Main:z from

Percentage Accurate: 91.8% → 97.5%

Time: 26.9s

Alternatives: 23

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.8% 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: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := t\_1 - \sqrt{z}\\ t_3 := \sqrt{1 + t}\\ t_4 := t\_3 - \sqrt{t}\\ t_5 := \left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\\ t_6 := t\_2 + t\_5\\ \mathbf{if}\;t\_6 \leq 0:\\ \;\;\;\;t\_4 + \left(t\_2 + 0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{x}}\right)\right)\\ \mathbf{elif}\;t\_6 \leq 2.005:\\ \;\;\;\;t\_4 + \left(t\_5 + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + t\right) - t}{\sqrt{t} + t\_3} + \left(2 + \left(t\_1 - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)\\ \end{array} \end{array} \]

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)))
        (t_2 (- t_1 (sqrt z)))
        (t_3 (sqrt (+ 1.0 t)))
        (t_4 (- t_3 (sqrt t)))
        (t_5 (- (- (sqrt (+ 1.0 y)) (sqrt y)) (- (sqrt x) (sqrt (+ x 1.0)))))
        (t_6 (+ t_2 t_5)))
   (if (<= t_6 0.0)
     (+ t_4 (+ t_2 (* 0.5 (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 x))))))
     (if (<= t_6 2.005)
       (+ t_4 (+ t_5 (* 0.5 (sqrt (/ 1.0 z)))))
       (+
        (/ (- (+ 1.0 t) t) (+ (sqrt t) t_3))
        (+ 2.0 (- t_1 (+ (sqrt x) (+ (sqrt z) (sqrt y))))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = t_1 - sqrt(z);
	double t_3 = sqrt((1.0 + t));
	double t_4 = t_3 - sqrt(t);
	double t_5 = (sqrt((1.0 + y)) - sqrt(y)) - (sqrt(x) - sqrt((x + 1.0)));
	double t_6 = t_2 + t_5;
	double tmp;
	if (t_6 <= 0.0) {
		tmp = t_4 + (t_2 + (0.5 * (sqrt((1.0 / y)) + sqrt((1.0 / x)))));
	} else if (t_6 <= 2.005) {
		tmp = t_4 + (t_5 + (0.5 * sqrt((1.0 / z))));
	} else {
		tmp = (((1.0 + t) - t) / (sqrt(t) + t_3)) + (2.0 + (t_1 - (sqrt(x) + (sqrt(z) + sqrt(y)))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = t_1 - sqrt(z)
    t_3 = sqrt((1.0d0 + t))
    t_4 = t_3 - sqrt(t)
    t_5 = (sqrt((1.0d0 + y)) - sqrt(y)) - (sqrt(x) - sqrt((x + 1.0d0)))
    t_6 = t_2 + t_5
    if (t_6 <= 0.0d0) then
        tmp = t_4 + (t_2 + (0.5d0 * (sqrt((1.0d0 / y)) + sqrt((1.0d0 / x)))))
    else if (t_6 <= 2.005d0) then
        tmp = t_4 + (t_5 + (0.5d0 * sqrt((1.0d0 / z))))
    else
        tmp = (((1.0d0 + t) - t) / (sqrt(t) + t_3)) + (2.0d0 + (t_1 - (sqrt(x) + (sqrt(z) + sqrt(y)))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = t_1 - Math.sqrt(z);
	double t_3 = Math.sqrt((1.0 + t));
	double t_4 = t_3 - Math.sqrt(t);
	double t_5 = (Math.sqrt((1.0 + y)) - Math.sqrt(y)) - (Math.sqrt(x) - Math.sqrt((x + 1.0)));
	double t_6 = t_2 + t_5;
	double tmp;
	if (t_6 <= 0.0) {
		tmp = t_4 + (t_2 + (0.5 * (Math.sqrt((1.0 / y)) + Math.sqrt((1.0 / x)))));
	} else if (t_6 <= 2.005) {
		tmp = t_4 + (t_5 + (0.5 * Math.sqrt((1.0 / z))));
	} else {
		tmp = (((1.0 + t) - t) / (Math.sqrt(t) + t_3)) + (2.0 + (t_1 - (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y)))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = t_1 - math.sqrt(z)
	t_3 = math.sqrt((1.0 + t))
	t_4 = t_3 - math.sqrt(t)
	t_5 = (math.sqrt((1.0 + y)) - math.sqrt(y)) - (math.sqrt(x) - math.sqrt((x + 1.0)))
	t_6 = t_2 + t_5
	tmp = 0
	if t_6 <= 0.0:
		tmp = t_4 + (t_2 + (0.5 * (math.sqrt((1.0 / y)) + math.sqrt((1.0 / x)))))
	elif t_6 <= 2.005:
		tmp = t_4 + (t_5 + (0.5 * math.sqrt((1.0 / z))))
	else:
		tmp = (((1.0 + t) - t) / (math.sqrt(t) + t_3)) + (2.0 + (t_1 - (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y)))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = Float64(t_1 - sqrt(z))
	t_3 = sqrt(Float64(1.0 + t))
	t_4 = Float64(t_3 - sqrt(t))
	t_5 = Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) - Float64(sqrt(x) - sqrt(Float64(x + 1.0))))
	t_6 = Float64(t_2 + t_5)
	tmp = 0.0
	if (t_6 <= 0.0)
		tmp = Float64(t_4 + Float64(t_2 + Float64(0.5 * Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / x))))));
	elseif (t_6 <= 2.005)
		tmp = Float64(t_4 + Float64(t_5 + Float64(0.5 * sqrt(Float64(1.0 / z)))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + t) - t) / Float64(sqrt(t) + t_3)) + Float64(2.0 + Float64(t_1 - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y))))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = t_1 - sqrt(z);
	t_3 = sqrt((1.0 + t));
	t_4 = t_3 - sqrt(t);
	t_5 = (sqrt((1.0 + y)) - sqrt(y)) - (sqrt(x) - sqrt((x + 1.0)));
	t_6 = t_2 + t_5;
	tmp = 0.0;
	if (t_6 <= 0.0)
		tmp = t_4 + (t_2 + (0.5 * (sqrt((1.0 / y)) + sqrt((1.0 / x)))));
	elseif (t_6 <= 2.005)
		tmp = t_4 + (t_5 + (0.5 * sqrt((1.0 / z))));
	else
		tmp = (((1.0 + t) - t) / (sqrt(t) + t_3)) + (2.0 + (t_1 - (sqrt(x) + (sqrt(z) + sqrt(y)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$2 + t$95$5), $MachinePrecision]}, If[LessEqual[t$95$6, 0.0], N[(t$95$4 + N[(t$95$2 + N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.005], N[(t$95$4 + N[(t$95$5 + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + t), $MachinePrecision] - t), $MachinePrecision] / N[(N[Sqrt[t], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0

   1. Initial program 61.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 76.7%

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

   if 0.0 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.0049999999999999

   1. Initial program 97.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 53.8

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

   5. Applied rewrites 53.8%

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

   if 2.0049999999999999 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 93.1

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

   5. Applied rewrites 93.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 93.1

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 93.1

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 88.4%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right) \leq 0:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{x}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right) \leq 2.005:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + t\right) - t}{\sqrt{t} + \sqrt{1 + t}} + \left(2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + z}\\ \mathbf{if}\;t\_2 - \sqrt{x} \leq 0.004:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.0390625, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \sqrt{x} \cdot 0.5\right)\right)\right)}{x}\right) + \left(t\_3 - \sqrt{z}\right)\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - t\_2\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + t\_3}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (+ 1.0 z))))
   (if (<= (- t_2 (sqrt x)) 0.004)
     (+
      (+
       (fma
        0.5
        (sqrt (/ 1.0 y))
        (/
         (fma
          -0.125
          (sqrt (/ 1.0 x))
          (fma
           (sqrt (/ 1.0 (pow x 5.0)))
           -0.0390625
           (fma 0.0625 (sqrt (/ 1.0 (* x (* x x)))) (* (sqrt x) 0.5))))
         x))
       (- t_3 (sqrt z)))
      t_1)
     (+
      t_1
      (+
       (- (- (sqrt (+ 1.0 y)) (sqrt y)) (- (sqrt x) t_2))
       (/ (- (+ 1.0 z) z) (+ (sqrt z) t_3)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((1.0 + z));
	double tmp;
	if ((t_2 - sqrt(x)) <= 0.004) {
		tmp = (fma(0.5, sqrt((1.0 / y)), (fma(-0.125, sqrt((1.0 / x)), fma(sqrt((1.0 / pow(x, 5.0))), -0.0390625, fma(0.0625, sqrt((1.0 / (x * (x * x)))), (sqrt(x) * 0.5)))) / x)) + (t_3 - sqrt(z))) + t_1;
	} else {
		tmp = t_1 + (((sqrt((1.0 + y)) - sqrt(y)) - (sqrt(x) - t_2)) + (((1.0 + z) - z) / (sqrt(z) + t_3)));
	}
	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 + t)) - sqrt(t))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (Float64(t_2 - sqrt(x)) <= 0.004)
		tmp = Float64(Float64(fma(0.5, sqrt(Float64(1.0 / y)), Float64(fma(-0.125, sqrt(Float64(1.0 / x)), fma(sqrt(Float64(1.0 / (x ^ 5.0))), -0.0390625, fma(0.0625, sqrt(Float64(1.0 / Float64(x * Float64(x * x)))), Float64(sqrt(x) * 0.5)))) / x)) + Float64(t_3 - sqrt(z))) + t_1);
	else
		tmp = Float64(t_1 + Float64(Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) - Float64(sqrt(x) - t_2)) + Float64(Float64(Float64(1.0 + z) - z) / Float64(sqrt(z) + t_3))));
	end
	return 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 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.004], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[(N[(-0.125 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.0390625 + N[(0.0625 * N[Sqrt[N[(1.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$1 + N[(N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + z), $MachinePrecision] - z), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0040000000000000001

