r- 57.0%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in x around 0 33.2%

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

   6. Taylor expanded in y around 0 12.5%

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

   7. Taylor expanded in x around inf 15.3%

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

   8. Taylor expanded in t around 0 19.0%

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

   if 1.5799999999999999e-24 < z

   1. Initial program 84.5%

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

      1. +-commutative 84.5%

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

      2. associate-+r+ 84.5%

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

      3. associate-+r- 62.3%

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

      4. associate-+l- 48.3%

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

      5. associate-+r- 47.8%

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

   3. Simplified 26.7%

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

   5. Taylor expanded in t around inf 4.4%

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

      1. associate--l+ 21.6%

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

      2. associate--l+ 24.0%

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

      3. associate-+r+ 24.0%

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

   7. Simplified 24.0%

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

   8. Taylor expanded in x around 0 3.9%

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

      1. associate--l+ 28.4%

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

      2. associate--l+ 29.7%

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

      3. associate-+r+ 29.7%

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

      4. +-commutative 29.7%

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

      5. associate--l- 29.7%

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

      6. associate--l- 29.7%

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

      7. +-commutative 29.7%

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

      8. associate-+r+ 29.7%

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

      9. associate--r+ 29.7%

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

   10. Simplified 29.7%

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

   11. Taylor expanded in z around inf 34.4%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.58 \cdot 10^{-24}:\\ \;\;\;\;\left(2 + \left(\sqrt{z + 1} + 1\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\left(\sqrt{x} + \sqrt{y}\right) - \sqrt{y + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{-24}:\\ \;\;\;\;\left(2 + \left(\sqrt{z + 1} + 1\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.8e-24)
   (- (+ 2.0 (+ (sqrt (+ z 1.0)) 1.0)) (sqrt x))
   (+ (sqrt (+ x 1.0)) (- (sqrt (+ y 1.0)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.8e-24) {
		tmp = (2.0 + (sqrt((z + 1.0)) + 1.0)) - sqrt(x);
	} else {
		tmp = sqrt((x + 1.0)) + (sqrt((y + 1.0)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 3.8d-24) then
        tmp = (2.0d0 + (sqrt((z + 1.0d0)) + 1.0d0)) - sqrt(x)
    else
        tmp = sqrt((x + 1.0d0)) + (sqrt((y + 1.0d0)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.8e-24) {
		tmp = (2.0 + (Math.sqrt((z + 1.0)) + 1.0)) - Math.sqrt(x);
	} else {
		tmp = Math.sqrt((x + 1.0)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 3.8e-24:
		tmp = (2.0 + (math.sqrt((z + 1.0)) + 1.0)) - math.sqrt(x)
	else:
		tmp = math.sqrt((x + 1.0)) + (math.sqrt((y + 1.0)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.8e-24)
		tmp = Float64(Float64(2.0 + Float64(sqrt(Float64(z + 1.0)) + 1.0)) - sqrt(x));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3.8e-24)
		tmp = (2.0 + (sqrt((z + 1.0)) + 1.0)) - sqrt(x);
	else
		tmp = sqrt((x + 1.0)) + (sqrt((y + 1.0)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 3.8e-24], N[(N[(2.0 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.80000000000000026e-24

   1. Initial program 96.7%

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

      1. +-commutative 96.7%

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

      2. associate-+r+ 96.7%

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

      3. associate-+r- 77.9%

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

      4. associate-+l- 70.8%

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

      5. associate-+r- 57.0%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in x around 0 33.2%

