Main:z from

Percentage Accurate: 91.4% → 99.3%

Time: 35.4s

Alternatives: 27

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: 91.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ t_2 := \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ t_3 := \sqrt{\frac{1}{y}}\\ t_4 := \sqrt{z + 1} - \sqrt{z}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\left(0.5 \cdot \left(\sqrt{\frac{1}{x}} + t\_3\right) + t\_4\right) + t\_1\\ \mathbf{elif}\;t\_2 \leq 1.005:\\ \;\;\;\;\left(\left(\frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{1 + x}} + 0.5 \cdot t\_3\right) + t\_4\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_2 (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))))
        (t_3 (sqrt (/ 1.0 y)))
        (t_4 (- (sqrt (+ z 1.0)) (sqrt z))))
   (if (<= t_2 0.0)
     (+ (+ (* 0.5 (+ (sqrt (/ 1.0 x)) t_3)) t_4) t_1)
     (if (<= t_2 1.005)
       (+
        (+
         (+ (/ (- (+ 1.0 x) x) (+ (sqrt x) (sqrt (+ 1.0 x)))) (* 0.5 t_3))
         t_4)
        t_1)
       (+
        1.0
        (-
         (+
          (+ (sqrt (+ 1.0 y)) (/ 1.0 (+ (sqrt t) (sqrt (+ 1.0 t)))))
          (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))))
         (+ (sqrt y) (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0)) - sqrt(t);
	double t_2 = (sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	double t_3 = sqrt((1.0 / y));
	double t_4 = sqrt((z + 1.0)) - sqrt(z);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = ((0.5 * (sqrt((1.0 / x)) + t_3)) + t_4) + t_1;
	} else if (t_2 <= 1.005) {
		tmp = (((((1.0 + x) - x) / (sqrt(x) + sqrt((1.0 + x)))) + (0.5 * t_3)) + t_4) + t_1;
	} else {
		tmp = 1.0 + (((sqrt((1.0 + y)) + (1.0 / (sqrt(t) + sqrt((1.0 + t))))) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) - (sqrt(y) + sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((t + 1.0d0)) - sqrt(t)
    t_2 = (sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))
    t_3 = sqrt((1.0d0 / y))
    t_4 = sqrt((z + 1.0d0)) - sqrt(z)
    if (t_2 <= 0.0d0) then
        tmp = ((0.5d0 * (sqrt((1.0d0 / x)) + t_3)) + t_4) + t_1
    else if (t_2 <= 1.005d0) then
        tmp = (((((1.0d0 + x) - x) / (sqrt(x) + sqrt((1.0d0 + x)))) + (0.5d0 * t_3)) + t_4) + t_1
    else
        tmp = 1.0d0 + (((sqrt((1.0d0 + y)) + (1.0d0 / (sqrt(t) + sqrt((1.0d0 + t))))) + (1.0d0 / (sqrt(z) + sqrt((1.0d0 + z))))) - (sqrt(y) + sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_2 = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y));
	double t_3 = Math.sqrt((1.0 / y));
	double t_4 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = ((0.5 * (Math.sqrt((1.0 / x)) + t_3)) + t_4) + t_1;
	} else if (t_2 <= 1.005) {
		tmp = (((((1.0 + x) - x) / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (0.5 * t_3)) + t_4) + t_1;
	} else {
		tmp = 1.0 + (((Math.sqrt((1.0 + y)) + (1.0 / (Math.sqrt(t) + Math.sqrt((1.0 + t))))) + (1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z))))) - (Math.sqrt(y) + Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_2 = (math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))
	t_3 = math.sqrt((1.0 / y))
	t_4 = math.sqrt((z + 1.0)) - math.sqrt(z)
	tmp = 0
	if t_2 <= 0.0:
		tmp = ((0.5 * (math.sqrt((1.0 / x)) + t_3)) + t_4) + t_1
	elif t_2 <= 1.005:
		tmp = (((((1.0 + x) - x) / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (0.5 * t_3)) + t_4) + t_1
	else:
		tmp = 1.0 + (((math.sqrt((1.0 + y)) + (1.0 / (math.sqrt(t) + math.sqrt((1.0 + t))))) + (1.0 / (math.sqrt(z) + math.sqrt((1.0 + z))))) - (math.sqrt(y) + math.sqrt(x)))
	return tmp

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((t + 1.0)) - sqrt(t);
	t_2 = (sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	t_3 = sqrt((1.0 / y));
	t_4 = sqrt((z + 1.0)) - sqrt(z);
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = ((0.5 * (sqrt((1.0 / x)) + t_3)) + t_4) + t_1;
	elseif (t_2 <= 1.005)
		tmp = (((((1.0 + x) - x) / (sqrt(x) + sqrt((1.0 + x)))) + (0.5 * t_3)) + t_4) + t_1;
	else
		tmp = 1.0 + (((sqrt((1.0 + y)) + (1.0 / (sqrt(t) + sqrt((1.0 + t))))) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) - (sqrt(y) + sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 1.005], N[(N[(N[(N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$1), $MachinePrecision], N[(1.0 + N[(N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $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 82.0%

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

   3. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   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))) < 1.0049999999999999

   1. Initial program 94.8%

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

   3. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   if 1.0049999999999999 < (+.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 98.5%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ t_2 := \frac{1}{\sqrt{z} + \sqrt{1 + z}}\\ t_3 := \sqrt{\frac{1}{y}}\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \left(\sqrt{\frac{1}{x}} + t\_3\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_1 \leq 1.005:\\ \;\;\;\;t\_4 + \left(0.5 \cdot t\_3 - \sqrt{x}\right)\\ \mathbf{elif}\;t\_1 \leq 1.99999998:\\ \;\;\;\;t\_4 + \left(\left(\sqrt{1 + y} + t\_2\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\left(\frac{1}{\sqrt{t} + \sqrt{1 + t}} + t\_2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))))
        (t_2 (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))))
        (t_3 (sqrt (/ 1.0 y)))
        (t_4 (sqrt (+ 1.0 x))))
   (if (<= t_1 5e-5)
     (+
      (+ (* 0.5 (+ (sqrt (/ 1.0 x)) t_3)) (- (sqrt (+ z 1.0)) (sqrt z)))
      (- (sqrt (+ t 1.0)) (sqrt t)))
     (if (<= t_1 1.005)
       (+ t_4 (- (* 0.5 t_3) (sqrt x)))
       (if (<= t_1 1.99999998)
         (+ t_4 (- (+ (sqrt (+ 1.0 y)) t_2) (+ (sqrt y) (sqrt x))))
         (+
          2.0
          (-
           (+ (/ 1.0 (+ (sqrt t) (sqrt (+ 1.0 t)))) t_2)
           (+ (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((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	double t_2 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	double t_3 = sqrt((1.0 / y));
	double t_4 = sqrt((1.0 + x));
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = ((0.5 * (sqrt((1.0 / x)) + t_3)) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	} else if (t_1 <= 1.005) {
		tmp = t_4 + ((0.5 * t_3) - sqrt(x));
	} else if (t_1 <= 1.99999998) {
		tmp = t_4 + ((sqrt((1.0 + y)) + t_2) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = 2.0 + (((1.0 / (sqrt(t) + sqrt((1.0 + t)))) + t_2) - (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) :: tmp
    t_1 = (sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))
    t_2 = 1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))
    t_3 = sqrt((1.0d0 / y))
    t_4 = sqrt((1.0d0 + x))
    if (t_1 <= 5d-5) then
        tmp = ((0.5d0 * (sqrt((1.0d0 / x)) + t_3)) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    else if (t_1 <= 1.005d0) then
        tmp = t_4 + ((0.5d0 * t_3) - sqrt(x))
    else if (t_1 <= 1.99999998d0) then
        tmp = t_4 + ((sqrt((1.0d0 + y)) + t_2) - (sqrt(y) + sqrt(x)))
    else
        tmp = 2.0d0 + (((1.0d0 / (sqrt(t) + sqrt((1.0d0 + t)))) + t_2) - (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((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y));
	double t_2 = 1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)));
	double t_3 = Math.sqrt((1.0 / y));
	double t_4 = Math.sqrt((1.0 + x));
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = ((0.5 * (Math.sqrt((1.0 / x)) + t_3)) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	} else if (t_1 <= 1.005) {
		tmp = t_4 + ((0.5 * t_3) - Math.sqrt(x));
	} else if (t_1 <= 1.99999998) {
		tmp = t_4 + ((Math.sqrt((1.0 + y)) + t_2) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = 2.0 + (((1.0 / (Math.sqrt(t) + Math.sqrt((1.0 + t)))) + t_2) - (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((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))
	t_2 = 1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))
	t_3 = math.sqrt((1.0 / y))
	t_4 = math.sqrt((1.0 + x))
	tmp = 0
	if t_1 <= 5e-5:
		tmp = ((0.5 * (math.sqrt((1.0 / x)) + t_3)) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	elif t_1 <= 1.005:
		tmp = t_4 + ((0.5 * t_3) - math.sqrt(x))
	elif t_1 <= 1.99999998:
		tmp = t_4 + ((math.sqrt((1.0 + y)) + t_2) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = 2.0 + (((1.0 / (math.sqrt(t) + math.sqrt((1.0 + t)))) + t_2) - (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(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)))
	t_2 = Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))
	t_3 = sqrt(Float64(1.0 / y))
	t_4 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= 5e-5)
		tmp = Float64(Float64(Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + t_3)) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	elseif (t_1 <= 1.005)
		tmp = Float64(t_4 + Float64(Float64(0.5 * t_3) - sqrt(x)));
	elseif (t_1 <= 1.99999998)
		tmp = Float64(t_4 + Float64(Float64(sqrt(Float64(1.0 + y)) + t_2) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(2.0 + Float64(Float64(Float64(1.0 / Float64(sqrt(t) + sqrt(Float64(1.0 + t)))) + t_2) - 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((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	t_2 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	t_3 = sqrt((1.0 / y));
	t_4 = sqrt((1.0 + x));
	tmp = 0.0;
	if (t_1 <= 5e-5)
		tmp = ((0.5 * (sqrt((1.0 / x)) + t_3)) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	elseif (t_1 <= 1.005)
		tmp = t_4 + ((0.5 * t_3) - sqrt(x));
	elseif (t_1 <= 1.99999998)
		tmp = t_4 + ((sqrt((1.0 + y)) + t_2) - (sqrt(y) + sqrt(x)));
	else
		tmp = 2.0 + (((1.0 / (sqrt(t) + sqrt((1.0 + t)))) + t_2) - (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[(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]}, Block[{t$95$2 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 5e-5], N[(N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$3), $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], If[LessEqual[t$95$1, 1.005], N[(t$95$4 + N[(N[(0.5 * t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.99999998], N[(t$95$4 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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 80.9%