   1. Initial program 90.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 42.7

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

   5. Applied rewrites 42.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{-5}{128} \cdot \sqrt{\frac{1}{{x}^{5}}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 43.7%

      \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.0390625, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \sqrt{x} \cdot 0.5\right)\right)\right)}{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0040000000000000001 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 98.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 98.6

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.6

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

   4. Applied rewrites 98.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.004:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.0390625, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \sqrt{x} \cdot 0.5\right)\right)\right)}{x}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{x + 1}\\ t_4 := \sqrt{1 + z}\\ t_5 := t\_2 + \left(\left(t\_4 - \sqrt{z}\right) + \left(\left(t\_1 - \sqrt{y}\right) - \left(\sqrt{x} - t\_3\right)\right)\right)\\ t_6 := \sqrt{z} + t\_4\\ \mathbf{if}\;t\_5 \leq 1.005:\\ \;\;\;\;t\_3 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{t\_6}\right) - \sqrt{x}\right)\\ \mathbf{elif}\;t\_5 \leq 3.0000005:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{-1}{t\_6}, t\_1 + \left(\left(1 - \sqrt{x}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(\left(t\_4 + 2\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

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 y)))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (sqrt (+ 1.0 z)))
        (t_5
         (+ t_2 (+ (- t_4 (sqrt z)) (- (- t_1 (sqrt y)) (- (sqrt x) t_3)))))
        (t_6 (+ (sqrt z) t_4)))
   (if (<= t_5 1.005)
     (+ t_3 (- (fma 0.5 (sqrt (/ 1.0 y)) (/ 1.0 t_6)) (sqrt x)))
     (if (<= t_5 3.0000005)
       (fma -1.0 (/ -1.0 t_6) (+ t_1 (- (- 1.0 (sqrt x)) (sqrt y))))
       (+ t_2 (- (+ t_4 2.0) (+ (sqrt x) (+ (sqrt z) (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double t_3 = sqrt((x + 1.0));
	double t_4 = sqrt((1.0 + z));
	double t_5 = t_2 + ((t_4 - sqrt(z)) + ((t_1 - sqrt(y)) - (sqrt(x) - t_3)));
	double t_6 = sqrt(z) + t_4;
	double tmp;
	if (t_5 <= 1.005) {
		tmp = t_3 + (fma(0.5, sqrt((1.0 / y)), (1.0 / t_6)) - sqrt(x));
	} else if (t_5 <= 3.0000005) {
		tmp = fma(-1.0, (-1.0 / t_6), (t_1 + ((1.0 - sqrt(x)) - sqrt(y))));
	} else {
		tmp = t_2 + ((t_4 + 2.0) - (sqrt(x) + (sqrt(z) + sqrt(y))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = sqrt(Float64(1.0 + z))
	t_5 = Float64(t_2 + Float64(Float64(t_4 - sqrt(z)) + Float64(Float64(t_1 - sqrt(y)) - Float64(sqrt(x) - t_3))))
	t_6 = Float64(sqrt(z) + t_4)
	tmp = 0.0
	if (t_5 <= 1.005)
		tmp = Float64(t_3 + Float64(fma(0.5, sqrt(Float64(1.0 / y)), Float64(1.0 / t_6)) - sqrt(x)));
	elseif (t_5 <= 3.0000005)
		tmp = fma(-1.0, Float64(-1.0 / t_6), Float64(t_1 + Float64(Float64(1.0 - sqrt(x)) - sqrt(y))));
	else
		tmp = Float64(t_2 + Float64(Float64(t_4 + 2.0) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 + N[(N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[z], $MachinePrecision] + t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, 1.005], N[(t$95$3 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[(1.0 / t$95$6), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 3.0000005], N[(-1.0 * N[(-1.0 / t$95$6), $MachinePrecision] + N[(t$95$1 + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(t$95$4 + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.0049999999999999

   1. Initial program 86.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 86.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 86.0

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

   4. Applied rewrites 86.0%

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

   5. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 N/A

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

   7. Applied rewrites 4.6%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 4.6%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 29.9%

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

   if 1.0049999999999999 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.00000050000000007

   1. Initial program 97.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 98.3

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 N/A

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

   7. Applied rewrites 25.2%

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

   9. Applied rewrites 39.0%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 26.4%

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

   if 3.00000050000000007 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.8%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right)\right) \leq 1.005:\\ \;\;\;\;\sqrt{x + 1} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \sqrt{x}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right)\right) \leq 3.0000005:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{-1}{\sqrt{z} + \sqrt{1 + z}}, \sqrt{1 + y} + \left(\left(1 - \sqrt{x}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{x + 1}\\ t_4 := \sqrt{1 + z}\\ t_5 := t\_2 + \left(\left(t\_4 - \sqrt{z}\right) + \left(\left(t\_1 - \sqrt{y}\right) - \left(\sqrt{x} - t\_3\right)\right)\right)\\ \mathbf{if}\;t\_5 \leq 1:\\ \;\;\;\;\left(t\_3 - \sqrt{x}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{elif}\;t\_5 \leq 3.0000005:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{-1}{\sqrt{z} + t\_4}, t\_1 + \left(\left(1 - \sqrt{x}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(\left(t\_4 + 2\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

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 y)))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (sqrt (+ 1.0 z)))
        (t_5
         (+ t_2 (+ (- t_4 (sqrt z)) (- (- t_1 (sqrt y)) (- (sqrt x) t_3))))))
   (if (<= t_5 1.0)
     (+ (- t_3 (sqrt x)) (* 0.5 (sqrt (/ 1.0 t))))
     (if (<= t_5 3.0000005)
       (fma
        -1.0
        (/ -1.0 (+ (sqrt z) t_4))
        (+ t_1 (- (- 1.0 (sqrt x)) (sqrt y))))
       (+ t_2 (- (+ t_4 2.0) (+ (sqrt x) (+ (sqrt z) (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double t_3 = sqrt((x + 1.0));
	double t_4 = sqrt((1.0 + z));
	double t_5 = t_2 + ((t_4 - sqrt(z)) + ((t_1 - sqrt(y)) - (sqrt(x) - t_3)));
	double tmp;
	if (t_5 <= 1.0) {
		tmp = (t_3 - sqrt(x)) + (0.5 * sqrt((1.0 / t)));
	} else if (t_5 <= 3.0000005) {
		tmp = fma(-1.0, (-1.0 / (sqrt(z) + t_4)), (t_1 + ((1.0 - sqrt(x)) - sqrt(y))));
	} else {
		tmp = t_2 + ((t_4 + 2.0) - (sqrt(x) + (sqrt(z) + sqrt(y))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = sqrt(Float64(1.0 + z))
	t_5 = Float64(t_2 + Float64(Float64(t_4 - sqrt(z)) + Float64(Float64(t_1 - sqrt(y)) - Float64(sqrt(x) - t_3))))
	tmp = 0.0
	if (t_5 <= 1.0)
		tmp = Float64(Float64(t_3 - sqrt(x)) + Float64(0.5 * sqrt(Float64(1.0 / t))));
	elseif (t_5 <= 3.0000005)
		tmp = fma(-1.0, Float64(-1.0 / Float64(sqrt(z) + t_4)), Float64(t_1 + Float64(Float64(1.0 - sqrt(x)) - sqrt(y))));
	else
		tmp = Float64(t_2 + Float64(Float64(t_4 + 2.0) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 + N[(N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 1.0], N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 3.0000005], N[(-1.0 * N[(-1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(t$95$4 + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   1. Initial program 86.6%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 38.7%