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

   6. Taylor expanded in y around 0 12.5%

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

   7. Taylor expanded in x around inf 15.3%

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

   8. Taylor expanded in t around 0 19.0%

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

   if 3.80000000000000026e-24 < z

   1. Initial program 84.5%

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

      1. +-commutative 84.5%

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

      2. associate-+r+ 84.5%

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

      3. associate-+r- 62.3%

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

      4. associate-+l- 48.3%

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

      5. associate-+r- 47.8%

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

   3. Simplified 26.7%

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

   5. Taylor expanded in t around inf 4.4%

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

      1. associate--l+ 21.6%

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

      2. associate--l+ 24.0%

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

      3. associate-+r+ 24.0%

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

   7. Simplified 24.0%

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

   8. Taylor expanded in y around inf 31.1%

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

      1. mul-1-neg 31.1%

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

   10. Simplified 31.1%

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

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{-24}:\\ \;\;\;\;\left(2 + \left(\sqrt{z + 1} + 1\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.8:\\ \;\;\;\;\left(2 + \left(\sqrt{z + 1} + 1\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \end{array} \]

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 0.8)
   (- (+ 2.0 (+ (sqrt (+ z 1.0)) 1.0)) (sqrt x))
   (- (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 <= 0.8) {
		tmp = (2.0 + (sqrt((z + 1.0)) + 1.0)) - sqrt(x);
	} 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 <= 0.8d0) then
        tmp = (2.0d0 + (sqrt((z + 1.0d0)) + 1.0d0)) - sqrt(x)
    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 <= 0.8) {
		tmp = (2.0 + (Math.sqrt((z + 1.0)) + 1.0)) - Math.sqrt(x);
	} 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 <= 0.8:
		tmp = (2.0 + (math.sqrt((z + 1.0)) + 1.0)) - math.sqrt(x)
	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 <= 0.8)
		tmp = Float64(Float64(2.0 + Float64(sqrt(Float64(z + 1.0)) + 1.0)) - sqrt(x));
	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 <= 0.8)
		tmp = (2.0 + (sqrt((z + 1.0)) + 1.0)) - sqrt(x);
	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, 0.8], N[(N[(2.0 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $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 < 0.80000000000000004

   1. Initial program 96.7%

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

      1. +-commutative 96.7%

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

      2. associate-+r+ 96.7%

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

      3. associate-+r- 76.9%

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

      4. associate-+l- 70.2%

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

      5. associate-+r- 56.8%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in x around 0 32.2%

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

   6. Taylor expanded in y around 0 12.7%

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

   7. Taylor expanded in x around inf 15.3%

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

   8. Taylor expanded in t around 0 19.0%

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

   if 0.80000000000000004 < z

   1. Initial program 83.9%

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

      1. +-commutative 83.9%

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

      2. associate-+r+ 83.9%

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

      3. associate-+r- 62.3%

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

      4. associate-+l- 47.6%

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

      5. associate-+r- 47.4%

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

   3. Simplified 25.3%

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

   5. Taylor expanded in t around inf 4.3%

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

      1. associate--l+ 22.1%

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

      2. associate--l+ 24.6%

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

      3. associate-+r+ 24.6%

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

   7. Simplified 24.6%

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

   8. Taylor expanded in y around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. distribute-rgt-neg-in 0.0%

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

   10. Simplified 27.0%

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

   11. Taylor expanded in x around inf 23.1%

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

   12. Taylor expanded in x around inf 21.8%

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

       1. mul-1-neg 21.8%

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

   14. Simplified 21.8%

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

4. Final simplification 20.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.8:\\ \;\;\;\;\left(2 + \left(\sqrt{z + 1} + 1\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