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

   3. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   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))) < 1.0049999999999999

   1. Initial program 95.8%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]

   if 1.0049999999999999 < (+.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))) < 1.9999999799999999

   1. Initial program 97.8%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   5. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 1.9999999799999999 < (+.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 98.7%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   10. Taylor expanded in y around 0 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ t_2 := \sqrt{\frac{1}{y}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \left(\sqrt{\frac{1}{x}} + t\_2\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_1 \leq 1.005:\\ \;\;\;\;\sqrt{1 + x} + \left(0.5 \cdot t\_2 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))))
        (t_2 (sqrt (/ 1.0 y))))
   (if (<= t_1 5e-5)
     (+
      (+ (* 0.5 (+ (sqrt (/ 1.0 x)) t_2)) (- (sqrt (+ z 1.0)) (sqrt z)))
      (- (sqrt (+ t 1.0)) (sqrt t)))
     (if (<= t_1 1.005)
       (+ (sqrt (+ 1.0 x)) (- (* 0.5 t_2) (sqrt x)))
       (+
        1.0
        (-
         (+
          (+ (sqrt (+ 1.0 y)) (/ 1.0 (+ (sqrt t) (sqrt (+ 1.0 t)))))
          (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))))
         (+ (sqrt y) (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	double t_2 = sqrt((1.0 / y));
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = ((0.5 * (sqrt((1.0 / x)) + t_2)) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	} else if (t_1 <= 1.005) {
		tmp = sqrt((1.0 + x)) + ((0.5 * t_2) - sqrt(x));
	} else {
		tmp = 1.0 + (((sqrt((1.0 + y)) + (1.0 / (sqrt(t) + sqrt((1.0 + t))))) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) - (sqrt(y) + sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))
    t_2 = sqrt((1.0d0 / y))
    if (t_1 <= 5d-5) then
        tmp = ((0.5d0 * (sqrt((1.0d0 / x)) + t_2)) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    else if (t_1 <= 1.005d0) then
        tmp = sqrt((1.0d0 + x)) + ((0.5d0 * t_2) - sqrt(x))
    else
        tmp = 1.0d0 + (((sqrt((1.0d0 + y)) + (1.0d0 / (sqrt(t) + sqrt((1.0d0 + t))))) + (1.0d0 / (sqrt(z) + sqrt((1.0d0 + z))))) - (sqrt(y) + sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y));
	double t_2 = Math.sqrt((1.0 / y));
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = ((0.5 * (Math.sqrt((1.0 / x)) + t_2)) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	} else if (t_1 <= 1.005) {
		tmp = Math.sqrt((1.0 + x)) + ((0.5 * t_2) - Math.sqrt(x));
	} else {
		tmp = 1.0 + (((Math.sqrt((1.0 + y)) + (1.0 / (Math.sqrt(t) + Math.sqrt((1.0 + t))))) + (1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z))))) - (Math.sqrt(y) + Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = (math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))
	t_2 = math.sqrt((1.0 / y))
	tmp = 0
	if t_1 <= 5e-5:
		tmp = ((0.5 * (math.sqrt((1.0 / x)) + t_2)) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	elif t_1 <= 1.005:
		tmp = math.sqrt((1.0 + x)) + ((0.5 * t_2) - math.sqrt(x))
	else:
		tmp = 1.0 + (((math.sqrt((1.0 + y)) + (1.0 / (math.sqrt(t) + math.sqrt((1.0 + t))))) + (1.0 / (math.sqrt(z) + math.sqrt((1.0 + z))))) - (math.sqrt(y) + math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)))
	t_2 = sqrt(Float64(1.0 / y))
	tmp = 0.0
	if (t_1 <= 5e-5)
		tmp = Float64(Float64(Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + t_2)) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	elseif (t_1 <= 1.005)
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(Float64(0.5 * t_2) - sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(sqrt(Float64(1.0 + y)) + Float64(1.0 / Float64(sqrt(t) + sqrt(Float64(1.0 + t))))) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))) - Float64(sqrt(y) + sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = (sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	t_2 = sqrt((1.0 / y));
	tmp = 0.0;
	if (t_1 <= 5e-5)
		tmp = ((0.5 * (sqrt((1.0 / x)) + t_2)) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	elseif (t_1 <= 1.005)
		tmp = sqrt((1.0 + x)) + ((0.5 * t_2) - sqrt(x));
	else
		tmp = 1.0 + (((sqrt((1.0 + y)) + (1.0 / (sqrt(t) + sqrt((1.0 + t))))) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) - (sqrt(y) + sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(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]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 5e-5], N[(N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$2), $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], If[LessEqual[t$95$1, 1.005], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $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))) < 5.00000000000000024e-5

   1. Initial program 80.9%

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

   3. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   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))) < 1.0049999999999999