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

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.00000050000000007

   1. Initial program 97.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 97.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 N/A

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

   7. Applied rewrites 24.9%

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

   9. Applied rewrites 39.0%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 26.1%

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

   if 3.00000050000000007 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.8%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right)\right) \leq 1:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right)\right) \leq 3.0000005:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{-1}{\sqrt{z} + \sqrt{1 + z}}, \sqrt{1 + y} + \left(\left(1 - \sqrt{x}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{x + 1}\\ t_4 := \sqrt{1 + z}\\ t_5 := t\_2 + \left(\left(t\_4 - \sqrt{z}\right) + \left(\left(t\_1 - \sqrt{y}\right) - \left(\sqrt{x} - t\_3\right)\right)\right)\\ \mathbf{if}\;t\_5 \leq 1:\\ \;\;\;\;\left(t\_3 - \sqrt{x}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{elif}\;t\_5 \leq 3.0000005:\\ \;\;\;\;\left(1 + \left(t\_1 + \frac{1}{\sqrt{z} + t\_4}\right)\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(\left(t\_4 + 2\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

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 y)))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (sqrt (+ 1.0 z)))
        (t_5
         (+ t_2 (+ (- t_4 (sqrt z)) (- (- t_1 (sqrt y)) (- (sqrt x) t_3))))))
   (if (<= t_5 1.0)
     (+ (- t_3 (sqrt x)) (* 0.5 (sqrt (/ 1.0 t))))
     (if (<= t_5 3.0000005)
       (- (+ 1.0 (+ t_1 (/ 1.0 (+ (sqrt z) t_4)))) (sqrt y))
       (+ t_2 (- (+ t_4 2.0) (+ (sqrt x) (+ (sqrt z) (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double t_3 = sqrt((x + 1.0));
	double t_4 = sqrt((1.0 + z));
	double t_5 = t_2 + ((t_4 - sqrt(z)) + ((t_1 - sqrt(y)) - (sqrt(x) - t_3)));
	double tmp;
	if (t_5 <= 1.0) {
		tmp = (t_3 - sqrt(x)) + (0.5 * sqrt((1.0 / t)));
	} else if (t_5 <= 3.0000005) {
		tmp = (1.0 + (t_1 + (1.0 / (sqrt(z) + t_4)))) - sqrt(y);
	} else {
		tmp = t_2 + ((t_4 + 2.0) - (sqrt(x) + (sqrt(z) + sqrt(y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + t)) - sqrt(t)
    t_3 = sqrt((x + 1.0d0))
    t_4 = sqrt((1.0d0 + z))
    t_5 = t_2 + ((t_4 - sqrt(z)) + ((t_1 - sqrt(y)) - (sqrt(x) - t_3)))
    if (t_5 <= 1.0d0) then
        tmp = (t_3 - sqrt(x)) + (0.5d0 * sqrt((1.0d0 / t)))
    else if (t_5 <= 3.0000005d0) then
        tmp = (1.0d0 + (t_1 + (1.0d0 / (sqrt(z) + t_4)))) - sqrt(y)
    else
        tmp = t_2 + ((t_4 + 2.0d0) - (sqrt(x) + (sqrt(z) + sqrt(y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_3 = Math.sqrt((x + 1.0));
	double t_4 = Math.sqrt((1.0 + z));
	double t_5 = t_2 + ((t_4 - Math.sqrt(z)) + ((t_1 - Math.sqrt(y)) - (Math.sqrt(x) - t_3)));
	double tmp;
	if (t_5 <= 1.0) {
		tmp = (t_3 - Math.sqrt(x)) + (0.5 * Math.sqrt((1.0 / t)));
	} else if (t_5 <= 3.0000005) {
		tmp = (1.0 + (t_1 + (1.0 / (Math.sqrt(z) + t_4)))) - Math.sqrt(y);
	} else {
		tmp = t_2 + ((t_4 + 2.0) - (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_3 = math.sqrt((x + 1.0))
	t_4 = math.sqrt((1.0 + z))
	t_5 = t_2 + ((t_4 - math.sqrt(z)) + ((t_1 - math.sqrt(y)) - (math.sqrt(x) - t_3)))
	tmp = 0
	if t_5 <= 1.0:
		tmp = (t_3 - math.sqrt(x)) + (0.5 * math.sqrt((1.0 / t)))
	elif t_5 <= 3.0000005:
		tmp = (1.0 + (t_1 + (1.0 / (math.sqrt(z) + t_4)))) - math.sqrt(y)
	else:
		tmp = t_2 + ((t_4 + 2.0) - (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = sqrt(Float64(1.0 + z))
	t_5 = Float64(t_2 + Float64(Float64(t_4 - sqrt(z)) + Float64(Float64(t_1 - sqrt(y)) - Float64(sqrt(x) - t_3))))
	tmp = 0.0
	if (t_5 <= 1.0)
		tmp = Float64(Float64(t_3 - sqrt(x)) + Float64(0.5 * sqrt(Float64(1.0 / t))));
	elseif (t_5 <= 3.0000005)
		tmp = Float64(Float64(1.0 + Float64(t_1 + Float64(1.0 / Float64(sqrt(z) + t_4)))) - sqrt(y));
	else
		tmp = Float64(t_2 + Float64(Float64(t_4 + 2.0) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((1.0 + t)) - sqrt(t);
	t_3 = sqrt((x + 1.0));
	t_4 = sqrt((1.0 + z));
	t_5 = t_2 + ((t_4 - sqrt(z)) + ((t_1 - sqrt(y)) - (sqrt(x) - t_3)));
	tmp = 0.0;
	if (t_5 <= 1.0)
		tmp = (t_3 - sqrt(x)) + (0.5 * sqrt((1.0 / t)));
	elseif (t_5 <= 3.0000005)
		tmp = (1.0 + (t_1 + (1.0 / (sqrt(z) + t_4)))) - sqrt(y);
	else
		tmp = t_2 + ((t_4 + 2.0) - (sqrt(x) + (sqrt(z) + sqrt(y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + 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[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 + N[(N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 1.0], N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 3.0000005], N[(N[(1.0 + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(t$95$4 + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   1. Initial program 86.6%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 38.7%

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

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.00000050000000007

   1. Initial program 97.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 97.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 N/A

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

   7. Applied rewrites 24.9%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 23.8%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 43.6%

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

   if 3.00000050000000007 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.8%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right)\right) \leq 1:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right)\right) \leq 3.0000005:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + z}\\ t_4 := \left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(t\_3 - \sqrt{z}\right) + \left(\left(t\_1 - \sqrt{y}\right) - \left(\sqrt{x} - t\_2\right)\right)\right)\\ \mathbf{if}\;t\_4 \leq 0.9999999999999899:\\ \;\;\;\;\left(t\_2 - \sqrt{x}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{elif}\;t\_4 \leq 1.9999999999979958:\\ \;\;\;\;1 + \left(\left(t\_1 - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(t\_3 - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