   1. +-commutative 90.1%

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

   2. associate-+r+ 90.1%

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

   3. associate-+r- 69.3%

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

   4. associate-+l- 58.5%

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

   5. associate-+r- 52.0%

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

3. Simplified 40.4%

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

5. Taylor expanded in t around inf 14.5%

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

   1. associate--l+ 25.7%

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

   2. associate--l+ 27.0%

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

   3. associate-+r+ 27.0%

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

7. Simplified 27.0%

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

8. Taylor expanded in y around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. *-commutative 0.0%

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

   3. distribute-rgt-neg-in 0.0%

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

10. Simplified 27.5%

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

11. Taylor expanded in x around inf 24.9%

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

12. Taylor expanded in x around inf 17.0%

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

    1. mul-1-neg 17.0%

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

14. Simplified 17.0%

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

15. Final simplification 17.0%

    \[\leadsto \sqrt{x + 1} - \sqrt{x} \]
16. Add Preprocessing

Alternative 15: 34.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{x + 1} + y \cdot 0 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (+ (sqrt (+ x 1.0)) (* y 0.0)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt((x + 1.0)) + (y * 0.0);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt((x + 1.0d0)) + (y * 0.0d0)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.sqrt((x + 1.0)) + (y * 0.0);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt((x + 1.0)) + (y * 0.0)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(sqrt(Float64(x + 1.0)) + Float64(y * 0.0))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt((x + 1.0)) + (y * 0.0);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(y * 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.1%

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

   1. +-commutative 90.1%

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

   2. associate-+r+ 90.1%

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

   3. associate-+r- 69.3%

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

   4. associate-+l- 58.5%

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

   5. associate-+r- 52.0%

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

3. Simplified 40.4%

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

5. Taylor expanded in t around inf 14.5%

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

   1. associate--l+ 25.7%

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

   2. associate--l+ 27.0%

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

   3. associate-+r+ 27.0%

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

7. Simplified 27.0%

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

8. Taylor expanded in y around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. *-commutative 0.0%

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

   3. distribute-rgt-neg-in 0.0%

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

10. Simplified 27.5%

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

11. Taylor expanded in y around inf 17.6%

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

    1. distribute-rgt1-in 17.6%

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

    2. metadata-eval 17.6%

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

    3. mul0-lft 17.6%

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

13. Simplified 17.6%

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

14. Final simplification 17.6%

    \[\leadsto \sqrt{x + 1} + y \cdot 0 \]
15. Add Preprocessing

Alternative 16: 6.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 0.5 \cdot \sqrt{\frac{1}{t}} \end{array} \]

FPCore

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

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.5 * sqrt((1.0 / t));
}

Fortran

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

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 0.5 * Math.sqrt((1.0 / t));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 0.5 * math.sqrt((1.0 / t))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.5 * sqrt(Float64(1.0 / t)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 0.5 * sqrt((1.0 / t));
end

Wolfram

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

Derivation

1. Initial program 90.1%

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

   1. +-commutative 90.1%

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

   2. associate-+r+ 90.1%

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

   3. associate-+r- 69.3%

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

   4. associate-+l- 58.5%

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

   5. associate-+r- 52.0%

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

3. Simplified 40.4%

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

5. Taylor expanded in x around 0 22.3%

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

6. Taylor expanded in y around 0 7.8%

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

7. Taylor expanded in t around inf 11.0%

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

   1. associate--r+ 11.0%

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

   2. associate-+r+ 11.0%

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

   3. +-commutative 11.0%

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

9. Simplified 11.0%

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

10. Taylor expanded in t around 0 9.3%

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} \]

11. Final simplification 9.3%

    \[\leadsto 0.5 \cdot \sqrt{\frac{1}{t}} \]
12. Add Preprocessing

Alternative 17: 7.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 0.5 \cdot \sqrt{\frac{1}{z}} \end{array} \]

FPCore

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

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.5 * sqrt((1.0 / z));
}

Fortran

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

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 0.5 * Math.sqrt((1.0 / z));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 0.5 * math.sqrt((1.0 / z))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.5 * sqrt(Float64(1.0 / z)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 0.5 * sqrt((1.0 / z));
end

Wolfram

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

Derivation

1. Initial program 90.1%

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

   1. +-commutative 90.1%

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

   2. associate-+r+ 90.1%

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

   3. associate-+r- 69.3%

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

   4. associate-+l- 58.5%

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

   5. associate-+r- 52.0%

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

3. Simplified 40.4%

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

5. Taylor expanded in x around 0 22.3%

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

   1. flip-- 22.3%

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

   2. add-sqr-sqrt 17.0%

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

   3. add-sqr-sqrt 22.2%

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

   4. +-commutative 22.2%

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

   5. +-commutative 22.2%

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

7. Applied egg-rr 22.2%

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

   1. associate--r+ 22.4%

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

   2. +-inverses 22.4%

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

   3. metadata-eval 22.4%

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

9. Simplified 22.4%

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

10. Taylor expanded in z around inf 18.5%

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

11. Taylor expanded in z around 0 7.1%

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}} \]

