Main:z from

Percentage Accurate: 91.7% → 96.2%

Time: 24.8s

Alternatives: 16

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.7% 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: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1} + \sqrt{y}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \frac{-1}{t_1}\\ t_4 := \sqrt{x + 1}\\ t_5 := \frac{1}{t_1} - \sqrt{x}\\ t_6 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 5.5 \cdot 10^{+40}:\\ \;\;\;\;\left(t_4 + t_5\right) + \left(t_2 + \frac{1}{\sqrt{z} + t_6}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x + \left(\sqrt{x} + t_3\right) \cdot t_5\right)}{\sqrt{x} + \left(t_4 + t_3\right)} + \left(t_2 + \left(t_6 - \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt (+ y 1.0)) (sqrt y)))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_3 (/ -1.0 t_1))
        (t_4 (sqrt (+ x 1.0)))
        (t_5 (- (/ 1.0 t_1) (sqrt x)))
        (t_6 (sqrt (+ 1.0 z))))
   (if (<= y 5.5e+40)
     (+ (+ t_4 t_5) (+ t_2 (/ 1.0 (+ (sqrt z) t_6))))
     (+
      (/ (+ 1.0 (+ x (* (+ (sqrt x) t_3) t_5))) (+ (sqrt x) (+ t_4 t_3)))
      (+ t_2 (- t_6 (sqrt z)))))))

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)) + sqrt(y);
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double t_3 = -1.0 / t_1;
	double t_4 = sqrt((x + 1.0));
	double t_5 = (1.0 / t_1) - sqrt(x);
	double t_6 = sqrt((1.0 + z));
	double tmp;
	if (y <= 5.5e+40) {
		tmp = (t_4 + t_5) + (t_2 + (1.0 / (sqrt(z) + t_6)));
	} else {
		tmp = ((1.0 + (x + ((sqrt(x) + t_3) * t_5))) / (sqrt(x) + (t_4 + t_3))) + (t_2 + (t_6 - sqrt(z)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0)) + sqrt(y)
    t_2 = sqrt((1.0d0 + t)) - sqrt(t)
    t_3 = (-1.0d0) / t_1
    t_4 = sqrt((x + 1.0d0))
    t_5 = (1.0d0 / t_1) - sqrt(x)
    t_6 = sqrt((1.0d0 + z))
    if (y <= 5.5d+40) then
        tmp = (t_4 + t_5) + (t_2 + (1.0d0 / (sqrt(z) + t_6)))
    else
        tmp = ((1.0d0 + (x + ((sqrt(x) + t_3) * t_5))) / (sqrt(x) + (t_4 + t_3))) + (t_2 + (t_6 - sqrt(z)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0)) + Math.sqrt(y);
	double t_2 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_3 = -1.0 / t_1;
	double t_4 = Math.sqrt((x + 1.0));
	double t_5 = (1.0 / t_1) - Math.sqrt(x);
	double t_6 = Math.sqrt((1.0 + z));
	double tmp;
	if (y <= 5.5e+40) {
		tmp = (t_4 + t_5) + (t_2 + (1.0 / (Math.sqrt(z) + t_6)));
	} else {
		tmp = ((1.0 + (x + ((Math.sqrt(x) + t_3) * t_5))) / (Math.sqrt(x) + (t_4 + t_3))) + (t_2 + (t_6 - Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0)) + math.sqrt(y)
	t_2 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_3 = -1.0 / t_1
	t_4 = math.sqrt((x + 1.0))
	t_5 = (1.0 / t_1) - math.sqrt(x)
	t_6 = math.sqrt((1.0 + z))
	tmp = 0
	if y <= 5.5e+40:
		tmp = (t_4 + t_5) + (t_2 + (1.0 / (math.sqrt(z) + t_6)))
	else:
		tmp = ((1.0 + (x + ((math.sqrt(x) + t_3) * t_5))) / (math.sqrt(x) + (t_4 + t_3))) + (t_2 + (t_6 - math.sqrt(z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(y + 1.0)) + sqrt(y))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = Float64(-1.0 / t_1)
	t_4 = sqrt(Float64(x + 1.0))
	t_5 = Float64(Float64(1.0 / t_1) - sqrt(x))
	t_6 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (y <= 5.5e+40)
		tmp = Float64(Float64(t_4 + t_5) + Float64(t_2 + Float64(1.0 / Float64(sqrt(z) + t_6))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x + Float64(Float64(sqrt(x) + t_3) * t_5))) / Float64(sqrt(x) + Float64(t_4 + t_3))) + Float64(t_2 + Float64(t_6 - sqrt(z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0)) + sqrt(y);
	t_2 = sqrt((1.0 + t)) - sqrt(t);
	t_3 = -1.0 / t_1;
	t_4 = sqrt((x + 1.0));
	t_5 = (1.0 / t_1) - sqrt(x);
	t_6 = sqrt((1.0 + z));
	tmp = 0.0;
	if (y <= 5.5e+40)
		tmp = (t_4 + t_5) + (t_2 + (1.0 / (sqrt(z) + t_6)));
	else
		tmp = ((1.0 + (x + ((sqrt(x) + t_3) * t_5))) / (sqrt(x) + (t_4 + t_3))) + (t_2 + (t_6 - sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(1.0 / t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 5.5e+40], N[(N[(t$95$4 + t$95$5), $MachinePrecision] + N[(t$95$2 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(x + N[(N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[(t$95$4 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(t$95$6 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.49999999999999974e40

