Main:z from

Percentage Accurate: 91.9% → 98.9%

Time: 36.6s

Alternatives: 21

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + z}\\ t_4 := t\_3 - \sqrt{z}\\ t_5 := \sqrt{1 + y}\\ t_6 := \left(t\_2 - \sqrt{x}\right) + \left(t\_5 - \sqrt{y}\right)\\ \mathbf{if}\;t\_6 \leq 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right) + \left(t\_4 + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\\ \mathbf{elif}\;t\_6 \leq 1.002:\\ \;\;\;\;\frac{\sqrt{y} \cdot 0.5 + y \cdot \frac{\left(x + 1\right) - x}{\sqrt{x} + t\_2}}{y} + \left(t\_4 + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t\_3 + \sqrt{z}} + t\_1\right) + \left(\left(1 + t\_5\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4 (- t_3 (sqrt z)))
        (t_5 (sqrt (+ 1.0 y)))
        (t_6 (+ (- t_2 (sqrt x)) (- t_5 (sqrt y)))))
   (if (<= t_6 0.0)
     (+
      (* 0.5 (+ (sqrt (/ 1.0 x)) (sqrt (/ 1.0 y))))
      (+ t_4 (* 0.5 (sqrt (/ 1.0 t)))))
     (if (<= t_6 1.002)
       (+
        (/ (+ (* (sqrt y) 0.5) (* y (/ (- (+ x 1.0) x) (+ (sqrt x) t_2)))) y)
        (+ t_4 t_1))
       (+
        (+ (/ 1.0 (+ t_3 (sqrt z))) t_1)
        (- (+ 1.0 t_5) (+ (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 + t)) - sqrt(t);
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((1.0 + z));
	double t_4 = t_3 - sqrt(z);
	double t_5 = sqrt((1.0 + y));
	double t_6 = (t_2 - sqrt(x)) + (t_5 - sqrt(y));
	double tmp;
	if (t_6 <= 0.0) {
		tmp = (0.5 * (sqrt((1.0 / x)) + sqrt((1.0 / y)))) + (t_4 + (0.5 * sqrt((1.0 / t))));
	} else if (t_6 <= 1.002) {
		tmp = (((sqrt(y) * 0.5) + (y * (((x + 1.0) - x) / (sqrt(x) + t_2)))) / y) + (t_4 + t_1);
	} else {
		tmp = ((1.0 / (t_3 + sqrt(z))) + t_1) + ((1.0 + t_5) - (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: 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 + t)) - sqrt(t)
    t_2 = sqrt((x + 1.0d0))
    t_3 = sqrt((1.0d0 + z))
    t_4 = t_3 - sqrt(z)
    t_5 = sqrt((1.0d0 + y))
    t_6 = (t_2 - sqrt(x)) + (t_5 - sqrt(y))
    if (t_6 <= 0.0d0) then
        tmp = (0.5d0 * (sqrt((1.0d0 / x)) + sqrt((1.0d0 / y)))) + (t_4 + (0.5d0 * sqrt((1.0d0 / t))))
    else if (t_6 <= 1.002d0) then
        tmp = (((sqrt(y) * 0.5d0) + (y * (((x + 1.0d0) - x) / (sqrt(x) + t_2)))) / y) + (t_4 + t_1)
    else
        tmp = ((1.0d0 / (t_3 + sqrt(z))) + t_1) + ((1.0d0 + t_5) - (sqrt(x) + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = Math.sqrt((1.0 + z));
	double t_4 = t_3 - Math.sqrt(z);
	double t_5 = Math.sqrt((1.0 + y));
	double t_6 = (t_2 - Math.sqrt(x)) + (t_5 - Math.sqrt(y));
	double tmp;
	if (t_6 <= 0.0) {
		tmp = (0.5 * (Math.sqrt((1.0 / x)) + Math.sqrt((1.0 / y)))) + (t_4 + (0.5 * Math.sqrt((1.0 / t))));
	} else if (t_6 <= 1.002) {
		tmp = (((Math.sqrt(y) * 0.5) + (y * (((x + 1.0) - x) / (Math.sqrt(x) + t_2)))) / y) + (t_4 + t_1);
	} else {
		tmp = ((1.0 / (t_3 + Math.sqrt(z))) + t_1) + ((1.0 + t_5) - (Math.sqrt(x) + 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 + t)) - math.sqrt(t)
	t_2 = math.sqrt((x + 1.0))
	t_3 = math.sqrt((1.0 + z))
	t_4 = t_3 - math.sqrt(z)
	t_5 = math.sqrt((1.0 + y))
	t_6 = (t_2 - math.sqrt(x)) + (t_5 - math.sqrt(y))
	tmp = 0
	if t_6 <= 0.0:
		tmp = (0.5 * (math.sqrt((1.0 / x)) + math.sqrt((1.0 / y)))) + (t_4 + (0.5 * math.sqrt((1.0 / t))))
	elif t_6 <= 1.002:
		tmp = (((math.sqrt(y) * 0.5) + (y * (((x + 1.0) - x) / (math.sqrt(x) + t_2)))) / y) + (t_4 + t_1)
	else:
		tmp = ((1.0 / (t_3 + math.sqrt(z))) + t_1) + ((1.0 + t_5) - (math.sqrt(x) + math.sqrt(y)))
	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))
	t_4 = Float64(t_3 - sqrt(z))
	t_5 = sqrt(Float64(1.0 + y))
	t_6 = Float64(Float64(t_2 - sqrt(x)) + Float64(t_5 - sqrt(y)))
	tmp = 0.0
	if (t_6 <= 0.0)
		tmp = Float64(Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + sqrt(Float64(1.0 / y)))) + Float64(t_4 + Float64(0.5 * sqrt(Float64(1.0 / t)))));
	elseif (t_6 <= 1.002)
		tmp = Float64(Float64(Float64(Float64(sqrt(y) * 0.5) + Float64(y * Float64(Float64(Float64(x + 1.0) - x) / Float64(sqrt(x) + t_2)))) / y) + Float64(t_4 + t_1));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_3 + sqrt(z))) + t_1) + Float64(Float64(1.0 + t_5) - Float64(sqrt(x) + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + t)) - sqrt(t);
	t_2 = sqrt((x + 1.0));
	t_3 = sqrt((1.0 + z));
	t_4 = t_3 - sqrt(z);
	t_5 = sqrt((1.0 + y));
	t_6 = (t_2 - sqrt(x)) + (t_5 - sqrt(y));
	tmp = 0.0;
	if (t_6 <= 0.0)
		tmp = (0.5 * (sqrt((1.0 / x)) + sqrt((1.0 / y)))) + (t_4 + (0.5 * sqrt((1.0 / t))));
	elseif (t_6 <= 1.002)
		tmp = (((sqrt(y) * 0.5) + (y * (((x + 1.0) - x) / (sqrt(x) + t_2)))) / y) + (t_4 + t_1);
	else
		tmp = ((1.0 / (t_3 + sqrt(z))) + t_1) + ((1.0 + t_5) - (sqrt(x) + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := 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]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 0.0], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.002], N[(N[(N[(N[(N[Sqrt[y], $MachinePrecision] * 0.5), $MachinePrecision] + N[(y * N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + N[(t$95$4 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(t$95$3 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[(1.0 + t$95$5), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.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))) < 0.0

   1. Initial program 86.5%

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

      1. associate-+l+ 86.5%

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

      2. associate-+l- 86.5%

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

      3. associate-+l- 86.5%

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

      4. +-commutative 86.5%

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

      5. +-commutative 86.5%

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

      6. +-commutative 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in y around inf 86.5%

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

   6. Taylor expanded in x around inf 91.1%

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

      1. distribute-lft-out 91.1%

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

   8. Simplified 91.1%

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

   9. Taylor expanded in t around inf 36.5%

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

   if 0.0 < (+.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))) < 1.002

   1. Initial program 96.0%

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

      1. associate-+l+ 96.0%

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

      2. associate-+l- 60.7%

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

      3. associate-+l- 96.0%

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

      4. +-commutative 96.0%

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

      5. +-commutative 96.0%

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

      6. +-commutative 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in y around inf 57.9%

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

   6. Taylor expanded in y around 0 57.1%

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

      1. +-commutative 57.1%

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

      2. flip-- 57.0%

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

      3. add-sqr-sqrt 56.3%

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

      4. add-sqr-sqrt 57.5%

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

   8. Applied egg-rr 57.5%

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

   if 1.002 < (+.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)))

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. associate-+l- 97.1%

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

      3. associate-+l- 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

      1. flip-- 97.2%

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

      2. add-sqr-sqrt 75.5%

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

      3. add-sqr-sqrt 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. associate--l+ 97.8%