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 y)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4
         (+
          (- (sqrt (+ 1.0 t)) (sqrt t))
          (+ (- t_3 (sqrt z)) (- (- t_1 (sqrt y)) (- (sqrt x) t_2))))))
   (if (<= t_4 0.9999999999999899)
     (+ (- t_2 (sqrt x)) (* 0.5 (sqrt (/ 1.0 t))))
     (if (<= t_4 1.9999999999979958)
       (+ 1.0 (- (- t_1 (sqrt x)) (sqrt y)))
       (+ 2.0 (- t_3 (+ (sqrt x) (+ (sqrt z) (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((1.0 + z));
	double t_4 = (sqrt((1.0 + t)) - sqrt(t)) + ((t_3 - sqrt(z)) + ((t_1 - sqrt(y)) - (sqrt(x) - t_2)));
	double tmp;
	if (t_4 <= 0.9999999999999899) {
		tmp = (t_2 - sqrt(x)) + (0.5 * sqrt((1.0 / t)));
	} else if (t_4 <= 1.9999999999979958) {
		tmp = 1.0 + ((t_1 - sqrt(x)) - sqrt(y));
	} else {
		tmp = 2.0 + (t_3 - (sqrt(x) + (sqrt(z) + sqrt(y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((x + 1.0d0))
    t_3 = sqrt((1.0d0 + z))
    t_4 = (sqrt((1.0d0 + t)) - sqrt(t)) + ((t_3 - sqrt(z)) + ((t_1 - sqrt(y)) - (sqrt(x) - t_2)))
    if (t_4 <= 0.9999999999999899d0) then
        tmp = (t_2 - sqrt(x)) + (0.5d0 * sqrt((1.0d0 / t)))
    else if (t_4 <= 1.9999999999979958d0) then
        tmp = 1.0d0 + ((t_1 - sqrt(x)) - sqrt(y))
    else
        tmp = 2.0d0 + (t_3 - (sqrt(x) + (sqrt(z) + sqrt(y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = Math.sqrt((1.0 + z));
	double t_4 = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) + ((t_3 - Math.sqrt(z)) + ((t_1 - Math.sqrt(y)) - (Math.sqrt(x) - t_2)));
	double tmp;
	if (t_4 <= 0.9999999999999899) {
		tmp = (t_2 - Math.sqrt(x)) + (0.5 * Math.sqrt((1.0 / t)));
	} else if (t_4 <= 1.9999999999979958) {
		tmp = 1.0 + ((t_1 - Math.sqrt(x)) - Math.sqrt(y));
	} else {
		tmp = 2.0 + (t_3 - (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((x + 1.0))
	t_3 = math.sqrt((1.0 + z))
	t_4 = (math.sqrt((1.0 + t)) - math.sqrt(t)) + ((t_3 - math.sqrt(z)) + ((t_1 - math.sqrt(y)) - (math.sqrt(x) - t_2)))
	tmp = 0
	if t_4 <= 0.9999999999999899:
		tmp = (t_2 - math.sqrt(x)) + (0.5 * math.sqrt((1.0 / t)))
	elif t_4 <= 1.9999999999979958:
		tmp = 1.0 + ((t_1 - math.sqrt(x)) - math.sqrt(y))
	else:
		tmp = 2.0 + (t_3 - (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = sqrt(Float64(1.0 + z))
	t_4 = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(t_3 - sqrt(z)) + Float64(Float64(t_1 - sqrt(y)) - Float64(sqrt(x) - t_2))))
	tmp = 0.0
	if (t_4 <= 0.9999999999999899)
		tmp = Float64(Float64(t_2 - sqrt(x)) + Float64(0.5 * sqrt(Float64(1.0 / t))));
	elseif (t_4 <= 1.9999999999979958)
		tmp = Float64(1.0 + Float64(Float64(t_1 - sqrt(x)) - sqrt(y)));
	else
		tmp = Float64(2.0 + Float64(t_3 - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((x + 1.0));
	t_3 = sqrt((1.0 + z));
	t_4 = (sqrt((1.0 + t)) - sqrt(t)) + ((t_3 - sqrt(z)) + ((t_1 - sqrt(y)) - (sqrt(x) - t_2)));
	tmp = 0.0;
	if (t_4 <= 0.9999999999999899)
		tmp = (t_2 - sqrt(x)) + (0.5 * sqrt((1.0 / t)));
	elseif (t_4 <= 1.9999999999979958)
		tmp = 1.0 + ((t_1 - sqrt(x)) - sqrt(y));
	else
		tmp = 2.0 + (t_3 - (sqrt(x) + (sqrt(z) + sqrt(y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.9999999999999899], N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1.9999999999979958], N[(1.0 + N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(t$95$3 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 0.999999999999989897

   1. Initial program 47.9%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 25.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 56.6%

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

   if 0.999999999999989897 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.9999999999979958

   1. Initial program 98.1%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 6.3

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 33.6%

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

   if 1.9999999999979958 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 97.8%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 22.0

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

   5. Applied rewrites 22.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 21.4%

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

   9. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \sqrt{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 1.7%

       \[\leadsto -\sqrt{x} \]

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 25.7%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right)\right) \leq 0.9999999999999899:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right)\right) \leq 1.9999999999979958:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + z}\\ t_4 := \left(t\_3 - \sqrt{z}\right) + \left(\left(t\_1 - \sqrt{y}\right) - \left(\sqrt{x} - t\_2\right)\right)\\ t_5 := \sqrt{x} + \sqrt{y}\\ \mathbf{if}\;t\_4 \leq 0.9999999999999899:\\ \;\;\;\;\left(t\_2 - \sqrt{x}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{elif}\;t\_4 \leq 2.005:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, t\_1\right) - t\_5\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\mathsf{fma}\left(y, 0.5, t\_3\right) - \left(\sqrt{z} + t\_5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4 (+ (- t_3 (sqrt z)) (- (- t_1 (sqrt y)) (- (sqrt x) t_2))))
        (t_5 (+ (sqrt x) (sqrt y))))
   (if (<= t_4 0.9999999999999899)
     (+ (- t_2 (sqrt x)) (* 0.5 (sqrt (/ 1.0 t))))
     (if (<= t_4 2.005)
       (+ 1.0 (- (fma 0.5 (sqrt (/ 1.0 z)) t_1) t_5))
       (+ 2.0 (- (fma y 0.5 t_3) (+ (sqrt z) t_5)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((1.0 + z));
	double t_4 = (t_3 - sqrt(z)) + ((t_1 - sqrt(y)) - (sqrt(x) - t_2));
	double t_5 = sqrt(x) + sqrt(y);
	double tmp;
	if (t_4 <= 0.9999999999999899) {
		tmp = (t_2 - sqrt(x)) + (0.5 * sqrt((1.0 / t)));
	} else if (t_4 <= 2.005) {
		tmp = 1.0 + (fma(0.5, sqrt((1.0 / z)), t_1) - t_5);
	} else {
		tmp = 2.0 + (fma(y, 0.5, t_3) - (sqrt(z) + t_5));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = sqrt(Float64(1.0 + z))
	t_4 = Float64(Float64(t_3 - sqrt(z)) + Float64(Float64(t_1 - sqrt(y)) - Float64(sqrt(x) - t_2)))
	t_5 = Float64(sqrt(x) + sqrt(y))
	tmp = 0.0
	if (t_4 <= 0.9999999999999899)
		tmp = Float64(Float64(t_2 - sqrt(x)) + Float64(0.5 * sqrt(Float64(1.0 / t))));
	elseif (t_4 <= 2.005)
		tmp = Float64(1.0 + Float64(fma(0.5, sqrt(Float64(1.0 / z)), t_1) - t_5));
	else
		tmp = Float64(2.0 + Float64(fma(y, 0.5, t_3) - Float64(sqrt(z) + t_5)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.9999999999999899], N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.005], N[(1.0 + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$5), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(N[(y * 0.5 + t$95$3), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.999999999999989897

   1. Initial program 71.7%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 15.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 32.3%

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

   if 0.999999999999989897 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.0049999999999999

   1. Initial program 97.6%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 10.9

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

   5. Applied rewrites 10.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 22.7%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 23.1%

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

   if 2.0049999999999999 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.6%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.4%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 49.2%

       \[\leadsto 2 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right) \leq 0.9999999999999899:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right) \leq 2.005:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{x + 1}\\ t_4 := \left(t\_1 - \sqrt{z}\right) + \left(\left(t\_2 - \sqrt{y}\right) - \left(\sqrt{x} - t\_3\right)\right)\\ \mathbf{if}\;t\_4 \leq 0.9999999999999899:\\ \;\;\;\;\left(t\_3 - \sqrt{x}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{elif}\;t\_4 \leq 1.9999999999979958:\\ \;\;\;\;1 + \left(\left(t\_2 - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\mathsf{fma}\left(y, 0.5, t\_1\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

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)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (+ (- t_1 (sqrt z)) (- (- t_2 (sqrt y)) (- (sqrt x) t_3)))))
   (if (<= t_4 0.9999999999999899)
     (+ (- t_3 (sqrt x)) (* 0.5 (sqrt (/ 1.0 t))))
     (if (<= t_4 1.9999999999979958)
       (+ 1.0 (- (- t_2 (sqrt x)) (sqrt y)))
       (+ 2.0 (- (fma y 0.5 t_1) (+ (sqrt z) (+ (sqrt x) (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((x + 1.0));
	double t_4 = (t_1 - sqrt(z)) + ((t_2 - sqrt(y)) - (sqrt(x) - t_3));
	double tmp;
	if (t_4 <= 0.9999999999999899) {
		tmp = (t_3 - sqrt(x)) + (0.5 * sqrt((1.0 / t)));
	} else if (t_4 <= 1.9999999999979958) {
		tmp = 1.0 + ((t_2 - sqrt(x)) - sqrt(y));
	} else {
		tmp = 2.0 + (fma(y, 0.5, t_1) - (sqrt(z) + (sqrt(x) + sqrt(y))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(Float64(t_1 - sqrt(z)) + Float64(Float64(t_2 - sqrt(y)) - Float64(sqrt(x) - t_3)))
	tmp = 0.0
	if (t_4 <= 0.9999999999999899)
		tmp = Float64(Float64(t_3 - sqrt(x)) + Float64(0.5 * sqrt(Float64(1.0 / t))));
	elseif (t_4 <= 1.9999999999979958)
		tmp = Float64(1.0 + Float64(Float64(t_2 - sqrt(x)) - sqrt(y)));
	else
		tmp = Float64(2.0 + Float64(fma(y, 0.5, t_1) - Float64(sqrt(z) + Float64(sqrt(x) + sqrt(y)))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.9999999999999899], N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1.9999999999979958], N[(1.0 + N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(N[(y * 0.5 + t$95$1), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.999999999999989897