   1. Initial program 95.8%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]

   if 1.0049999999999999 < (+.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 98.5%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ t_2 := \sqrt{z + 1} - \sqrt{z}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right) + t\_2\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_1 \leq 1.998:\\ \;\;\;\;\left(t\_1 + t\_2\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\left(\left(\frac{1}{\sqrt{t} + \sqrt{1 + t}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + y \cdot 0.5\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))))
        (t_2 (- (sqrt (+ z 1.0)) (sqrt z))))
   (if (<= t_1 5e-5)
     (+
      (+ (* 0.5 (+ (sqrt (/ 1.0 x)) (sqrt (/ 1.0 y)))) t_2)
      (- (sqrt (+ t 1.0)) (sqrt t)))
     (if (<= t_1 1.998)
       (+ (+ t_1 t_2) (* 0.5 (sqrt (/ 1.0 t))))
       (+
        2.0
        (-
         (+
          (+
           (/ 1.0 (+ (sqrt t) (sqrt (+ 1.0 t))))
           (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))))
          (* y 0.5))
         (+ (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((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	double t_2 = sqrt((z + 1.0)) - sqrt(z);
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = ((0.5 * (sqrt((1.0 / x)) + sqrt((1.0 / y)))) + t_2) + (sqrt((t + 1.0)) - sqrt(t));
	} else if (t_1 <= 1.998) {
		tmp = (t_1 + t_2) + (0.5 * sqrt((1.0 / t)));
	} else {
		tmp = 2.0 + ((((1.0 / (sqrt(t) + sqrt((1.0 + t)))) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) + (y * 0.5)) - (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) :: tmp
    t_1 = (sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))
    t_2 = sqrt((z + 1.0d0)) - sqrt(z)
    if (t_1 <= 5d-5) then
        tmp = ((0.5d0 * (sqrt((1.0d0 / x)) + sqrt((1.0d0 / y)))) + t_2) + (sqrt((t + 1.0d0)) - sqrt(t))
    else if (t_1 <= 1.998d0) then
        tmp = (t_1 + t_2) + (0.5d0 * sqrt((1.0d0 / t)))
    else
        tmp = 2.0d0 + ((((1.0d0 / (sqrt(t) + sqrt((1.0d0 + t)))) + (1.0d0 / (sqrt(z) + sqrt((1.0d0 + z))))) + (y * 0.5d0)) - (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((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y));
	double t_2 = Math.sqrt((z + 1.0)) - 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) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	} else if (t_1 <= 1.998) {
		tmp = (t_1 + t_2) + (0.5 * Math.sqrt((1.0 / t)));
	} else {
		tmp = 2.0 + ((((1.0 / (Math.sqrt(t) + Math.sqrt((1.0 + t)))) + (1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z))))) + (y * 0.5)) - (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((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))
	t_2 = math.sqrt((z + 1.0)) - 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) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	elif t_1 <= 1.998:
		tmp = (t_1 + t_2) + (0.5 * math.sqrt((1.0 / t)))
	else:
		tmp = 2.0 + ((((1.0 / (math.sqrt(t) + math.sqrt((1.0 + t)))) + (1.0 / (math.sqrt(z) + math.sqrt((1.0 + z))))) + (y * 0.5)) - (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(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)))
	t_2 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	tmp = 0.0
	if (t_1 <= 5e-5)
		tmp = Float64(Float64(Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + sqrt(Float64(1.0 / y)))) + t_2) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	elseif (t_1 <= 1.998)
		tmp = Float64(Float64(t_1 + t_2) + Float64(0.5 * sqrt(Float64(1.0 / t))));
	else
		tmp = Float64(2.0 + Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(t) + sqrt(Float64(1.0 + t)))) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))) + Float64(y * 0.5)) - 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((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	t_2 = sqrt((z + 1.0)) - sqrt(z);
	tmp = 0.0;
	if (t_1 <= 5e-5)
		tmp = ((0.5 * (sqrt((1.0 / x)) + sqrt((1.0 / y)))) + t_2) + (sqrt((t + 1.0)) - sqrt(t));
	elseif (t_1 <= 1.998)
		tmp = (t_1 + t_2) + (0.5 * sqrt((1.0 / t)));
	else
		tmp = 2.0 + ((((1.0 / (sqrt(t) + sqrt((1.0 + t)))) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) + (y * 0.5)) - (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[(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]}, Block[{t$95$2 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-5], N[(N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.998], N[(N[(t$95$1 + t$95$2), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(N[(N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $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))) < 5.00000000000000024e-5

   1. Initial program 80.9%

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

   3. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   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))) < 1.998

   1. Initial program 95.9%

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

   3. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 1.998 < (+.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 98.5%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   10. Taylor expanded in y around 0 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ t_2 := \sqrt{\frac{1}{y}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \left(\sqrt{\frac{1}{x}} + t\_2\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_1 \leq 1.2:\\ \;\;\;\;\sqrt{1 + x} + \left(0.5 \cdot t\_2 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\left(\left(\frac{1}{\sqrt{t} + \sqrt{1 + t}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + y \cdot 0.5\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))))
        (t_2 (sqrt (/ 1.0 y))))
   (if (<= t_1 5e-5)
     (+
      (+ (* 0.5 (+ (sqrt (/ 1.0 x)) t_2)) (- (sqrt (+ z 1.0)) (sqrt z)))
      (- (sqrt (+ t 1.0)) (sqrt t)))
     (if (<= t_1 1.2)
       (+ (sqrt (+ 1.0 x)) (- (* 0.5 t_2) (sqrt x)))
       (+
        2.0
        (-
         (+
          (+
           (/ 1.0 (+ (sqrt t) (sqrt (+ 1.0 t))))
           (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))))
          (* y 0.5))
         (+ (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((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	double t_2 = sqrt((1.0 / y));
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = ((0.5 * (sqrt((1.0 / x)) + t_2)) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	} else if (t_1 <= 1.2) {
		tmp = sqrt((1.0 + x)) + ((0.5 * t_2) - sqrt(x));
	} else {
		tmp = 2.0 + ((((1.0 / (sqrt(t) + sqrt((1.0 + t)))) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) + (y * 0.5)) - (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) :: tmp
    t_1 = (sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))
    t_2 = sqrt((1.0d0 / y))
    if (t_1 <= 5d-5) then
        tmp = ((0.5d0 * (sqrt((1.0d0 / x)) + t_2)) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    else if (t_1 <= 1.2d0) then
        tmp = sqrt((1.0d0 + x)) + ((0.5d0 * t_2) - sqrt(x))
    else
        tmp = 2.0d0 + ((((1.0d0 / (sqrt(t) + sqrt((1.0d0 + t)))) + (1.0d0 / (sqrt(z) + sqrt((1.0d0 + z))))) + (y * 0.5d0)) - (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((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y));
	double t_2 = Math.sqrt((1.0 / y));
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = ((0.5 * (Math.sqrt((1.0 / x)) + t_2)) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	} else if (t_1 <= 1.2) {
		tmp = Math.sqrt((1.0 + x)) + ((0.5 * t_2) - Math.sqrt(x));
	} else {
		tmp = 2.0 + ((((1.0 / (Math.sqrt(t) + Math.sqrt((1.0 + t)))) + (1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z))))) + (y * 0.5)) - (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((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))
	t_2 = math.sqrt((1.0 / y))
	tmp = 0
	if t_1 <= 5e-5:
		tmp = ((0.5 * (math.sqrt((1.0 / x)) + t_2)) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	elif t_1 <= 1.2:
		tmp = math.sqrt((1.0 + x)) + ((0.5 * t_2) - math.sqrt(x))
	else:
		tmp = 2.0 + ((((1.0 / (math.sqrt(t) + math.sqrt((1.0 + t)))) + (1.0 / (math.sqrt(z) + math.sqrt((1.0 + z))))) + (y * 0.5)) - (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(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)))
	t_2 = sqrt(Float64(1.0 / y))
	tmp = 0.0
	if (t_1 <= 5e-5)
		tmp = Float64(Float64(Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + t_2)) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	elseif (t_1 <= 1.2)
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(Float64(0.5 * t_2) - sqrt(x)));
	else
		tmp = Float64(2.0 + Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(t) + sqrt(Float64(1.0 + t)))) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))) + Float64(y * 0.5)) - 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((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	t_2 = sqrt((1.0 / y));
	tmp = 0.0;
	if (t_1 <= 5e-5)
		tmp = ((0.5 * (sqrt((1.0 / x)) + t_2)) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	elseif (t_1 <= 1.2)
		tmp = sqrt((1.0 + x)) + ((0.5 * t_2) - sqrt(x));
	else
		tmp = 2.0 + ((((1.0 / (sqrt(t) + sqrt((1.0 + t)))) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) + (y * 0.5)) - (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[(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]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 5e-5], N[(N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$2), $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], If[LessEqual[t$95$1, 1.2], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(N[(N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $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))) < 5.00000000000000024e-5

   1. Initial program 80.9%

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

   3. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   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))) < 1.19999999999999996