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

      2. +-inverses 97.8%

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

      3. metadata-eval 97.8%

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

   8. Simplified 97.8%

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

   9. Taylor expanded in x around 0 93.8%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\\ \mathbf{elif}\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 1.002:\\ \;\;\;\;\frac{\sqrt{y} \cdot 0.5 + y \cdot \frac{\left(x + 1\right) - x}{\sqrt{x} + \sqrt{x + 1}}}{y} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ t_2 := \sqrt{1 + t} + \sqrt{t}\\ t_3 := \sqrt{1 + y} - \sqrt{y}\\ t_4 := \sqrt{1 + z}\\ t_5 := t\_4 + \sqrt{z}\\ \mathbf{if}\;t\_1 + t\_3 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right) + \left(\left(t\_4 - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(t\_3 + \frac{t\_2 + t\_5}{t\_2 \cdot t\_5}\right)\\ \end{array} \end{array} \]

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)) (sqrt x)))
        (t_2 (+ (sqrt (+ 1.0 t)) (sqrt t)))
        (t_3 (- (sqrt (+ 1.0 y)) (sqrt y)))
        (t_4 (sqrt (+ 1.0 z)))
        (t_5 (+ t_4 (sqrt z))))
   (if (<= (+ t_1 t_3) 5e-5)
     (+
      (* 0.5 (+ (sqrt (/ 1.0 x)) (sqrt (/ 1.0 y))))
      (+ (- t_4 (sqrt z)) (* 0.5 (sqrt (/ 1.0 t)))))
     (+ t_1 (+ t_3 (/ (+ t_2 t_5) (* t_2 t_5)))))))

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)) - sqrt(x);
	double t_2 = sqrt((1.0 + t)) + sqrt(t);
	double t_3 = sqrt((1.0 + y)) - sqrt(y);
	double t_4 = sqrt((1.0 + z));
	double t_5 = t_4 + sqrt(z);
	double tmp;
	if ((t_1 + t_3) <= 5e-5) {
		tmp = (0.5 * (sqrt((1.0 / x)) + sqrt((1.0 / y)))) + ((t_4 - sqrt(z)) + (0.5 * sqrt((1.0 / t))));
	} else {
		tmp = t_1 + (t_3 + ((t_2 + t_5) / (t_2 * t_5)));
	}
	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)) - sqrt(x)
    t_2 = sqrt((1.0d0 + t)) + sqrt(t)
    t_3 = sqrt((1.0d0 + y)) - sqrt(y)
    t_4 = sqrt((1.0d0 + z))
    t_5 = t_4 + sqrt(z)
    if ((t_1 + t_3) <= 5d-5) then
        tmp = (0.5d0 * (sqrt((1.0d0 / x)) + sqrt((1.0d0 / y)))) + ((t_4 - sqrt(z)) + (0.5d0 * sqrt((1.0d0 / t))))
    else
        tmp = t_1 + (t_3 + ((t_2 + t_5) / (t_2 * t_5)))
    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)) - Math.sqrt(x);
	double t_2 = Math.sqrt((1.0 + t)) + Math.sqrt(t);
	double t_3 = Math.sqrt((1.0 + y)) - Math.sqrt(y);
	double t_4 = Math.sqrt((1.0 + z));
	double t_5 = t_4 + Math.sqrt(z);
	double tmp;
	if ((t_1 + t_3) <= 5e-5) {
		tmp = (0.5 * (Math.sqrt((1.0 / x)) + Math.sqrt((1.0 / y)))) + ((t_4 - Math.sqrt(z)) + (0.5 * Math.sqrt((1.0 / t))));
	} else {
		tmp = t_1 + (t_3 + ((t_2 + t_5) / (t_2 * t_5)));
	}
	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)) - math.sqrt(x)
	t_2 = math.sqrt((1.0 + t)) + math.sqrt(t)
	t_3 = math.sqrt((1.0 + y)) - math.sqrt(y)
	t_4 = math.sqrt((1.0 + z))
	t_5 = t_4 + math.sqrt(z)
	tmp = 0
	if (t_1 + t_3) <= 5e-5:
		tmp = (0.5 * (math.sqrt((1.0 / x)) + math.sqrt((1.0 / y)))) + ((t_4 - math.sqrt(z)) + (0.5 * math.sqrt((1.0 / t))))
	else:
		tmp = t_1 + (t_3 + ((t_2 + t_5) / (t_2 * t_5)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_2 = Float64(sqrt(Float64(1.0 + t)) + sqrt(t))
	t_3 = Float64(sqrt(Float64(1.0 + y)) - sqrt(y))
	t_4 = sqrt(Float64(1.0 + z))
	t_5 = Float64(t_4 + sqrt(z))
	tmp = 0.0
	if (Float64(t_1 + t_3) <= 5e-5)
		tmp = Float64(Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + sqrt(Float64(1.0 / y)))) + Float64(Float64(t_4 - sqrt(z)) + Float64(0.5 * sqrt(Float64(1.0 / t)))));
	else
		tmp = Float64(t_1 + Float64(t_3 + Float64(Float64(t_2 + t_5) / Float64(t_2 * t_5))));
	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)) - sqrt(x);
	t_2 = sqrt((1.0 + t)) + sqrt(t);
	t_3 = sqrt((1.0 + y)) - sqrt(y);
	t_4 = sqrt((1.0 + z));
	t_5 = t_4 + sqrt(z);
	tmp = 0.0;
	if ((t_1 + t_3) <= 5e-5)
		tmp = (0.5 * (sqrt((1.0 / x)) + sqrt((1.0 / y)))) + ((t_4 - sqrt(z)) + (0.5 * sqrt((1.0 / t))));
	else
		tmp = t_1 + (t_3 + ((t_2 + t_5) / (t_2 * t_5)));
	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[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $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[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + t$95$3), $MachinePrecision], 5e-5], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(t$95$3 + N[(N[(t$95$2 + t$95$5), $MachinePrecision] / N[(t$95$2 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.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))) < 5.00000000000000024e-5

   1. Initial program 85.9%

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

      1. associate-+l+ 85.9%

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

      2. associate-+l- 85.0%

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

      3. associate-+l- 85.9%

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

      4. +-commutative 85.9%

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

      5. +-commutative 85.9%

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

      6. +-commutative 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in y around inf 85.0%

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

   6. Taylor expanded in x around inf 90.8%

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

      1. distribute-lft-out 90.8%

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

   8. Simplified 90.8%

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

   9. Taylor expanded in t around inf 36.5%

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

   if 5.00000000000000024e-5 < (+.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)))

   1. Initial program 96.7%

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

      1. associate-+l+ 96.7%

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

      2. associate-+l+ 96.7%

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

      3. +-commutative 96.7%

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

      4. +-commutative 96.7%

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

      5. associate-+l- 78.2%

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

      6. +-commutative 78.2%

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

      7. +-commutative 78.2%

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

   3. Simplified 78.2%

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

      1. associate--r- 96.7%

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

      2. +-commutative 96.7%

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

      3. flip-- 96.7%

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

      4. flip-- 96.9%

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

      5. frac-add 96.8%

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

   6. Applied egg-rr 97.3%

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

   8. Simplified 98.4%

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

4. Final simplification 82.6%

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

Alternative 3: 97.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.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))) < 5.00000000000000024e-5

   1. Initial program 85.9%

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

      1. associate-+l+ 85.9%

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

      2. associate-+l- 85.0%

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

      3. associate-+l- 85.9%

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

      4. +-commutative 85.9%

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

      5. +-commutative 85.9%

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

      6. +-commutative 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in y around inf 85.0%

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

   6. Taylor expanded in x around inf 90.8%

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

      1. distribute-lft-out 90.8%

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

   8. Simplified 90.8%

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

   9. Taylor expanded in t around inf 36.5%

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

   if 5.00000000000000024e-5 < (+.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)))

   1. Initial program 96.7%

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

      1. associate-+l+ 96.7%

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

      2. associate-+l- 74.3%

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

      3. associate-+l- 96.7%

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

      4. +-commutative 96.7%

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

      5. +-commutative 96.7%

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

      6. +-commutative 96.7%

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

   3. Simplified 96.7%

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

      1. flip-- 96.8%

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

      2. add-sqr-sqrt 75.9%

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

      3. add-sqr-sqrt 96.8%

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

   6. Applied egg-rr 96.8%

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

      1. associate--l+ 97.2%

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

      2. +-inverses 97.2%

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

      3. metadata-eval 97.2%

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

   8. Simplified 97.2%

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

4. Final simplification 81.8%

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

Alternative 4: 91.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 5.00000000000000024e-5

   1. Initial program 91.4%

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

      1. associate-+l+ 91.4%

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

      2. associate-+l- 55.1%

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

      3. associate-+l- 91.4%

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

      4. +-commutative 91.4%

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

      5. +-commutative 91.4%

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

      6. +-commutative 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in y around inf 50.7%