   1. Initial program 71.7%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 15.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 32.3%

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

   if 0.999999999999989897 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.9999999999979958

   1. Initial program 97.7%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 7.5

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

   5. Applied rewrites 7.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.2%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 22.9%

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

   if 1.9999999999979958 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 27.4

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

   5. Applied rewrites 27.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 26.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 29.7%

       \[\leadsto 2 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right) \leq 0.9999999999999899:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right) \leq 1.9999999999979958:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + t}\\ t_4 := \sqrt{1 + y}\\ \mathbf{if}\;\left(t\_2 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{y}\right) - \left(\sqrt{x} - t\_1\right)\right) \leq 2:\\ \;\;\;\;\left(t\_3 - \sqrt{t}\right) + \left(\left(\frac{1}{\sqrt{y} + t\_4} + t\_1\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + t\right) - t}{\sqrt{t} + t\_3} + \left(2 + \left(t\_2 - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (sqrt (+ 1.0 t)))
        (t_4 (sqrt (+ 1.0 y))))
   (if (<= (+ (- t_2 (sqrt z)) (- (- t_4 (sqrt y)) (- (sqrt x) t_1))) 2.0)
     (+ (- t_3 (sqrt t)) (- (+ (/ 1.0 (+ (sqrt y) t_4)) t_1) (sqrt x)))
     (+
      (/ (- (+ 1.0 t) t) (+ (sqrt t) t_3))
      (+ 2.0 (- t_2 (+ (sqrt x) (+ (sqrt z) (sqrt y)))))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2

   1. Initial program 94.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 94.6

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 94.6

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 45.6

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

   7. Applied rewrites 45.6%

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

   if 2 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   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. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 89.8

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

   5. Applied rewrites 89.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 89.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 89.8

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

   7. Applied rewrites 89.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 81.5%

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

4. Final simplification 51.5%

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

Alternative 10: 92.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + z} - \sqrt{z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + y}\\ t_5 := t\_4 - \sqrt{y}\\ \mathbf{if}\;t\_2 + \left(t\_5 - \left(\sqrt{x} - t\_1\right)\right) \leq 1.9965:\\ \;\;\;\;t\_3 + \left(\left(\frac{1}{\sqrt{y} + t\_4} + t\_1\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_2 + \left(t\_5 + \left(1 - \sqrt{x}\right)\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 y)))
        (t_5 (- t_4 (sqrt y))))
   (if (<= (+ t_2 (- t_5 (- (sqrt x) t_1))) 1.9965)
     (+ t_3 (- (+ (/ 1.0 (+ (sqrt y) t_4)) t_1) (sqrt x)))
     (+ t_3 (+ t_2 (+ t_5 (- 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((x + 1.0));
	double t_2 = sqrt((1.0 + z)) - sqrt(z);
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = sqrt((1.0 + y));
	double t_5 = t_4 - sqrt(y);
	double tmp;
	if ((t_2 + (t_5 - (sqrt(x) - t_1))) <= 1.9965) {
		tmp = t_3 + (((1.0 / (sqrt(y) + t_4)) + t_1) - sqrt(x));
	} else {
		tmp = t_3 + (t_2 + (t_5 + (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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((1.0d0 + z)) - sqrt(z)
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    t_4 = sqrt((1.0d0 + y))
    t_5 = t_4 - sqrt(y)
    if ((t_2 + (t_5 - (sqrt(x) - t_1))) <= 1.9965d0) then
        tmp = t_3 + (((1.0d0 / (sqrt(y) + t_4)) + t_1) - sqrt(x))
    else
        tmp = t_3 + (t_2 + (t_5 + (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((x + 1.0));
	double t_2 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_4 = Math.sqrt((1.0 + y));
	double t_5 = t_4 - Math.sqrt(y);
	double tmp;
	if ((t_2 + (t_5 - (Math.sqrt(x) - t_1))) <= 1.9965) {
		tmp = t_3 + (((1.0 / (Math.sqrt(y) + t_4)) + t_1) - Math.sqrt(x));
	} else {
		tmp = t_3 + (t_2 + (t_5 + (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((x + 1.0))
	t_2 = math.sqrt((1.0 + z)) - math.sqrt(z)
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_4 = math.sqrt((1.0 + y))
	t_5 = t_4 - math.sqrt(y)
	tmp = 0
	if (t_2 + (t_5 - (math.sqrt(x) - t_1))) <= 1.9965:
		tmp = t_3 + (((1.0 / (math.sqrt(y) + t_4)) + t_1) - math.sqrt(x))
	else:
		tmp = t_3 + (t_2 + (t_5 + (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(x + 1.0))
	t_2 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = sqrt(Float64(1.0 + y))
	t_5 = Float64(t_4 - sqrt(y))
	tmp = 0.0
	if (Float64(t_2 + Float64(t_5 - Float64(sqrt(x) - t_1))) <= 1.9965)
		tmp = Float64(t_3 + Float64(Float64(Float64(1.0 / Float64(sqrt(y) + t_4)) + t_1) - sqrt(x)));
	else
		tmp = Float64(t_3 + Float64(t_2 + Float64(t_5 + Float64(1.0 - sqrt(x)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	t_2 = sqrt((1.0 + z)) - sqrt(z);
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	t_4 = sqrt((1.0 + y));
	t_5 = t_4 - sqrt(y);
	tmp = 0.0;
	if ((t_2 + (t_5 - (sqrt(x) - t_1))) <= 1.9965)
		tmp = t_3 + (((1.0 / (sqrt(y) + t_4)) + t_1) - sqrt(x));
	else
		tmp = t_3 + (t_2 + (t_5 + (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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 + N[(t$95$5 - N[(N[Sqrt[x], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.9965], N[(t$95$3 + N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$2 + N[(t$95$5 + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.99649999999999994

   1. Initial program 91.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 91.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 91.9

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

   4. Applied rewrites 91.9%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 47.4

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

   7. Applied rewrites 47.4%

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

   if 1.99649999999999994 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 77.3

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

   5. Applied rewrites 77.3%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right) \leq 1.9965:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{x + 1}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + y}\\ \mathbf{if}\;\left(t\_2 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{y}\right) - \left(\sqrt{x} - t\_1\right)\right) \leq 1.9999999999979958:\\ \;\;\;\;t\_3 + \left(\left(\frac{1}{\sqrt{y} + t\_4} + t\_1\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(1 + \left(\left(t\_2 - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right) + t\_4\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 y))))
   (if (<=
        (+ (- t_2 (sqrt z)) (- (- t_4 (sqrt y)) (- (sqrt x) t_1)))
        1.9999999999979958)
     (+ t_3 (- (+ (/ 1.0 (+ (sqrt y) t_4)) t_1) (sqrt x)))
     (+ t_3 (+ 1.0 (+ (- t_2 (+ (sqrt x) (+ (sqrt z) (sqrt y)))) t_4))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = sqrt((1.0 + y));
	double tmp;
	if (((t_2 - sqrt(z)) + ((t_4 - sqrt(y)) - (sqrt(x) - t_1))) <= 1.9999999999979958) {
		tmp = t_3 + (((1.0 / (sqrt(y) + t_4)) + t_1) - sqrt(x));
	} else {
		tmp = t_3 + (1.0 + ((t_2 - (sqrt(x) + (sqrt(z) + sqrt(y)))) + t_4));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((1.0d0 + z))
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    t_4 = sqrt((1.0d0 + y))
    if (((t_2 - sqrt(z)) + ((t_4 - sqrt(y)) - (sqrt(x) - t_1))) <= 1.9999999999979958d0) then
        tmp = t_3 + (((1.0d0 / (sqrt(y) + t_4)) + t_1) - sqrt(x))
    else
        tmp = t_3 + (1.0d0 + ((t_2 - (sqrt(x) + (sqrt(z) + sqrt(y)))) + t_4))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_4 = Math.sqrt((1.0 + y));
	double tmp;
	if (((t_2 - Math.sqrt(z)) + ((t_4 - Math.sqrt(y)) - (Math.sqrt(x) - t_1))) <= 1.9999999999979958) {
		tmp = t_3 + (((1.0 / (Math.sqrt(y) + t_4)) + t_1) - Math.sqrt(x));
	} else {
		tmp = t_3 + (1.0 + ((t_2 - (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y)))) + t_4));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	t_2 = math.sqrt((1.0 + z))
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_4 = math.sqrt((1.0 + y))
	tmp = 0
	if ((t_2 - math.sqrt(z)) + ((t_4 - math.sqrt(y)) - (math.sqrt(x) - t_1))) <= 1.9999999999979958:
		tmp = t_3 + (((1.0 / (math.sqrt(y) + t_4)) + t_1) - math.sqrt(x))
	else:
		tmp = t_3 + (1.0 + ((t_2 - (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y)))) + t_4))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_4 - sqrt(y)) - Float64(sqrt(x) - t_1))) <= 1.9999999999979958)
		tmp = Float64(t_3 + Float64(Float64(Float64(1.0 / Float64(sqrt(y) + t_4)) + t_1) - sqrt(x)));
	else
		tmp = Float64(t_3 + Float64(1.0 + Float64(Float64(t_2 - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))) + t_4)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	t_2 = sqrt((1.0 + z));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	t_4 = sqrt((1.0 + y));
	tmp = 0.0;
	if (((t_2 - sqrt(z)) + ((t_4 - sqrt(y)) - (sqrt(x) - t_1))) <= 1.9999999999979958)
		tmp = t_3 + (((1.0 / (sqrt(y) + t_4)) + t_1) - sqrt(x));
	else
		tmp = t_3 + (1.0 + ((t_2 - (sqrt(x) + (sqrt(z) + sqrt(y)))) + t_4));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.9999999999979958], N[(t$95$3 + N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(1.0 + N[(N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.9999999999979958