12. Final simplification 7.1%

    \[\leadsto 0.5 \cdot \sqrt{\frac{1}{z}} \]
13. Add Preprocessing

Alternative 18: 6.1% accurate, 7.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{x} \cdot \left(--2\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (* (sqrt x) (- -2.0)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt(x) * -(-2.0);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt(x) * -(-2.0d0)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.sqrt(x) * -(-2.0);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt(x) * -(-2.0)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(sqrt(x) * Float64(-(-2.0)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt(x) * -(-2.0);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[Sqrt[x], $MachinePrecision] * (--2.0)), $MachinePrecision]

Derivation

1. Initial program 90.1%

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

   1. +-commutative 90.1%

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

   2. associate-+r+ 90.1%

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

   3. associate-+r- 69.3%

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

   4. associate-+l- 58.5%

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

   5. associate-+r- 52.0%

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

3. Simplified 40.4%

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

5. Taylor expanded in t around inf 14.5%

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

   1. associate--l+ 25.7%

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

   2. associate--l+ 27.0%

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

   3. associate-+r+ 27.0%

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

7. Simplified 27.0%

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

8. Taylor expanded in y around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. *-commutative 0.0%

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

   3. distribute-rgt-neg-in 0.0%

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

10. Simplified 27.5%

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

11. Taylor expanded in x around inf 24.9%

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

12. Taylor expanded in x around -inf 0.0%

    \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{x} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
13. Step-by-step derivation

    1. *-commutative 0.0%

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

    2. unpow2 0.0%

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

    3. rem-square-sqrt 6.7%

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

    4. mul-1-neg 6.7%

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

14. Simplified 6.7%

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

15. Final simplification 6.7%

    \[\leadsto \sqrt{x} \cdot \left(--2\right) \]
16. Add Preprocessing

Alternative 19: 3.1% accurate, 274.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ x \cdot 0 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (* x 0.0))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return x * 0.0;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * 0.0d0
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return x * 0.0;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return x * 0.0

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(x * 0.0)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = x * 0.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(x * 0.0), $MachinePrecision]

Derivation

1. Initial program 90.1%

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

   1. +-commutative 90.1%

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

   2. associate-+r+ 90.1%

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

   3. associate-+r- 69.3%

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

   4. associate-+l- 58.5%

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

   5. associate-+r- 52.0%

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

3. Simplified 40.4%

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

5. Taylor expanded in t around inf 14.5%

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

   1. associate--l+ 25.7%

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

   2. associate--l+ 27.0%

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

   3. associate-+r+ 27.0%

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

7. Simplified 27.0%

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

8. Taylor expanded in y around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. *-commutative 0.0%

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

   3. distribute-rgt-neg-in 0.0%

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

10. Simplified 27.5%

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

11. Taylor expanded in x around inf 3.1%

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

    1. distribute-rgt1-in 3.1%

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

    2. metadata-eval 3.1%

       \[\leadsto x \cdot \left(\color{blue}{0} \cdot \sqrt{\frac{1}{x}}\right) \]

    3. mul0-lft 3.1%

       \[\leadsto x \cdot \color{blue}{0} \]

13. Simplified 3.1%

    \[\leadsto \color{blue}{x \cdot 0} \]

14. Final simplification 3.1%

    \[\leadsto x \cdot 0 \]
15. Add Preprocessing

Developer target: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (+ (+ (+ (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))) (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y)))) (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z)))) (- (sqrt (+ t 1.0)) (sqrt t)))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))