   1. Initial program 95.8%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]

   if 1.19999999999999996 < (+.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 98.6%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   10. Taylor expanded in y around 0 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ t_2 := \sqrt{t + 1}\\ \mathbf{if}\;t\_1 \leq 0.0005:\\ \;\;\;\;\left(\left(\left(-0.125 \cdot \sqrt{\frac{\frac{1}{x}}{x \cdot x}} + 0.0625 \cdot \sqrt{\frac{1}{{x}^{5}}}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(t\_2 - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \frac{\left(t + 1\right) - t}{t\_2 + \sqrt{t}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ x 1.0)) (sqrt x))) (t_2 (sqrt (+ t 1.0))))
   (if (<= t_1 0.0005)
     (+
      (+
       (+
        (+
         (* -0.125 (sqrt (/ (/ 1.0 x) (* x x))))
         (* 0.0625 (sqrt (/ 1.0 (pow x 5.0)))))
        (* 0.5 (+ (sqrt (/ 1.0 x)) (sqrt (/ 1.0 y)))))
       (- (sqrt (+ z 1.0)) (sqrt z)))
      (- t_2 (sqrt t)))
     (+
      (+
       (+ t_1 (- (sqrt (+ y 1.0)) (sqrt y)))
       (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))))
      (/ (- (+ t 1.0) t) (+ t_2 (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0)) - sqrt(x);
	double t_2 = sqrt((t + 1.0));
	double tmp;
	if (t_1 <= 0.0005) {
		tmp = ((((-0.125 * sqrt(((1.0 / x) / (x * x)))) + (0.0625 * sqrt((1.0 / pow(x, 5.0))))) + (0.5 * (sqrt((1.0 / x)) + sqrt((1.0 / y))))) + (sqrt((z + 1.0)) - sqrt(z))) + (t_2 - sqrt(t));
	} else {
		tmp = ((t_1 + (sqrt((y + 1.0)) - sqrt(y))) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) + (((t + 1.0) - t) / (t_2 + sqrt(t)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0)) - sqrt(x)
    t_2 = sqrt((t + 1.0d0))
    if (t_1 <= 0.0005d0) then
        tmp = (((((-0.125d0) * sqrt(((1.0d0 / x) / (x * x)))) + (0.0625d0 * sqrt((1.0d0 / (x ** 5.0d0))))) + (0.5d0 * (sqrt((1.0d0 / x)) + sqrt((1.0d0 / y))))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (t_2 - sqrt(t))
    else
        tmp = ((t_1 + (sqrt((y + 1.0d0)) - sqrt(y))) + (1.0d0 / (sqrt(z) + sqrt((1.0d0 + z))))) + (((t + 1.0d0) - t) / (t_2 + sqrt(t)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double t_2 = Math.sqrt((t + 1.0));
	double tmp;
	if (t_1 <= 0.0005) {
		tmp = ((((-0.125 * Math.sqrt(((1.0 / x) / (x * x)))) + (0.0625 * Math.sqrt((1.0 / Math.pow(x, 5.0))))) + (0.5 * (Math.sqrt((1.0 / x)) + Math.sqrt((1.0 / y))))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (t_2 - Math.sqrt(t));
	} else {
		tmp = ((t_1 + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z))))) + (((t + 1.0) - t) / (t_2 + Math.sqrt(t)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0)) - math.sqrt(x)
	t_2 = math.sqrt((t + 1.0))
	tmp = 0
	if t_1 <= 0.0005:
		tmp = ((((-0.125 * math.sqrt(((1.0 / x) / (x * x)))) + (0.0625 * math.sqrt((1.0 / math.pow(x, 5.0))))) + (0.5 * (math.sqrt((1.0 / x)) + math.sqrt((1.0 / y))))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (t_2 - math.sqrt(t))
	else:
		tmp = ((t_1 + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (1.0 / (math.sqrt(z) + math.sqrt((1.0 + z))))) + (((t + 1.0) - t) / (t_2 + math.sqrt(t)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_2 = sqrt(Float64(t + 1.0))
	tmp = 0.0
	if (t_1 <= 0.0005)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.125 * sqrt(Float64(Float64(1.0 / x) / Float64(x * x)))) + Float64(0.0625 * sqrt(Float64(1.0 / (x ^ 5.0))))) + Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + sqrt(Float64(1.0 / y))))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(t_2 - sqrt(t)));
	else
		tmp = Float64(Float64(Float64(t_1 + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))) + Float64(Float64(Float64(t + 1.0) - t) / Float64(t_2 + sqrt(t))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0)) - sqrt(x);
	t_2 = sqrt((t + 1.0));
	tmp = 0.0;
	if (t_1 <= 0.0005)
		tmp = ((((-0.125 * sqrt(((1.0 / x) / (x * x)))) + (0.0625 * sqrt((1.0 / (x ^ 5.0))))) + (0.5 * (sqrt((1.0 / x)) + sqrt((1.0 / y))))) + (sqrt((z + 1.0)) - sqrt(z))) + (t_2 - sqrt(t));
	else
		tmp = ((t_1 + (sqrt((y + 1.0)) - sqrt(y))) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) + (((t + 1.0) - t) / (t_2 + sqrt(t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0005], N[(N[(N[(N[(N[(-0.125 * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.0625 * N[Sqrt[N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t + 1.0), $MachinePrecision] - t), $MachinePrecision] / N[(t$95$2 + N[Sqrt[t], $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)) < 5.0000000000000001e-4

   1. Initial program 88.1%

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

   3. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   if 5.0000000000000001e-4 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 97.0%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   6. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ t_2 := \sqrt{t + 1}\\ \mathbf{if}\;t\_1 \leq 0.0005:\\ \;\;\;\;\left(\left(\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \left(0.5 \cdot \sqrt{x} + 0.0625 \cdot \sqrt{\frac{\frac{1}{x}}{x \cdot x}}\right)}{x} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(t\_2 - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \frac{\left(t + 1\right) - t}{t\_2 + \sqrt{t}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ x 1.0)) (sqrt x))) (t_2 (sqrt (+ t 1.0))))
   (if (<= t_1 0.0005)
     (+
      (+
       (+
        (/
         (+
          (* -0.125 (sqrt (/ 1.0 x)))
          (+ (* 0.5 (sqrt x)) (* 0.0625 (sqrt (/ (/ 1.0 x) (* x x))))))
         x)
        (* 0.5 (sqrt (/ 1.0 y))))
       (- (sqrt (+ z 1.0)) (sqrt z)))
      (- t_2 (sqrt t)))
     (+
      (+
       (+ t_1 (- (sqrt (+ y 1.0)) (sqrt y)))
       (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))))
      (/ (- (+ t 1.0) t) (+ t_2 (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0)) - sqrt(x);
	double t_2 = sqrt((t + 1.0));
	double tmp;
	if (t_1 <= 0.0005) {
		tmp = (((((-0.125 * sqrt((1.0 / x))) + ((0.5 * sqrt(x)) + (0.0625 * sqrt(((1.0 / x) / (x * x)))))) / x) + (0.5 * sqrt((1.0 / y)))) + (sqrt((z + 1.0)) - sqrt(z))) + (t_2 - sqrt(t));
	} else {
		tmp = ((t_1 + (sqrt((y + 1.0)) - sqrt(y))) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) + (((t + 1.0) - t) / (t_2 + sqrt(t)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0)) - sqrt(x)
    t_2 = sqrt((t + 1.0d0))
    if (t_1 <= 0.0005d0) then
        tmp = ((((((-0.125d0) * sqrt((1.0d0 / x))) + ((0.5d0 * sqrt(x)) + (0.0625d0 * sqrt(((1.0d0 / x) / (x * x)))))) / x) + (0.5d0 * sqrt((1.0d0 / y)))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (t_2 - sqrt(t))
    else
        tmp = ((t_1 + (sqrt((y + 1.0d0)) - sqrt(y))) + (1.0d0 / (sqrt(z) + sqrt((1.0d0 + z))))) + (((t + 1.0d0) - t) / (t_2 + sqrt(t)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double t_2 = Math.sqrt((t + 1.0));
	double tmp;
	if (t_1 <= 0.0005) {
		tmp = (((((-0.125 * Math.sqrt((1.0 / x))) + ((0.5 * Math.sqrt(x)) + (0.0625 * Math.sqrt(((1.0 / x) / (x * x)))))) / x) + (0.5 * Math.sqrt((1.0 / y)))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (t_2 - Math.sqrt(t));
	} else {
		tmp = ((t_1 + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z))))) + (((t + 1.0) - t) / (t_2 + Math.sqrt(t)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0)) - math.sqrt(x)
	t_2 = math.sqrt((t + 1.0))
	tmp = 0
	if t_1 <= 0.0005:
		tmp = (((((-0.125 * math.sqrt((1.0 / x))) + ((0.5 * math.sqrt(x)) + (0.0625 * math.sqrt(((1.0 / x) / (x * x)))))) / x) + (0.5 * math.sqrt((1.0 / y)))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (t_2 - math.sqrt(t))
	else:
		tmp = ((t_1 + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (1.0 / (math.sqrt(z) + math.sqrt((1.0 + z))))) + (((t + 1.0) - t) / (t_2 + math.sqrt(t)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_2 = sqrt(Float64(t + 1.0))
	tmp = 0.0
	if (t_1 <= 0.0005)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-0.125 * sqrt(Float64(1.0 / x))) + Float64(Float64(0.5 * sqrt(x)) + Float64(0.0625 * sqrt(Float64(Float64(1.0 / x) / Float64(x * x)))))) / x) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(t_2 - sqrt(t)));
	else
		tmp = Float64(Float64(Float64(t_1 + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))) + Float64(Float64(Float64(t + 1.0) - t) / Float64(t_2 + sqrt(t))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0)) - sqrt(x);
	t_2 = sqrt((t + 1.0));
	tmp = 0.0;
	if (t_1 <= 0.0005)
		tmp = (((((-0.125 * sqrt((1.0 / x))) + ((0.5 * sqrt(x)) + (0.0625 * sqrt(((1.0 / x) / (x * x)))))) / x) + (0.5 * sqrt((1.0 / y)))) + (sqrt((z + 1.0)) - sqrt(z))) + (t_2 - sqrt(t));
	else
		tmp = ((t_1 + (sqrt((y + 1.0)) - sqrt(y))) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) + (((t + 1.0) - t) / (t_2 + sqrt(t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0005], N[(N[(N[(N[(N[(N[(-0.125 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.0625 * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t + 1.0), $MachinePrecision] - t), $MachinePrecision] / N[(t$95$2 + N[Sqrt[t], $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)) < 5.0000000000000001e-4