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 92.2

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 92.2

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

   4. Applied rewrites 92.2%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 45.5

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

   7. Applied rewrites 45.5%

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

   if 1.9999999999979958 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 59.1

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

   5. Applied rewrites 59.1%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right) \leq 1.9999999999979958:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{x + 1}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right) + \sqrt{1 + y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   1. Initial program 86.6%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 38.7%

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

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 98.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 N/A

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

   7. Applied rewrites 24.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 23.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 40.9%

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

4. Final simplification 40.3%

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

Alternative 13: 84.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(t\_2 - \sqrt{z}\right) + \left(\left(t\_1 - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right)\right) \leq 1.9999999999979958:\\ \;\;\;\;1 + \left(\left(t\_1 - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(t\_2 - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

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 y))) (t_2 (sqrt (+ 1.0 z))))
   (if (<=
        (+
         (- (sqrt (+ 1.0 t)) (sqrt t))
         (+
          (- t_2 (sqrt z))
          (- (- t_1 (sqrt y)) (- (sqrt x) (sqrt (+ x 1.0))))))
        1.9999999999979958)
     (+ 1.0 (- (- t_1 (sqrt x)) (sqrt y)))
     (+ 2.0 (- t_2 (+ (sqrt x) (+ (sqrt z) (sqrt y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + z));
	double tmp;
	if (((sqrt((1.0 + t)) - sqrt(t)) + ((t_2 - sqrt(z)) + ((t_1 - sqrt(y)) - (sqrt(x) - sqrt((x + 1.0)))))) <= 1.9999999999979958) {
		tmp = 1.0 + ((t_1 - sqrt(x)) - sqrt(y));
	} else {
		tmp = 2.0 + (t_2 - (sqrt(x) + (sqrt(z) + sqrt(y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + z))
    if (((sqrt((1.0d0 + t)) - sqrt(t)) + ((t_2 - sqrt(z)) + ((t_1 - sqrt(y)) - (sqrt(x) - sqrt((x + 1.0d0)))))) <= 1.9999999999979958d0) then
        tmp = 1.0d0 + ((t_1 - sqrt(x)) - sqrt(y))
    else
        tmp = 2.0d0 + (t_2 - (sqrt(x) + (sqrt(z) + sqrt(y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((1.0 + z));
	double tmp;
	if (((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + ((t_2 - Math.sqrt(z)) + ((t_1 - Math.sqrt(y)) - (Math.sqrt(x) - Math.sqrt((x + 1.0)))))) <= 1.9999999999979958) {
		tmp = 1.0 + ((t_1 - Math.sqrt(x)) - Math.sqrt(y));
	} else {
		tmp = 2.0 + (t_2 - (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((1.0 + z))
	tmp = 0
	if ((math.sqrt((1.0 + t)) - math.sqrt(t)) + ((t_2 - math.sqrt(z)) + ((t_1 - math.sqrt(y)) - (math.sqrt(x) - math.sqrt((x + 1.0)))))) <= 1.9999999999979958:
		tmp = 1.0 + ((t_1 - math.sqrt(x)) - math.sqrt(y))
	else:
		tmp = 2.0 + (t_2 - (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_1 - sqrt(y)) - Float64(sqrt(x) - sqrt(Float64(x + 1.0)))))) <= 1.9999999999979958)
		tmp = Float64(1.0 + Float64(Float64(t_1 - sqrt(x)) - sqrt(y)));
	else
		tmp = Float64(2.0 + Float64(t_2 - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((1.0 + z));
	tmp = 0.0;
	if (((sqrt((1.0 + t)) - sqrt(t)) + ((t_2 - sqrt(z)) + ((t_1 - sqrt(y)) - (sqrt(x) - sqrt((x + 1.0)))))) <= 1.9999999999979958)
		tmp = 1.0 + ((t_1 - sqrt(x)) - sqrt(y));
	else
		tmp = 2.0 + (t_2 - (sqrt(x) + (sqrt(z) + sqrt(y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.9999999999979958], N[(1.0 + N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.9999999999979958

   1. Initial program 88.5%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 5.9

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

   5. Applied rewrites 5.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 29.7%

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

   if 1.9999999999979958 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 97.8%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 22.0

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

   5. Applied rewrites 22.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 21.4%

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

   9. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \sqrt{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 1.7%

       \[\leadsto -\sqrt{x} \]

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 25.7%

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

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right)\right) \leq 1.9999999999979958:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_2 - \sqrt{x} \leq 0:\\ \;\;\;\;t\_3 + \left(\left(t\_1 - \sqrt{z}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{x}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - t\_2\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + t\_1}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= (- t_2 (sqrt x)) 0.0)
     (+ t_3 (+ (- t_1 (sqrt z)) (* 0.5 (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 x))))))
     (+
      t_3
      (+
       (- (- (sqrt (+ 1.0 y)) (sqrt y)) (- (sqrt x) t_2))
       (/ (- (+ 1.0 z) z) (+ (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 t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if ((t_2 - sqrt(x)) <= 0.0) {
		tmp = t_3 + ((t_1 - sqrt(z)) + (0.5 * (sqrt((1.0 / y)) + sqrt((1.0 / x)))));
	} else {
		tmp = t_3 + (((sqrt((1.0 + y)) - sqrt(y)) - (sqrt(x) - t_2)) + (((1.0 + z) - z) / (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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((x + 1.0d0))
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    if ((t_2 - sqrt(x)) <= 0.0d0) then
        tmp = t_3 + ((t_1 - sqrt(z)) + (0.5d0 * (sqrt((1.0d0 / y)) + sqrt((1.0d0 / x)))))
    else
        tmp = t_3 + (((sqrt((1.0d0 + y)) - sqrt(y)) - (sqrt(x) - t_2)) + (((1.0d0 + z) - z) / (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 t_2 = Math.sqrt((x + 1.0));
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if ((t_2 - Math.sqrt(x)) <= 0.0) {
		tmp = t_3 + ((t_1 - Math.sqrt(z)) + (0.5 * (Math.sqrt((1.0 / y)) + Math.sqrt((1.0 / x)))));
	} else {
		tmp = t_3 + (((Math.sqrt((1.0 + y)) - Math.sqrt(y)) - (Math.sqrt(x) - t_2)) + (((1.0 + z) - z) / (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))
	t_2 = math.sqrt((x + 1.0))
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if (t_2 - math.sqrt(x)) <= 0.0:
		tmp = t_3 + ((t_1 - math.sqrt(z)) + (0.5 * (math.sqrt((1.0 / y)) + math.sqrt((1.0 / x)))))
	else:
		tmp = t_3 + (((math.sqrt((1.0 + y)) - math.sqrt(y)) - (math.sqrt(x) - t_2)) + (((1.0 + z) - z) / (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))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (Float64(t_2 - sqrt(x)) <= 0.0)
		tmp = Float64(t_3 + Float64(Float64(t_1 - sqrt(z)) + Float64(0.5 * Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / x))))));
	else
		tmp = Float64(t_3 + Float64(Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) - Float64(sqrt(x) - t_2)) + Float64(Float64(Float64(1.0 + z) - z) / 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));
	t_2 = sqrt((x + 1.0));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if ((t_2 - sqrt(x)) <= 0.0)
		tmp = t_3 + ((t_1 - sqrt(z)) + (0.5 * (sqrt((1.0 / y)) + sqrt((1.0 / x)))));
	else
		tmp = t_3 + (((sqrt((1.0 + y)) - sqrt(y)) - (sqrt(x) - t_2)) + (((1.0 + z) - z) / (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]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.0], N[(t$95$3 + N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + z), $MachinePrecision] - z), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0