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

   6. Taylor expanded in x around inf 52.5%

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

      1. distribute-lft-out 52.5%

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

   8. Simplified 52.5%

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

   9. Taylor expanded in t around inf 22.8%

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

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 96.3%

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

      1. associate-+l+ 96.3%

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

      2. associate-+l+ 96.3%

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

      3. +-commutative 96.3%

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

      4. +-commutative 96.3%

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

      5. associate-+l- 79.5%

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

      6. +-commutative 79.5%

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

      7. +-commutative 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in t around inf 57.7%

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

4. Final simplification 41.3%

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

Alternative 5: 94.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{1}{t\_2 + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 + t\_1\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(t\_2 - \sqrt{z}\right) + \frac{1}{\sqrt{y} + t\_1}\right)\\ \end{array} \end{array} \]

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 y))) (t_2 (sqrt (+ 1.0 z))))
   (if (<= y 5.2e-15)
     (+
      (+ (/ 1.0 (+ t_2 (sqrt z))) (- (sqrt (+ 1.0 t)) (sqrt t)))
      (- (+ 1.0 t_1) (+ (sqrt x) (sqrt y))))
     (+
      (- (sqrt (+ x 1.0)) (sqrt x))
      (+ (- t_2 (sqrt z)) (/ 1.0 (+ (sqrt y) t_1)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + z));
	double tmp;
	if (y <= 5.2e-15) {
		tmp = ((1.0 / (t_2 + sqrt(z))) + (sqrt((1.0 + t)) - sqrt(t))) + ((1.0 + t_1) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((t_2 - sqrt(z)) + (1.0 / (sqrt(y) + t_1)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + z))
    if (y <= 5.2d-15) then
        tmp = ((1.0d0 / (t_2 + sqrt(z))) + (sqrt((1.0d0 + t)) - sqrt(t))) + ((1.0d0 + t_1) - (sqrt(x) + sqrt(y)))
    else
        tmp = (sqrt((x + 1.0d0)) - sqrt(x)) + ((t_2 - sqrt(z)) + (1.0d0 / (sqrt(y) + t_1)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((1.0 + z));
	double tmp;
	if (y <= 5.2e-15) {
		tmp = ((1.0 / (t_2 + Math.sqrt(z))) + (Math.sqrt((1.0 + t)) - Math.sqrt(t))) + ((1.0 + t_1) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + ((t_2 - Math.sqrt(z)) + (1.0 / (Math.sqrt(y) + t_1)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((1.0 + z))
	tmp = 0
	if y <= 5.2e-15:
		tmp = ((1.0 / (t_2 + math.sqrt(z))) + (math.sqrt((1.0 + t)) - math.sqrt(t))) + ((1.0 + t_1) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = (math.sqrt((x + 1.0)) - math.sqrt(x)) + ((t_2 - math.sqrt(z)) + (1.0 / (math.sqrt(y) + t_1)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (y <= 5.2e-15)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_2 + sqrt(z))) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) + Float64(Float64(1.0 + t_1) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(Float64(t_2 - sqrt(z)) + Float64(1.0 / Float64(sqrt(y) + t_1))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((1.0 + z));
	tmp = 0.0;
	if (y <= 5.2e-15)
		tmp = ((1.0 / (t_2 + sqrt(z))) + (sqrt((1.0 + t)) - sqrt(t))) + ((1.0 + t_1) - (sqrt(x) + sqrt(y)));
	else
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((t_2 - sqrt(z)) + (1.0 / (sqrt(y) + t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 5.2e-15], N[(N[(N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + t$95$1), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.20000000000000009e-15

   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. Step-by-step derivation

      1. associate-+l+ 97.4%

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

      2. associate-+l- 62.2%

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

      3. associate-+l- 97.4%

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

      4. +-commutative 97.4%

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

      5. +-commutative 97.4%

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

      6. +-commutative 97.4%

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

   3. Simplified 97.4%

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

      1. flip-- 97.4%

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

      2. add-sqr-sqrt 74.9%

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

      3. add-sqr-sqrt 97.4%

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

   6. Applied egg-rr 97.4%

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

      1. associate--l+ 97.9%

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

      2. +-inverses 97.9%

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

      3. metadata-eval 97.9%

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

   8. Simplified 97.9%

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

   9. Taylor expanded in x around 0 51.7%

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

   if 5.20000000000000009e-15 < y

   1. Initial program 91.1%

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

      1. associate-+l+ 91.1%

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

      2. associate-+l+ 91.1%

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

      3. +-commutative 91.1%

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

      4. +-commutative 91.1%

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

      5. associate-+l- 74.0%

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

      6. +-commutative 74.0%

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

      7. +-commutative 74.0%

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

   3. Simplified 74.0%

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

      1. flip-- 74.1%

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

      2. add-sqr-sqrt 45.9%

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

      3. add-sqr-sqrt 74.4%

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

   6. Applied egg-rr 74.4%

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

      1. associate--l+ 77.2%

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

      2. +-inverses 77.2%

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

      3. metadata-eval 77.2%

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

   8. Simplified 77.2%

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

   9. Taylor expanded in t around inf 46.2%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 400000:\\ \;\;\;\;\left(\left(1 + t\_2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(t\_1 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_1 + \frac{1}{\sqrt{y} + t\_2}\right)\\ \end{array} \end{array} \]

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)) (sqrt z))) (t_2 (sqrt (+ 1.0 y))))
   (if (<= z 400000.0)
     (+
      (- (+ 1.0 t_2) (+ (sqrt x) (sqrt y)))
      (+ t_1 (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t)))))
     (+ (- (sqrt (+ x 1.0)) (sqrt x)) (+ t_1 (/ 1.0 (+ (sqrt y) t_2)))))))

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 + y));
	double tmp;
	if (z <= 400000.0) {
		tmp = ((1.0 + t_2) - (sqrt(x) + sqrt(y))) + (t_1 + (1.0 / (sqrt((1.0 + t)) + sqrt(t))));
	} else {
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + (t_1 + (1.0 / (sqrt(y) + t_2)));
	}
	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 + y))
    if (z <= 400000.0d0) then
        tmp = ((1.0d0 + t_2) - (sqrt(x) + sqrt(y))) + (t_1 + (1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t))))
    else
        tmp = (sqrt((x + 1.0d0)) - sqrt(x)) + (t_1 + (1.0d0 / (sqrt(y) + t_2)))
    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 + y));
	double tmp;
	if (z <= 400000.0) {
		tmp = ((1.0 + t_2) - (Math.sqrt(x) + Math.sqrt(y))) + (t_1 + (1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))));
	} else {
		tmp = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (t_1 + (1.0 / (Math.sqrt(y) + t_2)));
	}
	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 + y))
	tmp = 0
	if z <= 400000.0:
		tmp = ((1.0 + t_2) - (math.sqrt(x) + math.sqrt(y))) + (t_1 + (1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))))
	else:
		tmp = (math.sqrt((x + 1.0)) - math.sqrt(x)) + (t_1 + (1.0 / (math.sqrt(y) + t_2)))
	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(1.0 + y))
	tmp = 0.0
	if (z <= 400000.0)
		tmp = Float64(Float64(Float64(1.0 + t_2) - Float64(sqrt(x) + sqrt(y))) + Float64(t_1 + Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t)))));
	else
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_1 + Float64(1.0 / Float64(sqrt(y) + t_2))));
	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 + y));
	tmp = 0.0;
	if (z <= 400000.0)
		tmp = ((1.0 + t_2) - (sqrt(x) + sqrt(y))) + (t_1 + (1.0 / (sqrt((1.0 + t)) + sqrt(t))));
	else
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + (t_1 + (1.0 / (sqrt(y) + t_2)));
	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[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 400000.0], N[(N[(N[(1.0 + t$95$2), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 4e5