   1. Initial program 88.1%

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

   3. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   if 5.0000000000000001e-4 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 97.0%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   6. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 91.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ t_2 := \sqrt{1 + z}\\ \mathbf{if}\;t\_1 \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + t\_2}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_2 + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{z}\right) + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ t 1.0)) (sqrt t))) (t_2 (sqrt (+ 1.0 z))))
   (if (<= t_1 1.5e-8)
     (+
      (sqrt (+ 1.0 x))
      (- (+ (sqrt (+ 1.0 y)) (/ 1.0 (+ (sqrt z) t_2))) (+ (sqrt y) (sqrt x))))
     (+ (- (- (+ t_2 2.0) (+ (sqrt x) (sqrt y))) (sqrt z)) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0)) - sqrt(t);
	double t_2 = sqrt((1.0 + z));
	double tmp;
	if (t_1 <= 1.5e-8) {
		tmp = sqrt((1.0 + x)) + ((sqrt((1.0 + y)) + (1.0 / (sqrt(z) + t_2))) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = (((t_2 + 2.0) - (sqrt(x) + sqrt(y))) - sqrt(z)) + t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((t + 1.0d0)) - sqrt(t)
    t_2 = sqrt((1.0d0 + z))
    if (t_1 <= 1.5d-8) then
        tmp = sqrt((1.0d0 + x)) + ((sqrt((1.0d0 + y)) + (1.0d0 / (sqrt(z) + t_2))) - (sqrt(y) + sqrt(x)))
    else
        tmp = (((t_2 + 2.0d0) - (sqrt(x) + sqrt(y))) - sqrt(z)) + t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_2 = Math.sqrt((1.0 + z));
	double tmp;
	if (t_1 <= 1.5e-8) {
		tmp = Math.sqrt((1.0 + x)) + ((Math.sqrt((1.0 + y)) + (1.0 / (Math.sqrt(z) + t_2))) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = (((t_2 + 2.0) - (Math.sqrt(x) + Math.sqrt(y))) - Math.sqrt(z)) + t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_2 = math.sqrt((1.0 + z))
	tmp = 0
	if t_1 <= 1.5e-8:
		tmp = math.sqrt((1.0 + x)) + ((math.sqrt((1.0 + y)) + (1.0 / (math.sqrt(z) + t_2))) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = (((t_2 + 2.0) - (math.sqrt(x) + math.sqrt(y))) - math.sqrt(z)) + t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_2 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (t_1 <= 1.5e-8)
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(Float64(sqrt(Float64(1.0 + y)) + Float64(1.0 / Float64(sqrt(z) + t_2))) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(Float64(Float64(Float64(t_2 + 2.0) - Float64(sqrt(x) + sqrt(y))) - sqrt(z)) + t_1);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((t + 1.0)) - sqrt(t);
	t_2 = sqrt((1.0 + z));
	tmp = 0.0;
	if (t_1 <= 1.5e-8)
		tmp = sqrt((1.0 + x)) + ((sqrt((1.0 + y)) + (1.0 / (sqrt(z) + t_2))) - (sqrt(y) + sqrt(x)));
	else
		tmp = (((t_2 + 2.0) - (sqrt(x) + sqrt(y))) - sqrt(z)) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 1.5e-8], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$2 + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)) < 1.49999999999999987e-8

   1. Initial program 88.5%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   5. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 1.49999999999999987e-8 < (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))

   1. Initial program 95.1%

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

   3. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;t\_2 \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;1 + \left(\left(\frac{1}{\sqrt{z} + t\_1} + \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_1 + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{z}\right) + t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))) (t_2 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= t_2 1.5e-8)
     (+
      1.0
      (- (+ (/ 1.0 (+ (sqrt z) t_1)) (sqrt (+ 1.0 y))) (+ (sqrt y) (sqrt x))))
     (+ (- (- (+ t_1 2.0) (+ (sqrt x) (sqrt y))) (sqrt z)) 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));
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if (t_2 <= 1.5e-8) {
		tmp = 1.0 + (((1.0 / (sqrt(z) + t_1)) + sqrt((1.0 + y))) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = (((t_1 + 2.0) - (sqrt(x) + sqrt(y))) - sqrt(z)) + 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))
    t_2 = sqrt((t + 1.0d0)) - sqrt(t)
    if (t_2 <= 1.5d-8) then
        tmp = 1.0d0 + (((1.0d0 / (sqrt(z) + t_1)) + sqrt((1.0d0 + y))) - (sqrt(y) + sqrt(x)))
    else
        tmp = (((t_1 + 2.0d0) - (sqrt(x) + sqrt(y))) - sqrt(z)) + 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));
	double t_2 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double tmp;
	if (t_2 <= 1.5e-8) {
		tmp = 1.0 + (((1.0 / (Math.sqrt(z) + t_1)) + Math.sqrt((1.0 + y))) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = (((t_1 + 2.0) - (Math.sqrt(x) + Math.sqrt(y))) - Math.sqrt(z)) + 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))
	t_2 = math.sqrt((t + 1.0)) - math.sqrt(t)
	tmp = 0
	if t_2 <= 1.5e-8:
		tmp = 1.0 + (((1.0 / (math.sqrt(z) + t_1)) + math.sqrt((1.0 + y))) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = (((t_1 + 2.0) - (math.sqrt(x) + math.sqrt(y))) - math.sqrt(z)) + t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (t_2 <= 1.5e-8)
		tmp = Float64(1.0 + Float64(Float64(Float64(1.0 / Float64(sqrt(z) + t_1)) + sqrt(Float64(1.0 + y))) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(Float64(Float64(Float64(t_1 + 2.0) - Float64(sqrt(x) + sqrt(y))) - sqrt(z)) + 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));
	t_2 = sqrt((t + 1.0)) - sqrt(t);
	tmp = 0.0;
	if (t_2 <= 1.5e-8)
		tmp = 1.0 + (((1.0 / (sqrt(z) + t_1)) + sqrt((1.0 + y))) - (sqrt(y) + sqrt(x)));
	else
		tmp = (((t_1 + 2.0) - (sqrt(x) + sqrt(y))) - sqrt(z)) + 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[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1.5e-8], N[(1.0 + N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$1 + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)) < 1.49999999999999987e-8

   1. Initial program 88.5%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   10. Taylor expanded in t around inf 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]

   if 1.49999999999999987e-8 < (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))

   1. Initial program 95.1%

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

   3. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 0.999995:\\ \;\;\;\;1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + t\_1\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(2 + t\_1\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \end{array} \]

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))))
   (if (<= (- (sqrt (+ z 1.0)) (sqrt z)) 0.999995)
     (+
      1.0
      (- (+ (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))) t_1) (+ (sqrt y) (sqrt x))))
     (+
      (- (- (+ 2.0 t_1) (+ (sqrt x) (sqrt y))) (sqrt z))
      (- (sqrt (+ t 1.0)) (sqrt t))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double tmp;
	if ((sqrt((z + 1.0)) - sqrt(z)) <= 0.999995) {
		tmp = 1.0 + (((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + t_1) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = (((2.0 + t_1) - (sqrt(x) + sqrt(y))) - sqrt(z)) + (sqrt((t + 1.0)) - sqrt(t));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    if ((sqrt((z + 1.0d0)) - sqrt(z)) <= 0.999995d0) then
        tmp = 1.0d0 + (((1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))) + t_1) - (sqrt(y) + sqrt(x)))
    else
        tmp = (((2.0d0 + t_1) - (sqrt(x) + sqrt(y))) - sqrt(z)) + (sqrt((t + 1.0d0)) - sqrt(t))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double tmp;
	if ((Math.sqrt((z + 1.0)) - Math.sqrt(z)) <= 0.999995) {
		tmp = 1.0 + (((1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)))) + t_1) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = (((2.0 + t_1) - (Math.sqrt(x) + Math.sqrt(y))) - Math.sqrt(z)) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	tmp = 0
	if (math.sqrt((z + 1.0)) - math.sqrt(z)) <= 0.999995:
		tmp = 1.0 + (((1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))) + t_1) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = (((2.0 + t_1) - (math.sqrt(x) + math.sqrt(y))) - math.sqrt(z)) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) <= 0.999995)
		tmp = Float64(1.0 + Float64(Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z)))) + t_1) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 + t_1) - Float64(sqrt(x) + sqrt(y))) - sqrt(z)) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	tmp = 0.0;
	if ((sqrt((z + 1.0)) - sqrt(z)) <= 0.999995)
		tmp = 1.0 + (((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + t_1) - (sqrt(y) + sqrt(x)));
	else
		tmp = (((2.0 + t_1) - (sqrt(x) + sqrt(y))) - sqrt(z)) + (sqrt((t + 1.0)) - sqrt(t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.999995], N[(1.0 + N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 + t$95$1), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 0.99999499999999997