   1. Initial program 95.2%

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

      1. associate-+l+ 95.2%

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

      2. associate-+l- 66.0%

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

      3. +-commutative 66.0%

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

      4. sub-neg 66.0%

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

      5. sub-neg 66.0%

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

      6. +-commutative 66.0%

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

      7. +-commutative 66.0%

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

   3. Simplified 66.0%

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

      1. flip-- 66.0%

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

      2. add-sqr-sqrt 65.7%

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

      3. add-sqr-sqrt 66.6%

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

   5. Applied egg-rr 66.6%

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

      1. associate--l+ 66.8%

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

      2. +-inverses 66.8%

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

      3. metadata-eval 66.8%

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

   7. Simplified 66.8%

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

      1. flip-- 66.9%

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

      2. add-sqr-sqrt 56.8%

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

      3. +-commutative 56.8%

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

      4. add-sqr-sqrt 67.3%

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

      5. +-commutative 67.3%

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

   9. Applied egg-rr 67.3%

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

   10. Taylor expanded in z around 0 67.8%

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

   if 5.49999999999999974e40 < y

   1. Initial program 86.4%

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

      1. associate-+l+ 86.4%

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

      2. associate-+l- 86.4%

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

      3. +-commutative 86.4%

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

      4. sub-neg 86.4%

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

      5. sub-neg 86.4%

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

      6. +-commutative 86.4%

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

      7. +-commutative 86.4%

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

   3. Simplified 86.4%

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

      1. flip-- 86.4%

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

      2. add-sqr-sqrt 58.7%

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

      3. add-sqr-sqrt 86.4%

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

   5. Applied egg-rr 86.4%

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

      1. associate--l+ 86.4%

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

      2. +-inverses 86.4%

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

      3. metadata-eval 86.4%

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

   7. Simplified 86.4%

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

      1. flip-- 86.4%

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

      2. add-sqr-sqrt 70.1%

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

   9. Applied egg-rr 70.1%

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

   11. Simplified 70.2%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+40}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} - \sqrt{x}\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x + \left(\sqrt{x} + \frac{-1}{\sqrt{y + 1} + \sqrt{y}}\right) \cdot \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} - \sqrt{x}\right)\right)}{\sqrt{x} + \left(\sqrt{x + 1} + \frac{-1}{\sqrt{y + 1} + \sqrt{y}}\right)} + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \end{array} \]

Alternative 2: 94.3% accurate, 1.0× speedup

Math

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

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 1.0)) (- (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))) (sqrt x)))
  (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (sqrt((x + 1.0)) + ((1.0 / (sqrt((y + 1.0)) + sqrt(y))) - sqrt(x))) + ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 / (sqrt(z) + 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 = (sqrt((x + 1.0d0)) + ((1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y))) - sqrt(x))) + ((sqrt((1.0d0 + t)) - sqrt(t)) + (1.0d0 / (sqrt(z) + 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 (Math.sqrt((x + 1.0)) + ((1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y))) - Math.sqrt(x))) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)))));
}

Python

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

Julia

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

Derivation

1. Initial program 91.6%

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

   1. associate-+l+ 91.6%

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

   2. associate-+l- 74.3%

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

   3. +-commutative 74.3%

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

   4. sub-neg 74.3%

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

   5. sub-neg 74.3%

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

   6. +-commutative 74.3%

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

   7. +-commutative 74.3%

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

3. Simplified 74.3%

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

   1. flip-- 74.3%

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

   2. add-sqr-sqrt 62.9%

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

   3. add-sqr-sqrt 74.7%

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

5. Applied egg-rr 74.7%

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

   1. associate--l+ 74.8%

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

   2. +-inverses 74.8%

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

   3. metadata-eval 74.8%

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

7. Simplified 74.8%

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

   1. flip-- 75.1%

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

   2. add-sqr-sqrt 61.6%

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

   3. +-commutative 61.6%

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

   4. add-sqr-sqrt 75.5%

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

   5. +-commutative 75.5%

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

9. Applied egg-rr 75.5%

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

10. Taylor expanded in z around 0 77.3%

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

11. Final simplification 77.3%

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

Alternative 3: 94.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;2 + \left(t_1 + \left(\sqrt{1 + t} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} - \sqrt{x}\right)\right) + \frac{1}{\sqrt{z} + t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))))
   (if (<= t 2.5e+14)
     (+ 2.0 (+ t_1 (- (sqrt (+ 1.0 t)) (+ (sqrt z) (sqrt t)))))
     (+
      (+ (sqrt (+ x 1.0)) (- (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))) (sqrt x)))
      (/ 1.0 (+ (sqrt z) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double tmp;
	if (t <= 2.5e+14) {
		tmp = 2.0 + (t_1 + (sqrt((1.0 + t)) - (sqrt(z) + sqrt(t))));
	} else {
		tmp = (sqrt((x + 1.0)) + ((1.0 / (sqrt((y + 1.0)) + sqrt(y))) - sqrt(x))) + (1.0 / (sqrt(z) + t_1));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if (t <= 2.5d+14) then
        tmp = 2.0d0 + (t_1 + (sqrt((1.0d0 + t)) - (sqrt(z) + sqrt(t))))
    else
        tmp = (sqrt((x + 1.0d0)) + ((1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y))) - sqrt(x))) + (1.0d0 / (sqrt(z) + t_1))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double tmp;
	if (t <= 2.5e+14) {
		tmp = 2.0 + (t_1 + (Math.sqrt((1.0 + t)) - (Math.sqrt(z) + Math.sqrt(t))));
	} else {
		tmp = (Math.sqrt((x + 1.0)) + ((1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y))) - Math.sqrt(x))) + (1.0 / (Math.sqrt(z) + t_1));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	tmp = 0
	if t <= 2.5e+14:
		tmp = 2.0 + (t_1 + (math.sqrt((1.0 + t)) - (math.sqrt(z) + math.sqrt(t))))
	else:
		tmp = (math.sqrt((x + 1.0)) + ((1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y))) - math.sqrt(x))) + (1.0 / (math.sqrt(z) + t_1))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (t <= 2.5e+14)
		tmp = Float64(2.0 + Float64(t_1 + Float64(sqrt(Float64(1.0 + t)) - Float64(sqrt(z) + sqrt(t)))));
	else
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) + Float64(Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y))) - sqrt(x))) + Float64(1.0 / Float64(sqrt(z) + t_1)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	tmp = 0.0;
	if (t <= 2.5e+14)
		tmp = 2.0 + (t_1 + (sqrt((1.0 + t)) - (sqrt(z) + sqrt(t))));
	else
		tmp = (sqrt((x + 1.0)) + ((1.0 / (sqrt((y + 1.0)) + sqrt(y))) - sqrt(x))) + (1.0 / (sqrt(z) + t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 2.5e+14], N[(2.0 + N[(t$95$1 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.5e14

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. sub-neg 96.2%

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

      3. associate-+l+ 81.8%

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

      4. associate-+l+ 56.4%

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

      5. +-commutative 56.4%

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

      6. neg-sub0 56.4%

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

      7. associate-+l- 56.4%

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

      8. neg-sub0 56.4%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in y around 0 35.3%

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

   5. Taylor expanded in x around 0 25.9%

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

      1. associate--l+ 45.3%

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

      2. +-commutative 45.3%

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

      3. associate--l+ 45.3%

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

      4. +-commutative 45.3%

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

   7. Simplified 45.3%

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

   if 2.5e14 < t

   1. Initial program 86.2%

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

      1. associate-+l+ 86.2%

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

      2. associate-+l- 65.5%

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

      3. +-commutative 65.5%

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

      4. sub-neg 65.5%

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

      5. sub-neg 65.5%

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

      6. +-commutative 65.5%

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

      7. +-commutative 65.5%

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

   3. Simplified 65.5%

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

      1. flip-- 65.5%

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

      2. add-sqr-sqrt 55.9%

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

      3. add-sqr-sqrt 65.7%

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

   5. Applied egg-rr 65.7%

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

      1. associate--l+ 65.7%

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

      2. +-inverses 65.7%

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

      3. metadata-eval 65.7%

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

   7. Simplified 65.7%

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

      1. flip-- 66.0%

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

      2. add-sqr-sqrt 56.8%

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

      3. +-commutative 56.8%

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

      4. add-sqr-sqrt 66.8%

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

      5. +-commutative 66.8%

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

   9. Applied egg-rr 66.8%

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

   10. Taylor expanded in z around 0 70.6%

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

   11. Taylor expanded in t around inf 70.6%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} + \left(\sqrt{1 + t} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} - \sqrt{x}\right)\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\\ \end{array} \]