   1. Initial program 96.6%

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

      1. associate-+l+ 96.6%

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

      2. associate-+l- 79.4%

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

      3. associate-+l- 96.6%

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

      4. +-commutative 96.6%

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

      5. +-commutative 96.6%

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

      6. +-commutative 96.6%

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

   3. Simplified 96.6%

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

      1. flip-- 96.7%

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

      2. add-sqr-sqrt 76.3%

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

      3. add-sqr-sqrt 97.3%

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

   6. Applied egg-rr 97.3%

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

      1. associate--l+ 97.7%

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

      2. +-inverses 97.7%

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

      3. metadata-eval 97.7%

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

   8. Simplified 97.7%

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

   9. Taylor expanded in x around 0 41.6%

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

   if 4e5 < z

   1. Initial program 91.0%

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

      1. associate-+l+ 91.0%

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

      2. associate-+l+ 91.0%

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

      3. +-commutative 91.0%

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

      4. +-commutative 91.0%

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

      5. associate-+l- 90.6%

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

      6. +-commutative 90.6%

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

      7. +-commutative 90.6%

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

   3. Simplified 90.6%

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

      1. flip-- 90.8%

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

      2. add-sqr-sqrt 73.7%

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

      3. add-sqr-sqrt 91.1%

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

   6. Applied egg-rr 91.1%

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

      1. associate--l+ 93.7%

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

      2. +-inverses 93.7%

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

      3. metadata-eval 93.7%

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

   8. Simplified 93.7%

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

   9. Taylor expanded in t around inf 50.4%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 400000:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z} - \sqrt{z}\\ \mathbf{if}\;x \leq 3.35 \cdot 10^{-164}:\\ \;\;\;\;\left(t\_2 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 + \left(t\_1 - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;x \leq 115000000:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_2 + \frac{1}{\sqrt{y} + t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right) + \left(t\_2 + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\\ \end{array} \end{array} \]

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)) (sqrt z))))
   (if (<= x 3.35e-164)
     (+
      (+ t_2 (- (sqrt (+ 1.0 t)) (sqrt t)))
      (+ 1.0 (- t_1 (+ (sqrt x) (sqrt y)))))
     (if (<= x 115000000.0)
       (+ (- (sqrt (+ x 1.0)) (sqrt x)) (+ t_2 (/ 1.0 (+ (sqrt y) t_1))))
       (+
        (* 0.5 (+ (sqrt (/ 1.0 x)) (sqrt (/ 1.0 y))))
        (+ t_2 (* 0.5 (sqrt (/ 1.0 t)))))))))

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)) - sqrt(z);
	double tmp;
	if (x <= 3.35e-164) {
		tmp = (t_2 + (sqrt((1.0 + t)) - sqrt(t))) + (1.0 + (t_1 - (sqrt(x) + sqrt(y))));
	} else if (x <= 115000000.0) {
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + (t_2 + (1.0 / (sqrt(y) + t_1)));
	} else {
		tmp = (0.5 * (sqrt((1.0 / x)) + sqrt((1.0 / y)))) + (t_2 + (0.5 * sqrt((1.0 / t))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + z)) - sqrt(z)
    if (x <= 3.35d-164) then
        tmp = (t_2 + (sqrt((1.0d0 + t)) - sqrt(t))) + (1.0d0 + (t_1 - (sqrt(x) + sqrt(y))))
    else if (x <= 115000000.0d0) then
        tmp = (sqrt((x + 1.0d0)) - sqrt(x)) + (t_2 + (1.0d0 / (sqrt(y) + t_1)))
    else
        tmp = (0.5d0 * (sqrt((1.0d0 / x)) + sqrt((1.0d0 / y)))) + (t_2 + (0.5d0 * sqrt((1.0d0 / t))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double tmp;
	if (x <= 3.35e-164) {
		tmp = (t_2 + (Math.sqrt((1.0 + t)) - Math.sqrt(t))) + (1.0 + (t_1 - (Math.sqrt(x) + Math.sqrt(y))));
	} else if (x <= 115000000.0) {
		tmp = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (t_2 + (1.0 / (Math.sqrt(y) + t_1)));
	} else {
		tmp = (0.5 * (Math.sqrt((1.0 / x)) + Math.sqrt((1.0 / y)))) + (t_2 + (0.5 * Math.sqrt((1.0 / t))));
	}
	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)) - math.sqrt(z)
	tmp = 0
	if x <= 3.35e-164:
		tmp = (t_2 + (math.sqrt((1.0 + t)) - math.sqrt(t))) + (1.0 + (t_1 - (math.sqrt(x) + math.sqrt(y))))
	elif x <= 115000000.0:
		tmp = (math.sqrt((x + 1.0)) - math.sqrt(x)) + (t_2 + (1.0 / (math.sqrt(y) + t_1)))
	else:
		tmp = (0.5 * (math.sqrt((1.0 / x)) + math.sqrt((1.0 / y)))) + (t_2 + (0.5 * math.sqrt((1.0 / t))))
	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 + z)) - sqrt(z))
	tmp = 0.0
	if (x <= 3.35e-164)
		tmp = Float64(Float64(t_2 + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) + Float64(1.0 + Float64(t_1 - Float64(sqrt(x) + sqrt(y)))));
	elseif (x <= 115000000.0)
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_2 + Float64(1.0 / Float64(sqrt(y) + t_1))));
	else
		tmp = Float64(Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + sqrt(Float64(1.0 / y)))) + Float64(t_2 + Float64(0.5 * sqrt(Float64(1.0 / t)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((1.0 + z)) - sqrt(z);
	tmp = 0.0;
	if (x <= 3.35e-164)
		tmp = (t_2 + (sqrt((1.0 + t)) - sqrt(t))) + (1.0 + (t_1 - (sqrt(x) + sqrt(y))));
	elseif (x <= 115000000.0)
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + (t_2 + (1.0 / (sqrt(y) + t_1)));
	else
		tmp = (0.5 * (sqrt((1.0 / x)) + sqrt((1.0 / y)))) + (t_2 + (0.5 * sqrt((1.0 / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.35e-164], N[(N[(t$95$2 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 115000000.0], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.35e-164

   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. Step-by-step derivation

      1. associate-+l+ 97.3%

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

      2. associate-+l- 97.3%

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

      3. associate-+l- 97.3%

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

      4. +-commutative 97.3%

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

      5. +-commutative 97.3%

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

      6. +-commutative 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around 0 56.8%

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

      1. associate--l+ 97.3%

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

   7. Simplified 97.3%

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

   if 3.35e-164 < x < 1.15e8

   1. Initial program 95.3%

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

      1. associate-+l+ 95.3%

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

      2. associate-+l+ 95.3%

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

      3. +-commutative 95.3%

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

      4. +-commutative 95.3%

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

      5. associate-+l- 76.2%

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

      6. +-commutative 76.2%

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

      7. +-commutative 76.2%

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

   3. Simplified 76.2%

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

      1. flip-- 76.4%

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

      2. add-sqr-sqrt 61.8%

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

      3. add-sqr-sqrt 76.4%

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

   6. Applied egg-rr 76.4%

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

      1. associate--l+ 78.0%

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

      2. +-inverses 78.0%

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

      3. metadata-eval 78.0%

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

   8. Simplified 78.0%

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

   9. Taylor expanded in t around inf 62.3%

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

   if 1.15e8 < x

   1. Initial program 91.4%

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

      1. associate-+l+ 91.4%

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

      2. associate-+l- 55.1%

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

      3. associate-+l- 91.4%

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

      4. +-commutative 91.4%

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

      5. +-commutative 91.4%

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

      6. +-commutative 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in y around inf 50.7%

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

   6. Taylor expanded in x around inf 52.5%

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

      1. distribute-lft-out 52.5%

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

   8. Simplified 52.5%

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

   9. Taylor expanded in t around inf 22.8%

      \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.35 \cdot 10^{-164}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;x \leq 115000000:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 7.2 \cdot 10^{+14}:\\ \;\;\;\;\left(2 + \left(t\_1 + \sqrt{1 + t}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(t\_1 - \sqrt{z}\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))))
   (if (<= t 7.2e+14)
     (-
      (+ 2.0 (+ t_1 (sqrt (+ 1.0 t))))
      (+ (sqrt t) (+ (sqrt x) (+ (sqrt y) (sqrt z)))))
     (+
      (- (sqrt (+ x 1.0)) (sqrt x))
      (+ (- t_1 (sqrt z)) (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 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 tmp;
	if (t <= 7.2e+14) {
		tmp = (2.0 + (t_1 + sqrt((1.0 + t)))) - (sqrt(t) + (sqrt(x) + (sqrt(y) + sqrt(z))));
	} else {
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((t_1 - sqrt(z)) + (1.0 / (sqrt(y) + sqrt((1.0 + y)))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if (t <= 7.2d+14) then
        tmp = (2.0d0 + (t_1 + sqrt((1.0d0 + t)))) - (sqrt(t) + (sqrt(x) + (sqrt(y) + sqrt(z))))
    else
        tmp = (sqrt((x + 1.0d0)) - sqrt(x)) + ((t_1 - sqrt(z)) + (1.0d0 / (sqrt(y) + sqrt((1.0d0 + 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 tmp;
	if (t <= 7.2e+14) {
		tmp = (2.0 + (t_1 + Math.sqrt((1.0 + t)))) - (Math.sqrt(t) + (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z))));
	} else {
		tmp = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + ((t_1 - Math.sqrt(z)) + (1.0 / (Math.sqrt(y) + Math.sqrt((1.0 + 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))
	tmp = 0
	if t <= 7.2e+14:
		tmp = (2.0 + (t_1 + math.sqrt((1.0 + t)))) - (math.sqrt(t) + (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z))))
	else:
		tmp = (math.sqrt((x + 1.0)) - math.sqrt(x)) + ((t_1 - math.sqrt(z)) + (1.0 / (math.sqrt(y) + math.sqrt((1.0 + 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))
	tmp = 0.0
	if (t <= 7.2e+14)
		tmp = Float64(Float64(2.0 + Float64(t_1 + sqrt(Float64(1.0 + t)))) - Float64(sqrt(t) + Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))));
	else
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(Float64(t_1 - sqrt(z)) + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + 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));
	tmp = 0.0;
	if (t <= 7.2e+14)
		tmp = (2.0 + (t_1 + sqrt((1.0 + t)))) - (sqrt(t) + (sqrt(x) + (sqrt(y) + sqrt(z))));
	else
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((t_1 - sqrt(z)) + (1.0 / (sqrt(y) + sqrt((1.0 + 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]}, If[LessEqual[t, 7.2e+14], N[(N[(2.0 + N[(t$95$1 + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 7.2e14