   1. Initial program 87.2%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   10. Taylor expanded in t around inf 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]

   if 0.99999499999999997 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   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. Add Preprocessing

   3. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in z around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 0.999995:\\ \;\;\;\;1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + t\_1\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + \left(t\_1 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \end{array} \]

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))))
   (if (<= (- (sqrt (+ z 1.0)) (sqrt z)) 0.999995)
     (+
      1.0
      (- (+ (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))) t_1) (+ (sqrt y) (sqrt x))))
     (+
      (+ 2.0 (- t_1 (+ (sqrt x) (+ (sqrt y) (sqrt z)))))
      (- (sqrt (+ t 1.0)) (sqrt t))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double tmp;
	if ((sqrt((z + 1.0)) - sqrt(z)) <= 0.999995) {
		tmp = 1.0 + (((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + t_1) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = (2.0 + (t_1 - (sqrt(x) + (sqrt(y) + sqrt(z))))) + (sqrt((t + 1.0)) - sqrt(t));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    if ((sqrt((z + 1.0d0)) - sqrt(z)) <= 0.999995d0) then
        tmp = 1.0d0 + (((1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))) + t_1) - (sqrt(y) + sqrt(x)))
    else
        tmp = (2.0d0 + (t_1 - (sqrt(x) + (sqrt(y) + sqrt(z))))) + (sqrt((t + 1.0d0)) - sqrt(t))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double tmp;
	if ((Math.sqrt((z + 1.0)) - Math.sqrt(z)) <= 0.999995) {
		tmp = 1.0 + (((1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)))) + t_1) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = (2.0 + (t_1 - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z))))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	tmp = 0
	if (math.sqrt((z + 1.0)) - math.sqrt(z)) <= 0.999995:
		tmp = 1.0 + (((1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))) + t_1) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = (2.0 + (t_1 - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z))))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) <= 0.999995)
		tmp = Float64(1.0 + Float64(Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z)))) + t_1) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_1 - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	tmp = 0.0;
	if ((sqrt((z + 1.0)) - sqrt(z)) <= 0.999995)
		tmp = 1.0 + (((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + t_1) - (sqrt(y) + sqrt(x)));
	else
		tmp = (2.0 + (t_1 - (sqrt(x) + (sqrt(y) + sqrt(z))))) + (sqrt((t + 1.0)) - sqrt(t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.999995], N[(1.0 + N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 0.99999499999999997

   1. Initial program 87.2%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   10. Taylor expanded in t around inf 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]

   if 0.99999499999999997 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   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. Add Preprocessing

   3. Taylor expanded in z around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 94.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \frac{1}{\sqrt{z} + \sqrt{1 + z}}\\ \mathbf{if}\;y \leq 6.2 \cdot 10^{-22}:\\ \;\;\;\;2 + \left(\left(\frac{1}{\sqrt{t} + \sqrt{1 + t}} + t\_2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;y \leq 31000000000:\\ \;\;\;\;t\_1 + \left(\left(\sqrt{1 + y} + t\_2\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))) (t_2 (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z))))))
   (if (<= y 6.2e-22)
     (+
      2.0
      (- (+ (/ 1.0 (+ (sqrt t) (sqrt (+ 1.0 t)))) t_2) (+ (sqrt x) (sqrt y))))
     (if (<= y 31000000000.0)
       (+ t_1 (- (+ (sqrt (+ 1.0 y)) t_2) (+ (sqrt y) (sqrt x))))
       (+ t_1 (- (* 0.5 (sqrt (/ 1.0 y))) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	double tmp;
	if (y <= 6.2e-22) {
		tmp = 2.0 + (((1.0 / (sqrt(t) + sqrt((1.0 + t)))) + t_2) - (sqrt(x) + sqrt(y)));
	} else if (y <= 31000000000.0) {
		tmp = t_1 + ((sqrt((1.0 + y)) + t_2) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = t_1 + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = 1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))
    if (y <= 6.2d-22) then
        tmp = 2.0d0 + (((1.0d0 / (sqrt(t) + sqrt((1.0d0 + t)))) + t_2) - (sqrt(x) + sqrt(y)))
    else if (y <= 31000000000.0d0) then
        tmp = t_1 + ((sqrt((1.0d0 + y)) + t_2) - (sqrt(y) + sqrt(x)))
    else
        tmp = t_1 + ((0.5d0 * sqrt((1.0d0 / y))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = 1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)));
	double tmp;
	if (y <= 6.2e-22) {
		tmp = 2.0 + (((1.0 / (Math.sqrt(t) + Math.sqrt((1.0 + t)))) + t_2) - (Math.sqrt(x) + Math.sqrt(y)));
	} else if (y <= 31000000000.0) {
		tmp = t_1 + ((Math.sqrt((1.0 + y)) + t_2) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = t_1 + ((0.5 * Math.sqrt((1.0 / y))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = 1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))
	tmp = 0
	if y <= 6.2e-22:
		tmp = 2.0 + (((1.0 / (math.sqrt(t) + math.sqrt((1.0 + t)))) + t_2) - (math.sqrt(x) + math.sqrt(y)))
	elif y <= 31000000000.0:
		tmp = t_1 + ((math.sqrt((1.0 + y)) + t_2) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = t_1 + ((0.5 * math.sqrt((1.0 / y))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))
	tmp = 0.0
	if (y <= 6.2e-22)
		tmp = Float64(2.0 + Float64(Float64(Float64(1.0 / Float64(sqrt(t) + sqrt(Float64(1.0 + t)))) + t_2) - Float64(sqrt(x) + sqrt(y))));
	elseif (y <= 31000000000.0)
		tmp = Float64(t_1 + Float64(Float64(sqrt(Float64(1.0 + y)) + t_2) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(t_1 + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	tmp = 0.0;
	if (y <= 6.2e-22)
		tmp = 2.0 + (((1.0 / (sqrt(t) + sqrt((1.0 + t)))) + t_2) - (sqrt(x) + sqrt(y)));
	elseif (y <= 31000000000.0)
		tmp = t_1 + ((sqrt((1.0 + y)) + t_2) - (sqrt(y) + sqrt(x)));
	else
		tmp = t_1 + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.2e-22], N[(2.0 + N[(N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 31000000000.0], N[(t$95$1 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.20000000000000025e-22

   1. Initial program 97.5%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   10. Taylor expanded in y around 0 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]

   if 6.20000000000000025e-22 < y < 3.1e10

   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. Add Preprocessing

   4. Applied egg-rr 0

      \[\leadsto expr\]

   5. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 3.1e10 < y

   1. Initial program 87.3%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 88.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 11000000000:\\ \;\;\;\;1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 11000000000.0)
   (+
    1.0
    (-
     (+ (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))) (sqrt (+ 1.0 y)))
     (+ (sqrt y) (sqrt x))))
   (+ (sqrt (+ 1.0 x)) (- (* 0.5 (sqrt (/ 1.0 y))) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 11000000000.0) {
		tmp = 1.0 + (((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + sqrt((1.0 + y))) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = sqrt((1.0 + x)) + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 11000000000.0d0) then
        tmp = 1.0d0 + (((1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))) + sqrt((1.0d0 + y))) - (sqrt(y) + sqrt(x)))
    else
        tmp = sqrt((1.0d0 + x)) + ((0.5d0 * sqrt((1.0d0 / y))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 11000000000.0) {
		tmp = 1.0 + (((1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)))) + Math.sqrt((1.0 + y))) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = Math.sqrt((1.0 + x)) + ((0.5 * Math.sqrt((1.0 / y))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 11000000000.0:
		tmp = 1.0 + (((1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))) + math.sqrt((1.0 + y))) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = math.sqrt((1.0 + x)) + ((0.5 * math.sqrt((1.0 / y))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 11000000000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z)))) + sqrt(Float64(1.0 + y))) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 11000000000.0)
		tmp = 1.0 + (((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + sqrt((1.0 + y))) - (sqrt(y) + sqrt(x)));
	else
		tmp = sqrt((1.0 + x)) + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 11000000000.0], N[(1.0 + N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e10