Alternative 4: 94.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;2 + \left(t_1 + \left(\sqrt{1 + t} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \left(\frac{1}{\sqrt{z} + t_1} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))))
   (if (<= t 2.5e+14)
     (+ 2.0 (+ t_1 (- (sqrt (+ 1.0 t)) (+ (sqrt z) (sqrt t)))))
     (+
      (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y)))
      (+ (/ 1.0 (+ (sqrt z) t_1)) (- (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 t_1 = sqrt((1.0 + z));
	double tmp;
	if (t <= 2.5e+14) {
		tmp = 2.0 + (t_1 + (sqrt((1.0 + t)) - (sqrt(z) + sqrt(t))));
	} else {
		tmp = (1.0 / (sqrt((y + 1.0)) + sqrt(y))) + ((1.0 / (sqrt(z) + t_1)) + (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) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if (t <= 2.5d+14) then
        tmp = 2.0d0 + (t_1 + (sqrt((1.0d0 + t)) - (sqrt(z) + sqrt(t))))
    else
        tmp = (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y))) + ((1.0d0 / (sqrt(z) + t_1)) + (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 t_1 = Math.sqrt((1.0 + z));
	double tmp;
	if (t <= 2.5e+14) {
		tmp = 2.0 + (t_1 + (Math.sqrt((1.0 + t)) - (Math.sqrt(z) + Math.sqrt(t))));
	} else {
		tmp = (1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y))) + ((1.0 / (Math.sqrt(z) + t_1)) + (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):
	t_1 = math.sqrt((1.0 + z))
	tmp = 0
	if t <= 2.5e+14:
		tmp = 2.0 + (t_1 + (math.sqrt((1.0 + t)) - (math.sqrt(z) + math.sqrt(t))))
	else:
		tmp = (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y))) + ((1.0 / (math.sqrt(z) + t_1)) + (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)
	t_1 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (t <= 2.5e+14)
		tmp = Float64(2.0 + Float64(t_1 + Float64(sqrt(Float64(1.0 + t)) - Float64(sqrt(z) + sqrt(t)))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y))) + Float64(Float64(1.0 / Float64(sqrt(z) + t_1)) + 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)
	t_1 = sqrt((1.0 + z));
	tmp = 0.0;
	if (t <= 2.5e+14)
		tmp = 2.0 + (t_1 + (sqrt((1.0 + t)) - (sqrt(z) + sqrt(t))));
	else
		tmp = (1.0 / (sqrt((y + 1.0)) + sqrt(y))) + ((1.0 / (sqrt(z) + t_1)) + (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_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 2.5e+14], N[(2.0 + N[(t$95$1 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.5e14

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. sub-neg 96.2%

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

      3. associate-+l+ 81.8%

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

      4. associate-+l+ 56.4%

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

      5. +-commutative 56.4%

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

      6. neg-sub0 56.4%

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

      7. associate-+l- 56.4%

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

      8. neg-sub0 56.4%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in y around 0 35.3%

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

   5. Taylor expanded in x around 0 25.9%

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

      1. associate--l+ 45.3%

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

      2. +-commutative 45.3%

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

      3. associate--l+ 45.3%

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

      4. +-commutative 45.3%

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

   7. Simplified 45.3%

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

   if 2.5e14 < t

   1. Initial program 86.2%

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

      1. associate-+l+ 86.2%

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

      2. associate-+l- 65.5%

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

      3. +-commutative 65.5%

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

      4. sub-neg 65.5%

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

      5. sub-neg 65.5%

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

      6. +-commutative 65.5%

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

      7. +-commutative 65.5%

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

   3. Simplified 65.5%

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

      1. flip-- 65.5%

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

      2. add-sqr-sqrt 55.9%

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

      3. add-sqr-sqrt 65.7%

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

   5. Applied egg-rr 65.7%

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

      1. associate--l+ 65.7%

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

      2. +-inverses 65.7%

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

      3. metadata-eval 65.7%

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

   7. Simplified 65.7%

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

      1. flip-- 66.0%

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

      2. add-sqr-sqrt 56.8%

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

      3. +-commutative 56.8%

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

      4. add-sqr-sqrt 66.8%

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

      5. +-commutative 66.8%

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

   9. Applied egg-rr 66.8%

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

   10. Taylor expanded in z around 0 70.6%

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

   11. Taylor expanded in t around inf 54.0%

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

       1. associate--l+ 66.3%

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

       2. +-commutative 66.3%

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

       3. +-commutative 66.3%

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

       4. associate--l+ 93.8%

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

       5. +-commutative 93.8%

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

   13. Simplified 93.8%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} + \left(\sqrt{1 + t} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 5: 92.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + ((1.0 / (sqrt((y + 1.0)) + sqrt(y))) + ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 / (sqrt(z) + 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 = 1.0d0 + ((1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y))) + ((sqrt((1.0d0 + t)) - sqrt(t)) + (1.0d0 / (sqrt(z) + 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 1.0 + ((1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y))) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z))))));
}