   1. Initial program 96.3%

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

      1. associate-+l+ 96.3%

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

      2. associate-+l- 82.5%

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

      3. associate-+l- 96.3%

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

      4. +-commutative 96.3%

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

      5. +-commutative 96.3%

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

      6. +-commutative 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in y around 0 15.5%

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

      1. associate-+r+ 15.5%

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

      2. +-commutative 15.5%

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

   7. Simplified 15.5%

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

   8. Taylor expanded in x around 0 13.4%

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

   if 7.2e14 < t

   1. Initial program 91.2%

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

      1. associate-+l+ 91.2%

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

      2. associate-+l+ 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

      5. associate-+l- 50.7%

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

      6. +-commutative 50.7%

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

      7. +-commutative 50.7%

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

   3. Simplified 50.7%

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

      1. flip-- 50.8%

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

      2. add-sqr-sqrt 44.0%

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

      3. add-sqr-sqrt 51.2%

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

   6. Applied egg-rr 51.2%

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

      1. associate--l+ 53.7%

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

      2. +-inverses 53.7%

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

      3. metadata-eval 53.7%

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

   8. Simplified 53.7%

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

   9. Taylor expanded in t around inf 93.9%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{+14}:\\ \;\;\;\;\left(2 + \left(\sqrt{1 + z} + \sqrt{1 + t}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 7.2 \cdot 10^{+14}:\\ \;\;\;\;\left(2 + \left(t\_1 + \sqrt{1 + t}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(t\_1 - \sqrt{z}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))))
   (if (<= t 7.2e+14)
     (-
      (+ 2.0 (+ t_1 (sqrt (+ 1.0 t))))
      (+ (sqrt t) (+ (sqrt x) (+ (sqrt y) (sqrt z)))))
     (+
      (- (sqrt (+ x 1.0)) (sqrt x))
      (+ (- t_1 (sqrt z)) (- (sqrt (+ 1.0 y)) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double tmp;
	if (t <= 7.2e+14) {
		tmp = (2.0 + (t_1 + sqrt((1.0 + t)))) - (sqrt(t) + (sqrt(x) + (sqrt(y) + sqrt(z))));
	} else {
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((t_1 - sqrt(z)) + (sqrt((1.0 + y)) - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if (t <= 7.2d+14) then
        tmp = (2.0d0 + (t_1 + sqrt((1.0d0 + t)))) - (sqrt(t) + (sqrt(x) + (sqrt(y) + sqrt(z))))
    else
        tmp = (sqrt((x + 1.0d0)) - sqrt(x)) + ((t_1 - sqrt(z)) + (sqrt((1.0d0 + y)) - sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double tmp;
	if (t <= 7.2e+14) {
		tmp = (2.0 + (t_1 + Math.sqrt((1.0 + t)))) - (Math.sqrt(t) + (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z))));
	} else {
		tmp = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + ((t_1 - Math.sqrt(z)) + (Math.sqrt((1.0 + y)) - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	tmp = 0
	if t <= 7.2e+14:
		tmp = (2.0 + (t_1 + math.sqrt((1.0 + t)))) - (math.sqrt(t) + (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z))))
	else:
		tmp = (math.sqrt((x + 1.0)) - math.sqrt(x)) + ((t_1 - math.sqrt(z)) + (math.sqrt((1.0 + y)) - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (t <= 7.2e+14)
		tmp = Float64(Float64(2.0 + Float64(t_1 + sqrt(Float64(1.0 + t)))) - Float64(sqrt(t) + Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))));
	else
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(Float64(t_1 - sqrt(z)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	tmp = 0.0;
	if (t <= 7.2e+14)
		tmp = (2.0 + (t_1 + sqrt((1.0 + t)))) - (sqrt(t) + (sqrt(x) + (sqrt(y) + sqrt(z))));
	else
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((t_1 - sqrt(z)) + (sqrt((1.0 + y)) - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 7.2e+14], N[(N[(2.0 + N[(t$95$1 + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 7.2e14

   1. Initial program 96.3%

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

      1. associate-+l+ 96.3%

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

      2. associate-+l- 82.5%

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

      3. associate-+l- 96.3%

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

      4. +-commutative 96.3%

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

      5. +-commutative 96.3%

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

      6. +-commutative 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in y around 0 15.5%

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

      1. associate-+r+ 15.5%

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

      2. +-commutative 15.5%

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

   7. Simplified 15.5%

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

   8. Taylor expanded in x around 0 13.4%

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

   if 7.2e14 < t

   1. Initial program 91.2%

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

      1. associate-+l+ 91.2%

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

      2. associate-+l+ 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

      5. associate-+l- 50.7%

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

      6. +-commutative 50.7%

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

      7. +-commutative 50.7%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in t around inf 91.2%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{+14}:\\ \;\;\;\;\left(2 + \left(\sqrt{1 + z} + \sqrt{1 + t}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} + \sqrt{y}\\ \mathbf{if}\;z \leq 7400000:\\ \;\;\;\;2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \left(\sqrt{z} + t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt x) (sqrt y))))
   (if (<= z 7400000.0)
     (+ 2.0 (- (hypot 1.0 (sqrt z)) (+ (sqrt z) t_1)))
     (+
      (sqrt (+ x 1.0))
      (- (+ (sqrt (+ 1.0 y)) (* 0.5 (sqrt (/ 1.0 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(x) + sqrt(y);
	double tmp;
	if (z <= 7400000.0) {
		tmp = 2.0 + (hypot(1.0, sqrt(z)) - (sqrt(z) + t_1));
	} else {
		tmp = sqrt((x + 1.0)) + ((sqrt((1.0 + y)) + (0.5 * sqrt((1.0 / z)))) - t_1);
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(x) + Math.sqrt(y);
	double tmp;
	if (z <= 7400000.0) {
		tmp = 2.0 + (Math.hypot(1.0, Math.sqrt(z)) - (Math.sqrt(z) + t_1));
	} else {
		tmp = Math.sqrt((x + 1.0)) + ((Math.sqrt((1.0 + y)) + (0.5 * Math.sqrt((1.0 / 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(x) + math.sqrt(y)
	tmp = 0
	if z <= 7400000.0:
		tmp = 2.0 + (math.hypot(1.0, math.sqrt(z)) - (math.sqrt(z) + t_1))
	else:
		tmp = math.sqrt((x + 1.0)) + ((math.sqrt((1.0 + y)) + (0.5 * math.sqrt((1.0 / z)))) - t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) + sqrt(y))
	tmp = 0.0
	if (z <= 7400000.0)
		tmp = Float64(2.0 + Float64(hypot(1.0, sqrt(z)) - Float64(sqrt(z) + t_1)));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(Float64(sqrt(Float64(1.0 + y)) + Float64(0.5 * sqrt(Float64(1.0 / 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(x) + sqrt(y);
	tmp = 0.0;
	if (z <= 7400000.0)
		tmp = 2.0 + (hypot(1.0, sqrt(z)) - (sqrt(z) + t_1));
	else
		tmp = sqrt((x + 1.0)) + ((sqrt((1.0 + y)) + (0.5 * sqrt((1.0 / 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[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 7400000.0], N[(2.0 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[z], $MachinePrecision] ^ 2], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 7.4e6

   1. Initial program 96.5%

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

      1. associate-+l+ 96.5%

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

      2. associate-+l- 79.5%

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

      3. associate-+l- 96.5%

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

      4. +-commutative 96.5%

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

      5. +-commutative 96.5%

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

      6. +-commutative 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in t around inf 19.5%