   1. Initial program 97.4%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   10. Taylor expanded in t around inf 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]

   if 1.1e10 < y

   1. Initial program 87.3%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 84.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 8.8 \cdot 10^{+14}:\\ \;\;\;\;\left(\left(\left(\left(1 + t\_1\right) + \sqrt{1 + z}\right) - \sqrt{z}\right) - \sqrt{y}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(t\_1 - \sqrt{x}\right) - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))))
   (if (<= z 8.8e+14)
     (- (- (- (+ (+ 1.0 t_1) (sqrt (+ 1.0 z))) (sqrt z)) (sqrt y)) (sqrt x))
     (+ (sqrt (+ 1.0 x)) (- (- t_1 (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 + y));
	double tmp;
	if (z <= 8.8e+14) {
		tmp = ((((1.0 + t_1) + sqrt((1.0 + z))) - sqrt(z)) - sqrt(y)) - sqrt(x);
	} else {
		tmp = sqrt((1.0 + x)) + ((t_1 - 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) :: tmp
    t_1 = sqrt((1.0d0 + y))
    if (z <= 8.8d+14) then
        tmp = ((((1.0d0 + t_1) + sqrt((1.0d0 + z))) - sqrt(z)) - sqrt(y)) - sqrt(x)
    else
        tmp = sqrt((1.0d0 + x)) + ((t_1 - 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 + y));
	double tmp;
	if (z <= 8.8e+14) {
		tmp = ((((1.0 + t_1) + Math.sqrt((1.0 + z))) - Math.sqrt(z)) - Math.sqrt(y)) - Math.sqrt(x);
	} else {
		tmp = Math.sqrt((1.0 + x)) + ((t_1 - 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 + y))
	tmp = 0
	if z <= 8.8e+14:
		tmp = ((((1.0 + t_1) + math.sqrt((1.0 + z))) - math.sqrt(z)) - math.sqrt(y)) - math.sqrt(x)
	else:
		tmp = math.sqrt((1.0 + x)) + ((t_1 - 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 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 8.8e+14)
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 + t_1) + sqrt(Float64(1.0 + z))) - sqrt(z)) - sqrt(y)) - sqrt(x));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(Float64(t_1 - 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 + y));
	tmp = 0.0;
	if (z <= 8.8e+14)
		tmp = ((((1.0 + t_1) + sqrt((1.0 + z))) - sqrt(z)) - sqrt(y)) - sqrt(x);
	else
		tmp = sqrt((1.0 + x)) + ((t_1 - 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[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 8.8e+14], N[(N[(N[(N[(N[(1.0 + t$95$1), $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 8.8e14

   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. Add Preprocessing

   3. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   if 8.8e14 < z

   1. Initial program 86.8%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 82.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;z \leq 1.05:\\ \;\;\;\;1 + \left(\left(t\_1 + t\_2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(\left(t\_1 - \sqrt{x}\right) - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= z 1.05)
     (+ 1.0 (- (+ t_1 t_2) (+ (sqrt x) (+ (sqrt y) (sqrt z)))))
     (+ t_2 (- (- t_1 (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 + y));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (z <= 1.05) {
		tmp = 1.0 + ((t_1 + t_2) - (sqrt(x) + (sqrt(y) + sqrt(z))));
	} else {
		tmp = t_2 + ((t_1 - 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) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + x))
    if (z <= 1.05d0) then
        tmp = 1.0d0 + ((t_1 + t_2) - (sqrt(x) + (sqrt(y) + sqrt(z))))
    else
        tmp = t_2 + ((t_1 - 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 + y));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (z <= 1.05) {
		tmp = 1.0 + ((t_1 + t_2) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z))));
	} else {
		tmp = t_2 + ((t_1 - 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 + y))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if z <= 1.05:
		tmp = 1.0 + ((t_1 + t_2) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z))))
	else:
		tmp = t_2 + ((t_1 - 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 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (z <= 1.05)
		tmp = Float64(1.0 + Float64(Float64(t_1 + t_2) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))));
	else
		tmp = Float64(t_2 + Float64(Float64(t_1 - 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 + y));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (z <= 1.05)
		tmp = 1.0 + ((t_1 + t_2) - (sqrt(x) + (sqrt(y) + sqrt(z))));
	else
		tmp = t_2 + ((t_1 - 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[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1.05], N[(1.0 + N[(N[(t$95$1 + t$95$2), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.05000000000000004

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   if 1.05000000000000004 < z

   1. Initial program 87.2%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 66.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 29000000:\\ \;\;\;\;t\_1 + \left(\frac{\left(1 + y\right) - x}{\sqrt{1 + y} + \sqrt{x}} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))))
   (if (<= y 29000000.0)
     (+ t_1 (- (/ (- (+ 1.0 y) x) (+ (sqrt (+ 1.0 y)) (sqrt x))) (sqrt y)))
     (+ t_1 (- (* 0.5 (sqrt (/ 1.0 y))) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double tmp;
	if (y <= 29000000.0) {
		tmp = t_1 + ((((1.0 + y) - x) / (sqrt((1.0 + y)) + sqrt(x))) - sqrt(y));
	} else {
		tmp = t_1 + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    if (y <= 29000000.0d0) then
        tmp = t_1 + ((((1.0d0 + y) - x) / (sqrt((1.0d0 + y)) + sqrt(x))) - sqrt(y))
    else
        tmp = t_1 + ((0.5d0 * sqrt((1.0d0 / y))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 29000000.0) {
		tmp = t_1 + ((((1.0 + y) - x) / (Math.sqrt((1.0 + y)) + Math.sqrt(x))) - Math.sqrt(y));
	} else {
		tmp = t_1 + ((0.5 * Math.sqrt((1.0 / y))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 29000000.0:
		tmp = t_1 + ((((1.0 + y) - x) / (math.sqrt((1.0 + y)) + math.sqrt(x))) - math.sqrt(y))
	else:
		tmp = t_1 + ((0.5 * math.sqrt((1.0 / y))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 29000000.0)
		tmp = Float64(t_1 + Float64(Float64(Float64(Float64(1.0 + y) - x) / Float64(sqrt(Float64(1.0 + y)) + sqrt(x))) - sqrt(y)));
	else
		tmp = Float64(t_1 + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 29000000.0)
		tmp = t_1 + ((((1.0 + y) - x) / (sqrt((1.0 + y)) + sqrt(x))) - sqrt(y));
	else
		tmp = t_1 + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 29000000.0], N[(t$95$1 + N[(N[(N[(N[(1.0 + y), $MachinePrecision] - x), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9e7

   1. Initial program 97.6%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   10. Applied egg-rr 0

       \[\leadsto expr\]

   if 2.9e7 < y

   1. Initial program 87.2%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 66.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 70000000:\\ \;\;\;\;t\_1 + \left(\left(\sqrt{1 + y} - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))))
   (if (<= y 70000000.0)
     (+ t_1 (- (- (sqrt (+ 1.0 y)) (sqrt x)) (sqrt y)))
     (+ t_1 (- (* 0.5 (sqrt (/ 1.0 y))) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double tmp;
	if (y <= 70000000.0) {
		tmp = t_1 + ((sqrt((1.0 + y)) - sqrt(x)) - sqrt(y));
	} else {
		tmp = t_1 + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    if (y <= 70000000.0d0) then
        tmp = t_1 + ((sqrt((1.0d0 + y)) - sqrt(x)) - sqrt(y))
    else
        tmp = t_1 + ((0.5d0 * sqrt((1.0d0 / y))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 70000000.0) {
		tmp = t_1 + ((Math.sqrt((1.0 + y)) - Math.sqrt(x)) - Math.sqrt(y));
	} else {
		tmp = t_1 + ((0.5 * Math.sqrt((1.0 / y))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 70000000.0:
		tmp = t_1 + ((math.sqrt((1.0 + y)) - math.sqrt(x)) - math.sqrt(y))
	else:
		tmp = t_1 + ((0.5 * math.sqrt((1.0 / y))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 70000000.0)
		tmp = Float64(t_1 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(x)) - sqrt(y)));
	else
		tmp = Float64(t_1 + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 70000000.0)
		tmp = t_1 + ((sqrt((1.0 + y)) - sqrt(x)) - sqrt(y));
	else
		tmp = t_1 + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 70000000.0], N[(t$95$1 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 7e7