   1. Initial program 90.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 43.1

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

   5. Applied rewrites 43.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 44.1%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 98.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 98.3

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.3

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

   4. Applied rewrites 98.3%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{x}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + t}\\ \mathbf{if}\;t\_2 - \sqrt{x} \leq 0:\\ \;\;\;\;\left(t\_3 - \sqrt{t}\right) + \left(t\_1 + 0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{x}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - t\_2\right)\right)\right) + \frac{\left(1 + t\right) - t}{\sqrt{t} + t\_3}\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_3 (sqrt (+ 1.0 t))))
   (if (<= (- t_2 (sqrt x)) 0.0)
     (+ (- t_3 (sqrt t)) (+ t_1 (* 0.5 (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 x))))))
     (+
      (+ t_1 (- (- (sqrt (+ 1.0 y)) (sqrt y)) (- (sqrt x) t_2)))
      (/ (- (+ 1.0 t) t) (+ (sqrt t) t_3))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z)) - sqrt(z);
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((1.0 + t));
	double tmp;
	if ((t_2 - sqrt(x)) <= 0.0) {
		tmp = (t_3 - sqrt(t)) + (t_1 + (0.5 * (sqrt((1.0 / y)) + sqrt((1.0 / x)))));
	} else {
		tmp = (t_1 + ((sqrt((1.0 + y)) - sqrt(y)) - (sqrt(x) - t_2))) + (((1.0 + t) - t) / (sqrt(t) + t_3));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z)) - sqrt(z)
    t_2 = sqrt((x + 1.0d0))
    t_3 = sqrt((1.0d0 + t))
    if ((t_2 - sqrt(x)) <= 0.0d0) then
        tmp = (t_3 - sqrt(t)) + (t_1 + (0.5d0 * (sqrt((1.0d0 / y)) + sqrt((1.0d0 / x)))))
    else
        tmp = (t_1 + ((sqrt((1.0d0 + y)) - sqrt(y)) - (sqrt(x) - t_2))) + (((1.0d0 + t) - t) / (sqrt(t) + t_3))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = Math.sqrt((1.0 + t));
	double tmp;
	if ((t_2 - Math.sqrt(x)) <= 0.0) {
		tmp = (t_3 - Math.sqrt(t)) + (t_1 + (0.5 * (Math.sqrt((1.0 / y)) + Math.sqrt((1.0 / x)))));
	} else {
		tmp = (t_1 + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) - (Math.sqrt(x) - t_2))) + (((1.0 + t) - t) / (Math.sqrt(t) + t_3));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z)) - math.sqrt(z)
	t_2 = math.sqrt((x + 1.0))
	t_3 = math.sqrt((1.0 + t))
	tmp = 0
	if (t_2 - math.sqrt(x)) <= 0.0:
		tmp = (t_3 - math.sqrt(t)) + (t_1 + (0.5 * (math.sqrt((1.0 / y)) + math.sqrt((1.0 / x)))))
	else:
		tmp = (t_1 + ((math.sqrt((1.0 + y)) - math.sqrt(y)) - (math.sqrt(x) - t_2))) + (((1.0 + t) - t) / (math.sqrt(t) + t_3))
	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(x + 1.0))
	t_3 = sqrt(Float64(1.0 + t))
	tmp = 0.0
	if (Float64(t_2 - sqrt(x)) <= 0.0)
		tmp = Float64(Float64(t_3 - sqrt(t)) + Float64(t_1 + Float64(0.5 * Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / x))))));
	else
		tmp = Float64(Float64(t_1 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) - Float64(sqrt(x) - t_2))) + Float64(Float64(Float64(1.0 + t) - t) / Float64(sqrt(t) + t_3)));
	end
	return tmp
end

MATLAB

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

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + t), $MachinePrecision] - t), $MachinePrecision] / N[(N[Sqrt[t], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0

   1. Initial program 90.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x} - \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 43.1

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 43.1%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 44.1%

      \[\leadsto \left(0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 98.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)} \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\sqrt{t + 1}} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\sqrt{t + 1} \cdot \color{blue}{\sqrt{t + 1}} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right)} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{\sqrt{t}} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \sqrt{t} \cdot \color{blue}{\sqrt{t}}}{\sqrt{t + 1} + \sqrt{t}} \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right) - t}}{\sqrt{t + 1} + \sqrt{t}} \]

      11. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right)} - t}{\sqrt{t + 1} + \sqrt{t}} \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(1 + t\right)} - t}{\sqrt{t + 1} + \sqrt{t}} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(1 + t\right)} - t}{\sqrt{t + 1} + \sqrt{t}} \]

      14. lower-+.f64 98.2

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(1 + t\right) - t}{\color{blue}{\sqrt{t + 1} + \sqrt{t}}} \]

      15. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(1 + t\right) - t}{\sqrt{\color{blue}{t + 1}} + \sqrt{t}} \]

      16. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(1 + t\right) - t}{\sqrt{\color{blue}{1 + t}} + \sqrt{t}} \]

      17. lower-+.f64 98.2

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(1 + t\right) - t}{\sqrt{\color{blue}{1 + t}} + \sqrt{t}} \]

   4. Applied rewrites 98.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\left(1 + t\right) - t}{\sqrt{1 + t} + \sqrt{t}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{x}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right)\right) + \frac{\left(1 + t\right) - t}{\sqrt{t} + \sqrt{1 + t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 92.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_1 - \sqrt{z} \leq 0:\\ \;\;\;\;t\_2 + \left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{x + 1}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(\left(t\_1 - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right) + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))) (t_2 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= (- t_1 (sqrt z)) 0.0)
     (+
      t_2
      (- (+ (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y)))) (sqrt (+ x 1.0))) (sqrt x)))
     (+ t_2 (+ (- t_1 (+ (sqrt x) (+ (sqrt z) (sqrt y)))) 2.0)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if ((t_1 - sqrt(z)) <= 0.0) {
		tmp = t_2 + (((1.0 / (sqrt(y) + sqrt((1.0 + y)))) + sqrt((x + 1.0))) - sqrt(x));
	} else {
		tmp = t_2 + ((t_1 - (sqrt(x) + (sqrt(z) + sqrt(y)))) + 2.0);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + t)) - sqrt(t)
    if ((t_1 - sqrt(z)) <= 0.0d0) then
        tmp = t_2 + (((1.0d0 / (sqrt(y) + sqrt((1.0d0 + y)))) + sqrt((x + 1.0d0))) - sqrt(x))
    else
        tmp = t_2 + ((t_1 - (sqrt(x) + (sqrt(z) + sqrt(y)))) + 2.0d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if ((t_1 - Math.sqrt(z)) <= 0.0) {
		tmp = t_2 + (((1.0 / (Math.sqrt(y) + Math.sqrt((1.0 + y)))) + Math.sqrt((x + 1.0))) - Math.sqrt(x));
	} else {
		tmp = t_2 + ((t_1 - (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y)))) + 2.0);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if (t_1 - math.sqrt(z)) <= 0.0:
		tmp = t_2 + (((1.0 / (math.sqrt(y) + math.sqrt((1.0 + y)))) + math.sqrt((x + 1.0))) - math.sqrt(x))
	else:
		tmp = t_2 + ((t_1 - (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y)))) + 2.0)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (Float64(t_1 - sqrt(z)) <= 0.0)
		tmp = Float64(t_2 + Float64(Float64(Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y)))) + sqrt(Float64(x + 1.0))) - sqrt(x)));
	else
		tmp = Float64(t_2 + Float64(Float64(t_1 - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))) + 2.0));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if ((t_1 - sqrt(z)) <= 0.0)
		tmp = t_2 + (((1.0 / (sqrt(y) + sqrt((1.0 + y)))) + sqrt((x + 1.0))) - sqrt(x));
	else
		tmp = t_2 + ((t_1 - (sqrt(x) + (sqrt(z) + sqrt(y)))) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.0], N[(t$95$2 + N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 0.0

   1. Initial program 91.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-+.f64 92.2

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lower-+.f64 92.2

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 70.2

          \[\leadsto \left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-sqrt.f64 33.3

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 33.3%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 25.7%

      \[\leadsto \left(2 + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 0:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{x + 1}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right) + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;x \leq 2.2 \cdot 10^{+19}:\\ \;\;\;\;t\_2 + \left(t\_1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(t\_1 + 0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{x}}\right)\right)\\ \end{array} \end{array} \]