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

      1. associate--l+ 23.4%

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

   7. Simplified 23.4%

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

   8. Taylor expanded in y around 0 17.6%

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

      1. associate--l+ 26.6%

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

      2. +-commutative 26.6%

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

      3. +-commutative 26.6%

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

   10. Simplified 26.6%

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

   11. Taylor expanded in x around 0 15.6%

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

       1. associate--l+ 15.6%

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

       2. metadata-eval 15.6%

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

       3. rem-square-sqrt 15.6%

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

       4. hypot-undefine 15.6%

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

       5. associate-+r+ 15.6%

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

   13. Simplified 15.6%

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

   if 7.4e6 < z

   1. Initial program 91.0%

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

      1. associate-+l+ 91.0%

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

      2. associate-+l- 74.1%

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

      3. associate-+l- 91.0%

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

      4. +-commutative 91.0%

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

      5. +-commutative 91.0%

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

      6. +-commutative 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in t around inf 3.3%

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

      1. associate--l+ 22.3%

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

   7. Simplified 22.3%

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

   8. Taylor expanded in z around inf 35.1%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7400000:\\ \;\;\;\;2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} + \sqrt{y}\\ \mathbf{if}\;z \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \left(\sqrt{z} + t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} - t\_1\right) + \sqrt{x + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt x) (sqrt y))))
   (if (<= z 5.8e+14)
     (+ 2.0 (- (hypot 1.0 (sqrt z)) (+ (sqrt z) t_1)))
     (+ (- (sqrt (+ 1.0 y)) t_1) (sqrt (+ x 1.0))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(x) + sqrt(y);
	double tmp;
	if (z <= 5.8e+14) {
		tmp = 2.0 + (hypot(1.0, sqrt(z)) - (sqrt(z) + t_1));
	} else {
		tmp = (sqrt((1.0 + y)) - t_1) + sqrt((x + 1.0));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(x) + Math.sqrt(y);
	double tmp;
	if (z <= 5.8e+14) {
		tmp = 2.0 + (Math.hypot(1.0, Math.sqrt(z)) - (Math.sqrt(z) + t_1));
	} else {
		tmp = (Math.sqrt((1.0 + y)) - t_1) + Math.sqrt((x + 1.0));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(x) + math.sqrt(y)
	tmp = 0
	if z <= 5.8e+14:
		tmp = 2.0 + (math.hypot(1.0, math.sqrt(z)) - (math.sqrt(z) + t_1))
	else:
		tmp = (math.sqrt((1.0 + y)) - t_1) + math.sqrt((x + 1.0))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) + sqrt(y))
	tmp = 0.0
	if (z <= 5.8e+14)
		tmp = Float64(2.0 + Float64(hypot(1.0, sqrt(z)) - Float64(sqrt(z) + t_1)));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + y)) - t_1) + sqrt(Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(x) + sqrt(y);
	tmp = 0.0;
	if (z <= 5.8e+14)
		tmp = 2.0 + (hypot(1.0, sqrt(z)) - (sqrt(z) + t_1));
	else
		tmp = (sqrt((1.0 + y)) - t_1) + sqrt((x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 5.8e+14], N[(2.0 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[z], $MachinePrecision] ^ 2], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 5.8e14

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. associate-+l- 79.4%

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

      3. associate-+l- 96.2%

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

      4. +-commutative 96.2%

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

      5. +-commutative 96.2%

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

      6. +-commutative 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in t around inf 19.2%

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

      1. associate--l+ 23.2%

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

   7. Simplified 23.2%

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

   8. Taylor expanded in y around 0 17.3%

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

      1. associate--l+ 26.2%

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

      2. +-commutative 26.2%

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

      3. +-commutative 26.2%

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

   10. Simplified 26.2%

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

   11. Taylor expanded in x around 0 15.4%

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

       1. associate--l+ 15.4%

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

       2. metadata-eval 15.4%

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

       3. rem-square-sqrt 15.4%

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

       4. hypot-undefine 15.4%

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

       5. associate-+r+ 15.4%

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

   13. Simplified 15.4%

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

   if 5.8e14 < z

   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. Step-by-step derivation

      1. associate-+l+ 91.3%

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

      2. associate-+l- 74.1%

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

      3. associate-+l- 91.3%

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

      4. +-commutative 91.3%

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

      5. +-commutative 91.3%

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

      6. +-commutative 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in t around inf 3.3%

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

      1. associate--l+ 22.6%

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

   7. Simplified 22.6%

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

   8. Taylor expanded in z around inf 23.4%

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

      1. associate--l+ 35.6%

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

   10. Simplified 35.6%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \sqrt{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} + \sqrt{y}\\ \mathbf{if}\;z \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} + z \cdot \left(t\_1 \cdot \frac{-1}{z} - \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} - t\_1\right) + \sqrt{x + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt x) (sqrt y))))
   (if (<= z 5.8e+14)
     (+ 2.0 (+ (sqrt (+ 1.0 z)) (* z (- (* t_1 (/ -1.0 z)) (sqrt (/ 1.0 z))))))
     (+ (- (sqrt (+ 1.0 y)) t_1) (sqrt (+ x 1.0))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(x) + sqrt(y);
	double tmp;
	if (z <= 5.8e+14) {
		tmp = 2.0 + (sqrt((1.0 + z)) + (z * ((t_1 * (-1.0 / z)) - sqrt((1.0 / z)))));
	} else {
		tmp = (sqrt((1.0 + y)) - t_1) + sqrt((x + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(x) + sqrt(y)
    if (z <= 5.8d+14) then
        tmp = 2.0d0 + (sqrt((1.0d0 + z)) + (z * ((t_1 * ((-1.0d0) / z)) - sqrt((1.0d0 / z)))))
    else
        tmp = (sqrt((1.0d0 + y)) - t_1) + sqrt((x + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(x) + Math.sqrt(y);
	double tmp;
	if (z <= 5.8e+14) {
		tmp = 2.0 + (Math.sqrt((1.0 + z)) + (z * ((t_1 * (-1.0 / z)) - Math.sqrt((1.0 / z)))));
	} else {
		tmp = (Math.sqrt((1.0 + y)) - t_1) + Math.sqrt((x + 1.0));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(x) + math.sqrt(y)
	tmp = 0
	if z <= 5.8e+14:
		tmp = 2.0 + (math.sqrt((1.0 + z)) + (z * ((t_1 * (-1.0 / z)) - math.sqrt((1.0 / z)))))
	else:
		tmp = (math.sqrt((1.0 + y)) - t_1) + math.sqrt((x + 1.0))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) + sqrt(y))
	tmp = 0.0
	if (z <= 5.8e+14)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + z)) + Float64(z * Float64(Float64(t_1 * Float64(-1.0 / z)) - sqrt(Float64(1.0 / z))))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + y)) - t_1) + sqrt(Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(x) + sqrt(y);
	tmp = 0.0;
	if (z <= 5.8e+14)
		tmp = 2.0 + (sqrt((1.0 + z)) + (z * ((t_1 * (-1.0 / z)) - sqrt((1.0 / z)))));
	else
		tmp = (sqrt((1.0 + y)) - t_1) + sqrt((x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 5.8e+14], N[(2.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(z * N[(N[(t$95$1 * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 5.8e14

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. associate-+l- 79.4%

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

      3. associate-+l- 96.2%

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

      4. +-commutative 96.2%

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

      5. +-commutative 96.2%

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

      6. +-commutative 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in t around inf 19.2%

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

      1. associate--l+ 23.2%

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

   7. Simplified 23.2%

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

   8. Taylor expanded in y around 0 17.3%

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

      1. associate--l+ 26.2%

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

      2. +-commutative 26.2%

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

      3. +-commutative 26.2%

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

   10. Simplified 26.2%

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

   11. Taylor expanded in x around 0 15.4%

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

       1. associate--l+ 15.4%

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

       2. associate-+r+ 15.4%

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

   13. Simplified 15.4%

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

   14. Taylor expanded in z around inf 15.3%

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

       1. distribute-rgt-out 15.3%

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

       2. +-commutative 15.3%

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

   16. Simplified 15.3%

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

   if 5.8e14 < z

   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. Step-by-step derivation

      1. associate-+l+ 91.3%

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

      2. associate-+l- 74.1%

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

      3. associate-+l- 91.3%

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

      4. +-commutative 91.3%

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

      5. +-commutative 91.3%

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

      6. +-commutative 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in t around inf 3.3%