   1. Initial program 97.6%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   if 7e7 < y

   1. Initial program 87.2%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 66.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 28000000:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 + -0.125\right)\right)\right) + \left(\left(\sqrt{1 + y} - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 28000000.0)
   (+
    (+ 1.0 (* x (+ 0.5 (* x (+ (* x 0.0625) -0.125)))))
    (- (- (sqrt (+ 1.0 y)) (sqrt x)) (sqrt y)))
   (+ (sqrt (+ 1.0 x)) (- (* 0.5 (sqrt (/ 1.0 y))) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 28000000.0) {
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) + -0.125))))) + ((sqrt((1.0 + y)) - sqrt(x)) - sqrt(y));
	} else {
		tmp = sqrt((1.0 + x)) + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 28000000.0d0) then
        tmp = (1.0d0 + (x * (0.5d0 + (x * ((x * 0.0625d0) + (-0.125d0)))))) + ((sqrt((1.0d0 + y)) - sqrt(x)) - sqrt(y))
    else
        tmp = sqrt((1.0d0 + x)) + ((0.5d0 * sqrt((1.0d0 / y))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 28000000.0) {
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) + -0.125))))) + ((Math.sqrt((1.0 + y)) - Math.sqrt(x)) - Math.sqrt(y));
	} else {
		tmp = Math.sqrt((1.0 + x)) + ((0.5 * Math.sqrt((1.0 / y))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 28000000.0:
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) + -0.125))))) + ((math.sqrt((1.0 + y)) - math.sqrt(x)) - math.sqrt(y))
	else:
		tmp = math.sqrt((1.0 + x)) + ((0.5 * math.sqrt((1.0 / y))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 28000000.0)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.0625) + -0.125))))) + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(x)) - sqrt(y)));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 28000000.0)
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) + -0.125))))) + ((sqrt((1.0 + y)) - sqrt(x)) - sqrt(y));
	else
		tmp = sqrt((1.0 + x)) + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 28000000.0], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.0625), $MachinePrecision] + -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8e7

   1. Initial program 97.6%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]

   if 2.8e7 < y

   1. Initial program 87.2%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 66.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 68000000:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) + \left(\left(\sqrt{1 + y} - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 68000000.0)
   (+
    (+ 1.0 (* x (+ 0.5 (* x -0.125))))
    (- (- (sqrt (+ 1.0 y)) (sqrt x)) (sqrt y)))
   (+ (sqrt (+ 1.0 x)) (- (* 0.5 (sqrt (/ 1.0 y))) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 68000000.0) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) + ((sqrt((1.0 + y)) - sqrt(x)) - sqrt(y));
	} else {
		tmp = sqrt((1.0 + x)) + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 68000000.0d0) then
        tmp = (1.0d0 + (x * (0.5d0 + (x * (-0.125d0))))) + ((sqrt((1.0d0 + y)) - sqrt(x)) - sqrt(y))
    else
        tmp = sqrt((1.0d0 + x)) + ((0.5d0 * sqrt((1.0d0 / y))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 68000000.0) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) + ((Math.sqrt((1.0 + y)) - Math.sqrt(x)) - Math.sqrt(y));
	} else {
		tmp = Math.sqrt((1.0 + x)) + ((0.5 * Math.sqrt((1.0 / y))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 68000000.0:
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) + ((math.sqrt((1.0 + y)) - math.sqrt(x)) - math.sqrt(y))
	else:
		tmp = math.sqrt((1.0 + x)) + ((0.5 * math.sqrt((1.0 / y))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 68000000.0)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * -0.125)))) + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(x)) - sqrt(y)));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 68000000.0)
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) + ((sqrt((1.0 + y)) - sqrt(x)) - sqrt(y));
	else
		tmp = sqrt((1.0 + x)) + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 68000000.0], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.8e7

   1. Initial program 97.6%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]

   if 6.8e7 < y

   1. Initial program 87.2%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 66.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 38000000:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) + x \cdot 0.5\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 38000000.0)
   (- (+ (+ 1.0 (sqrt (+ 1.0 y))) (* x 0.5)) (+ (sqrt x) (sqrt y)))
   (+ (sqrt (+ 1.0 x)) (- (* 0.5 (sqrt (/ 1.0 y))) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 38000000.0) {
		tmp = ((1.0 + sqrt((1.0 + y))) + (x * 0.5)) - (sqrt(x) + sqrt(y));
	} else {
		tmp = sqrt((1.0 + x)) + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 38000000.0d0) then
        tmp = ((1.0d0 + sqrt((1.0d0 + y))) + (x * 0.5d0)) - (sqrt(x) + sqrt(y))
    else
        tmp = sqrt((1.0d0 + x)) + ((0.5d0 * sqrt((1.0d0 / y))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 38000000.0) {
		tmp = ((1.0 + Math.sqrt((1.0 + y))) + (x * 0.5)) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = Math.sqrt((1.0 + x)) + ((0.5 * Math.sqrt((1.0 / y))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 38000000.0:
		tmp = ((1.0 + math.sqrt((1.0 + y))) + (x * 0.5)) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = math.sqrt((1.0 + x)) + ((0.5 * math.sqrt((1.0 / y))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 38000000.0)
		tmp = Float64(Float64(Float64(1.0 + sqrt(Float64(1.0 + y))) + Float64(x * 0.5)) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 38000000.0)
		tmp = ((1.0 + sqrt((1.0 + y))) + (x * 0.5)) - (sqrt(x) + sqrt(y));
	else
		tmp = sqrt((1.0 + x)) + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 38000000.0], N[(N[(N[(1.0 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.8e7

   1. Initial program 97.6%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]

   if 3.8e7 < y

   1. Initial program 87.2%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 66.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 55000000:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 55000000.0)
   (+ 1.0 (- (- (sqrt (+ 1.0 y)) (sqrt x)) (sqrt y)))
   (+ (sqrt (+ 1.0 x)) (- (* 0.5 (sqrt (/ 1.0 y))) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 55000000.0) {
		tmp = 1.0 + ((sqrt((1.0 + y)) - sqrt(x)) - sqrt(y));
	} else {
		tmp = sqrt((1.0 + x)) + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 55000000.0d0) then
        tmp = 1.0d0 + ((sqrt((1.0d0 + y)) - sqrt(x)) - sqrt(y))
    else
        tmp = sqrt((1.0d0 + x)) + ((0.5d0 * sqrt((1.0d0 / y))) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 55000000.0) {
		tmp = 1.0 + ((Math.sqrt((1.0 + y)) - Math.sqrt(x)) - Math.sqrt(y));
	} else {
		tmp = Math.sqrt((1.0 + x)) + ((0.5 * Math.sqrt((1.0 / y))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 55000000.0:
		tmp = 1.0 + ((math.sqrt((1.0 + y)) - math.sqrt(x)) - math.sqrt(y))
	else:
		tmp = math.sqrt((1.0 + x)) + ((0.5 * math.sqrt((1.0 / y))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 55000000.0)
		tmp = Float64(1.0 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(x)) - sqrt(y)));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 55000000.0)
		tmp = 1.0 + ((sqrt((1.0 + y)) - sqrt(x)) - sqrt(y));
	else
		tmp = sqrt((1.0 + x)) + ((0.5 * sqrt((1.0 / y))) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 55000000.0], N[(1.0 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5e7

   1. Initial program 97.6%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]

   if 5.5e7 < y

   1. Initial program 87.2%

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

   3. Taylor expanded in z around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 64.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in z around inf 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]

6. Taylor expanded in t around inf 0

   \[\leadsto expr\]

8. Simplified 0

   \[\leadsto expr\]

9. Taylor expanded in x around 0 0

   \[\leadsto expr\]

11. Simplified 0

    \[\leadsto expr\]
12. Add Preprocessing

Alternative 23: 35.4% accurate, 4.0× speedup

Math

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

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 (+ 1.0 x)) (sqrt x)))

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in z around inf 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]

6. Taylor expanded in t around inf 0

   \[\leadsto expr\]

8. Simplified 0

   \[\leadsto expr\]

9. Taylor expanded in y around inf 0

   \[\leadsto expr\]

11. Simplified 0

    \[\leadsto expr\]
12. Add Preprocessing

Alternative 24: 34.3% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

6. Applied egg-rr 0

   \[\leadsto expr\]

7. Taylor expanded in x around 0 0

   \[\leadsto expr\]

9. Simplified 0

   \[\leadsto expr\]

10. Taylor expanded in x around inf 0

    \[\leadsto expr\]

12. Simplified 0

    \[\leadsto expr\]
13. Add Preprocessing

Alternative 25: 7.9% accurate, 7.8× speedup

Math

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

FPCore

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

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.5 * 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 = 0.5d0 * 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 0.5 * Math.sqrt((1.0 / y));
}

Python

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

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.5 * 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 = 0.5 * 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[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in y around inf 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]

6. Taylor expanded in y around 0 0

   \[\leadsto expr\]

8. Simplified 0

   \[\leadsto expr\]
9. Add Preprocessing

Alternative 26: 1.9% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

6. Applied egg-rr 0

   \[\leadsto expr\]

7. Taylor expanded in x around 0 0

   \[\leadsto expr\]

9. Simplified 0

   \[\leadsto expr\]

10. Taylor expanded in x around inf 0

    \[\leadsto expr\]

12. Simplified 0

    \[\leadsto expr\]
13. Add Preprocessing

Alternative 27: 1.5% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in z around 0 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]

6. Taylor expanded in z around inf 0

   \[\leadsto expr\]

8. Simplified 0

   \[\leadsto expr\]

10. Applied egg-rr 0

    \[\leadsto expr\]
11. Add Preprocessing

Developer target: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024110 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

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

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