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 (+ 1.0 t)) (sqrt t))))
   (if (<= x 2.2e+19)
     (+
      t_2
      (+ t_1 (- (- (sqrt (+ 1.0 y)) (sqrt y)) (- (sqrt x) (sqrt (+ x 1.0))))))
     (+ t_2 (+ t_1 (* 0.5 (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 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((1.0 + t)) - sqrt(t);
	double tmp;
	if (x <= 2.2e+19) {
		tmp = t_2 + (t_1 + ((sqrt((1.0 + y)) - sqrt(y)) - (sqrt(x) - sqrt((x + 1.0)))));
	} else {
		tmp = t_2 + (t_1 + (0.5 * (sqrt((1.0 / y)) + sqrt((1.0 / x)))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z)) - sqrt(z)
    t_2 = sqrt((1.0d0 + t)) - sqrt(t)
    if (x <= 2.2d+19) then
        tmp = t_2 + (t_1 + ((sqrt((1.0d0 + y)) - sqrt(y)) - (sqrt(x) - sqrt((x + 1.0d0)))))
    else
        tmp = t_2 + (t_1 + (0.5d0 * (sqrt((1.0d0 / y)) + sqrt((1.0d0 / x)))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double t_2 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (x <= 2.2e+19) {
		tmp = t_2 + (t_1 + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) - (Math.sqrt(x) - Math.sqrt((x + 1.0)))));
	} else {
		tmp = t_2 + (t_1 + (0.5 * (Math.sqrt((1.0 / y)) + Math.sqrt((1.0 / 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((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if x <= 2.2e+19:
		tmp = t_2 + (t_1 + ((math.sqrt((1.0 + y)) - math.sqrt(y)) - (math.sqrt(x) - math.sqrt((x + 1.0)))))
	else:
		tmp = t_2 + (t_1 + (0.5 * (math.sqrt((1.0 / y)) + math.sqrt((1.0 / 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 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (x <= 2.2e+19)
		tmp = Float64(t_2 + Float64(t_1 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) - Float64(sqrt(x) - sqrt(Float64(x + 1.0))))));
	else
		tmp = Float64(t_2 + Float64(t_1 + Float64(0.5 * Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / x))))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z)) - sqrt(z);
	t_2 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (x <= 2.2e+19)
		tmp = t_2 + (t_1 + ((sqrt((1.0 + y)) - sqrt(y)) - (sqrt(x) - sqrt((x + 1.0)))));
	else
		tmp = t_2 + (t_1 + (0.5 * (sqrt((1.0 / y)) + sqrt((1.0 / x)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.2e+19], N[(t$95$2 + N[(t$95$1 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(t$95$1 + N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2e19

   1. Initial program 98.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   if 2.2e19 < x

   1. Initial program 90.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x} - \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 43.1

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 43.1%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 44.1%

      \[\leadsto \left(0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+19}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{x}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;x \leq 1.46:\\ \;\;\;\;t\_2 + \left(t\_1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(t\_1 + 0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{x}}\right)\right)\\ \end{array} \end{array} \]

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 (+ 1.0 t)) (sqrt t))))
   (if (<= x 1.46)
     (+
      t_2
      (+ t_1 (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (fma x 0.5 (- 1.0 (sqrt x))))))
     (+ t_2 (+ t_1 (* 0.5 (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 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((1.0 + t)) - sqrt(t);
	double tmp;
	if (x <= 1.46) {
		tmp = t_2 + (t_1 + ((sqrt((1.0 + y)) - sqrt(y)) + fma(x, 0.5, (1.0 - sqrt(x)))));
	} else {
		tmp = t_2 + (t_1 + (0.5 * (sqrt((1.0 / y)) + sqrt((1.0 / 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 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (x <= 1.46)
		tmp = Float64(t_2 + Float64(t_1 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + fma(x, 0.5, Float64(1.0 - sqrt(x))))));
	else
		tmp = Float64(t_2 + Float64(t_1 + Float64(0.5 * Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / x))))));
	end
	return 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[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.46], N[(t$95$2 + N[(t$95$1 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(x * 0.5 + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(t$95$1 + N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.46

   1. Initial program 98.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 97.5

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 97.5%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.46 < x

   1. Initial program 90.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x} - \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 42.8

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 42.8%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \left(0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.46:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{x}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 91.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)\right)\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+
  (- (sqrt (+ 1.0 t)) (sqrt t))
  (+
   (- (sqrt (+ 1.0 z)) (sqrt z))
   (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (fma x 0.5 (- 1.0 (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (sqrt((1.0 + t)) - sqrt(t)) + ((sqrt((1.0 + z)) - sqrt(z)) + ((sqrt((1.0 + y)) - sqrt(y)) + fma(x, 0.5, (1.0 - sqrt(x)))));
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + fma(x, 0.5, Float64(1.0 - sqrt(x))))))
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(x * 0.5 + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate--l+ N/A

      \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. *-commutative N/A

      \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. lower--.f64 N/A

      \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. lower-sqrt.f64 55.7

      \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

5. Applied rewrites 55.7%

   \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

6. Final simplification 55.7%

   \[\leadsto \left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)\right)\right) \]
7. Add Preprocessing

Alternative 20: 64.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + \left(\left(\sqrt{1 + y} - \sqrt{x}\right) - \sqrt{y}\right) \end{array} \]

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 (+ 1.0 y)) (sqrt x)) (sqrt y))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + ((sqrt((1.0 + y)) - sqrt(x)) - 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((1.0d0 + y)) - sqrt(x)) - 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((1.0 + y)) - Math.sqrt(x)) - Math.sqrt(y));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 + ((math.sqrt((1.0 + y)) - math.sqrt(x)) - math.sqrt(y))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(x)) - 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((1.0 + y)) - sqrt(x)) - 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[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   13. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

   15. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

   16. lower-sqrt.f64 16.7

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Applied rewrites 16.7%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 27.3%

   \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)} \]

9. Taylor expanded in z around inf

   \[\leadsto 1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]
10. Step-by-step derivation

11. Applied rewrites 22.7%

    \[\leadsto 1 + \left(\left(\sqrt{1 + y} - \sqrt{x}\right) - \sqrt{y}\right) \]
12. Add Preprocessing

Alternative 21: 35.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ 1.0 (- (sqrt (+ 1.0 z)) (+ (sqrt x) (sqrt z)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + (sqrt((1.0 + z)) - (sqrt(x) + sqrt(z)));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 + (sqrt((1.0d0 + z)) - (sqrt(x) + sqrt(z)))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(x) + Math.sqrt(z)));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 + (math.sqrt((1.0 + z)) - (math.sqrt(x) + math.sqrt(z)))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(x) + sqrt(z))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 + (sqrt((1.0 + z)) - (sqrt(x) + sqrt(z)));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   13. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

   15. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

   16. lower-sqrt.f64 16.7

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Applied rewrites 16.7%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 27.3%

   \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)} \]

9. Taylor expanded in y around inf

   \[\leadsto 1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\sqrt{z}}\right)\right) \]
10. Step-by-step derivation

11. Applied rewrites 26.8%

    \[\leadsto 1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\sqrt{z}}\right)\right) \]
12. Add Preprocessing

Alternative 22: 35.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + \left(-\sqrt{x}\right) \end{array} \]

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 x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + -sqrt(x);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 + -sqrt(x)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 + -Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 + -math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 + Float64(-sqrt(x)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 + -sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 + (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 94.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   13. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

   15. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

   16. lower-sqrt.f64 16.7

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Applied rewrites 16.7%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 27.3%

   \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)} \]

9. Taylor expanded in x around inf

   \[\leadsto 1 + -1 \cdot \sqrt{x} \]
10. Step-by-step derivation

11. Applied rewrites 15.7%

    \[\leadsto 1 + \left(-\sqrt{x}\right) \]
12. Add Preprocessing

Alternative 23: 1.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ -\sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return -sqrt(x);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -sqrt(x)
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);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return -math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(-sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = -sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := (-N[Sqrt[x], $MachinePrecision])

Derivation

1. Initial program 94.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   13. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

   15. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

   16. lower-sqrt.f64 16.7

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Applied rewrites 16.7%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 27.3%

   \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)} \]

9. Taylor expanded in x around inf

   \[\leadsto -1 \cdot \sqrt{x} \]
10. Step-by-step derivation

11. Applied rewrites 1.6%

    \[\leadsto -\sqrt{x} \]
12. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.8× 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 2024219 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (/ 1 (+ (sqrt (+ x 1)) (sqrt x))) (/ 1 (+ (sqrt (+ y 1)) (sqrt y)))) (/ 1 (+ (sqrt (+ z 1)) (sqrt z)))) (- (sqrt (+ t 1)) (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))))