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

      1. associate--l+ 22.6%

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

   7. Simplified 22.6%

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

   8. Taylor expanded in z around inf 23.4%

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

      1. associate--l+ 35.6%

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

   10. Simplified 35.6%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} + z \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \frac{-1}{z} - \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \sqrt{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} + \sqrt{y}\\ \mathbf{if}\;z \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} - t\_1\right) + \sqrt{x + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt x) (sqrt y))))
   (if (<= z 5.8e+14)
     (+ 2.0 (- (sqrt (+ 1.0 z)) (+ (sqrt z) t_1)))
     (+ (- (sqrt (+ 1.0 y)) t_1) (sqrt (+ x 1.0))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(x) + sqrt(y);
	double tmp;
	if (z <= 5.8e+14) {
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(z) + t_1));
	} else {
		tmp = (sqrt((1.0 + y)) - t_1) + sqrt((x + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(x) + sqrt(y)
    if (z <= 5.8d+14) then
        tmp = 2.0d0 + (sqrt((1.0d0 + z)) - (sqrt(z) + t_1))
    else
        tmp = (sqrt((1.0d0 + y)) - t_1) + sqrt((x + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(x) + Math.sqrt(y);
	double tmp;
	if (z <= 5.8e+14) {
		tmp = 2.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(z) + t_1));
	} else {
		tmp = (Math.sqrt((1.0 + y)) - t_1) + Math.sqrt((x + 1.0));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(x) + math.sqrt(y)
	tmp = 0
	if z <= 5.8e+14:
		tmp = 2.0 + (math.sqrt((1.0 + z)) - (math.sqrt(z) + t_1))
	else:
		tmp = (math.sqrt((1.0 + y)) - t_1) + math.sqrt((x + 1.0))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) + sqrt(y))
	tmp = 0.0
	if (z <= 5.8e+14)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(z) + t_1)));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + y)) - t_1) + sqrt(Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(x) + sqrt(y);
	tmp = 0.0;
	if (z <= 5.8e+14)
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(z) + t_1));
	else
		tmp = (sqrt((1.0 + y)) - t_1) + sqrt((x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 5.8e+14], N[(2.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 5.8e14

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. associate-+l- 79.4%

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

      3. associate-+l- 96.2%

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

      4. +-commutative 96.2%

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

      5. +-commutative 96.2%

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

      6. +-commutative 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in t around inf 19.2%

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

      1. associate--l+ 23.2%

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

   7. Simplified 23.2%

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

   8. Taylor expanded in y around 0 17.3%

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

      1. associate--l+ 26.2%

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

      2. +-commutative 26.2%

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

      3. +-commutative 26.2%

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

   10. Simplified 26.2%

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

   11. Taylor expanded in x around 0 15.4%

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

       1. associate--l+ 15.4%

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

       2. associate-+r+ 15.4%

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

   13. Simplified 15.4%

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

   if 5.8e14 < z

   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. Step-by-step derivation

      1. associate-+l+ 91.3%

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

      2. associate-+l- 74.1%

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

      3. associate-+l- 91.3%

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

      4. +-commutative 91.3%

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

      5. +-commutative 91.3%

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

      6. +-commutative 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in t around inf 3.3%

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

      1. associate--l+ 22.6%

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

   7. Simplified 22.6%

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

   8. Taylor expanded in z around inf 23.4%

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

      1. associate--l+ 35.6%

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

   10. Simplified 35.6%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \sqrt{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.0)
   (+ 2.0 (- (sqrt (+ 1.0 z)) (+ (sqrt z) (+ (sqrt x) (sqrt y)))))
   (- (sqrt (+ x 1.0)) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.0) {
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(z) + (sqrt(x) + sqrt(y))));
	} else {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 3.0d0) then
        tmp = 2.0d0 + (sqrt((1.0d0 + z)) - (sqrt(z) + (sqrt(x) + sqrt(y))))
    else
        tmp = sqrt((x + 1.0d0)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.0) {
		tmp = 2.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(z) + (Math.sqrt(x) + Math.sqrt(y))));
	} else {
		tmp = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 3.0:
		tmp = 2.0 + (math.sqrt((1.0 + z)) - (math.sqrt(z) + (math.sqrt(x) + math.sqrt(y))))
	else:
		tmp = math.sqrt((x + 1.0)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.0)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(z) + Float64(sqrt(x) + sqrt(y)))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3.0)
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(z) + (sqrt(x) + sqrt(y))));
	else
		tmp = sqrt((x + 1.0)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 3.0], N[(2.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3

   1. Initial program 97.5%

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

      1. associate-+l+ 97.5%

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

      2. associate-+l- 62.9%

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

      3. associate-+l- 97.5%

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

      4. +-commutative 97.5%

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

      5. +-commutative 97.5%

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

      6. +-commutative 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in t around inf 20.5%

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

      1. associate--l+ 24.9%

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

   7. Simplified 24.9%

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

   8. Taylor expanded in y around 0 20.0%

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

      1. associate--l+ 38.3%

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

      2. +-commutative 38.3%

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

      3. +-commutative 38.3%

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

   10. Simplified 38.3%

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

   11. Taylor expanded in x around 0 17.9%

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

       1. associate--l+ 35.9%

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

       2. associate-+r+ 35.9%

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

   13. Simplified 35.9%

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

   if 3 < y

   1. Initial program 90.7%

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

      1. associate-+l+ 90.7%

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

      2. associate-+l- 90.2%

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

      3. associate-+l- 90.7%

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

      4. +-commutative 90.7%

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

      5. +-commutative 90.7%

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

      6. +-commutative 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 21.0%

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

   7. Simplified 21.0%

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

   8. Taylor expanded in x around inf 20.2%

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

      1. neg-mul-1 20.2%

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

   10. Simplified 20.2%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 81.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.45)
   (+ (- (sqrt (+ 1.0 z)) (sqrt z)) 2.0)
   (- (sqrt (+ x 1.0)) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.45) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	} else {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.45d0) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + 2.0d0
    else
        tmp = sqrt((x + 1.0d0)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.45) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + 2.0;
	} else {
		tmp = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 1.45:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + 2.0
	else:
		tmp = math.sqrt((x + 1.0)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.45)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + 2.0);
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.45)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	else
		tmp = sqrt((x + 1.0)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 1.45], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.44999999999999996

   1. Initial program 97.5%

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

      1. associate-+l+ 97.5%

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

      2. associate-+l- 62.9%

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

      3. associate-+l- 97.5%

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

      4. +-commutative 97.5%

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

      5. +-commutative 97.5%

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

      6. +-commutative 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in t around inf 20.5%

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

      1. associate--l+ 24.9%

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

   7. Simplified 24.9%

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

   8. Taylor expanded in y around 0 20.0%

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

      1. associate--l+ 38.3%

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

      2. +-commutative 38.3%

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

      3. +-commutative 38.3%

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

   10. Simplified 38.3%

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

   11. Taylor expanded in x around 0 17.9%

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

       1. associate--l+ 35.9%

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

       2. associate-+r+ 35.9%

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

   13. Simplified 35.9%

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

   14. Taylor expanded in z around inf 53.7%

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

   if 1.44999999999999996 < y

   1. Initial program 90.7%

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

      1. associate-+l+ 90.7%

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

      2. associate-+l- 90.2%

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

      3. associate-+l- 90.7%

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

      4. +-commutative 90.7%

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

      5. +-commutative 90.7%

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

      6. +-commutative 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 21.0%

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

   7. Simplified 21.0%

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

   8. Taylor expanded in x around inf 20.2%

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

      1. neg-mul-1 20.2%

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

   10. Simplified 20.2%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.2% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 97.5%

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

      1. associate-+l+ 97.5%

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

      2. associate-+l- 62.9%

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

      3. associate-+l- 97.5%

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

      4. +-commutative 97.5%

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

      5. +-commutative 97.5%

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

      6. +-commutative 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in t around inf 20.5%

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

      1. associate--l+ 24.9%

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

   7. Simplified 24.9%

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

   8. Taylor expanded in y around 0 20.0%

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

      1. associate--l+ 38.3%

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

      2. +-commutative 38.3%

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

      3. +-commutative 38.3%

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

   10. Simplified 38.3%

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

   11. Taylor expanded in x around 0 17.9%

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

       1. associate--l+ 35.9%

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

       2. associate-+r+ 35.9%

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

   13. Simplified 35.9%

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

   14. Taylor expanded in z around inf 21.9%

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

       1. +-commutative 21.9%

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

   16. Simplified 21.9%

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

   if 1 < y

   1. Initial program 90.7%

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

      1. associate-+l+ 90.7%

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

      2. associate-+l- 90.2%

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

      3. associate-+l- 90.7%

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

      4. +-commutative 90.7%

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

      5. +-commutative 90.7%

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

      6. +-commutative 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 21.0%

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

   7. Simplified 21.0%

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

   8. Taylor expanded in x around inf 20.2%

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

      1. neg-mul-1 20.2%

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

   10. Simplified 20.2%

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

4. Final simplification 21.0%

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

Alternative 17: 43.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.25:\\ \;\;\;\;2 - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \sqrt{\frac{1}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 0.25) (- 2.0 (+ (sqrt x) (sqrt y))) (+ 1.0 (* y (sqrt (/ 1.0 y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.25) {
		tmp = 2.0 - (sqrt(x) + sqrt(y));
	} else {
		tmp = 1.0 + (y * sqrt((1.0 / y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 0.25d0) then
        tmp = 2.0d0 - (sqrt(x) + sqrt(y))
    else
        tmp = 1.0d0 + (y * sqrt((1.0d0 / y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.25) {
		tmp = 2.0 - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = 1.0 + (y * Math.sqrt((1.0 / y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 0.25:
		tmp = 2.0 - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = 1.0 + (y * math.sqrt((1.0 / y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 0.25)
		tmp = Float64(2.0 - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(1.0 + Float64(y * sqrt(Float64(1.0 / y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 0.25)
		tmp = 2.0 - (sqrt(x) + sqrt(y));
	else
		tmp = 1.0 + (y * sqrt((1.0 / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 0.25], N[(2.0 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.25