Python

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

Julia

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

Derivation

1. Initial program 91.6%

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

   1. associate-+l+ 91.6%

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

   2. associate-+l- 74.3%

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

   3. +-commutative 74.3%

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

   4. sub-neg 74.3%

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

   5. sub-neg 74.3%

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

   6. +-commutative 74.3%

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

   7. +-commutative 74.3%

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

3. Simplified 74.3%

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

   1. flip-- 74.3%

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

   2. add-sqr-sqrt 62.9%

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

   3. add-sqr-sqrt 74.7%

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

5. Applied egg-rr 74.7%

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

   1. associate--l+ 74.8%

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

   2. +-inverses 74.8%

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

   3. metadata-eval 74.8%

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

7. Simplified 74.8%

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

   1. flip-- 75.1%

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

   2. add-sqr-sqrt 61.6%

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

   3. +-commutative 61.6%

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

   4. add-sqr-sqrt 75.5%

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

   5. +-commutative 75.5%

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

9. Applied egg-rr 75.5%

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

10. Taylor expanded in z around 0 77.3%

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

11. Taylor expanded in x around 0 35.0%

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

    1. associate--l+ 51.3%

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

    2. associate--l+ 58.1%

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

    3. +-commutative 58.1%

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

    4. associate-+r- 58.6%

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

13. Simplified 58.6%

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

14. Final simplification 58.6%

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

Alternative 6: 92.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;2 + \left(t_1 + \left(\sqrt{1 + t} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \left(\left(t_1 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))))
   (if (<= t 2.5e+14)
     (+ 2.0 (+ t_1 (- (sqrt (+ 1.0 t)) (+ (sqrt z) (sqrt t)))))
     (+
      (sqrt (+ x 1.0))
      (+
       (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y)))
       (- (- t_1 (sqrt z)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double tmp;
	if (t <= 2.5e+14) {
		tmp = 2.0 + (t_1 + (sqrt((1.0 + t)) - (sqrt(z) + sqrt(t))));
	} else {
		tmp = sqrt((x + 1.0)) + ((1.0 / (sqrt((y + 1.0)) + sqrt(y))) + ((t_1 - sqrt(z)) - sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if (t <= 2.5d+14) then
        tmp = 2.0d0 + (t_1 + (sqrt((1.0d0 + t)) - (sqrt(z) + sqrt(t))))
    else
        tmp = sqrt((x + 1.0d0)) + ((1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y))) + ((t_1 - sqrt(z)) - sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double tmp;
	if (t <= 2.5e+14) {
		tmp = 2.0 + (t_1 + (Math.sqrt((1.0 + t)) - (Math.sqrt(z) + Math.sqrt(t))));
	} else {
		tmp = Math.sqrt((x + 1.0)) + ((1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y))) + ((t_1 - Math.sqrt(z)) - Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	tmp = 0
	if t <= 2.5e+14:
		tmp = 2.0 + (t_1 + (math.sqrt((1.0 + t)) - (math.sqrt(z) + math.sqrt(t))))
	else:
		tmp = math.sqrt((x + 1.0)) + ((1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y))) + ((t_1 - math.sqrt(z)) - math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (t <= 2.5e+14)
		tmp = Float64(2.0 + Float64(t_1 + Float64(sqrt(Float64(1.0 + t)) - Float64(sqrt(z) + sqrt(t)))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y))) + Float64(Float64(t_1 - sqrt(z)) - sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	tmp = 0.0;
	if (t <= 2.5e+14)
		tmp = 2.0 + (t_1 + (sqrt((1.0 + t)) - (sqrt(z) + sqrt(t))));
	else
		tmp = sqrt((x + 1.0)) + ((1.0 / (sqrt((y + 1.0)) + sqrt(y))) + ((t_1 - sqrt(z)) - sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 2.5e+14], N[(2.0 + N[(t$95$1 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.5e14

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. sub-neg 96.2%

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

      3. associate-+l+ 81.8%

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

      4. associate-+l+ 56.4%

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

      5. +-commutative 56.4%

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

      6. neg-sub0 56.4%

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

      7. associate-+l- 56.4%

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

      8. neg-sub0 56.4%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in y around 0 35.3%

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

   5. Taylor expanded in x around 0 25.9%

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

      1. associate--l+ 45.3%

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

      2. +-commutative 45.3%

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

      3. associate--l+ 45.3%

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

      4. +-commutative 45.3%

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

   7. Simplified 45.3%

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

   if 2.5e14 < t

   1. Initial program 86.2%

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

      1. associate-+l+ 86.2%

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

      2. +-commutative 86.2%

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

      3. associate-+r- 65.5%

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

      4. associate-+l- 53.1%

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

      5. +-commutative 53.1%

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

      6. associate--l+ 53.1%

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

      7. +-commutative 53.1%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in t around inf 50.9%

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

      1. +-commutative 50.9%

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

      2. +-commutative 50.9%

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

      3. associate--l+ 53.1%

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

   6. Simplified 53.1%

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

      1. flip-- 65.5%

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

      2. add-sqr-sqrt 55.9%

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

      3. add-sqr-sqrt 65.7%

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

   8. Applied egg-rr 53.3%

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

      1. associate--l+ 65.7%

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

      2. +-inverses 65.7%

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

      3. metadata-eval 65.7%

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

   10. Simplified 53.3%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} + \left(\sqrt{1 + t} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 7: 92.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{y + 1}\\ \mathbf{if}\;y \leq 1.4:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + t_1\right) + \left(\left(1 + t_2\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1}{t_2 + \sqrt{y}} + \left(t_1 - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 z)) (sqrt z))) (t_2 (sqrt (+ y 1.0))))
   (if (<= y 1.4)
     (+ (+ (- (sqrt (+ 1.0 t)) (sqrt t)) t_1) (- (+ 1.0 t_2) (sqrt y)))
     (+ (sqrt (+ x 1.0)) (+ (/ 1.0 (+ t_2 (sqrt y))) (- t_1 (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z)) - sqrt(z);
	double t_2 = sqrt((y + 1.0));
	double tmp;
	if (y <= 1.4) {
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) + t_1) + ((1.0 + t_2) - sqrt(y));
	} else {
		tmp = sqrt((x + 1.0)) + ((1.0 / (t_2 + sqrt(y))) + (t_1 - sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z)) - sqrt(z)
    t_2 = sqrt((y + 1.0d0))
    if (y <= 1.4d0) then
        tmp = ((sqrt((1.0d0 + t)) - sqrt(t)) + t_1) + ((1.0d0 + t_2) - sqrt(y))
    else
        tmp = sqrt((x + 1.0d0)) + ((1.0d0 / (t_2 + sqrt(y))) + (t_1 - sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double t_2 = Math.sqrt((y + 1.0));
	double tmp;
	if (y <= 1.4) {
		tmp = ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + t_1) + ((1.0 + t_2) - Math.sqrt(y));
	} else {
		tmp = Math.sqrt((x + 1.0)) + ((1.0 / (t_2 + Math.sqrt(y))) + (t_1 - Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z)) - math.sqrt(z)
	t_2 = math.sqrt((y + 1.0))
	tmp = 0
	if y <= 1.4:
		tmp = ((math.sqrt((1.0 + t)) - math.sqrt(t)) + t_1) + ((1.0 + t_2) - math.sqrt(y))
	else:
		tmp = math.sqrt((x + 1.0)) + ((1.0 / (t_2 + math.sqrt(y))) + (t_1 - math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_2 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (y <= 1.4)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + t_1) + Float64(Float64(1.0 + t_2) - sqrt(y)));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(Float64(1.0 / Float64(t_2 + sqrt(y))) + Float64(t_1 - sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z)) - sqrt(z);
	t_2 = sqrt((y + 1.0));
	tmp = 0.0;
	if (y <= 1.4)
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) + t_1) + ((1.0 + t_2) - sqrt(y));
	else
		tmp = sqrt((x + 1.0)) + ((1.0 / (t_2 + sqrt(y))) + (t_1 - sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.4], N[(N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[(1.0 + t$95$2), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3999999999999999