   1. Initial program 97.5%

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

      1. associate-+l+ 97.5%

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

      2. associate-+l- 62.9%

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

      3. associate-+l- 97.5%

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

      4. +-commutative 97.5%

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

      5. +-commutative 97.5%

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

      6. +-commutative 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in t around inf 20.5%

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

      1. associate--l+ 24.9%

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

   7. Simplified 24.9%

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

   8. Taylor expanded in y around 0 20.0%

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

      1. associate--l+ 38.3%

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

      2. +-commutative 38.3%

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

      3. +-commutative 38.3%

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

   10. Simplified 38.3%

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

   11. Taylor expanded in x around 0 17.9%

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

       1. associate--l+ 35.9%

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

       2. associate-+r+ 35.9%

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

   13. Simplified 35.9%

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

   14. Taylor expanded in z around inf 21.9%

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

       1. +-commutative 21.9%

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

   16. Simplified 21.9%

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

   if 0.25 < y

   1. Initial program 90.7%

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

      1. associate-+l+ 90.7%

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

      2. associate-+l- 90.2%

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

      3. associate-+l- 90.7%

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

      4. +-commutative 90.7%

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

      5. +-commutative 90.7%

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

      6. +-commutative 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 21.0%

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

   7. Simplified 21.0%

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

   8. Taylor expanded in y around 0 1.9%

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

      1. associate--l+ 22.4%

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

      2. +-commutative 22.4%

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

      3. +-commutative 22.4%

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

   10. Simplified 22.4%

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

   11. Taylor expanded in y around inf 22.4%

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

       1. distribute-lft-out 21.7%

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

       2. +-commutative 21.7%

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

       3. associate-*r/ 18.1%

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

       4. *-rgt-identity 18.1%

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

       5. *-commutative 18.1%

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

   13. Simplified 18.1%

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

   14. Taylor expanded in y around -inf 0.0%

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

       1. mul-1-neg 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 6.9%

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

       5. distribute-lft-neg-in 6.9%

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

       6. metadata-eval 6.9%

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

       7. *-lft-identity 6.9%

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

   16. Simplified 6.9%

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

4. Final simplification 14.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.25:\\ \;\;\;\;2 - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \sqrt{\frac{1}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 15.3% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 + (y * sqrt((1.0 / 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[(y * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.0%

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

   1. associate-+l+ 94.0%

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

   2. associate-+l- 77.0%

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

   3. associate-+l- 94.0%

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

   4. +-commutative 94.0%

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

   5. +-commutative 94.0%

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

   6. +-commutative 94.0%

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

3. Simplified 94.0%

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

5. Taylor expanded in t around inf 11.9%

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

   1. associate--l+ 22.9%

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

7. Simplified 22.9%

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

8. Taylor expanded in y around 0 10.7%

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

   1. associate--l+ 30.1%

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

   2. +-commutative 30.1%

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

   3. +-commutative 30.1%

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

10. Simplified 30.1%

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

11. Taylor expanded in y around inf 27.8%

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

    1. distribute-lft-out 27.1%

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

    2. +-commutative 27.1%

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

    3. associate-*r/ 24.5%

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

    4. *-rgt-identity 24.5%

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

    5. *-commutative 24.5%

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

13. Simplified 24.5%

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

14. Taylor expanded in y around -inf 0.0%

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

    1. mul-1-neg 0.0%

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

    2. *-commutative 0.0%

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

    3. unpow2 0.0%

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

    4. rem-square-sqrt 16.1%

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

    5. distribute-lft-neg-in 16.1%

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

    6. metadata-eval 16.1%

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

    7. *-lft-identity 16.1%

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

16. Simplified 16.1%

    \[\leadsto 1 + y \cdot \color{blue}{\sqrt{\frac{1}{y}}} \]
17. Add Preprocessing

Alternative 19: 2.5% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 94.0%

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

   1. associate-+l+ 94.0%

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

   2. associate-+l- 77.0%

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

   3. associate-+l- 94.0%

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

   4. +-commutative 94.0%

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

   5. +-commutative 94.0%

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

   6. +-commutative 94.0%

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

3. Simplified 94.0%

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

5. Taylor expanded in y around 0 10.0%

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

   1. associate-+r+ 10.0%

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

   2. +-commutative 10.0%

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

7. Simplified 10.0%

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

8. Taylor expanded in y around inf 1.6%

   \[\leadsto \color{blue}{-1 \cdot \sqrt{y}} \]
9. Step-by-step derivation

   1. mul-1-neg 1.6%

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

10. Simplified 1.6%

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

    1. expm1-log1p-u 0.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-\sqrt{y}\right)\right)} \]

    2. expm1-undefine 1.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\sqrt{y}\right)} - 1} \]

12. Applied egg-rr 1.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\sqrt{y}\right)} - 1} \]
13. Step-by-step derivation

    1. sub-neg 1.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\sqrt{y}\right)} + \left(-1\right)} \]

    2. log1p-undefine 1.4%

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

    3. rem-exp-log 2.1%

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

    4. unsub-neg 2.1%

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

    5. metadata-eval 2.1%

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

14. Simplified 2.1%

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

    1. associate-+l- 2.1%

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

16. Applied egg-rr 2.1%

    \[\leadsto \color{blue}{1 - \left(\sqrt{y} - -1\right)} \]
17. Add Preprocessing

Alternative 20: 2.5% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. associate-+l+ 94.0%

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

   2. associate-+l- 77.0%

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

   3. associate-+l- 94.0%

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

   4. +-commutative 94.0%

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

   5. +-commutative 94.0%

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

   6. +-commutative 94.0%

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

3. Simplified 94.0%

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

5. Taylor expanded in y around 0 10.0%

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

   1. associate-+r+ 10.0%

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

   2. +-commutative 10.0%

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

7. Simplified 10.0%

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

8. Taylor expanded in y around inf 1.6%

   \[\leadsto \color{blue}{-1 \cdot \sqrt{y}} \]
9. Step-by-step derivation

   1. mul-1-neg 1.6%

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

10. Simplified 1.6%

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

    1. expm1-log1p-u 0.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-\sqrt{y}\right)\right)} \]

    2. expm1-undefine 1.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\sqrt{y}\right)} - 1} \]

12. Applied egg-rr 1.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\sqrt{y}\right)} - 1} \]
13. Step-by-step derivation

    1. sub-neg 1.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\sqrt{y}\right)} + \left(-1\right)} \]

    2. log1p-undefine 1.4%

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

    3. rem-exp-log 2.1%

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

    4. unsub-neg 2.1%

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

    5. metadata-eval 2.1%

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

14. Simplified 2.1%

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

15. Final simplification 2.1%

    \[\leadsto -1 + \left(1 - \sqrt{y}\right) \]
16. Add Preprocessing

Alternative 21: 1.9% accurate, 8.1× 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 94.0%

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

   1. associate-+l+ 94.0%

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

   2. associate-+l- 77.0%

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

   3. associate-+l- 94.0%

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

   4. +-commutative 94.0%

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

   5. +-commutative 94.0%

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

   6. +-commutative 94.0%

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

3. Simplified 94.0%

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

5. Taylor expanded in t around inf 11.9%

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

   1. associate--l+ 22.9%

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

7. Simplified 22.9%

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

8. Taylor expanded in y around 0 10.7%

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

   1. associate--l+ 30.1%

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

   2. +-commutative 30.1%

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

   3. +-commutative 30.1%

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

10. Simplified 30.1%

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

11. Taylor expanded in x around 0 9.4%

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

    1. associate--l+ 21.1%

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

    2. associate-+r+ 21.1%

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

13. Simplified 21.1%

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

14. Taylor expanded in x around inf 1.6%

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

    1. mul-1-neg 1.6%

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

16. Simplified 1.6%

    \[\leadsto \color{blue}{-\sqrt{x}} \]
17. Add Preprocessing

Developer Target 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024147 
(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))))