   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. associate-+l+ 97.5%

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

      2. associate-+l- 64.6%

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

      3. +-commutative 64.6%

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

      4. sub-neg 64.6%

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

      5. sub-neg 64.6%

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

      6. +-commutative 64.6%

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

      7. +-commutative 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in x around 0 61.4%

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

   if 1.3999999999999999 < y

   1. Initial program 85.4%

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

      1. associate-+l+ 85.4%

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

      2. +-commutative 85.4%

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

      3. associate-+r- 84.7%

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

      4. associate-+l- 50.5%

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

      5. +-commutative 50.5%

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

      6. associate--l+ 50.5%

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

      7. +-commutative 50.5%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in t around inf 28.1%

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

      1. +-commutative 28.1%

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

      2. +-commutative 28.1%

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

      3. associate--l+ 28.2%

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

   6. Simplified 28.2%

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

      1. flip-- 84.7%

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

      2. add-sqr-sqrt 61.0%

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

      3. add-sqr-sqrt 85.4%

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

   8. Applied egg-rr 28.4%

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

      1. associate--l+ 85.6%

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

      2. +-inverses 85.6%

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

      3. metadata-eval 85.6%

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

   10. Simplified 28.4%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\left(1 + \sqrt{y + 1}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 8: 91.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 1.16 \cdot 10^{+15}:\\ \;\;\;\;2 + \left(t_1 + \left(\sqrt{1 + t} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(t_1 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))))
   (if (<= t 1.16e+15)
     (+ 2.0 (+ t_1 (- (sqrt (+ 1.0 t)) (+ (sqrt z) (sqrt t)))))
     (+
      (sqrt (+ x 1.0))
      (+ (- (sqrt (+ y 1.0)) (sqrt y)) (- (- t_1 (sqrt z)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double tmp;
	if (t <= 1.16e+15) {
		tmp = 2.0 + (t_1 + (sqrt((1.0 + t)) - (sqrt(z) + sqrt(t))));
	} else {
		tmp = sqrt((x + 1.0)) + ((sqrt((y + 1.0)) - sqrt(y)) + ((t_1 - sqrt(z)) - sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if (t <= 1.16d+15) then
        tmp = 2.0d0 + (t_1 + (sqrt((1.0d0 + t)) - (sqrt(z) + sqrt(t))))
    else
        tmp = sqrt((x + 1.0d0)) + ((sqrt((y + 1.0d0)) - sqrt(y)) + ((t_1 - sqrt(z)) - sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double tmp;
	if (t <= 1.16e+15) {
		tmp = 2.0 + (t_1 + (Math.sqrt((1.0 + t)) - (Math.sqrt(z) + Math.sqrt(t))));
	} else {
		tmp = Math.sqrt((x + 1.0)) + ((Math.sqrt((y + 1.0)) - Math.sqrt(y)) + ((t_1 - Math.sqrt(z)) - Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	tmp = 0
	if t <= 1.16e+15:
		tmp = 2.0 + (t_1 + (math.sqrt((1.0 + t)) - (math.sqrt(z) + math.sqrt(t))))
	else:
		tmp = math.sqrt((x + 1.0)) + ((math.sqrt((y + 1.0)) - math.sqrt(y)) + ((t_1 - math.sqrt(z)) - math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (t <= 1.16e+15)
		tmp = Float64(2.0 + Float64(t_1 + Float64(sqrt(Float64(1.0 + t)) - Float64(sqrt(z) + sqrt(t)))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) + Float64(Float64(t_1 - sqrt(z)) - sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	tmp = 0.0;
	if (t <= 1.16e+15)
		tmp = 2.0 + (t_1 + (sqrt((1.0 + t)) - (sqrt(z) + sqrt(t))));
	else
		tmp = sqrt((x + 1.0)) + ((sqrt((y + 1.0)) - sqrt(y)) + ((t_1 - sqrt(z)) - sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1.16e+15], N[(2.0 + N[(t$95$1 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.16e15

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. sub-neg 96.2%

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

      3. associate-+l+ 81.8%

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

      4. associate-+l+ 56.4%

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

      5. +-commutative 56.4%

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

      6. neg-sub0 56.4%

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

      7. associate-+l- 56.4%

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

      8. neg-sub0 56.4%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in y around 0 35.3%

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

   5. Taylor expanded in x around 0 25.9%

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

      1. associate--l+ 45.3%

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

      2. +-commutative 45.3%

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

      3. associate--l+ 45.3%

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

      4. +-commutative 45.3%

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

   7. Simplified 45.3%

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

   if 1.16e15 < t

   1. Initial program 86.2%

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

      1. associate-+l+ 86.2%

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

      2. +-commutative 86.2%

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

      3. associate-+r- 65.5%

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

      4. associate-+l- 53.1%

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

      5. +-commutative 53.1%

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

      6. associate--l+ 53.1%

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

      7. +-commutative 53.1%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in t around inf 50.9%

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

      1. +-commutative 50.9%

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

      2. +-commutative 50.9%

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

      3. associate--l+ 53.1%

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

   6. Simplified 53.1%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.16 \cdot 10^{+15}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} + \left(\sqrt{1 + t} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 9: 86.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 2.5 \cdot 10^{+16}:\\ \;\;\;\;1 + \left(t_1 - \left(\left(\sqrt{y} + \sqrt{z}\right) - \sqrt{1 + z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(t_1 - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))))
   (if (<= z 2.5e+16)
     (+ 1.0 (- t_1 (- (+ (sqrt y) (sqrt z)) (sqrt (+ 1.0 z)))))
     (+ (sqrt (+ x 1.0)) (- (- t_1 (sqrt y)) (sqrt x))))))

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 <= 2.5e+16) {
		tmp = 1.0 + (t_1 - ((sqrt(y) + sqrt(z)) - sqrt((1.0 + z))));
	} else {
		tmp = sqrt((x + 1.0)) + ((t_1 - sqrt(y)) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    if (z <= 2.5d+16) then
        tmp = 1.0d0 + (t_1 - ((sqrt(y) + sqrt(z)) - sqrt((1.0d0 + z))))
    else
        tmp = sqrt((x + 1.0d0)) + ((t_1 - sqrt(y)) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double tmp;
	if (z <= 2.5e+16) {
		tmp = 1.0 + (t_1 - ((Math.sqrt(y) + Math.sqrt(z)) - Math.sqrt((1.0 + z))));
	} else {
		tmp = Math.sqrt((x + 1.0)) + ((t_1 - Math.sqrt(y)) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	tmp = 0
	if z <= 2.5e+16:
		tmp = 1.0 + (t_1 - ((math.sqrt(y) + math.sqrt(z)) - math.sqrt((1.0 + z))))
	else:
		tmp = math.sqrt((x + 1.0)) + ((t_1 - math.sqrt(y)) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 2.5e+16)
		tmp = Float64(1.0 + Float64(t_1 - Float64(Float64(sqrt(y) + sqrt(z)) - sqrt(Float64(1.0 + z)))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(Float64(t_1 - sqrt(y)) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	tmp = 0.0;
	if (z <= 2.5e+16)
		tmp = 1.0 + (t_1 - ((sqrt(y) + sqrt(z)) - sqrt((1.0 + z))));
	else
		tmp = sqrt((x + 1.0)) + ((t_1 - sqrt(y)) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 2.5e+16], N[(1.0 + N[(t$95$1 - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 2.5e16

   1. Initial program 96.0%

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

      1. associate-+l+ 96.0%

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

      2. +-commutative 96.0%

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

      3. associate-+r- 76.1%

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

      4. associate-+l- 56.6%

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

      5. +-commutative 56.6%

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

      6. associate--l+ 56.6%

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

      7. +-commutative 56.6%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in t around inf 33.5%

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

      1. +-commutative 33.5%

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

      2. +-commutative 33.5%

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

      3. associate--l+ 33.5%

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

   6. Simplified 33.5%

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

   7. Taylor expanded in x around 0 35.4%

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

      1. associate--l+ 46.9%

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

      2. +-commutative 46.9%

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

      3. associate--l+ 46.9%

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

      4. +-commutative 46.9%

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

   9. Simplified 46.9%

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

   if 2.5e16 < z

   1. Initial program 86.1%

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

      1. associate-+l+ 86.1%

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

      2. +-commutative 86.1%

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

      3. associate-+r- 72.1%

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

      4. associate-+l- 52.8%

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

      5. +-commutative 52.8%

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

      6. associate--l+ 52.8%

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

      7. +-commutative 52.8%

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

   3. Simplified 18.8%

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

   4. Taylor expanded in t around inf 26.9%

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

      1. +-commutative 26.9%

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

      2. +-commutative 26.9%

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

      3. associate--l+ 28.7%

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

   6. Simplified 28.7%

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

   7. Taylor expanded in z around inf 28.7%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{+16}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \left(\left(\sqrt{y} + \sqrt{z}\right) - \sqrt{1 + z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 10: 90.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{1 + t} + 3\right) - \left(\sqrt{z} + \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 2.5e+14)
   (- (+ (sqrt (+ 1.0 t)) 3.0) (+ (sqrt z) (sqrt t)))
   (+ 1.0 (- (sqrt (+ y 1.0)) (+ (sqrt y) (- (sqrt z) (sqrt (+ 1.0 z))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.5e+14) {
		tmp = (sqrt((1.0 + t)) + 3.0) - (sqrt(z) + sqrt(t));
	} else {
		tmp = 1.0 + (sqrt((y + 1.0)) - (sqrt(y) + (sqrt(z) - sqrt((1.0 + z)))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 2.5d+14) then
        tmp = (sqrt((1.0d0 + t)) + 3.0d0) - (sqrt(z) + sqrt(t))
    else
        tmp = 1.0d0 + (sqrt((y + 1.0d0)) - (sqrt(y) + (sqrt(z) - sqrt((1.0d0 + z)))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.5e+14) {
		tmp = (Math.sqrt((1.0 + t)) + 3.0) - (Math.sqrt(z) + Math.sqrt(t));
	} else {
		tmp = 1.0 + (Math.sqrt((y + 1.0)) - (Math.sqrt(y) + (Math.sqrt(z) - Math.sqrt((1.0 + z)))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 2.5e+14:
		tmp = (math.sqrt((1.0 + t)) + 3.0) - (math.sqrt(z) + math.sqrt(t))
	else:
		tmp = 1.0 + (math.sqrt((y + 1.0)) - (math.sqrt(y) + (math.sqrt(z) - math.sqrt((1.0 + z)))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 2.5e+14)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) + 3.0) - Float64(sqrt(z) + sqrt(t)));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(y) + Float64(sqrt(z) - sqrt(Float64(1.0 + z))))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 2.5e+14)
		tmp = (sqrt((1.0 + t)) + 3.0) - (sqrt(z) + sqrt(t));
	else
		tmp = 1.0 + (sqrt((y + 1.0)) - (sqrt(y) + (sqrt(z) - sqrt((1.0 + z)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 2.5e+14], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.5e14

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. sub-neg 96.2%

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

      3. associate-+l+ 81.8%

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

      4. associate-+l+ 56.4%

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

      5. +-commutative 56.4%

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

      6. neg-sub0 56.4%

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

      7. associate-+l- 56.4%

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

      8. neg-sub0 56.4%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in y around 0 35.3%

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

   5. Taylor expanded in z around 0 21.8%

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

   6. Taylor expanded in x around 0 23.6%

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

   if 2.5e14 < t

   1. Initial program 86.2%

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

      1. associate-+l+ 86.2%

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

      2. +-commutative 86.2%

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

      3. associate-+r- 65.5%

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

      4. associate-+l- 53.1%

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

      5. +-commutative 53.1%

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

      6. associate--l+ 53.1%

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

      7. +-commutative 53.1%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in t around inf 50.9%

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

      1. +-commutative 50.9%

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

      2. +-commutative 50.9%

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

      3. associate--l+ 53.1%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in x around 0 25.0%

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

      1. associate--l+ 53.4%

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

      2. +-commutative 53.4%

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

      3. +-commutative 53.4%

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

   9. Applied egg-rr 53.4%

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

       1. associate--l+ 49.8%

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

       2. +-commutative 49.8%

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

       3. associate--r+ 54.8%

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

   11. Simplified 54.8%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{1 + t} + 3\right) - \left(\sqrt{z} + \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\right)\\ \end{array} \]

Alternative 11: 90.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 2.55 \cdot 10^{+14}:\\ \;\;\;\;2 + \left(t_1 + \left(\sqrt{1 + t} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \left(\sqrt{y} + \left(\sqrt{z} - t_1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))))
   (if (<= t 2.55e+14)
     (+ 2.0 (+ t_1 (- (sqrt (+ 1.0 t)) (+ (sqrt z) (sqrt t)))))
     (+ 1.0 (- (sqrt (+ y 1.0)) (+ (sqrt y) (- (sqrt z) t_1)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double tmp;
	if (t <= 2.55e+14) {
		tmp = 2.0 + (t_1 + (sqrt((1.0 + t)) - (sqrt(z) + sqrt(t))));
	} else {
		tmp = 1.0 + (sqrt((y + 1.0)) - (sqrt(y) + (sqrt(z) - t_1)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if (t <= 2.55d+14) then
        tmp = 2.0d0 + (t_1 + (sqrt((1.0d0 + t)) - (sqrt(z) + sqrt(t))))
    else
        tmp = 1.0d0 + (sqrt((y + 1.0d0)) - (sqrt(y) + (sqrt(z) - t_1)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double tmp;
	if (t <= 2.55e+14) {
		tmp = 2.0 + (t_1 + (Math.sqrt((1.0 + t)) - (Math.sqrt(z) + Math.sqrt(t))));
	} else {
		tmp = 1.0 + (Math.sqrt((y + 1.0)) - (Math.sqrt(y) + (Math.sqrt(z) - t_1)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	tmp = 0
	if t <= 2.55e+14:
		tmp = 2.0 + (t_1 + (math.sqrt((1.0 + t)) - (math.sqrt(z) + math.sqrt(t))))
	else:
		tmp = 1.0 + (math.sqrt((y + 1.0)) - (math.sqrt(y) + (math.sqrt(z) - t_1)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (t <= 2.55e+14)
		tmp = Float64(2.0 + Float64(t_1 + Float64(sqrt(Float64(1.0 + t)) - Float64(sqrt(z) + sqrt(t)))));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(y) + Float64(sqrt(z) - t_1))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	tmp = 0.0;
	if (t <= 2.55e+14)
		tmp = 2.0 + (t_1 + (sqrt((1.0 + t)) - (sqrt(z) + sqrt(t))));
	else
		tmp = 1.0 + (sqrt((y + 1.0)) - (sqrt(y) + (sqrt(z) - t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 2.55e+14], N[(2.0 + N[(t$95$1 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.55e14

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. sub-neg 96.2%

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

      3. associate-+l+ 81.8%

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

      4. associate-+l+ 56.4%

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

      5. +-commutative 56.4%

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

      6. neg-sub0 56.4%

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

      7. associate-+l- 56.4%

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

      8. neg-sub0 56.4%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in y around 0 35.3%

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

   5. Taylor expanded in x around 0 25.9%

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

      1. associate--l+ 45.3%

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

      2. +-commutative 45.3%

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

      3. associate--l+ 45.3%

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

      4. +-commutative 45.3%

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

   7. Simplified 45.3%

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

   if 2.55e14 < t

   1. Initial program 86.2%

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

      1. associate-+l+ 86.2%

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

      2. +-commutative 86.2%

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

      3. associate-+r- 65.5%

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

      4. associate-+l- 53.1%

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

      5. +-commutative 53.1%

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

      6. associate--l+ 53.1%

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

      7. +-commutative 53.1%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in t around inf 50.9%

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

      1. +-commutative 50.9%

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

      2. +-commutative 50.9%

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

      3. associate--l+ 53.1%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in x around 0 25.0%

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

      1. associate--l+ 53.4%

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

      2. +-commutative 53.4%

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

      3. +-commutative 53.4%

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

   9. Applied egg-rr 53.4%

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

       1. associate--l+ 49.8%

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

       2. +-commutative 49.8%

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

       3. associate--r+ 54.8%

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

   11. Simplified 54.8%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.55 \cdot 10^{+14}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} + \left(\sqrt{1 + t} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\right)\\ \end{array} \]

Alternative 12: 85.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.85e+15)
   (- (+ (sqrt (+ 1.0 z)) 2.0) (sqrt z))
   (+ (sqrt (+ x 1.0)) (- (- (sqrt (+ y 1.0)) (sqrt y)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.85e+15) {
		tmp = (sqrt((1.0 + z)) + 2.0) - sqrt(z);
	} else {
		tmp = sqrt((x + 1.0)) + ((sqrt((y + 1.0)) - sqrt(y)) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.85d+15) then
        tmp = (sqrt((1.0d0 + z)) + 2.0d0) - sqrt(z)
    else
        tmp = sqrt((x + 1.0d0)) + ((sqrt((y + 1.0d0)) - sqrt(y)) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.85e+15) {
		tmp = (Math.sqrt((1.0 + z)) + 2.0) - Math.sqrt(z);
	} else {
		tmp = Math.sqrt((x + 1.0)) + ((Math.sqrt((y + 1.0)) - Math.sqrt(y)) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.85e+15:
		tmp = (math.sqrt((1.0 + z)) + 2.0) - math.sqrt(z)
	else:
		tmp = math.sqrt((x + 1.0)) + ((math.sqrt((y + 1.0)) - math.sqrt(y)) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.85e+15)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) + 2.0) - sqrt(z));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.85e+15)
		tmp = (sqrt((1.0 + z)) + 2.0) - sqrt(z);
	else
		tmp = sqrt((x + 1.0)) + ((sqrt((y + 1.0)) - sqrt(y)) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 1.85e+15], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.85e15

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. +-commutative 96.2%

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

      3. associate-+r- 76.2%

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

      4. associate-+l- 56.5%

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

      5. +-commutative 56.5%

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

      6. associate--l+ 56.5%

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

      7. +-commutative 56.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in t around inf 33.6%

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

      1. +-commutative 33.6%

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

      2. +-commutative 33.6%

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

      3. associate--l+ 33.6%

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

   6. Simplified 33.6%

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

   7. Taylor expanded in x around 0 35.6%

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

   8. Taylor expanded in y around 0 46.5%

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

   if 1.85e15 < z

   1. Initial program 85.9%

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

      1. associate-+l+ 85.9%

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

      2. +-commutative 85.9%

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

      3. associate-+r- 72.0%

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

      4. associate-+l- 52.9%

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

      5. +-commutative 52.9%

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

      6. associate--l+ 52.9%

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

      7. +-commutative 52.9%

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

   3. Simplified 19.2%

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

   4. Taylor expanded in t around inf 26.9%

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

      1. +-commutative 26.9%

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

      2. +-commutative 26.9%

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

      3. associate--l+ 28.6%

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

   6. Simplified 28.6%

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

   7. Taylor expanded in z around inf 28.6%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 13: 84.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{+15}:\\ \;\;\;\;2 - \left(\sqrt{z} - \sqrt{1 + z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \end{array} \end{array} \]

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 1.85e+15)
   (- 2.0 (- (sqrt z) (sqrt (+ 1.0 z))))
   (+ 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 <= 1.85e+15) {
		tmp = 2.0 - (sqrt(z) - sqrt((1.0 + z)));
	} else {
		tmp = 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 <= 1.85d+15) then
        tmp = 2.0d0 - (sqrt(z) - sqrt((1.0d0 + z)))
    else
        tmp = 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 <= 1.85e+15) {
		tmp = 2.0 - (Math.sqrt(z) - Math.sqrt((1.0 + z)));
	} else {
		tmp = 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 <= 1.85e+15:
		tmp = 2.0 - (math.sqrt(z) - math.sqrt((1.0 + z)))
	else:
		tmp = 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 <= 1.85e+15)
		tmp = Float64(2.0 - Float64(sqrt(z) - sqrt(Float64(1.0 + z))));
	else
		tmp = Float64(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 <= 1.85e+15)
		tmp = 2.0 - (sqrt(z) - sqrt((1.0 + z)));
	else
		tmp = 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, 1.85e+15], N[(2.0 - N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + 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 < 1.85e15

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. +-commutative 96.2%

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

      3. associate-+r- 76.2%

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

      4. associate-+l- 56.5%

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

      5. +-commutative 56.5%

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

      6. associate--l+ 56.5%

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

      7. +-commutative 56.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in t around inf 33.6%

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

      1. +-commutative 33.6%

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

      2. +-commutative 33.6%

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

      3. associate--l+ 33.6%

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

   6. Simplified 33.6%

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

   7. Taylor expanded in x around 0 35.6%

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

   8. Taylor expanded in y around 0 46.5%

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

      1. associate--l+ 46.5%

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

   10. Simplified 46.5%

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

   if 1.85e15 < z

   1. Initial program 85.9%

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

      1. associate-+l+ 85.9%

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

      2. +-commutative 85.9%

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

      3. associate-+r- 72.0%

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

      4. associate-+l- 52.9%

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

      5. +-commutative 52.9%

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

      6. associate--l+ 52.9%

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

      7. +-commutative 52.9%

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

   3. Simplified 19.2%

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

   4. Taylor expanded in t around inf 26.9%

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

      1. +-commutative 26.9%

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

      2. +-commutative 26.9%

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

      3. associate--l+ 28.6%

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

   6. Simplified 28.6%

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

   7. Taylor expanded in x around inf 4.0%

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

      1. associate--l+ 17.9%

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

      2. +-commutative 17.9%

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

   9. Simplified 17.9%

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

   10. Taylor expanded in z around 0 27.6%

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

       1. associate--l+ 55.3%

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

   12. Simplified 55.3%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{+15}:\\ \;\;\;\;2 - \left(\sqrt{z} - \sqrt{1 + z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \end{array} \]

Alternative 14: 84.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \end{array} \end{array} \]

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 1.85e+15)
   (- (+ (sqrt (+ 1.0 z)) 2.0) (sqrt z))
   (+ 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 <= 1.85e+15) {
		tmp = (sqrt((1.0 + z)) + 2.0) - sqrt(z);
	} else {
		tmp = 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 <= 1.85d+15) then
        tmp = (sqrt((1.0d0 + z)) + 2.0d0) - sqrt(z)
    else
        tmp = 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 <= 1.85e+15) {
		tmp = (Math.sqrt((1.0 + z)) + 2.0) - Math.sqrt(z);
	} else {
		tmp = 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 <= 1.85e+15:
		tmp = (math.sqrt((1.0 + z)) + 2.0) - math.sqrt(z)
	else:
		tmp = 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 <= 1.85e+15)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) + 2.0) - sqrt(z));
	else
		tmp = Float64(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 <= 1.85e+15)
		tmp = (sqrt((1.0 + z)) + 2.0) - sqrt(z);
	else
		tmp = 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, 1.85e+15], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.85e15

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. +-commutative 96.2%

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

      3. associate-+r- 76.2%

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

      4. associate-+l- 56.5%

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

      5. +-commutative 56.5%

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

      6. associate--l+ 56.5%

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

      7. +-commutative 56.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in t around inf 33.6%

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

      1. +-commutative 33.6%

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

      2. +-commutative 33.6%

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

      3. associate--l+ 33.6%

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

   6. Simplified 33.6%

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

   7. Taylor expanded in x around 0 35.6%

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

   8. Taylor expanded in y around 0 46.5%

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

   if 1.85e15 < z

   1. Initial program 85.9%

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

      1. associate-+l+ 85.9%

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

      2. +-commutative 85.9%

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

      3. associate-+r- 72.0%

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

      4. associate-+l- 52.9%

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

      5. +-commutative 52.9%

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

      6. associate--l+ 52.9%

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

      7. +-commutative 52.9%

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

   3. Simplified 19.2%

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

   4. Taylor expanded in t around inf 26.9%

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

      1. +-commutative 26.9%

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

      2. +-commutative 26.9%

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

      3. associate--l+ 28.6%

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

   6. Simplified 28.6%

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

   7. Taylor expanded in x around inf 4.0%

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

      1. associate--l+ 17.9%

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

      2. +-commutative 17.9%

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

   9. Simplified 17.9%

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

   10. Taylor expanded in z around 0 27.6%

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

       1. associate--l+ 55.3%

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

   12. Simplified 55.3%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \end{array} \]

Alternative 15: 64.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. associate-+l+ 91.6%

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

   2. +-commutative 91.6%

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

   3. associate-+r- 74.3%

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

   4. associate-+l- 54.9%

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

   5. +-commutative 54.9%

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

   6. associate--l+ 54.9%

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

   7. +-commutative 54.9%

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

3. Simplified 37.7%

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

4. Taylor expanded in t around inf 30.6%

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

   1. +-commutative 30.6%

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

   2. +-commutative 30.6%

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

   3. associate--l+ 31.4%

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

6. Simplified 31.4%

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

7. Taylor expanded in x around inf 16.3%

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

   1. associate--l+ 22.4%

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

   2. +-commutative 22.4%

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

9. Simplified 22.4%

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

10. Taylor expanded in z around 0 27.0%

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

    1. associate--l+ 45.7%

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

12. Simplified 45.7%

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

13. Final simplification 45.7%

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

Alternative 16: 13.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. associate-+l+ 91.6%

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

   2. +-commutative 91.6%

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

   3. associate-+r- 74.3%

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

   4. associate-+l- 54.9%

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

   5. +-commutative 54.9%

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

   6. associate--l+ 54.9%

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

   7. +-commutative 54.9%

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

3. Simplified 37.7%

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

4. Taylor expanded in t around inf 30.6%

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

   1. +-commutative 30.6%

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

   2. +-commutative 30.6%

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

   3. associate--l+ 31.4%

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

6. Simplified 31.4%

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

7. Taylor expanded in x around inf 16.3%

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

   1. associate--l+ 22.4%

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

   2. +-commutative 22.4%

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

9. Simplified 22.4%

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

10. Taylor expanded in z around inf 14.4%

    \[\leadsto \color{blue}{\sqrt{1 + y} - \sqrt{y}} \]

11. Final simplification 14.4%

    \[\leadsto \sqrt{y + 1} - \sqrt{y} \]

Developer target: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023229 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :herbie-target
  (+ (+ (+ (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))) (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y)))) (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z)))) (- (sqrt (+ t 1.0)) (sqrt t)))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))